Constitutive Modeling of Triaxially Loaded Concrete Considering Large Compressive Stresses: Application of Pull-out Tests of Ahchor Bolts [PDF]

Zitiervorschau

DOCTORAL

THESIS

CONSTITUTIVE MODELING OF TRIAXIALLY LOADED CONCRETE CONSIDERING LARGE COMPRESSIVE STRESSES: APPLICATION TO PULL-OUT TESTS OF ANCHOR BOLTS

DISSERTATION KONSTITUTIVES MODELLIEREN TRIAXIAL ¨ BEANSPRUCHTEN BETONS BEI BERUCKSICHTIGUNG GROSSER DRUCKSPANNUNGEN: ANWENDUNG AUF BOLZENAUSZIEHVERSUCHE

ausgef¨ uhrt zum Zwecke der Erlangung des akademischen Grades eines Doktors der technischen Wissenschaften eingereicht an der Technischen Universit¨at Wien Fakult¨at f¨ ur Bauingenieurwesen von Dipl.-Ing. Peter Pivonka Matrikelnummer 8925239 ¨ Haydnstrasse 52, 2333 Leopoldsdorf, Osterreich

Referent:

O.Univ.Prof. Dipl.-Ing. Dr.techn. Dr.h.c. Herbert Mang, Ph.D. Institut f¨ ur Festigkeitslehre, Technische Universit¨at Wien ¨ Karlsplatz 13/202, 1040 Wien, Osterreich

Koreferent:

O.Univ.Prof. Dipl.-Ing. Dr.-Ing. habil. Kaspar Willam, Ph.D. Department of Civil, Environmental and Architectural Engineering University of Colorado at Boulder, Boulder, CO 80309-0428, USA

Wien, im Dezember 2001

..............................

Acknowledgement The present thesis emerged during my work as a Research and University Assistant at the Institute of Strength of Materials at the Vienna University of Technology. In the following I would like to thank numerous people for their support during that periode. First of all I would like to express my gratitude to Prof. Herbert Mang who initiated this research project. His advice and guidance provided the foundation for the successful outcome of this work. I am also grateful to Prof. Kaspar Willam for his willingness to serve on the doctoral commitee. Special thanks to my colleague and friend Roman Lackner for the fruitful discussions we had during lunch and coffee-breaks, not to forget his entertaining ”funny” jokes in difficult situations. Furthermore, I would like to mention the wonderful working enviroment and the great infrastructure of the Institute, which are consequences of the good cooperation of all colleagues. For this I would like to thank former and present members of the Institute, Christian Hellmich, Peter Helnwein, Thomas Huemer, Juergen Macht, Yvonne Spira, Christian Schranz, and all the others. I would like to express great thanks to my parents, who supported me all the time and gave me the chance to obtain a good education. Last but not least, I would like to thank all friends and relatives, who shared this time with me. Some of them have seen how time-consuming Computational Mechanics can be. Thanks to all of you for your patience and comprehension.

Kurzfassung Das moderne Ingenieurwesen ist durch eine hohe Komplexit¨at von Design, Funktion und Konstruktion gekennzeichnet. An die dabei verwendeten Materialien werden große Anforderungen in Bezug auf Festigkeit und Dauerhaftigkeit gestellt. Um den Sicherheitsanspr¨ uchen unserer Gesellschaft Rechnung zu tragen, muss das Werkstoffverhalten der verwendeten Materialien genau analysiert werden. Numerische Berechnungsverfahren, wie z.B. die Methode der Finiten Elemente, erlauben es zusammen mit der Verwendung komplexer Werkstoff Modelle das Tragverhalten einer Struktur zu analysieren und m¨ogliches Versagen der Struktur zu prognostizieren. Einer der h¨aufigst verwendeten Werkstoffe im konstruktiven Ingenieur-Bau ist Beton. Das Werkstoffverhalten von Beton ist durch stark unterschiedliches Materialverhalten bei Zugund Druck-Beanspruchung gekennzeichnet. Das Verhalten von Beton unter Zugbeanspruchung ist durch spr¨odes Versagen charakterisiert, wobei die Zugfestigkeit in Vergleich zur Druckfestigkeit klein ist. Das Verhalten von Beton unter Druck-Beanspruchung ist durch duktiles Versagen charakterisiert. Aus Versuchen mit unterschiedlichem Lateraldruck kann festgestellt werden, daß die Festigkeit von Beton mit zunehmendem Lateraldruck steigt. Die vorliegende Arbeit beinhaltet die Entwicklung zweier dreidimensionaler elastoplastischer konstitutiver Modelle f¨ ur Beton, welche f¨ ur ein großes Belastungsspektrum, wie Zug, Druck und hohe Druck-Beanspruchung geeignet sind. Das erste Modell ist ein Einfl¨achenplastizit¨atsmodell, welches die Abh¨angikeit der Festigkeit vom Lode-Winkel ber¨ ucksichtigt. Das duktile Verhalten von Beton wird durch eine Duktilit¨atsfunktion, die u ¨ ber den hydrostatischen Druck gesteuert wird, ber¨ ucksichtigt. Das zweite Werkstoff Modell ist ein Mehrfl¨achenplastizit¨atsmodell, welches aus drei Rankinefl¨achen zur Beschreibung des ZugVerhaltens und einer Drucker-Prager Fließfl¨ache zur Beschreibung des Druck-Verhaltens besteht. Das Drucker-Prager Kriterium wurde zur Beschreibung von Lateraldr¨ ucken erweitert. Die Beschreibung des in-elastischen Dilatanzverhaltens beim Einfl¨achenmodell und beim Drucker-Prager Kriterium erfolgt mittels einer nicht-assoziierten Fließregel. F¨ ur das Rankine Kriterium wird eine assoziierte Fließregel verwendet. Das Verhalten der Werkstoff Modelle auf konstitutiver Ebene wurde anhand einer großen Anzahl von Versuchen mit unterschiedlichen Belastungspfaden analysiert. Besonderes Interesse war an einer effizienten algorithmischen Umsetzung dieser Werkstoff Modelle f¨ ur FE Simulationen gegeben. Im Rahmen relativ großer numerischer FE Simulationen mit mehreren tausend Freiheitsgraden ist besonders auf Robustheit und Effizienz der zugrunde liegenden Algorithmen zu achten. Um den Versagens Mode einer Struktur beschreiben zu k¨onnen, ist es notwendig, Materialentfestigung, wie Reißen bzw. Zerstauchen von Beton, zu ber¨ ucksichtigen. Im Rahmen der vorliegenden Arbeit wurde Entfestigung unter Zugrundelegung des Bruchenergiekonzeptes formuliert. Weiters wurde das Verhalten der Werkstoff Modelle bei Lokalisierung untersucht.

Das Verhalten der Werkstoffmodelle auf Strukturebene wird im Rahmen einer umfangreichen numerischen Studie analysiert. Zun¨achst wird eine Betonscheibe unter Zugrundelegung eines ebenen Verzerrungszustandes unter Druck- und Zugbeanspruchung untersucht. Danach wird ein Spaltzugversuch berechnet und mit experimentellen Daten verglichen. Die letzten drei Simulationen beinhalten die Analyse des Tragverhaltens von Verankerungselementen im Beton. Derartige Systeme sind durch eine konzentrierte Krafteinleitung in einem relativ kleinen Bereich des Betons gekennzeichnet. Diese Bereiche sind stark nichtuniformen triaxialen Spannungszust¨anden unterworfen. Die ersten beiden Berechnungen besch¨aftigen sich mit der Analyse von Kopfbolzenversuchen. Die letzte Simulation umfaßt die Analyse des Tragverhaltens eines Hinterschnittd¨ ubels, der f¨ ur sehr hohe Traglasten konzipiert wurde. Bei dieser numerischen Berechnung wird sowohl der Setzvorgang des D¨ ubels, als auch der Auszugsvorgang desselben simuliert.

Abstract Modern structural engineering is characterized by great complexity as regards design, function and construction. Live cycle engineering includes extreme load scenarios of plain and reinforced concrete structures. In reinfored concrete structures special attention must be paid to extreme overload conditions leading to a complex redistribution of internal loading paths. Hence, high requirements with respect to the strength and durability of the employed materials are requested. Safety requirements of society give rise to analyze the constitutive behavior of the employed materials in detail. Numerical tools such as the Finite Element Method (FEM), together with the use of sophisticated constitutive models allow to monitor the development of structural failure and estimate the peak load of the system. Numerical and experimetal investigations provide the basis for the development of modern design codes. Plain concrete plays an important role in structural engineering because of its easy in situ installation and the rather low material costs. The constitutive behavior of concrete is characterized by different behavior under tensile and compressive loading. Tensile loading is characterized by brittle failure, whereas compressive loading leads to the development of ductile failure. The ratio of the uniaxial tensile strength to the uniaxial compressive strength is approximately 1/10. Triaxial compression experiments with different confining pressure clearly indicate the increase of compressive strength with increasing confinement. The present thesis deals with the development of two 3D elasto-plastic constitutive models for concrete. These models are capable of capturing the material behavior of concrete under a broad range of loading conditions such as tensile, low compressive and high compressive loading states. The first model is a single-surface model. Dependence of the concrete strength on the Lode angle is accounted for by means of an elliptic deviatoric shape function. Ductile behavior of concrete is controlled by means of pressure-dependent ductility functions. The second model is a multi-surface model consisting of a Drucker-Prager surface for the description of compressive failure of concrete and three Rankine surfaces for the description of tensile loading. The Drucker-Prager surface is reformulated to account for confined compressive stress states. Inelastic dilatational behavior of the single-surface and the Drucker-Prager surface is controlled by means of a non-associative flow rule. For the Rankine criterion an associative flow rule is employed. The performance of both material models on the constitutive level has been investigated for various loading paths. Because of the rather complex format of the proposed material models special emphasis has been laid on a robust and efficient algorithmic implementation in the context of relatively large FE simulations. Such simulations are characterized by several thousand degrees of freedom in 2D and ten to hundred thousand degrees of freedom in 3D. Constitutive models for concrete accounting for an appropriate description of structural failure must incorporate softening material behavior in the form of cracking and crushing of concrete. For the proposed models softening is formulated on the basis of the fracture energy concept. The localization behavior of the models is investigated by means of several loading paths.

The performance of the material models on the structural level is demonstrated by means of extensive numerical simulations, starting with a concrete panel loaded in plane strain tension and compression. The second problem deals with a cylinder splitting test together with comparison of experimental data. The last three simulations are concerned with investigations of the load-carrying behavior of anchor devices. Such devices are characterized by concentrated loads in the vicinity of the anchor head. Thus, in these regions concrete is subjected to strongly non-uniform triaxial stress states. The first two simulations are dealing with headed studs, whereas the third problem deals with the investigation of an undercut anchor. This anchor was designed for very large loads. For this simulation the setting of the anchor and the pull-out phase are simulated.

Contents 1 Introduction and Scope of Work

1

1.1

Literature review and scope of work . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Constitutive Models for Concrete 2.1

2.2

2.3

5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Remarks on the modeling process . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Modeling length scale . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.3

Experimental observations and conclusions . . . . . . . . . . . . . . .

6

2.1.4

Validation of constitutive models . . . . . . . . . . . . . . . . . . . .

11

Multi-surface plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.1

Thermodynamic framework . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.2

Differential consistency . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Material models for plain concrete . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3.1

Single-surface plasticity model . . . . . . . . . . . . . . . . . . . . . .

19

2.3.1.1

General characteristics of the Extended Leon Model (ELM)

19

2.3.1.2

Non-associative flow rule . . . . . . . . . . . . . . . . . . . .

21

2.3.1.3

Non-linear isotropic hardening law . . . . . . . . . . . . . .

22

2.3.1.4

Non-linear isotropic softening law . . . . . . . . . . . . . . .

23

2.3.1.5

Calibration of the ELM in the context of the fracture energy concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Multi-surface plasticity model . . . . . . . . . . . . . . . . . . . . . .

28

2.3.2.1

Yield surfaces . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.3.2.2

Evolution equations . . . . . . . . . . . . . . . . . . . . . .

29

2.3.2.3

Isotropic hardening/softening laws . . . . . . . . . . . . . .

30

2.3.2.4

Consideration of confinement . . . . . . . . . . . . . . . . .

31

2.3.2

CONTENTS 2.4

ii

Calibration and re-analyses of test results . . . . . . . . . . . . . . . . . . . .

34

2.4.1

Boulder experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.4.1.1

Model parameters . . . . . . . . . . . . . . . . . . . . . . .

35

2.4.2

Northwestern experiments . . . . . . . . . . . . . . . . . . . . . . . .

39

2.4.3

Toronto experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.4.4

Eindhoven experiments . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.4.5

Cachan experiments . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3 Numerical Integration of Material Laws

47

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2

Incremental constitutive relations . . . . . . . . . . . . . . . . . . . . . . . .

49

3.3

Return map algorithms (RMA) . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3.1

General formulation

. . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.2

Update algorithm for scalar hardening/softening law . . . . . . . . .

53

3.3.3

Two level return map algorithm . . . . . . . . . . . . . . . . . . . . .

55

3.3.4

Return map algorithm in the cone regions . . . . . . . . . . . . . . .

56

3.4

Consistent tangent moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.5

Numerical analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.5.1

Convergence properties of the algorithms used . . . . . . . . . . . . .

62

3.5.2

Accuracy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.5.3

FEM efficiency analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4 Finite Element Method

74

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2

Continuum mechanics

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.2.1

Elastoplastic boundary value problem . . . . . . . . . . . . . . . . . .

75

4.2.2

Displacement formulation . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2.3

Hu-Washizu formulation . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2.4

The Newton-Raphson scheme . . . . . . . . . . . . . . . . . . . . . .

77

Localization analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.3.1

Diffuse failure analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.3.2

Localized failure analysis . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3.2.1

Uniaxial tension test . . . . . . . . . . . . . . . . . . . . . .

84

4.3.2.2

Uniaxial compression test . . . . . . . . . . . . . . . . . . .

85

4.3

CONTENTS

4.3.3

iii 4.3.2.3

Confined compression test . . . . . . . . . . . . . . . . . . .

87

4.3.2.4

Plane strain test . . . . . . . . . . . . . . . . . . . . . . . .

88

4.3.2.5

Unconfined shear test . . . . . . . . . . . . . . . . . . . . .

90

4.3.2.6

Confined shear test . . . . . . . . . . . . . . . . . . . . . . .

91

4.3.2.7

Simple shear test . . . . . . . . . . . . . . . . . . . . . . . .

94

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5 Numerical Simulations 5.1

5.2

5.3

Plane strain test

97

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.1.1

Geometric dimensions and mesh generation

. . . . . . . . . . . . . . 100

5.1.2

Numerical results for the single-surface model . . . . . . . . . . . . . 101

5.1.3

Numerical results for the multi-surface model . . . . . . . . . . . . . 107

5.1.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Cylinder splitting test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.1

Geometric dimensions and material parameters . . . . . . . . . . . . 110

5.2.2

Numerical results for the ELM: convergence study . . . . . . . . . . . 111

5.2.3

Numerical results for the ELM: model parameter study . . . . . . . . 114

Fastening systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.3.2

Reformulation of fictitious crack concept for axisymmetric problems . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3.3

Pull-out test I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.4

5.3.5

5.3.3.1

Geometric dimensions and material parameters . . . . . . . 119

5.3.3.2

Numerical study I: influence of material model . . . . . . . . 119

5.3.3.3

Numerical study II: influence of boundary condition . . . . . 125

5.3.3.4

Numerical study III: parameters of the ELM . . . . . . . . . 127

5.3.3.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Pull-out test II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3.4.1

Geometric dimensions and material parameters . . . . . . . 132

5.3.4.2

Numerical study I: variation of a/d-ratio . . . . . . . . . . . 133

5.3.4.3

Numerical study II: variation of discretization . . . . . . . . 137

5.3.4.4

Numerical study III: variation of material model

5.3.4.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

High strength undercut anchor

. . . . . . 139

. . . . . . . . . . . . . . . . . . . . . 143

CONTENTS

iv 5.3.5.1

Geometric dimensions and material properties . . . . . . . . 144

5.3.5.2

Installation of the undercut anchor . . . . . . . . . . . . . . 144

5.3.5.3

Numerical results single-surface model . . . . . . . . . . . . 151

5.3.5.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Summary and Conclusions

157

A Transformation to Principal Axes

170

B Algorithmic Aspects

174

B.1 Update eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 B.2 Picard iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 B.3 Local tangent moduli: standard regions . . . . . . . . . . . . . . . . . . . . . 177 B.4 Local tangent moduli: cone regions . . . . . . . . . . . . . . . . . . . . . . . 180 B.5 Definition of error regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 C Derivatives of the ELM

184

C.1 Invariants of the stress tensor σ . . . . . . . . . . . . . . . . . . . . . . . . . 184 C.2 Derivatives of the invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 C.3 Yield function and yield potential . . . . . . . . . . . . . . . . . . . . . . . . 185 C.4 First derivatives of f and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 C.5 Second derivatives of f and Q . . . . . . . . . . . . . . . . . . . . . . . . . . 187 D Parametrization of Deviatoric Shape Function

189

Chapter

1

Introduction and Scope of Work For numerical simulations in modern structural engineering realistic material models are needed. As far as modeling of concrete structures is concerned, the complex material behavior requires special considerations. Concrete failure is governed by different degradation processes within the cement matrix-aggregate composite. In uniaxial tension experiments (Hurlbut, 1985), highly localized tensile cracks result in brittle failure. In the low confined compression regime, a region of transition from brittle to ductile fracture exists, separating brittle softening behavior from ductile failure regimes with little or no degradation of strength (see, e.g., Hurlbut (1985), Smith (1987)). Failure of concrete subjected to high triaxial stress states, however, is characterized by compaction of micro-pores (see Baˇzant et al. (1986)). High triaxial compression tests involving large strains and rotations can be found in (Brocca and Baˇzant, 2001). The rapid development of efficient mathematical algorithms and the increasing availability of powerful computer resources in the last decades have facilitated the development of realistic constitutive material models. The Finite Element Method (FEM) together with highly sophisticated constitutive models have become an indispensable tool in structural engineering for the prediction of the ultimate load and the corresponding failure mechanism. The bulk of existing constitutive models of concrete is designed and calibrated for the mathematical description of concrete under moderately large stresses. Certain engineering applications, such as, e.g., the anchorage of tendons in prestressed concrete structures, however, are characterized by highly concentrated loads. In order to cover the whole response spectrum of concrete under triaxial compression, two elasto-plasticity models have been extended to account for high compressive stress states.

Introduction and Scope of Work

1.1

1.1: Literature review and scope of work 2

Literature review and scope of work

Because of the large number of constitutive models for concrete only a broad and general overview of the main classes of constitutive theories, with special attention to those which can provide a basis for modeling of damage and fracture will be given. Attention is restricted to time-independent models for monotonic loading. A structure or specimen to be analyzed is usually considered as an assembly of certain elementary entities. The purpose of constitutive modeling is to describe the link between the deformation of these entities and the forces transmitted by them. Continuum models are represented by stress-strain laws or, if bending is involved, by generalized moment-curvature relationships. On the other hand, discrete models postulate relations between forces and relative displacements, whereas mixed models deal with a stress-strain law for the continuum part and a traction-separation law for the discontinuous part. Basically continuum models can be subdivided into nonlinear elastic models, elasto-plastic models, and damage models. Nonlinear elastic models are usually based on hyperelasticity, where the stress-strain relations are derived from an elastic potential, or hypoelasticity, characterized by an incremental form of these equations. The term ”elastic” refers to the incremental reversibility of these equations. Elasto-plasticity models are appealing for constitutive modeling. The theory of plasticity was first developed for metals. It proved to be so powerful that it was also applied to concrete (Chen, 1982). It provides a flexible and reliable basis for the description of nonlinear material behavior including path-dependent behavior, unloading and softening. Its most widely used form is the flow theory of plasticity formulated in stress space. This formulation will be discussed in detail in Chapter 2. Examples of elasto-plastic concrete models formulated in this manner can be found in work by Pramono and Willam (1989a), Etse and Willam (1994), Kang (1997). Alternative approaches include the formulation in the strain space (Pekau and Liu, 1992), the deformation (or total) theory of plasticity (Feenstra, 1993), the endochronic theory (Valanis, 1999), and hypoplasticity (Kolymbas, 1991). In plasticity theories the strain is usually decomposed into an elastic and a plastic part. In these theories it is assumed that the elastic stiffness remains constant. In theories based on the notion of damage irreversible processes leading to a progressive degradation of the stiffness moduli are described. Pioneering work in the area of continuum damage mechanics was done by Dougill (1979) and Kachanov (1986). The simplest version of the isotropic damage model presents the damage stiffness tensor as a scalar multiple of the initial elastic stiffness tensor, i.e., damage is characterized by a single scalar parameter. A general isotropic damage model should contain two scalar parameters corresponding to two independent elastic constants of standard isotropic elasticity. More refined theories take the anisotropic character of damage into account, by means of introducing a family of vectors (Krajcinovic and Fonseka, 1981), a second-order tensor (Papa and Taliercia, 1996), a fourth-order tensor (Meyer and Peng, 1997), or even eight-order tensors (Simo and Ju, 1987) (Ju, 1990).

Introduction and Scope of Work

1.1: Literature review and scope of work 3

Because nonlinear concrete behavior is a combination of damage and plastic slip, some researchers combined plasticity and damage models (see , e.g., Lackner (1995), Burlion (1997)). All classes of continuum models mentioned so far are of tensorial character in the sense of establishing a direct relationship between the strain tensor and the stress tensor, satisfying the requirement of frame indifference. Microplane models, however, postulate the relationship between vectors obtained by projecting the stress and strain tensor onto a plane of a given orientation using the principal of virtual work in the form of static or kinematic constraints. Tensorial stress-strain laws are then obtained by averaging over all possible orientations of the microplane. Microplane models used for concrete research can be found in Baˇzant and Oh (1985) Baˇzant and Prat (1988) Oˇzbolt (2001). In the following, the enhancement of the continuum description by displacement discontinuities corresponding to highly localized deformation patterns will be referred to as mixed models. The part of the body that remains continuous is described by a stress-strain law, whereas for consideration of internal discontinuities it is necessary to postulate an initiation criterion and a propagation criterion. A prominent example is the theory of linear elastic fracture mechanics, which deals with pre-existing cracks or notches with sharp tips. Propagation starts when the stress singularity at a tip reaches a critical level. The direction of propagation can be defined by various criteria based, e.g., on the maximum tangential stress, maximum energy release rate, or minimum strain energy density (Shah et al., 1995). The fictitious crack model of Hillerborg et al. (1976) introduced a traction-separation law that governs the progressive loss of cohesion across the crack line. In the discrete crack ˇ approach (Ngo and Scordelis, 1967) (Cervenka, 1994), cracks are modeled as discontinuities between finite elements representing the discretization of the structure. Unless the crack trajectory is known in advance, such an approach requires frequent remeshing. However, displacement discontinuities can also be placed into the interior of finite elements. This idea has been exploited in the embedded crack approach (Dvorkin et al., 1988) (Simo et al., 1993) (Tano et al., 1998) (Wells, 2001). In contrast to models that start from a constitutive description of a continuous medium and discretize the governing equations by the finite element method (or other discretization techniques), discrete models are assemblies of elementary entities of a finite size such as bars, beams, spheres or springs. The elementary entities often correspond to a macroscopic structural element such as a reinforced concrete member of a frame building. This choice, however, is not restricted to bars and beams. For example, the distinct element method (Cundall, 1990) originally dealt with rigid particles that interact by friction. Under the notion of particle models, this approach was extended to the study of microstructure and crack growth in cohesive geomaterials (Baˇzant and Kaxemi, 1990) (M¨ uhlhaus et al., 2001). Particle methods are closely related to lattice models (Vervuurt et al., 1994) (Van Mier et al., 1997), inspired by simulation techniques used in physics of distorted materials. The probabilistic model (Rossi et al., 1996) could also be interpreted as a particle-type model with triangular

Introduction and Scope of Work

1.2: Contents 4

elastic particles connected by damageable interfaces. The primary aim of the present thesis is the development of realistic concrete models for the description of concrete under a broad range of loading states with special emphasis on confined compressive stress states, applicable to ultimate load analyses on the structural level together with failure mode determination. The objectives of this thesis are: • investigation and development of material models for concrete subjected to triaxial stress states, • algorithmic formulation of the proposed concrete models in an efficient and robust manner, leading to a stable computation on the structural level, • use of localization analysis for the proposed models in order to study the influence of material and model parameters on the failure mode, • application of the concrete models to nonlinear FE-calculations with special emphasis on ultimate load analysis together with detection of the failure mode of anchor devices installed in plain concrete structures.

1.2

Contents

Chapter 2 contains a concise summary of constitutive modeling of plain concrete and a validation of models predominantly used for triaxial stress states. Based on this validation two elasto-plasticity models are proposed. Calibration and re-analyses of experimental data is also dealt with in this chapter. A review of the algorithmic formulation used for integration of the governing equations within each time increment of the incremental - iterative procedure of the FEM is given in Chapter 3. Algorithmic properties, e.g., convergence behavior, accuracy and efficiency are investigated. In Chapter 4, the fundamentals of the FEM and the localization properties of the proposed models on the constitutive level are described. In Chapter 5, the performance of the proposed constitutive models is demonstrated by means of five structural simulations. They consist of a concrete panel under plane strain conditions, a cylinder splitting test and three simulations dealing with anchor devices. The work is completed by summary and conclusions.

Chapter

2

Constitutive Models for Concrete 2.1

Introduction

This section is intended to give some insight about the possibilities of constitutive modeling of concrete. The description, analysis and control of physical quantities of concrete can be established in several ways. On the one hand, experiments and measurements are used to evaluate physical quantities directly on the object. On the other hand, numerical simulations of mathematical models are performed with the aid of computer programs. The fundamentals of these models are theories for the description of specific phenomena and their physical relations. Starting with a general description of the modeling process, modeling length scales and a comprehensive description of experimental observations of concrete, selected material models predominantly developed for the description of concrete behavior under triaxial compressive stress states will be investigated. Based on this investigation two material models for plain concrete will be proposed. In Section 2.2, the theory of multi-surface plasticity together with the basic concepts of thermodynamics will be presented. A detailed description of the material models will be given in Section 2.3. The proposed models will be calibrated and validated in Section 2.4. A substantial part of the constitutive models proposed in this chapter was published in the open literature (Pivonka and Mang, 1999a) (Pivonka and Mang, 1999b) (Pivonka et al., 2000).

2.1.1

Remarks on the modeling process

The term model has different meanings. In natural science and mathematics, where models and investigations of models play a central role, this term is understood as follows ¨ (Uberhuber, 1995a): A model is an artificially created object, representing a simplification of the main characteristics, structures and functions of the investigated object (original). Therefore, the process to obtain information is facilitated by means of the model. Figure

problem

model subject model behavior

model assumptions

information

analogies

model

solution of application problem

mathematical problem numerical

(b) application

(a)

original

2.1: Introduction 6

analytical

Constitutive Models for Concrete

interpretation mathematical solution

Figure 2.1: (a) relations of modeling process and (b) procedure from application problem to numerical solution 2.1(a) schematically represents the relations between the original, the model and the model subject (programmer of the model or model user). In order to obtain a numerical solution of an application problem, different modeling phases are employed. The procedure from the application problem to the numerical solution is schematically shown in Figure 2.1(b). The description, analysis and control of objects used in various scientific disciplines can be either made by experiments and measurements, where the relevant quantities are obtained from the object itself, or by simulations with the aid of computer programs. Depending on the expenses and the practical feasibility, one of the proposed paths will be taken.

2.1.2

Modeling length scale

Depending on the observation scale, concrete shows different physical and geometrical properties. Taking into account the chosen scale the structural inhomogeneities play a more or less dominant role and thus determine the condition under which concrete can be regarded as homogeneous. This is accompanied by the fact that, as the resolution length at a specific level is decreasing, the internal material structure is less identifiable up to the point when it is considered to be continuous. The observed size scales for concrete are typically subdivided into hierarchical levels, such as the atomic, micro-, meso- and macrolevel (see, e.g., D’Addetta et al. (2001)). The basic question regarding the mathematical description of concrete focuses on how the constitutive behavior can be described for different size scales. The range of applicability of different simulation models is directly related to the observation scale, as can be seen in Figure 2.2. Physical processes of the higher scale levels are governed by processes at lower scale levels. Material modeling on the macrolevel is often denoted as phenomenological constitutive modeling.

2.1.3

Experimental observations and conclusions

This survey intends to give an overall view on the basic Al properties of concrete under multiaxial loading conditions with special emphasis on compressive loading. A detailed literature review of concrete subjected to multiaxial loading is given in Van Geel (1995).

Constitutive Models for Concrete

2.1: Introduction 7 microlevel

atomistic level 10−8

10−7

10−6

10−5

10−4













































10−3

10−2

macrolevel

mesolevel 10−1

100

101

102

103 [m]

homogeneity

discontinuity atomistic molecular lattice models dynamics models discrete

particle microplane plasticity models models damage continuous

Figure 2.2: Scale levels of constitutive modeling according to D’Addetta et al. (2001) For an extensive description of uniaxial loading conditions see, e.g., Hurlbut (1985), Vonk (1992), and Lee and Willam (1997). Concrete can generally be regarded as a composite material made of cement, aggregates, and water. After chemical hardening the material consists of a mortar matrix including randomly distributed aggregates. In the following, all relations are considered at the macrolevel. Hence, they have been obtained by homogenization processes. While the stress-strain relation of both the mortar and the aggregate material (sand, gravel) is more or less linear up to the peak strength and brittle in the post-peak branch, concrete as a composite material shows pronounced nonlinear behavior even at low loading levels. After reaching the peak load, a descending branch can be observed under displacement control. This difference in the stress-strain behavior is caused by cracking at the microlevel. Similar to uniaxial tests, the relation between volume changes and cracking at the microlevel, and the one between stresses and strains determines the characteristic properties of concrete under multiaxial loading. Kotsovos and Newman (1977) distinguished three transitions in the stress-strain curve obtained from multiaxial tests. The pre-peak behavior of standard triaxial tests can be subdivided into an initial cracking stress level (see Kotsovos and Newman (1977)), where concrete behavior is similar in loading and unloading indicating elastic behavior. This first transition is also often denoted as ’onset of stable fracture propagation’ (OSFP). The second transition was denoted as final breakdown or ’onset of unstable fracture propagation’ (OUFP), which corresponds to the point of minimum volume change in the volume change diagram. Finally, the last transition was called ultimate stress level (peak stress), which corresponds to the bearable load often denoted as failure load (failure surface). Depending on the amount of lateral confinement, very large strains can occur, indicating a highly deformed specimen. Many researchers concluded that the peak stress and peak strain increase with increasing confinement (Jamet et al., 1984) (Smith, 1987). In general, it can be stated that in multiaxial tests the critical stress level (point of minimum

Constitutive Models for Concrete

2.1: Introduction 8 axial stress σ3 [N/mm2 ] experiment -80 σ1 =13.79 [Hurlbut, 1985] σ3 lateral -60 σ1 =6.89 stress: 2 σ1 [N/mm ]

r fcu

10

compressive meridian tensile meridian rc rt

8

Chinn Mills Richart Balmer

6

-40

4 2

rc

r = 2J2 p = I1√/3

rt

3p

fcu

0 0

-2

-4

(a)

-6

-8

-10

σ1 =3.44

σ3

√

-20

σ1 =0

lateral strain ε1 0.020

0.010

0

-0.010

σ1 =0.69 axial strain ε3 -0.020

(b)

Figure 2.3: Triaxial concrete experiments: (a) comparison of failure envelopes of the Leon criterion with test data (Kang, 1997), (b) stress-strain diagrams according to Smith (1987) for different levels of confinement volume change) is reached at relatively higher stress levels than in the uniaxial case. Figure 2.3(a) shows a comparison of the Leon failure envelopes with several triaxial test data (see Kang (1997)). From Figure 2.3(a) it can be seen that shape functions connecting the tensile and the compressive meridian generally deviate from circular form (Lode-angle dependence). Figure 2.3(b) shows stress-strain diagrams obtained from Smith (1987) for different levels of confinement. A special case of multiaxial test is the biaxial one. Detailed descriptions of such tests are given in Kupfer (1973), Gerstle et al. (1980) and including post-peak behavior, in Van Mier (1984). Special attention has to be paid to the post-peak behavior of concrete, often denoted as softening. The reason for concrete softening can be deduced from continuous crack-growth at the microlevel. Localization of deformations means that further deformations are concentrated in the vicinity of cracks at the macrolevel, while the remaining parts of the specimen exhibit decreasing deformations caused by unloading. The question if softening can be modeled within the framework of continuum mechanics arises. Generally, the theory of a homogeneous continuum is applicable to concrete when the size of the concrete volume is much greater than the characteristic size of the heterogeneity of the material. During cracking at the microlevel the theory of the homogeneous continuum is still applicable because these cracks are at the scale level of the heterogeneity of concrete. When cracks at the macrolevel start to grow, the scale level of the cracks rapidly rises to the scale level of the structure. Thus continuum theory is no longer applicable. Opening cracks have to be taken into account as discontinuities together with the continuum determining the behavior of the structure. The response of the structure becomes dependent on the size. Thus softening has to be viewed as a structural property rather than a material property (see, e.g., Kotsovos (1984)). The post-peak behavior of multiaxial tests was first investigated by Van Mier (1984). He concluded that the stress-strain behavior was strongly influenced by the smallest principal

Constitutive Models for Concrete

2.1: Introduction 9

stress and the intermediate principal stress. Based on these experiments two different failure modes are distinguished: • planar failure modes: A pronounced shear band fracture mode (localization of deformations) was observed in stress regions near the tensile meridian when a preferential direction of failure was present (two different confining stresses or plane strain tests). The presence of one larger tensile deformation results in this kind of failure mode; • cylindrical failure modes: A distributed failure mode was observed in stress regions near the compression meridian. It was caused by mutually crossing shear bands. The presence of two large tensile deformations results in this kind of failure mode. Van Mier also related the volume change of several tests to the observed stress-strain and failure behavior. The volume change curves appeared to be closely related to the distinguished failure modes. Extensive investigations concerning the softening behavior of concrete in compression was made by Vonk (1992), where the softening process, the influence of the boundary conditions and of the size effect were studied. Investigations concerning the influence of the confining pressure on the the post-peak behavior of concrete were reported by Jamet et al. (1984) and Smith (1987). These tests confirmed that at low confinement levels softening behavior was observed, whereas at higher confinement levels hardening occurred. This behavior was also reflected by the fracture modes where at low confining pressure shear bands were observed. At medium confinement levels, the specimens showed a smeared fracture pattern, and at higher confinement levels no visible damage could be detected at all. The brittle-ductile transition of concrete under triaxial compressive loading was investigated by Hurlbut (1985). Figure 2.4(a) shows the different fracture modes obtained for various compressive meridian

σ2 short inclined shear planes

σ1

σ1

ε1 σ2

transition tensile-shear band rupture

J2

(σ1 < σ2 = σ3 )

σ3 = βσ1

σ3 σ1

σ3

σ2 ε2

(a)

residual strength envelope

8 σ1

brittle softening

4

σ1

σ2

transition point (T P )

σ2

pronounced shear bands

tensile rupture

maximum strength envelope

(σ1 = σ2 < σ3 )

ε1

ε2 = 0

12

tensile meridian

continuous hardening

I1 0

0

-10

-20

-30

(b)

Figure 2.4: Triaxial concrete experiments: (a) failure modes obtained by Van Mier (1984) for different loading paths and (b) failure surface and residual strength envelope obtained by Hurlbut (1985)

Constitutive Models for Concrete

2.1: Introduction 10

loading cases according to Van Mier (1984). Figure 2.4(b) shows the failure surface together with the residual strength envelope indicating the transition from brittle to ductile post-peak behavior. Apart from these standard triaxial compression tests, only a few researchers investigated concrete behavior under very high compressive stress states. Among triaxial compression tests reported in the literature, the highest compressive stresses where apparently reached by Chinn and Zimmermann (1965), Jamet et al. (1984), Baˇzant et al. (1986) and recently by Burlion et al. (1997). Figure 2.5(a) shows the axial stress - axial strain behavior as observed in confined compression tests of small cylindrical concrete specimens loaded axially up to 2068 MPa, reported by Baˇzant et al. (1986). Figure 2.5(b) shows the results of Burlion (1997). In this investigation the differences between hydrostatic compression tests and oedometric tests were studied1 . (a)

axial stress σ1 [MPa]

-2200

(b) -600 -500

-1800

hydrostatic pressure p [MPa] hydrostatic test oedometric test

-400

-1400

-300

-1000

-200

-600 -200 0 0

0.04

axial -100 strain ε1 0 0.08 0.12 0

volumetric strain εv 0.10

0.20

0.30

Figure 2.5: Triaxial compression tests: (a) uniaxial strain test according to Baˇzant et al. (1986) and (b) oedometric and hydrostatic compression test obtained by Burlion (1997) Burlion concluded from the experiments that the elastic-plastic compaction of concrete is characterized by elastic- and plastic hardening. Elastic hardening corresponds to the increase of the unloading stiffness (not very pronounced) with increasing compaction, while plastic hardening is first linear and then nonlinear (pronounced) with an increasing compression modulus. Comparison between oedometric and hydrostatic tests showed very different response behavior. The oedometric test produces more stiffening and hardening compared to the hydrostatic compression test. An analysis with an electronic scanning microscope showed substantial differences in the micro structure of the material after compaction. Within the oedometric test a reorganization of the aggregate skeleton in the material is possible. It provokes a steeper drop of porosity but also micro-cracking in the material. Micro-cracks are perpendicular to the axis of the maximum compressive stress caused by deviatoric stresses 1

For the oedometric test, concrete specimens were fit into very stiff steel vessels in order to obtain almost uniaxial strain conditions.

Constitutive Models for Concrete

2.1: Introduction 11

emerging from the development of plastic strains. The granular organization of the material is preserved without totally closing the initial porosity of the material in the hydrostatic test. Figure 2.6(a) schematically shows the different loading paths for oedometric and hydrostatic compression tests. The deformation processes of the matrix-aggregate composite material obtained from these tests are shown in Figure 2.6(b). (a)

hydrostatic oedometric loading loading −σ1 σ1

=

σ2

=

−σ3

(b)

σ3

Oedometric test

hydrostatic test

−σ2 Figure 2.6: Triaxial compression tests: (a) schematic representation of the loading paths and (b) matrix-aggregate deformation process for oedometric and hydrostatic compression tests From the current state of triaxial experimentation it can be concluded that macroscopic quantities such as strength, stiffness, and cracks are the main characteristics of concrete. For an appropriate description of strength, the pronounced pressure sensitivity together with the influence of the Lode angle θ must be considered. The concrete stiffness is characterized by pressure sensitivity, hydrostatic-deviatoric coupling, brittle to ductile transition and elasticplastic interactions. Finally, for the description of concrete cracking it can be stated that softening behavior is a structural phenomenon rather than a material property. Strainsoftening diminishes with increasing confinement. Furthermore, the dependence of the failure mode on the loading path has to be considered.

2.1.4

Validation of constitutive models

This subsection provides the basis for the development of two constitutive models for concrete applicable to a broad range of loading states with special emphasis on compressive loading. First, a validation of some selected constitutive models will be given. In view of modeling of concrete subjected to compressive loading, continuum models seem most promising. As described in Chapter 1, mixed and discrete models are predominantly used for the description of cracking of concrete. Thus, the following investigation and validation of concrete models will be restricted to continuum models. Four models formulated within the framework of continuum theory will be described. The first model is a plasticity model proposed by Etse

Constitutive Models for Concrete

2.1: Introduction 12

and Willam (1994). The evolution of the loading surfaces in the pre-peak regime is shown in Figure 2.7(a). The second model was developed by Voyiadjis and Abu-Lebdeh (1993). It is based on damage theory. The loading surfaces are schematically shown in Figure 2.7(b). The microplane model, proposed by Baˇzant et al. (1996), is the third investigated model2 . A schematic representation of the constitutive relations prescribed on a microplane is shown in Figure 2.7(c). One of the latest investigations considering concrete under high compressive stress states is a combined plasticity-damage model proposed by Burlion et al. (1998). The loading surfaces of the respective model are shown in Figure 2.7(d). √

2J2 failure surface hardening

√ 2J2 mixed loading compressive loading −I /3 1

−I1 /3 (a) σij integral formula

kinematic constraint

εij

ei

(b)

microplane law

(c)

q

J2 /σM f∗ = 0

f ∗ = 0.3

I1 /σM

si

(d)

Figure 2.7: Selected material models for concrete: (a) plasticity model (Etse and Willam, 1994), (b) damage model (Voyiadjis and Abu-Lebdeh, 1993), (c) microplane model (Baˇzant et al., 1996) and (d) plasticity-damage model (Burlion et al., 1998)

2

The proposed microplane model was continuously improved over several years. A historic review of the development of microplane models for concrete can by found in (Jir´ asek, 1999)

Constitutive Models for Concrete

2.1: Introduction 13

For each model a short description of the underlying theory together with the control mechanism of the model will be given. Criteria for the validation of the considered models are the model behavior under different modes of loading. Further, the number of employed material and model parameters and the application to structural analysis were considered. • Plasticity model (M1): The model proposed by Etse and Willam (1994) is an isotropic plasticity model. The loading surfaces are described as fEW (p, r, θ, k, c) =

  

(1 − k)

"

p rg(θ) +√ fcu 6fcu

#2

+

s

2 3 rg(θ) 

2 fcu 

k2 m rg(θ) + − k 2 c = 0, p+ √ fcu 6 "

#

(2.1)

where p, r and θ are invariants of the stress tensor. The function g(θ) in Equation (2.1) defines the deviatoric shape of the yield surface. fcu is the uniaxial compressive strength. The normalized strength variable k controls the pre-peak regime, while the cohesion parameter c controls the post-peak behavior. The parameter m is called frictional parameter and defines the shape of the yield surface in the meridian plane of the softening regime. The dependence on the confining stress is introduced in the evolution equations for the hardening and softening variables. • Damage model (M2): The damage model proposed by Voyiadjis and Abu-Lebdeh (1993) uses the bounding surface concept. The bounding surface FV A and the loading surface fV A are described as follows q

¯ = aJ2 + λ J2 + bI1 − g(D) ¯ =0 FV A (σ, D) q

¯ = aJ2 + λK J2 + K 2 bI1 − K 2 g(D) ¯ = 0, fV A (σ, D)

(2.2)

¯ denotes the damage parameter and g(D) ¯ is a damage accumulation function. where D The parameters a, b, λ are constants computed according to Voyiadjis and Abu-Lebdeh (1993). I1 and J2 are invariants of the stress tensor and the deviatoric stress tensor, respectively. K in Equation (2.2) denotes the shape factor. The influence of damage on the compressive and tensile behavior of concrete is considered by means of different ¯ t ) and a comdamage kinematics. The damage behavior is described by a tensile (D ¯ c ) damage parameter together with two different damage loading surfaces. pressive (D The use of a bounding surface allows description of cyclic loading paths. • Microplane model (M3): The microplane model proposed by Baˇzant et al. (1996) is an improvement of previously developed microplane models, using the new concept of stress-strain boundaries. While in the classical approach, the constitutive model is defined by algebraic or differential relations between the stress tensor σ and the strain tensor ε, the microplane approach

Constitutive Models for Concrete

2.1: Introduction 14

is defined by relations between stresses and strains acting on a plane of arbitrary orientation the so-called microplane. The model uses the kinematic constraint, which defines the strain vector e on an arbitrary microplane with unit normal n as e = ε · n,

(2.3)

where the dot defines a tensor contraction. The microplane stress vector, s, is defined as the work-conjugate variable of the microplane strain vector, e. A formula linking the microplane stress vector to the macroscopic stress tensor follows from the principle of virtual work as (see Baˇzant et al. (1996)) σij =

3 Z (σN Nij + σM Mij + σL Lij )dΩ(N ), 2π Ω

(2.4)

where σN is the normal stress on the microplane and σM , σL are the shear components in a plane normal to n. Nij , Mij and Lij are symmetric tensors related to the direction of the microplane. For the formulation of constitutive relations on the microplanes the microplane strain vector e is decomposed into its normal part εN and its tangential part eT . Further, the mean normal strain εN is decomposed into a volumetric (εV = εkk /3) and a deviatoric part (εD = εN − εV ). By means of fitting of various types of test data for concrete, the functions and parameters defining the constitutive relations were identified. Because of the finite number of microplanes used for the numerical integration, the stress-strain diagram generally exhibits slope discontinuities (see Jir´asek (1999)). This aspect is improved in the most recent modification (Baˇzant et al., 2000), which also uses a thermodynamically consistent formulation. • Plasticity-damage model (M4): The model recently proposed by Burlion et al. (1998) is a combination of plasticity and damage. The model uses the yield function according to Needleman and Tvergaard (1984), which is a modification of the yield function proposed by Gurson (see, e.g., Mahnken (1999)). It is described as fN T (σ, σM , f ∗ ) =

I1 3J2 ) − (1 + (q3 f ∗ )2 ) = 0, + 2q1 f ∗ cosh (q2 2 σM 2σM

(2.5)

where I1 and J2 are invariants of the stress tensor. σM is the equivalent yield stress in the matrix and f ∗ represents the volume fraction of voids. q1 , q2 and q3 are model parameters. In the model, the decrease of the void volume fraction f ∗ is controlled by the plastic flow. Similar to the Gurson’s model the void evolution is controlled by the irreversible volumetric strain (see, e.g., Mahnken (1999)). While f ∗ increases with void development in tension, it decreases with void closure in compression. In the model damage growth is associated to the evolution of porosity and to the evolution of micro-cracking at the same time (plasticity-damage coupling). From the algorithmic point of view, an explicit Forward Euler integration scheme has been used for solving the evolution equations, thus only small step sizes should be applied.

Constitutive Models for Concrete

2.2: Multi-surface plasticity

15

Table 2.1 gives an overview of the performance of the investigated models on the integration point level and on the structural level .3 applications at the integration point level

M1

M2

uniaxial tensile loading uniaxial compressive loading confined compressive loading hydrostaticoedometric loading number of parameters

good

good

good good

–

good

good good

– – 15

– – > 10

applications at the structural level

few

no

good moderate

M3

M4

good good – good good good 16 15 no3

no

Table 2.1: Validation of constitutive models for concrete at the integration pointand the structural level

2.2

Multi-surface plasticity

2.2.1

Thermodynamic framework

The thermodynamic approach to the description of a continuum starts from the assumption that the current state of the material can be uniquely characterized by a suitable selected set of state variables (local state postulate). Each state variable is linked to a physical phenomenon on the microlevel of the material description. Some of the state variables, e.g., temperature or strain, can be observed and controlled on the macroscopic level; they are usually called the observable variables. The remaining state variables, e.g., plastic strain or damage, characterize the internal changes of the material and are called internal variables. They can sometimes be measured but they cannot be controlled from the outside. The following derivation is restricted to the geometrically linear theory, leading to an additive decomposition of the strain tensor into an elastic and a plastic part, ε = εe + εp . 3

(2.6)

The increase of computer power and developments concerning the algorithmic properties of microplane models have recently rendered the model applicable on the structural level (see, e.g., Oˇzbolt (2001)).

Constitutive Models for Concrete

2.2: Multi-surface plasticity

16

For the description of phenomena at the microlevel, so-called internal variables α are introduced in the material model. They are used to describe the micro structural change of the material. The energetically conjugated thermodynamic quantities are the hardening/softening forces q. They are related to the internal variables via the state equation q = q(α). The hardening forces represent the actual strength of the material, defining the space of admissible stress states, CE : σ ∈ CE ⇔ fk = fk (σ, q(α)) ≤ 0 ∀ k ∈ [1, 2, . . . , N ],

(2.7)

where σ represents the stress tensor and fk denotes the k-th yield function. N denotes the number of employed yield functions. The energetic state of an elementary system can be characterized by the free (Helmholtz) energy per unit volume. For the proposed concrete models Ψ can be expressed as Ψ = Ψ(ε, εp , α).

(2.8)

The strain tensor ε is an external state variable. εp and α denote the tensor of plastic strains and the vector of hardening/softening variables, respectively. With regards to macroscopic modeling, εp and α are linked to irreversible skeleton deformations resulting from microcracking. Concerning the structure of Ψ, an additive split into two parts is made (Ulm and Coussy, 1996) (Ulm, 1998), 1 Ψ = ψ(ε − εp ) + U (α) with ψ(ε − εp ) = (ε − εp ) : C : (ε − εp ), 2

(2.9)

where ψ is the elastic part of the strain energy, which can be recovered macroscopically by unloading. The energy U is the frozen energy resulting either from hardening or softening. It cannot be recovered macroscopically by unloading, but it can be increased by loading and recovered by unloading at the microlevel of the material (see Coussy (1995), Ulm and Coussy ˙ ≥ 0, (1996)). Inserting Equation (2.9) into the Clausius-Duhem inequality, D = σ : ε˙ − Ψ the dissipation function is obtained as (see, e.g., Simo and Hughes (1998)) D = (σ −

∂Ψ ∂Ψ ∂Ψ ˙ ≥ 0. ) : ε˙ − p : ε˙p − :α ˙ ∂ε ∂ε ∂α

(2.10)

˙ = 0 leads the elastic The assumption of elastic material response, i.e., D = 0, ε˙ p = 0 and α material law as σ=

∂Ψ = C : (ε − εp ). ∂ε

(2.11)

Constitutive Models for Concrete

2.2: Multi-surface plasticity

17

The evolution equations of plasticity are obtained by application of the principle of maximum plastic dissipation to the remaining part of D. The yield condition (2.7), f (σ, q) = 0, is considered by a Lagrangian multiplier γ, ˙ with γ˙ ≥ 0, giving the following extreme value 4 problem with a constraint condition : ˙ + γf L = −D + γf ˙ (σ, q) = −σ : ε˙ p − qα ˙ (σ, q)

→ stationary,

(2.12)

where ∂U/∂α = −q. Solving the extremal problem for L, using ∂σ L = 0 and ∂q L = 0, leads to the evolution equations for the plastic strain tensor and the internal hardening/softening variables: ε˙ p = γ˙

∂f ∂f ˙ = γ˙ = γ˙ n and α . ∂σ ∂q

(2.13)

The evolution equations in the format of Equation (2.13) are often denoted as associative flow rule and associative hardening law, respectively. A more general formulation of the evolution equations reads ε˙ p = γ˙

∂Q ∂H ˙ = γ˙ = γ˙ m and α , ∂σ ∂q

(2.14)

where Q and H are potentials depending on σ and q. The flow rule and hardening/softening law in Equation (2.14) are denoted as non-associative. Plasticity formulations using general potentials Q and H do not necessarily obey the principle of maximum plastic dissipation. ˙ However, for a consistent thermodynamic formulation only a positive dissipation, i.e., D>0 must be guaranteed. Remark 2.2.1 Higher order theories, e.g., non-local and gradient theories (Str¨omberg and Ristinmaa, 1996), (Pamin, 1994) introduce derivatives of state variables with respect to space. Thus, the evolution of an elementary system also depends on states of the neighboring elementary system. The use of such theories, however, implies the introduction of an internal length scale. Problems arising from such formulations are the definition of the internal length parameter, which can not be determined form experiments and the definition and satisfaction of boundary conditions for the internal variables.

2.2.2

Differential consistency

In the following, an analytical expression for the rate of the plastic multiplier γ and the differential constitutive law will be derived. The consistency parameter is assumed to obey 4

In the following derivation the case of single-surface plasticity, i.e., k = 1 is considered. The extension to multi-surface plasticity is straight forward.

Constitutive Models for Concrete

2.2: Multi-surface plasticity

18

the Kuhn-Tucker complementary conditions γ˙ ≥ 0, f (σ, q) ≤ 0, γ˙ f (σ, q) = 0.

(2.15)

In addition to the conditions (2.15), γ˙ satisfies the consistency requirement γ˙ f˙(σ, q) = 0.

(2.16)

The consistency parameter can be computed from Equation (2.16) by evaluating the time derivative of f as ∂f ∂f : σ˙ + · q˙ = 0. f˙ = ∂σ ∂q

(2.17)

Equation (2.17) together with σ˙ = C : (ε˙ − ε˙p ) = C : (ε˙ − γ˙ m)

(2.18)

and Equation (2.14), and use of the chain rule, yields ∂f ∂q ∂H · · = 0. f˙ = n : C : (ε˙ − γ˙ m) + γ˙ ∂q ∂α ∂q

(2.19)

Therefore, the consistency parameter can be expressed as γ˙ =

n : C : ε˙ n : C : ε˙ = . ∂f ∂q ∂H Ep + E n · · +n:C:m − ∂q ∂α ∂q

(2.20)

Inserting Equation (2.20) into Equation (2.18) results in the following expression of the stress rates in terms of the total strain rates: ˙ σ˙ = Cep : ε,

(2.21)

with the so-called elasto-plastic continuum tangent moduli Cep = C −

C:m⊗n:C . Ep + E n

(2.22)

From Equation (2.22) it follows that Cep is symmetric only if an associative flow rule is employed.

Constitutive Models for Concrete

2.3

2.3: Material models for plain concrete 19

Material models for plain concrete

The use of standard plasticity models such as the Mohr-Coulomb model or the DruckerPrager model for the description of concrete is restricted to moderately large stresses. However, because of the relatively simple formulation these models are commonly used in numerical analysis beyond their original range of applicability. The need for realistic material models covering a larger response spectrum of concrete under various stress states and loading paths is evident. In many engineering applications, structures are subjected to different modes of loading. In the following, two kinds of material models will be dealt with. They are referred to as single-surface and multi-surface models. They were developed in order to obtain realistic numerical results for concrete subjected to a wide range of triaxial stress states.

2.3.1

Single-surface plasticity model

From the single-surface models proposed in the open literature, the Extended Leon Model (ELM) (Etse and Willam, 1996) was chosen. The loading surface of the ELM was designed such that good agreement between numerical results and experimental data was obtained in a wide range of stress states. Figure 2.8 shows the loading surface of the ELM at different loading states.

Figure 2.8: Loading surface of the ELM in principal stress space for different loading states

2.3.1.1

General characteristics of the Extended Leon Model (ELM)

The overall concrete response of the ELM is divided into three regions, i.e., an initial linear elastic regime, a non-linear hardening pre-peak, and a non-linear softening post-peak regime. Unloading follows the initial elastic behavior.

Constitutive Models for Concrete

2.3: Material models for plain concrete 20

According to this model, the loading surface is formulated by means of the hydrostatic pressure p, the deviatoric radius r, the Lode angle θ, and the stress-like internal variables q h and qs :  

q¯h f (p, r, θ; qh , qs ) = 1−  fcu q¯h + fcu

!2

!"

p rg(θ, e) + √ fcu 6fcu

"

#

#2

+

p rg(θ, e) q¯h m(qs ) + √ − fcu fcu 6fcu

s

!2

2

3 rg(θ, e)  2 fcu 

q¯s = 0, ftu

(2.23)

with q¯h = fcy − qh

and q¯s = ftu − qs .

(2.24)

fcu and ftu denote the uniaxial compressive and tensile strength, respectively. fcy represents the elastic limit under compressive loading. The deviatoric shape of the loading surface is described by the elliptic function g(θ, e) =

4(1 − e2 ) cos2 θ + (2e − 1)2 q

2(1 − e2 ) cos θ + (2e − 1) 4(1 − e2 ) cos2 θ + 5e2 − 4e

,

(2.25)

where the parameter e = rt /rc is referred to as eccentricity. rt and rc denote the pressuredependent deviatoric strength for θ = 0o and θ = 60o , respectively (see Pramono (1988)). The parameter m(qs ) is called frictional parameter. It defines the slope of the loading surface. The elastic response is bounded by an initial loading surface, which grows isotropically with increasing inelastic deformations. The evolution of the loading surface is controlled by two stress-like internal variables, qh and qs . In the pre-peak regime the material is assumed to exhibit degrading stiffness without localized macro-defects. Degrading stiffness is modelled by means of strain-hardening resulting in an increasing compressive strength q¯h . During hardening, the stress-like softening parameter qs and the frictional parameter m(qs ) remain unchanged: q¯s = ftu

and m(qs ) = mo =

2 2 (fcu − ftu ) . fcu ftu

(2.26)

Softening is initiated when micro-defects at peak, characterized by q¯h = fcu , coalesce into localized macro-defects, i.e., when the concrete starts cracking. Softening is characterized by a decrease of the tensile strength q¯s resulting in an increase of the frictional parameter m(qs ) (for details see Subsection 2.3.1.4). Figures 2.9 and 2.10 show the loading surface of the ELM for the pre-peak (hardening) and the post-peak (softening) regime.

Constitutive Models for Concrete

2.3: Material models for plain concrete 21 r fcu

tension meridian

θ=0

4

q¯h =fcu

q¯h =fcu

3 2 1 -5

-4

-3

-2

-1

q¯h =fcu /10

-1 -2

θ= 34 π

θ= 23 π

-3

compression meridian

q¯h =fcu /10

p fcu

1

-4 (b)

(a)

Figure 2.9: Different locations of the loading surface of the ELM in the prepeak regime for an increase of the compressive strength from q¯h = fcu /10 to q¯h = fcu in consequence of strain-hardening: (a) meridian plane and (b) deviatoric plane 1.5

tension meridian

r fcu

θ=0 q¯s =ftu

1.0 0.5 pT P /fcu -1.0

p fcu

-0.5 -0.5

compression meridian

q¯s =ftu

q¯s =ftu /10

-1.0

TP

q¯s =ftu /10

θ= 34 π

θ= 23 π

-1.5 (a)

(b)

Figure 2.10: Different locations of the loading surface of the ELM in the post-peak regime for a decrease of the tensile strength q¯s from q¯s = ftu to q¯s = ftu /10 in consequence of strain-softening (T P : transition point): (a) meridian plane and (b) deviatoric plane 2.3.1.2

Non-associative flow rule

In elasto-plasticity, the flow rule defines the evolution of the plastic strains. In general, the direction of the plastic strain rates is related to the derivative of the yield surface with respect to the stress tensor (associative flow rule). Hence, the shape of the yield surface defines the plastic response. Smith et al. (1989) have conducted comprehensive experiments to determine the direction of the incremental plastic strains during strain-driven triaxial tests of concrete specimens. The results of this study have clearly shown that the assumption of associativeness is not valid for concrete subjected to triaxial states of stress. Therefore, in

Constitutive Models for Concrete

2.3: Material models for plain concrete 22

the framework of the Extended Leon Model a non-associative flow rule is used to define the direction of the plastic flow. A yield potential Q is introduced to modify the yield function f with respect to its volumetric part. The yield potential has the form q¯h Q(p, r, θ; qh , qs ) = f (p, r, θ; qh , qs ) + fcu

!2

mQ 1 p m(qs ) − m(qs ) fcu fcu

!

= 0,

(2.27)

with the modified frictional parameter mQ = mQ (p),

∂mQ = D exp (ER2 (p)) + F, ∂p

(2.28)

where R(p) =

p − ftu /3 . 2fcu

(2.29)

The parameters D, E, and F are calibrated from measurements of the plastic dilatancy obtained from three different experiments. One uniaxial tension test and one confined compression test each at a low- and a high-confinement level, respectively, are sufficient to determine these parameters. With respect to the deviatoric section, the yield potential Q and the yield function f are coinciding. Hence, the deviatoric components of the plastic strain tensor are governed by an associative law. The gradient of the yield potential is obtained as m=

∂Q ∂p ∂f ∂r ∂f ∂θ ∂Q = + + . ∂σ ∂p ∂σ ∂r ∂σ ∂θ ∂σ

(2.30)

The rate of the plastic strain tensor follows then from Equation (2.14). 2.3.1.3

Non-linear isotropic hardening law

The material behavior is assumed to be isotropic during the entire deformation history. Inelastic deformations occur when the elastic limit, which is defined by the initial loading surface (¯ qh = fcy ), is exceeded. In consequence of strain-hardening, the compressive strength increases until q¯h = fcu , i.e., until the failure surface is reached. The evolution of the loading surface from its initial location (¯ qh = fcy ) to the failure surface (¯ qh = fcu ) is controlled by p strain-hardening based on the equivalent plastic strain  . The rate of the equivalent plastic strain, ˙p , is defined as the Euclidean norm of the tensor of the rate of plastic strains: ˙p =

√

ε˙ p : ε˙ p = γ˙ kmk.

(2.31)

Constitutive Models for Concrete

2.3: Material models for plain concrete 23

The evolution equation for the strain-like internal variable αh governing the hardening behavior is defined as (Etse and Willam, 1994) α˙ h =

1 p ˙ , xh

(2.32)

where the confining pressure is accounted for by the ductility parameter xh . In order to capture concrete behavior under low and high confinement levels, two quadratic polynomials are employed for the definition of xh , yielding xh as a continuous function of the hydrostatic pressure p: xh = xh (p) =

(

Ah (p/fcu )2 + Bh (p/fcu ) + Ch Dh (p/fcu )2 + Eh (p/fcu ) + Fh

for low confinement, for high confinement.

(2.33)

The six coefficients in Equation (2.33) are determined from five experiments at different levels of confinement, together with the continuity condition. The value of αh defines the actual compressive strength of concrete, q¯h . The respective stress-like internal variable qh is expressed by a monotonically decreasing function of αh :  

q

−(fcu − fcy ) αh (2 − αh ) for hardening (αh < 1), qh = qh (αh ) =  −(f − f ) for softening (αh ≥ 1). cu cy

(2.34)

Figure 2.11 shows the dependence of the evolution of compressive strength, q¯h = fcy − qh , on confinement represented by xh . q¯h (αh (p , xh ))

αh = 1

fcu (1) low confinement (2) medium confinement fcy

(3) high confinement p

Figure 2.11: On the influence of confinement on the evolution of the compressive strength q¯h

2.3.1.4

Non-linear isotropic softening law

The most important consequence of material instability in the form of cracking is localization of the deformations. Localization occurs suddenly at a certain point of the loading history when the entire additional deformation is confined in narrow band-shaped parts of the body,

Constitutive Models for Concrete

2.3: Material models for plain concrete 24

while the remaining parts of the body exhibit unloading. Localization is usually accompanied by a decrease of the load-carrying capacity after reaching the peak load. Such a gradual decrease of stiffness and load-carrying capacity with the increase of deformation imposed on the body is called softening. Within the smeared-crack concept, commonly employed in the context of the FEM, softening is described in terms of the respective stress-strain relations. However, this concept is characterized by lack of objectivity of the numerical results with respect to the element size. In Section 4.3 more sophisticated techniques for regularization of the boundary value problem are discussed. The ELM is formulated on the basis of the smeared-crack approach regularized by the fracture energy concept (see Hillerborg et al. (1976), Hurlbut (1985), Baˇzant and Oh (1988), Oliver (1989)). The residual loading surface in the softening regime, fr , is given by 3 fr (p, r, θ) = 2

rg(θ, e) √ 6fcu

!2

+ mr

p rg(θ, e) + √ fcu 6fcu

!

= 0.

(2.35)

fr is obtained from the loading surface given in Equation (2.23) by setting q¯h = fcu , q¯s = 0, and m = mr . The evaluation of the residual frictional parameter mr is based on the location of the so-called transition point (T P ) of brittle to ductile failure (Willam et al., 1989). The location of T P is assumed to remain fixed in the stress space. The transition point T P is equal to the stress point on the failure surface characterized by a confined stress state given as σ1 = σ2 , σ3 = aT P σ1 (Smith, 1987). Inserting of this relation into the failure surface of the ELM leads the following expression for the hydrostatic pressure and the deviatoric radius at the transition point pT P

rT P

q

(2 + aT P )(mo + 4 − 8aT P + 4a2T P + m2o ) = − fcu 6(aT P − 1)2 q

(2 + aT P )(mo + 4 − 8aT P + 4a2T P + m2o ) √ 2 = fcu . 6(aT P + aT P − 2)

(2.36)

From the experiments it was found that for a concrete characterized by a compressive strength of fcu =22 N/mm2 , the value of aT P equals 8. For fcu =35 N/mm2 , aT P =6. The assumption of a fixed location of the transition point leads to an increase of the frictional parameter (m → mr ) for a decreasing tensile strength q¯s in consequence of softening5 . Once the location of T P is known, the residual frictional parameter mr can computed by inserting of Equation (2.36) into (2.35), using θ = 0◦ and rearranging terms as:

mr = 5

(2 + aT P )(mo +

q

4 − 8aT P + 4a2T P + m2o )

2(2 + aT P )

.

In the following a constant value of aT P is assumed, i.e., aT P =8.

(2.37)

Constitutive Models for Concrete

2.3: Material models for plain concrete 25

The function employed for the description of the friction parameter is given as m = m(qs ) =

(

mo mr − (mr − mo )¯ qs /ftu

for hardening (αh < 1), for softening (αh ≥ 1),

(2.38)

with q¯s = ftu − qs . Based on this definition, an intermediate state of the softening surface is defined as 3 f (p, r, θ; qs ) = 2

rg(θ, e) √ 6fcu

!2

!

rg(θ, e) p q¯s + √ = 0. − + m(qs ) fcu ftu 6fcu

(2.39)

The evolution of the softening surface from its initial location (¯ qs = ftu ) to its residual location (¯ qs = 0) is controlled by strain-softening. Therefore, the equivalent plastic strain s is introduced. It can be viewed as the damage metric. Rates of s should only be monitored if the existing micro-cracks are activated. Mathematically, this situation may be expressed by requiring that the tensor of plastic strain rates, ε˙ p , has at least one positive eigenvalue (Ortiz, 1985). Considering the principal components of ε˙ p and introducing the McAuley operator h•i = (• + | • |)/2, the equivalent plastic strain rate can be expressed as ˙s = khε˙ p ik =

q

hε˙ p i : hε˙ p i = γ˙ khmik.

(2.40)

Similar to the isotropic hardening formulation, a strain-like internal variable is introduced as: α˙ s =

1 s ˙ , xs (p)

(2.41)

where xs accounts for the ductile behavior of concrete in softening. xs is defined by a polynomial function of the hydrostatic pressure p: ¯ 4 (p) + Bs R ¯ 2 (p) + 1, xs = xs (p) = As R

(2.42)

¯ with R(p) = (p − ftu /3)/fcu . The value for xs is computed from the hydrostatic pressure at peak load and kept constant for the remaining part of the loading path. Hence, for uniaxial tensile loading characterized by p = ftu /3 at peak load, xs becomes equal to one. The parameters As and Bs are calibrated from low-confined and high-confined compression tests.

Constitutive Models for Concrete

2.3: Material models for plain concrete 26

q¯s (αs (s , xs )) ftu

(3) high confinement (2) medium confinement (1) low confinement s

Figure 2.12: On the influence of confinement on the evolution of the tensile strength q¯s 2.3.1.5

Calibration of the ELM in the context of the fracture energy concept

Softening material behavior leads to a decrease of the tensile strength q¯s . The respective stress-like internal variable, qs , is expressed by an exponential function of αs : qs = qs (αs ) =

(

0 for hardening (αh < 1), n ftu [1 − exp[−(αs /αu ) ]] for softening (αh ≥ 1),

(2.43)

where, according to (Hurlbut, 1985), n = 1. However, n = 1 leads to a discontinuous slope for the transition from the hardening regime (αh < 1) to the softening regime (αh ≥ 1), which was not confirmed by experimental data. The exponent n = 2 is characterized by a continuous slope. For this reason, n = 2 was chosen in the present work. The decrease of the tensile strength q¯s = ftu − qs with increasing s is illustrated in Figure 2.12 for different levels of confinement. In Equation (2.43), αu denotes a calibration parameter. It is computed from the fracture energy of mode-I cracking GIf , representing the energy released as one crack is opening. It is considered as a material parameter. GIf is obtained from GIf =

Z

∞ 0

σdus ,

(2.44)

where us denotes the crack opening displacement and σ is the applied tensile stress. In the context of the smeared-crack approach (see Figure 2.13), the integration over u s is replaced by the respective plastic strain measure s . Rewriting Equation (2.44) and setting σ equal to the tensile strength q¯s , yields GIf

=

Z

∞ 0

s

q¯s `t d = `t

Z

∞ 0

(ftu − qs )ds ,

(2.45)

where `t represents the width of the crack band in the context of the smeared-crack approach. Within the FEM, `t is related to the element size (see, e.g., Oliver (1989)). Inserting Equation (2.43) into (2.45) and considering α˙ s = ˙s (see Equation (2.41)) for uniaxial tensile loading,

At

σ

V σ σ

localization of deformation

us

`t

homogenization

σ

2.3: Material models for plain concrete 27 unloading

Constitutive Models for Concrete

s

`t V σ

σ GIf

GIf

`t s

us (a)

(b)

Figure 2.13: Homogenization of localization of deformations in consequence of cracking by means of the smeared-crack approach: (a) discrete and (b) smeared representation of a crack gives GIf = `t

Z

∞ 0

ftu exp[−αs /αu ]2 dαs .

(2.46)

Integration, followed by re-arranging terms, finally yields 2g I αu = √ f , πftu

with gfI = GIf /`t .

(2.47)

The extension of the fracture energy concept from uniaxial tensile loading to triaxial confined stress states is accounted for by xs . Replacing 1/xs in Equation (2.41) by GIf /GII f , where II Gf denotes the fracture energy in mixed mode cracking, one gets GIf s 1 s ˙ = II ˙ . α˙ s = xs (p) Gf (p)

(2.48)

Hence, xs can be identified as the increase of the fracture energy in consequence of increasing confinement (see, e.g., Etse (1992) for a similar identification). It represents the number of I opening cracks. For uniaxial tensile loading (GII f (p) = Gf ) it is equal to one and it is growing with increasing confinement.

Constitutive Models for Concrete

2.3.2

2.3: Material models for plain concrete 28

Multi-surface plasticity model

0

This subsection contains the reformulation of a multi-surface plasticity model described in Meschke (1996). It consists of a Drucker-Prager (DP) yield surface for the description of concrete subjected to compressive loading and three Rankine (RK) surfaces for the description of the tensile behavior of concrete (see Figure 2.14) . Generally, the Drucker-Prager criterion is calibrated by means of a uniaxial and a biaxial compression test. Based on this mode of calibration, the simulation of confined compression tests shows poor agreement with experimental data (see Section 2.4). Hence, a modification of the Drucker-Prager model is proposed accounting for the influence of confinement on the hardening/softening behavior. Confinement is represented by the major principal stress (see, e.g., Pramono (1988)).

fR

2RK

K

,2, 3

=

DP+2RK

DP 0 fDP =

p

elastic domain

3RK

fR

K

1RK DP

,1

=

0

DP+1RK Figure 2.14: Illustration of the Drucker-Prager - Rankine multi-surface plasticity model in the meridian plane: yield surfaces and associate regions

2.3.2.1

Yield surfaces

Cracking of concrete is modelled within the framework of the smeared-crack concept. The maximum tensile stress criterion (Rankine criterion) is used to determine the tensile strength of concrete for 3D states of cracks. In the principal stress space, the failure criterion is described as fRK,A (σA , qRK ) = σA − q¯RK ,

with q¯RK = ftu − qRK ,

(2.49)

where the subscript ”A”(A=1,2,3) refers to one of the three principal axes and qRK is an isotropic stress-like internal variable. The ductile material behavior of concrete subjected to a multiaxial state of compressive stresses is accounted for by a hardening/softening Drucker-Prager model reading fDP (σ, qDP ) =

q

J2 − κDP I1 −

q¯DP βDP

with q¯DP = fcy − qDP ,

(2.50)

Constitutive Models for Concrete

2.3: Material models for plain concrete 29

where fcy represents the elastic limit of concrete. The parameters κDP and βDP are calibrated from the peak strengths of uniaxial and biaxial loading. For fcb /fcu = 1.16, κDP = −0.07 and βDP = 1.97. 2.3.2.2

Evolution equations

The increase of the strain-like internal variables αRK and αDP is controlled by means of associate evolution equations, reading

α˙ RK =

3 X

γ˙ RK,A

A=1

∂fRK,A , ∂qRK

α˙ DP = γ˙ DP

∂fDP , ∂qDP

(2.51)

where γRK,A represents the plastic multiplier related to the Rankine surface fRK,A , and γDP is the plastic multiplier related to the Drucker-Prager yield surface fDP . In the context of multi-surface plasticity, plastic loading is characterized by γ˙ k > 0 and fk = 0. For elastic loading, γ˙ k = 0 and fk < 0. The flow rule for the evolution of the plastic strain tensor εp in the context of multi-surface plasticity is given by (Koiter, 1953) ε˙ p =

3 X

A=1

γ˙ RK,A

∂QDP ∂QRK,A + γ˙ DP , ∂σ ∂σ

(2.52)

with QRK,A and QDP denoting the yield potentials for the Rankine and the Drucker-Prager criterion, respectively. Whereas the assumption of an associative flow rule for tensile loading states is in good agreement with experimental results, strong deviations between experimental and numerical results are obtained for compressive loading states. Hence, an associative flow rule is chosen for the Rankine surfaces, i.e., QRK,A = fRK,A , whereas a non-associative flow rule is employed for the Drucker-Prager criterion. It is characterized by modification of the Drucker-Prager yield function with respect to its volumetric part, reading QDP (σ, qDP ) =

q

J2 − κ ¯ DP I1 −

q¯DP = 0. βDP

(2.53)

The gradient of the yield potential is obtained as

m=

q

∂ J2 ∂QDP = −κ ¯ DP 1I, ∂σ ∂σ

where 1I represents the second-order identity tensor.

(2.54)

Constitutive Models for Concrete 2.3.2.3

2.3: Material models for plain concrete 30

Isotropic hardening/softening laws

Within the framework of the Rankine criterion the softening behavior is accounted for by an exponential softening law (see Figure 2.15(a)), reading qRK = qRK (αRK ) = (ftu − ftr ) [1 − exp (−αRK /αRK,u )] ,

(2.55)

with ftu and ftr denoting the peak and the residual tensile strength, respectively. Based on the fracture energy concept, the parameter αRK,u is adjusted to the element size, represented by the characteristic length `t , and to the fracture energy in mode-I cracking, GIf , yielding αRK,u

GIf gfI I , with gf = . = ftu − ftr `t

(2.56)

The hardening/softening law for the Drucker-Prager criterion is expressed as (see Figure 2.15(b))   

qDP (αDP ) =

q

2 (fcy − fcu ) (αDP (2αDP,m − αDP )/αDP,m )



fcy − (fcu − fcr )exp −

 



 αDP −αDP,m 2 αDP,u



for αDP ≤ αDP,m ,

− fcr for αDP > αDP,m ,

(2.57)

where fcu and fcr are the peak and the residual compressive strength, respectively. The parameter αDP,m accounts for the ductile behavior. αDP,u is a calibration parameter for the softening regime. It is adjusted to the element size represented by the characteristic length II `t and the fracture energy GII f . Gf represents the fracture energy obtained from uniaxial compression tests. It is computed from (see Figure 2.15(b)) GII f

= −`t

Z

−∞ εm

p

(¯ qDP − fcr )dε = −`t

Z

−∞ εm

(fcy − qDP − fcr )dεp,

(2.58)

where εm represents the strain at peak stress, with εm < 0. For the uniaxial case the flow rule and the evolution law for the strain-like internal variable αDP can be expressed as ∂QDP 1 ε˙ = γ˙ = γ˙ − √ − κ ¯ DP ∂σ1 3 p

!

and α˙ DP = γ˙

1 ∂fDP = γ˙ , ∂qDP βDP

(2.59)

yielding ε˙p = −¯ c α˙ DP

√ with c¯ = βDP (1/ 3 + κ ¯ DP ).

(2.60)

Constitutive Models for Concrete

2.3: Material models for plain concrete 31

Inserting Equation (2.57) into Equation (2.58) and using (2.60) leads to GII ¯ `t f = c

Z

∞ αDP,m

i

h

(fcu − fcr )exp − ((αDP − αDP,m )/αDP,u )2 dαDP .

(2.61)

Finally, the required calibration parameter is obtained as αDP,u =

2 gfII √ , with gfII = GII f /`t . fcu π c¯

(2.62)

q¯DP (αDP )

q¯RK (αRK ) ftu

exponential softening

GII c`t ) f /(¯

GIf /`t ftr

exponential softening

fcu

fcy αRK

(a)

fcr αDP,m

αDP

(b)

Figure 2.15: Employed hardening/softening formulation for (a) the Rankine criterion and (b) the Drucker-Prager surface

2.3.2.4

Consideration of confinement

The influence of confinement on the peak and residual compressive strength of concrete, fcu and fcr , is considered by means of the major principal stress σ1 , with σ1 ≥σ2 ≥σ3 , yielding fcu =fcu (σ1 ) and fcr =fcr (σ1 ). In the following, the dependence of fcu and fcr on the major principal stress σ1 is investigated by means of experimental data. For this purpose, results obtained from triaxial tests conducted by Hurlbut (1985) and Smith (1987) are employed. These tests are characterized by constant lateral stresses, i.e., 0 ≥ σ1 = σ2 . As regards the influence of confinement on the peak strength, experimental data (Smith, 1987) (Hurlbut, 1985) show a linear dependence of the axial peak stress σ3 on confinement represented by the lateral stress σ1 (see Figure 2.16(a)): σ3 σ3 σ1 = (σ1 ) = −1 + c , fcu,0 fcu,0 fcu,0

(2.63)

where fcu,0 denotes the uniaxial compressive strength obtained from an unconfined test, i.e., σ1 = σ2 = 0. c is the confinement parameter. Regression analysis performed on the basis of the considered experimental data resulted in c=4.41. The experimental data together with the regression curve are shown in Figure 2.16(a). The peak strength fcu corresponding to an

Constitutive Models for Concrete

2.3: Material models for plain concrete 32 σ3 /fcu,0

σ3 /fcu,0 -4

-4

Hurlbut Smith

-3

-3 -2

-2

peak stress −1+4.41σ1 /fcu,0

-1 0

Hurlbut Smith

-1

σ1 /fcu,0 0

-0.2

-0.4 (a)

-0.6

residual stress −4.5(−σ1 /fcu,0 )0.57

0

-0.8

σ1 /fcu,0 0

-0.2

-0.4 (b)

-0.6

-0.8

Figure 2.16: Relation between the axial stress σ3 and the lateral stress σ1 for triaxial experiments obtained from Hurlbut (1985) and Smith (1987): (a) σ3 = peak stress and (b) σ3 = residual stress axial peak stress σ3 is obtained from the yield criterion using σ T =bσ1 , σ1 , σ3 (σ1 )c: fDP (σ) =

q

J2 (σ1 ) − κDP I1 (σ1 ) −

fcu (σ1 ) = 0. βDP

(2.64)

Reformulation of Equation (2.64) yields the peak strength of concrete as a function of the confinement represented by the lateral stress σ1 : fcu (σ1 ) = βDP

q

J2 (σ1 ) − κDP I1 (σ1 )



.

(2.65)

The residual strength fcr denotes the final strength of concrete in the post-peak regime. The post-peak regime is characterized by strain-softening. Experimental results have shown that softening only occurs at low levels of confinement. At high levels of confinement continuous hardening is observed. In order to distinguish between both modes of failure, i.e., between brittle softening and continuous hardening, the transition point (T P ) is introduced. According to Willam et al. (1989), the transition point lies on the loading path defined by σ3 /fcu,0 = 8σ1 /fcu,0 (where only the case of aT P = 8 is considered). Intersection of this loading path with the regression curve for the peak stress given in Equation (2.63), σ1 σ1 σ3 = −1 + c := 8 , fcu,0 fcu,0 fcu,0

(2.66)

yields the lateral stress σ1 at the transition point:

1 σ1 = = −0.28. fcu,0 T P c−8

(2.67)

Since no softening occurs for stress paths characterized by σ1 /fcu ≤ −0.28, the following

Constitutive Models for Concrete

2.3: Material models for plain concrete 33

considerations concerning the residual strength fcr are restricted to σ1 /fcu > −0.28. Figure 2.16(b) contains the residual axial stress of triaxial compression tests reported in (Hurlbut, 1985) (Smith, 1987). For the numerical simulation, the experimental data are approximated by the following regression function: σ3 σ1 σ3 = (σ1 ) = −a − fcu,0 fcu,0 fcu,0

!b

for 0 ≥

σ1 > −0.28, fcu,0

(2.68)

where a and b are calibration parameters. They are computed from the condition of a smooth transition from the regression curve for the residual stress (2.68) to the regression curve for the peak stress (2.63) at the transition point T P (see Figure 2.17(a)), yielding c b= 8



σ1 and a = 8 − fcu,0 T P

!1−b

.

(2.69)

For a given confinement represented by the lateral stress σ1 , with σ1 < 0, the residual strength fcr is obtained from the yield criterion using σ T =bσ1 , σ1 , σ3 (σ1 )c: fcr (σ1 ) = βDP

q

J2 (σ1 ) − κDP I1 (σ1 )



.

(2.70)

Figure 2.17(b) contains the regression functions employed for the approximation of the peak and residual values of the axial stress σ3 in consequence of confinement. Remarkably, up to now, only one additional parameter, namely c, was used for consideration of confinement. In addition to the increase of the peak and the residual strength in consequence of confinement, also an increase of the ductility of concrete in the pre-peak regime is observed. In general, the parameter αDP,m , which defines the ductile behavior of concrete, is computed from the σ3 /fcu,0 transition point (T P )

1 8

−a(−σ1 /fcu,0 )b (σ1 /fcu,0 )|T P (a)

c¯ 1

σ3 /fcu,0 transition -4 point (T P )

1 8

-3 residual stress -2 -1 σ1 /fcu,0

0

peak stress

brittle continuous softening hardening 0

-0.2

-0.4 (b)

σ1 /fcu,0 -0.6

-0.8

Figure 2.17: Assumed relations between the axial stress σ3 and the lateral stress σ1 : (a) regression function for the residual stress and (b) combination with the regression function for the peak stress

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2.4: Calibration and re-analyses of test results

34

strain at the peak load, εm , using Equation (2.60) αDP,m =

−(εm − εem ) −(εm + fcu (σ1 )/Ec ) −εpm = = , c¯ c¯ c¯

(2.71)

where Ec denotes Young’s modulus of concrete. The dependence of εm on confinement represented by the lateral stress σ1 is shown in Figure 2.18. A linear relation of the form εm [%]

-2.0

Hurlbut Smith

-1.5 -1.0 -0.5 -0.22 0

−0.0022+0.021σ1 /fcu,0 σ1 /fcu,0 0

-0.2

-0.4

-0.6

-0.8

Figure 2.18: Ductility function: dependence of the peak-strain εm on confinement represented by the lateral stress σ1

εm = εm (σ1 ) = εm,0 + d

σ1 , fcu,0

(2.72)

is chosen, where εm,0 = −0.0022 (see CEB-FIP (1990)) represents the peak-strain for no confinement. The parameter d is adjusted to experimental data, yielding d = 0.021.

2.4

Calibration and re-analyses of test results

This section deals with the calibration of the proposed material models. Independent of the calibration, five experiments were investigated in order to verify the behavior of the models under various loading conditions. The investigated experiments together with a short description of the loading path are given in Table 2.2:

2.4.1

Boulder experiments

This subsection deals with re-analysis of experimental results contained in Hurlbut (1985). These results refer to triaxial compression tests performed on cylindrical concrete specimen of 107.95 mm height and 53.98 mm diameter. Axisymmetric confinement, ranging from 0 to 69 N/mm2 , was applied. Thereafter, the axial load was increased. For the purpose of verification of the proposed models, five compression tests with confinement levels of p = 0/ − 0.69/ − 3.44/ − 6.89/ − 13.79 N/mm2 and one tension test were

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

no experiment

description

1

Boulder experiments (Hurlbut, 1985)

2

• triaxial compression tests under various levels of confinement • uniaxial tension • uniaxial strain test

Northwestern experiments (Baˇzant et al., 1986) Toronto experiments • triaxial compression tests (Imran and Pantazopoulou, 1996) under various levels of confinement Eindhoven experiments (Van Mier, 1984) • multiaxial compression tests (Van Geel, 1995) • confined plane strain tests Cachan experiments • hydrostatic and oedometric tests (Burlion, 1997)

3 4

5

35

Table 2.2: Investigated experiments on the integration point level considered. The numerical simulations were performed by means of a constitutive driver (see Pramono (1988)). The boundary conditions for the different tests were considered by prescribed values for the respective strain and stress components. The material properties for the experiments conducted in Hurlbut (1985) are listed in Table 2.3: material properties Young’s modulus Ec Poisson’s ratio ν uniaxial compressive strength fcu uniaxial tensile strength ftu tensile fracture energy GIf compressive fracture energy GII f characteristic length `t

19305.32 N/mm2 0.2 22.063 N/mm2 2.758 N/mm2 0.035 − 0.050 Nmm/mm2 4.938 Nmm/mm2 107.95 mm

Table 2.3: Material properties corresponding to the experimental data given in Hurlbut (1985)

2.4.1.1

Model parameters

For the single-surface model, the uniaxial compressive strength fcu and the uniaxial tensile strength ftu were employed to determine the failure surface at θ = 0◦ and θ = 60◦ . The initial elastic region is defined by the elastic limit fcy . The model parameters of the described non-associative flow rule D, E, and F were obtained from Etse (1992). The hardening law

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

36

for low and medium confinement was specified by the model parameters of ductility Ah , Bh , and Ch . For high confinement, the model parameters Dh , Eh , and Fh are required. The softening behavior was controlled by two ductility softening parameters As and Bs . The Drucker-Prager failure surface of the multi-surface model was determined by the uniaxial and the biaxial compressive strength, fcu and fbc , respectively, yielding κDP and βDP . The elastic limit, fcy , defines the initial elastic region. The dependence of the peak and the residual strength on the confinement is described by the confinement parameter c, and the ductile behavior by the peak strain εm,0 and the ductility parameter d. The non-associative flow rule is controlled by the value of κ ¯ DP . The model parameters are listed in Table 2.4. model parameters elastic limit: pre-peak:

post-peak: flow rule:

single-surface model

multi-surface model

fcy = 0.1 fcu Ah = −0.000425 Dh = 0.002210 Bh = −0.004950 Eh = −0.007388 Ch = 0.000212 Fh = −0.008870 As = 12.51717 Bs = 118.767 D = 8.675 + 5.115exp[x] E = −14.956 + 6.736exp[x] F = −6.3 with x = −5(1 − q¯h /fcu ) acc. to (Etse, 1992)

fcy = 0.4 fcu εm,0 = −0.0022 d = 0.021 c = 4.429 ftr = 0.2 N/mm2 κ ¯ DP = 0 and κDP

Table 2.4: Model parameters for the single-surface and the multi-surface plasticity model, respectively, used for re-analysis of experimental test data (Hurlbut, 1985) Figure 2.19 contains the numerical result obtained from the proposed single-surface model. The axial stress - axial strain curves for no confinement and low confinement, p = 0 and -0.69 N/mm2 , respectively, show good agreement with the respective experimental results. For high confinement, the axial peak stress is underestimated by the single-surface model. In accordance with experimental data, no softening is observed for stress paths corresponding to medium to high confinement. Such stress paths are characterized by ideally plastic behavior after the failure surface (αh = 1) has been reached. As regards the numerical response of the lateral strains, Figure 2.19 shows good agreement with experimentally obtained results. This fact reflects the high quality of the employed yield potential Q (see Equation (2.27)), which controls the evolution of the plastic strain tensor, and hence, the size of the lateral strains. Figure 2.20 contains the numerical results obtained from the multi-surface model characterized by disregard of confinement. As expected, only the axial stress - axial strain curve for the unconfined test shows good agreement with the respective experimental result. For

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

37

axial stress σ3 [N/mm2 ]

experiment single-surface model

-80 -70 -60 -50 -40 -30 -20

lateral strain ε1 0.025

0.020

axial strain ε3

-10 0.015

0.010

0.005

0

-0.005 -0.010 -0.015 -0.020 -0.025

Figure 2.19: Re-analysis of triaxial compression tests: stress-strain curves obtained from the single-surface model confined tests, however, the axial peak stress as well as the post-peak behavior show high deviations from the experimental results. The results obtained from the multi-surface model considering confinement are shown in Figure 2.21. They indicate good agreement of the experimentally obtained axial stress in the pre-peak as well as in the post-peak regime. As regards the lateral strain, the underlying associative flow rule (¯ κDP = κDP ) leads to underestimation of the lateral deformation for low confinement and to overestimation for high confinement.

axial stress σ3 [N/mm2 ]

experiment multi-surface model (associate flow rule, unconfined)

-80 -70 -60 -50 -40 -30 -20

lateral strain ε1 0.025

0.020

axial strain ε3

-10 0.015

0.010

0.005

0

-0.005 -0.010 -0.015 -0.020 -0.025

Figure 2.20: Re-analysis of triaxial compression tests: stress-strain curves obtained from the multi-surface model disregarding confinement (c = 0, d = 0) and using an associative flow rule (¯ κDP = κDP )

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

38

axial stress σ3 [N/mm2 ]

experiment multi-surface model (associate flow rule, confined)

-80 -70 -60 -50 -40 -30 -20

lateral strain ε1 0.025

0.020

axial strain ε3

-10 0.015

0.010

0.005

0

-0.005 -0.010 -0.015 -0.020 -0.025

Figure 2.21: Re-analysis of triaxial compression tests: stress-strain curves obtained from the multi-surface model considering confinement (c = 4.429, d = 0.0021) and using an associative flow rule (¯ κDP = κDP ) Use of a non-associative flow rule κ ¯ DP = 0 results in a decrease of the lateral strains (see Figure 2.22) and, hence, in an improvement of the numerical results for high levels of confinement. However, for low confinement, the lateral deformation is still underestimated. For a proper representation of lateral deformations the proposed yield potential given in Equation (2.53) has to be modified. Such a modification must account for the dependence of the volumetric plastic strains on the actual state of stress. In the analyses presented in Figures 2.21 and 2.22 the evolution of the volumetric plastic strain was assumed to be constant (¯ κDP = κDP ) and zero (¯ κDP = 0), respectively. experiment multi-surface model (non-associate flow rule, confined)

axial stress σ3 [N/mm2 ] -80 -70 -60 -50 -40 -30 -20

lateral strain ε1 0.025

0.020

axial strain ε3

-10 0.015

0.010

0.005

0

-0.005 -0.010 -0.015 -0.020 -0.025

Figure 2.22: Re-analysis of triaxial compression tests: stress-strain curves obtained from the multi-surface model considering confinement (c = 4.429, d = 0.0021) and using a non-associative flow rule (¯ κDP = 0)

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

39

Figure 2.23 contains the numerical results for a uniaxial tension test. The different definitions of the post-peak behavior (compare Equations (2.43) and (2.55)) are reflected by the obtained axial stress - axial strain curves for the single-surface model (Figure 2.23 (a)) and the multisurface model (Figure 2.23 (b)). axial stress [N/mm2 ]

3

1 0

3

experiment single-surface model (GIf =0.050 Nmm/mm2 )

2

axial strain 0

0.0005 (a)

axial stress [N/mm2 ]

0.0010

experiment multi-surface model (GIf =0.035 Nmm/mm2 )

2 1

axial strain

0

0

0.0005 (b)

0.0010

Figure 2.23: Re-analysis of a uniaxial tension test: stress-strain curves obtained from (a) a single-surface and (b) a multi-surface model

2.4.2

Northwestern experiments

The model behavior under high compressive stress states was calibrated according to experiments made at Northwestern University by Baˇzant et al. (1986). Cylindrical concrete specimens were tightly fit into a cavity in a pressure vessel and loaded axially by a hard piston up to 2000 N/mm2 . The lateral expansion of the specimens was kept so small that the strain state can approximately be assumed as uniaxial. The material properties of concrete conducted in Baˇzant et al. (1986) are listed in Table 2.5. Experimental Data Young’s modulus Poisson’s ratio uniaxial compressive strength

E ν fcu

36715.88 [N/mm2 ] 0.22 34.89 [N/mm2 ]

Table 2.5: Material properties of uniaxial compressive strain test corresponding to the experimental data given in Baˇzant et al. (1986) From the experimentally obtained stress-strain diagram the following conclusions can be drawn: The initial decrease of the slope in the axial stress - axial strain diagram is caused by breakage of pore walls and collapse of pores, whereas the subsequent stiffening is a consequence of the closure of pores. The minimum slope is achieved at roughly 345 N/mm2 axial compression.

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

40

In Figure 2.24 the experimental result is shown together with the numerical results obtained with the single-surface model using different formulations of the ductility function in the hardening regime xh (see Equation (2.33)). axial stress σz [N/mm2 ]

-2500

experiment single-surface model: no modification of xh

-2000

single-surface model: extension of xh for high compressive loading

-1500 -1000 -500

axial strain εz [10−1 ]

0 0

-0.2 -0.4 -0.6 -0.8 -1.0 -1.2

Figure 2.24: Re-analysis of uniaxial strain test: stress-strain curves obtained from single-surface model with and without modification of the ductility function xh in the high compression regime The axial stress - axial strain curve obtained from the ELM using the original ductility function (Etse and Willam, 1994) is shown in Figure 2.24. This ductility function xh is based on the first one of the two polynomial expressions in Equation (2.33), which is valid only for low to medium confining pressure. Hence, the high compression regime is neglected. The axial stress - axial strain curve based on this mode of calibration is far too stiff compared to the experimental data. On the one hand, the rapid increase of strength is obtained because of rather small values of the ductility function xh in the hardening regime and, on the other hand, because the loading path reaches the failure surface. The strong deviations of the response behavior of the ELM from the experimental results gave reason for extending the ductility function xh in the hardening regime to high compressive stress states. The numerical results obtained by the modified ductility function show good agreement with the experimentally obtained results. However, it should be noted that the formulation of the model is still restricted to small-strain plasticity. The formulation of the multi-surface model is only valid for states of moderately large confined compressive stresses. Thus, for the description of states of high compressive stresses as observed in the uniaxial strain test, an additional loading surface, of the form of a cap, is required (see, e.g., Hofstetter et al. (1993)).

2.4.3

Toronto experiments

A parametric experimental study of a series of 130 triaxial tests was undertaken at the University of Toronto. It was aimed at exploring and quantifying the influence of a number of significant variables on the triaxial behavior of concrete. The triaxial compression tests (see Imran and Pantazopoulou (1996)) were performed on

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

41

cylindrical specimens (l=107.95 mm) using a triaxial cell. One confinement test was chosen for verification. The concrete properties for the respective experiments are listed in Table 2.6. For the numerical simulations, the uniaxial tensile strength was taken as ftu = 0.1 fcu Experimental Data Young’s modulus Poisson’s ratio uniaxial compressive strength

E ν fcu

21250 [N/mm2 ] 0.21 21.00 [N/mm2 ]

Table 2.6: Material properties of triaxial compression tests corresponding to the experimental data given by Imran and Pantazopoulou (1996) and the fracture energy for uniaxial tensile loading was assumed as GIf = 0.07 [Nmm/mm2 ]. Because of the relatively high w/c value of 0.75, the ductility function xh was modified in the low to medium confinement region. The results obtained with this modification are shown in Figures 2.25 and 2.26. Figure 2.25(a) shows the numerical results obtained from the single-surface model for low confinement. The axial stress - axial strain diagram shows good agreement with the experimental data. However, deviations are observed for the volumetric strains. axial stress σz [N/mm2 ]

-40

experiment single-surface model

-40

p=-2.1

-20 -10

p=0.0 N/mm

0

-0.005

p=-1.0

2

axial strain εz

-0.010

-0.015

-0.020

-10 0

εvol [10−3 ]

0

0

-0.005

-0.010

4

p=-2.1

-0.005

p=0.0 N/mm2

axial strain εz -0.015

-0.020

εvol [10−3 ]

p=0.0 N/mm2 p=-1.0

2

p=-2.1

-20

p=-1.0

4

experiment multi-surface model

-30

-30

0

axial stress σz [N/mm2 ]

-0.010

-2

axial strain εz

-0.015

-0.020

2 0 -2

(a)

p=-1.0 p=0.0 N/mm

2

p=-2.1

-0.005

-0.010

axial strain εz -0.015

-0.020

(b)

Figure 2.25: Re-analysis of triaxial compression tests: axial stress - axial strain and volumetric strain curves for low confinement obtained from (a) single-surface and (b) multi-surface model

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

42

In Figure 2.25(b) the numerical results obtained from the multi-surface model for low confinement levels are presented. A comparison of the axial stress - axial strain diagram with the experimental data shows that the material strength is slightly underestimated. The volumetric strain - axial strain curve shows strong deviations for all considered confinement levels indicating use of an inaccurate plastic potential. The numerical results for high confinement, obtained from the single-surface model are shown in Figure 2.26(a). Comparison of the axial stress - axial strain curves with the experimental data indicates good agreement. However, rather strong deviations were obtained for the volumetric strains. Similar results were obtained from the analysis on the basis of the multi-surface model (see Figure 2.26(b)). axial stress σz [N/mm2 ]

experiment single-surface model

-80

p=-14.7

-60

p=-8.4

-40

0

axial strain εz 0

-0.010

-0.020

-0.030

-0.040

εvol [10−2 ]

3 2 1 0 -1 -2

experiment multi-surface model

-80 -60

p=-14.7

-40

p=-4.2 N/mm2

-20

axial stress σz [N/mm2 ]

p=-8.4 p=-4.2 N/mm2

-20 0

3 2 p=-8.4 p=-4.2 N/mm 1 -0.010 -0.020 -0.030 -0.040 0 axial -1 strain εz -2 (a)

0

-0.010

-0.020

-0.040

εvol [10−2 ]

p=-8.4

2

-0.030

axial strain εz

p=-8.4 p=-4.2 N/mm2

-0.010

-0.020

p=-14.7

-0.030

-0.040 axial strain εz

(b)

Figure 2.26: Re-analysis of triaxial compression tests: axial stress - axial strain and volumetric strain curves for high confinement obtained from (a) singlesurface and (b) multi-surface model

2.4.4

Eindhoven experiments

Since 1980 research has been carried out at Eindhoven University of Technology on the post-peak behavior of concrete under compressive loading. Van Mier (1984) carried out comprehensive servo-controlled multiaxial loading tests on cubic concrete specimens (l=100 mm) using brush platens. Two of these tests were chosen for verification of the current models. The concrete properties of the respective experiments are listed in Table 2.7. Standard values were chosen for the remaining material parameters. The numerical results based on

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

43

experimental data by Van Mier (1984) Young’s modulus Poisson’s ratio uniaxial compressive strength

E ν fcu

32000 [N/mm2 ] 0.2 45.90 [N/mm2 ]

experimental data by Van Geel (1995) Young’s modulus Poisson’s ratio uniaxial compressive strength

E ν fcu

33000 [N/mm2 ] 0.2 50.00 [N/mm2 ]

Table 2.7: Material properties of experimental data given by Van Mier (1984) and Van Geel (1995) the single-surface model were obtained by modification of model parameters of the calibration test (Hurlbut, 1985), whereas for the multi-surface model no such modifications were performed. In the first test reported by Van Mier (1984), the ratio of the lateral stresses and the axial stress was kept constant: σ2 /σ1 = 0.1, σ3 /σ1 = 0.05. Figure 2.27 shows the numerical results obtained from the single-surface model. Comparison with the experimental data (see Figure 2.27(a)) indicate that the pre-peak and post-peak behavior are captured quite well. A comparison of the multi-surface model (see Figure 2.27(b)) shows deviations with respect to the material strength and the post-peak behavior of the respective experimental data. σ1 , ε 1

axial stress2 σ1 [N/mm ]

σ3 , ε 3

-60 experiment -40 single-surface model -20 lateral strain ε3 0 0.020 0.010 0 (a)

axial stress2 σ1 [N/mm ]

σ2 , ε 2

axial strain ε1 -0.010 -0.020

-60 experiment -40 multi-surface model -20 lateral axial strain ε3 strain ε1 0 0.020 0.010 0 -0.010 -0.020 (b)

Figure 2.27: Re-analysis of mixed loading test (Van Mier, 1984) with σ2 /σ1 = 0.1, σ3 /σ1 = 0.05: stress-strain curves obtained from (a) single-surface model and (b) multi-surface model Figure 2.28 shows the numerical results corresponding to the second investigated experiment (Van Mier, 1984), where the ratios ε2 /ε1 and ε3 /ε1 where kept constant: ε2 /ε1 = 0.1, ε3 /ε1 = 0.05. Figure 2.28(a) shows good agreement of experimental and numerical results obtained from the single-surface model with respect to the peak strength as well as with re-

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

axial stress 2 -80 σ1 [N/mm ]

σ1 , ε 1

44

axial stress 2 -80 σ1 [N/mm ]

σ2 , ε 2 σ3 , ε 3

-60 experiment single-surface model

-60

experiment multi-surface -40 model -20 -20 axial lateral axial lateral strain ε1 strain ε strain ε3 strain ε 3 1 0 0 0.015 0.010 0.005 0 -0.005 -0.010 0.015 0.010 0.005 0 -0.005 -0.010 (b) (a) -40

Figure 2.28: Re-analysis of mixed loading test (Van Mier, 1984) with ε2 /ε1 = 0.1, ε3 /ε1 = 0.05: stress-strain curves obtained from (a) single-surface model and (b) multi-surface model spect to axial and lateral strain at peak strength. For the post-peak regime slight deviations were observed. The numerical results obtained from the multi-surface model are shown in Figure 2.28(b). In a research project, started in 1993, Van Geel (1995) investigated the behavior of concrete under plane strain conditions. Special emphasis was laid on compressive loading. From these test series two plane strain tests characterized by various confinement levels were chosen for a comparative study. Figure 2.29 shows the numerical results for the first plane strain test with a confinement level of σ3 /σ1 = 0.05. The axial stress - axial strain curve obtained from the single-surface model ε1 -100

axial stress σ1 [N/mm2 ]

σ3 /σ1 = 0.05

-80 -60 experiment single-surface-40 model lateral strain -20 ε3 [10−2 ] 0 1.5

1.0

0.5

ε2 = 0

-100

axial stress σ1 [N/mm2 ]

-80

experiment -60 multi-surface -40 model axial strain ε1 [10−2 ] 0 -0.5 -1.0 -1.5 (a)

lateral strain -20 axial strain ε3 [10−2 ] ε1 [10−2 ] 0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 (b)

Figure 2.29: Re-analysis of plane strain tests (Van Geel, 1995) with ε2 = 0, σ3 /σ1 = 0.05: stress-strain curves obtained from (a) single-surface model and (b) multi-surface model

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

45

(see Figure 2.29(a)) shows good agreement of the peak-strength and the pre-peak behavior. However, the brittleness of the post-peak behavior is underestimated. The numerical results obtained from the multi-surface model (see Figure 2.29(b)) show relatively large deviations of peak-strength and ductility behavior in the pre-peak regime. Thus, a modification of material parameters governing the strength and ductility behavior should be performed. For the second investigated plane strain test a problem with high confinement, i.e., σ3 /σ1 = 0.25, was chosen for verification. The numerical results are shown in Figure 2.30. Figure ε1 -250 -200 -150

axial stress σ1 [N/mm2 ]

experiment σ3 /σ1 = 0.25 single-surface model

-250 -200 -150

axial stress σ1 [N/mm2 ] experiment multi-surface model

-100

-100 lateral -50 strain ε3 0 0.005 0

ε2 = 0

lateral -50 strain ε3 0 0 -0.005 -0.010 -0.015 -0.020 0.005 (a) axial strain ε1

axial strain ε1 -0.005 -0.010 -0.015 -0.020 (b)

Figure 2.30: Re-analysis of plane strain tests (Van Geel, 1995) with ε2 = 0, σ3 /σ1 = 0.25: stress-strain curves obtained from (a) single-surface model and (b) multi-surface model 2.30(a) shows the numerical results obtained from the single-surface model. It indicates that the ductile behavior is reproduced fairly well. The numerical results on the basis of the multi-surface model (see Figure 2.30(b)) show strong deviations from the experimental results. The lack of a cap surface for the description of highly confined stress states seems to be the main reason for these strong deviations.

2.4.5

Cachan experiments

Several tests of concrete under high compaction were performed by Burlion (1997) at ENS Cachan. The experimental investigations were based on two different compression tests. The first test was an oedometric compression test. The second test was a purely hydrostatic compression test. The experimental data of these experiments are given in Table 2.8. For the numerical simulations the standard parameter set of the ELM was used. Because of the different concrete strength and ductility as compared to the uniaxial strain test (see Subsection 2.4.2), the hardening ductility function in the high confinement regime was modified with respect to the loading paths. The numerical results obtained from the ELM are shown in Figure 2.31.

Constitutive Models for Concrete

2.4: Calibration and re-analyses of test results

46

Experimental Data Young’s modulus Poisson’s ratio uniaxial compressive strength

E ν fcu

20050 [N/mm2 ] 0.2 45.00 [N/mm2 ]

Table 2.8: Material properties of hydrostatic and oedometric compression tests corresponding to the experimental data given by Burlion (1997) volumetric stress σv [N/mm2 ] experimental result single-surface model

-640

-480

oedometric loading

hydrostatic loading

-320

-160 volumetric strain εv

0 0

-0.04 -0.08 -0.12 -0.16 -0.20 -0.24 -0.28 -0.32 -0.36

Figure 2.31: Re-analysis of hydrostatic and oedometric test (Burlion, 1997): volumetric stress - volumetric strain curve obtained from single-surface model The unloading-reloading paths were not considered in the numerical simulation. The volumetric stress (σv =I1 /3) - volumetric strain curves indicate good agreement with the experimental results. Because of states of the high compressive stress no comparison with the multi-surface model was made. As previously mentioned an additional cap surface is required for the multi-surface model in order to obtain realistic numerical results for high compressive loading states.

Chapter

3

Numerical Integration of Material Laws 3.1

Introduction

From a mathematical point of view the constitutive equations of elasto-plasticity are differential equations with prescribed initial values. Thus, solving the initial value problem (IVP) is equivalent with integration of the governing equations together with enforcement of the plastic consistency condition. Because of the complexity of the differential equations this integration must be performed numerically, leading to a system of nonlinear algebraic equations standardly solved by means of Newton-type algorithms. Algorithms for numerical integration can be subdivided into single step and multi step algorithms. Either one can be subdivided into explicit and implicit methods. The fundamental concepts of numerical integration of the differential equations describing elasto-plastic problems are the consistency with these differential equations, numerical stability, convergence and incremental plastic consistency. • consistency The IVP is characterized by the field equations σ˙ = σ ∗ (ε, σ(ε)),

(3.1)

with the initial condition σ(ε0 ) = σ 0 .

(3.2)

The computing relation for a single step method reads σ n+1 = σ n + ∆t ψ(εn , σ n , ∆t).

(3.3)

Numerical Integration of Material Laws

3.1: Introduction 48

This method is consistent with the underlying differential equation, if σ n+1 − σ n ˙ n ) = ψ(εn , σ n , ∆t) = σ˙ = σ ∗ (ε, σ(ε)). = σ(ε ∆t →0 ∆t lim

(3.4)

Thus for ∆t → 0 the numerical integration algorithm results in the differential equation. • convergence The global discretization error ∆σ n+1 is defined as the difference between the discrete approximation σ n+1 and the exact value of the solution: ∆σ n+1 = σ n+1 − σ(tn+1 ).

(3.5)

A discretization method is called convergent, if k∆σ n+1 k → 0 as ∆t → 0.

(3.6)

It is called convergent of order p if k∆σ n+1 k = O(tp ).

(3.7)

• numerical stability Stability plays an important role in the approximation theory of IVPs. Consistency of a method does not imply its convergence. What is required in addition is that the global effect of local errors remains uniformly bounded for ∆t → 0. This property is called stability of the discretization scheme. It allows to study the effect of local error propagation. Accuracy and stability analyses of elasto-plastic constitutive relations have been made by Ortiz and Popov (1984). The main difference of explicit and implicit methods is their stability behavior. While implicit methods are unconditionally stable, i.e., independently of the step size ∆t, the numerical stability of explicit methods strongly depends on the step size. Consistency and stability guarantee the convergence of the integration method. • incremental plastic consistency Solving the elasto-plastic differential equations together with the fulfillment of the constraint condition, i.e., the yield criterion (or yield criteria in the case of multisurface plasticity) results in a so-called differential algebraic problem. The incremental form of the consistency condition can be written as ∆fn+1 = 0.

(3.8)

Numerical Integration of Material Laws

3.2

3.2: Incremental constitutive relations 49

Incremental constitutive relations

A wide range of elasto-plastic materials can be characterized by means of the following set of constitutive relations σ˙ = C : (ε˙ − ε˙ p ),

(3.9)

ε˙ p = γ˙ m,

(3.10)

˙ = γ˙ H : m, α

q = q(α)

with m = m(σ, q).

(3.11)

Equation (3.11) was obtained from Equation (2.14)2 by substituting ∂q H=H : m, which is valid for a broad class of constitutive models1 . Applying a single step method to Equation (3.9) leads to: ∆σ =

Z

∆t

σ˙ dt.

(3.12)

Thus, the integration of the elasto-plastic constitutive law yields the incremental constitutive relation as ∆σ =

Z

t+∆t t

C : ε˙ dt −

Z

t+∆t t

C : ε˙ p dt = C : ∆ε − ∆γ C : m.

(3.13)

At the time instant tn+1 , σ n+1 = σ tr n+1 − ∆γn+1 C : m,

(3.14)

σ tr n+1 = σ n + C : ∆εn+1 .

(3.15)

with

σ tr n+1 is the so-called elastic trial stress tensor. Integration of the internal variables leads to αn+1 = αn + ∆γn+1 H : m.

(3.16)

Equation (3.14) represents an elastic predictor - plastic corrector scheme, which is consistent with the additive structure of the constitutive law. In case of plastic loading, the incremental plastic consistency parameter ∆γn+1 will be computed from the condition f (σ n+1 , qn+1 ) = 0. Depending on the choice of time instant for m one can distinguish between explicit and 1

In the general case, where α is a tensor of first order, Equation (3.11) describes a tensor contraction of a 3 -order tensor (H) and a 2nd -order tensor (m). For the case of a scalar internal variable (α = α) Equation (3.11) defines a contraction of two 2nd -order tensors. rd

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

50

implicit integration methods. In the following, three typical forms of explicit methods will be listed: m = m(σ n , qn ),

Forward Euler Method

(3.17)

m = m(σ tr n+1 , qn ),

(3.18)

m = m(σ tr n+α , qn ),

(3.19)

where 0 < α ≤ 1 and f (σ n , qn ) = 0. Subsequently, three typical forms of implicit methods will be listed: m = m(σ n+1 , qn+1 ),

Backward Euler Method

(3.20)

m = m(σ n+α , qn+α ),

generalized midpoint rule

(3.21)

m = α m(σ n+1 , qn+1 ) + (1 − α)m(σ n , qn ). generalized trapezoidal rule (3.22) Because of their very poor numerical stability explicit methods are hardly used in computational plasticity. The most widely used integration scheme is the Backward Euler method (BE). According to Simo and Taylor (1986), the predictor-corrector schemes are often denoted as return map algorithms (RMA) or closest point projection algorithms (CPPA). The mathematical background of the CPPA for hardening plasticity was originally defined by Johnson (1978). A special case of the CPPA is the radial return map algorithm for the von Mises yield criterion, proposed by Wilkins (1964). The formulation of the CPPA in invariants of the stress tensor was made by Weihe (1989) to control the iteration procedure in the meridian and deviatoric plane separately. Numerical stability and accuracy of different integration schemes have been investigated e.g. by Ortiz and Popov (1984) and Runesson et al. (1988). Higher order integration schemes for visco-plasticity have been investigated e.g. by Kirchner and Simeon (1999). Because of the rather large strain increments used in general finite element calculations the BE method will be used in the following for the numerical integration of the proposed models.

3.3

Return map algorithms (RMA)

This section contains the algorithmic formulation of the proposed models on the basis of the stress-projection algorithm described in the previous section. The algorithmic formulation of the standard Drucker-Prager - Rankine model can be found in the paper by Meschke (1996). The modifications of the model with respect to confinement were performed in an explicit manner. Hence, the aforementioned formulation is still valid. To account for the rather complex format of the ELM, several additional considerations were made to ensure stability

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

51

and robustness of the algorithms. Starting with the general formulation of the return map algorithm in Subsection 3.3.1, the update algorithm used for the hardening/softening variable (see Subsection 3.3.2) together with a staggered projection algorithm (see Subsection 3.3.3) will be described. Special consideration concerning the projection algorithm in the cone regions of the ELM will be made in 3.3.4. The formulation of the consistent tangent moduli will be given in Section 3.4. Finally, numerical investigations of the proposed algorithmic formulations will be studied in Section 3.5.

3.3.1

General formulation

The following considerations are concerned with isotropic yield surfaces depending on a single scalar internal variable α. Rearrangement of Equations (3.14) and (3.16) results in the following system of nonlinear algebraic equations: Rσ = σ n+1 − σ tr n+1 − ∆γn+1 C :

∂Q = 0, ∂σ

Rα = αn+1 − αn − ∆αn+1 = 0,

(3.23) (3.24)

where ∆αn+1 in general depends on the current state of stress and plastic strain. Because of the assumption of isotropy the stress tensor is formulated in principal coordinates yielding, σ Tn+1 = bσ1 , σ2 , σ3 cn+1 . Equation (3.23) and (3.24) together with the incremental consistency condition fn+1 ≤ 0,

∆γn+1 ≥ 0,

∆γn+1 fn+1 = 0,

(3.25)

provide five equations for the five unknowns (σ1 , σ2 , σ3 , α, ∆γ)n+1 . The system of Equations (3.23), (3.24) and (3.25) can be solved by using the generalized Newton method. Hence, to solve R(X) = 0, an iterative procedure characterized by Xi+1 = Xi + ∆X,

with ∆X = −(Ji )−1 Ri ,

(3.26)

where    

σ n+1 X= αn+1    ∆γ n+1

      

Rσ and R = Rα    f    

      

,

(3.27)

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

52

denote the solution vector, and the residual vector, respectively. The Jacobian of this algebraic system is defined as ∂Rσ   ∂σ  

J=

        

∂Rσ ∂α

∂Rα ∂σ

∂Rα ∂α

∂f ∂σ

∂f ∂α

∂Rσ  ∂∆γ   

∂Rα ∂∆γ ∂f ∂∆γ

        

.

(3.28)

n+1

According to Weihe (1989), Rσ can also be formulated in terms of stress invariants, i.e., σ Tn+1 =bp, r, θcn+1 (p, r, and θ are defined in Equation (C.1) of the Appendix) as Rp = pn+1 − ptr n+1 + ∆γn+1 γK

∂Q , ∂p

(3.29)

tr tr Rr = rn+1 − rn+1 cos(θn+1 − θn+1 ) + ∆γn+1 2G tr tr Rθ = rn+1 rn+1 sin(θn+1 − θn+1 ) − ∆γn+1 2G

∂Q , ∂r

∂Q = 0, ∂θ

(3.30) (3.31)

where K is the bulk modulus and G is the shear modulus respectively. For the invariant formulation ∂Rσ /∂σ in Equation (3.28) is substituted as ∂Rp   ∂p 

∂Rp ∂r

∂Rr ∂p

∂Rr ∂r

∂Rp  ∂θ  

∂Rθ ∂p

∂Rθ ∂r

∂Rθ ∂θ

 ∗

J =

        



∂Rr ∂θ

        

.

(3.32)

n+1

Remark 3.3.1: The number of unknowns in the Newton scheme should be reduced as much as possible. The formulation in principal coordinates or in invariants reduces the dimension of the problem by 3. The drawback of the formulation in invariants is that anisotropic material behavior cannot be considered. In order to increase the convergence radius of the Newton method, a scalar variable, called the damping factor, can be introduced (see, e.g., Luenberger (1989)). The vector of unknowns X in Equation (3.26) is computed by an LU-decomposition together with back-substitution.

Numerical Integration of Material Laws

3.3.2

3.3: Return map algorithms (RMA)

53

Update algorithm for scalar hardening/softening law

To account for the rather complex format of the ELM, several additional considerations are made to ensure stability and robustness of the algorithm. For isotropic yield surfaces depending on a single scalar internal variable α, the CPPA is divided into two separate (i) iteration cycles (Weihe, 1995). To update the stresses σ n+1 , the trial stress is projected onto the yield surface which is kept fixed, i.e., α = const. Once the new stresses have been computed, the internal state parameters are updated by means of a Picard iteration together with inverse interpolation. In the following, the internal hardening and softening variables are described by one general internal state variable αn+1 =

(

h αn+1 s αn+1

for hardening for softening.

(3.33)

(1)

(1)

The update algorithm is started by initializing αn+1 , i.e., by setting αn+1 = αn . In the first iteration step, (1)

(1)

σ n+1 = P(σ tr n+1 , αn+1 ),

(3.34) (1)

(1)

is computed together with the consistency parameter ∆γn+1 and ∆αn+1 . P denotes the (2) projection operator. In the iteration step, i = 2, αn+1 is updated according to (1)

(1)

(2)

αn+1 = αn + ∆αn+1 = fP I (σ tr n+1 , αn+1 ),

(3.35)

with  

αnh + ∆γn+1 xh1(p) k m k for hardening fP I =  s αn + ∆γn+1 xs1(p) khmik for softening.

(3.36)

For subsequent iteration steps i, the closest point projection is performed with the internal (i) variable αn+1 computed at the preceding iteration step i − 1: (i)

(i)

σ n+1 = P(σ tr n+1 , αn+1 ).

(3.37)

The convergence of the sequence n

(1)

(1)

(2)

(2)

(i)

(i)

(σ n+1 , αn+1 ), (σ n+1 , αn+1 ), . . . , (σ n+1 , αn+1 )

o

(3.38)

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

54

is guaranteed if the following Lipschitz condition is fulfilled (i)

(i−1)

tr ||fP I (σ tr n+1 , αn+1 ) − fP I (σ n+1 , αn+1 )|| (i) ||αn+1

−

(i−1) αn+1 ||

(i−1)

< L, L < 1, ∀(σ tr n+1 , αn+1 ),

(3.39)

with L being the Lipschitz constant. Figure 3.1 schematically shows the convergence behavior of the fixed point iteration for different values of L. The rate of convergence is linear. As the α(i+1)

α(i+1)

α(i+1)

P

P

α(i)

α(i) (a)

(b)

α(i) (c)

Figure 3.1: Schematic representation of the Picard iteration: (a) L < 1, monotonic convergence, (b) L < 1 non-monotonic convergence and (c) L ≥ 1 divergence convergence rate of the Picard iteration is rather slow, a new function fAOII can be defined in terms of the difference of functions fP I from subsequent Picard iterations steps, expressed as (i)

(i)

(i−1)

tr tr fAOII (σ tr n+1 , αn+1 ) = fP I (σ n+1 , αn+1 ) − fP I (σ n+1 , αn+1 ).

(3.40)

As more discrete function values of Equation (3.40) become available, interpolation methods can be efficiently used to compute the solution of the hardening/softening problem. For two known evaluations of the function fk = fAOII (αk ) and fk−1 = fAOII (αk−1 ), the value of fk+1 = fAOII (αk+1 ) at any value of α can be obtained with first order accuracy by the linear interpolation2 : fk+1 = fk +

αk+1 − αk (fk−1 − fk ). αk−1 − αk

(3.41)

A specific function value can be obtained by inversion of Equation (3.41) provided fk and fk−1 are distinct values. This yields the first-order inverse interpolation method (secant-method) αk+1 = αk + 2

fk+1 − fk (αk−1 − αk ). fk−1 − fk

(3.42) (k)

For more concise notation, in the following the abbreviations αk = αn+1 and fAOII (αk ) = (k) fAOII (σ tr n+1 , αn+1 ) will be used.

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

55

If three function values are available, Muller’s method, using a quadratic interpolation scheme, leads to fk+1 = fk + (αk+1 − αk )a1 + (αk+1 − αk )(αk+1 − αk−1 )N,

(3.43)

with

N=

fk−2 −fk−1 αk−2 −αk−1

−

fk−1 −fk αk−1 −αk

αk−2 − αk−1

and a1 =

fk−1 − fk . αk−1 − αk

(3.44)

Inversion of Equation (3.43) leads to the second order inverse interpolation method 1 1 αk+1 = (αk + αk−1 ) − a1 ± 2 2N s

1 fk − fk+1 αk−1 1 a1 − (αk + αk−1 )2 − − αk αk−1 + a1 . 2N 2 N N (3.45)

Remark 3.3.2: The secant method of Equation (3.41) is a two step method of the form αk+1 = f (αk , αk−1 ). ¨ The order p of convergence is found to be p=1.618 (Uberhuber, 1995b). The Muller-method is a three step method of the form αk+1 = f (αk , αk−1 , αk−2 ). The order of convergence ¨ is p=1.839 (Uberhuber, 1995b). Higher order inverse interpolation schemes yield higher order equations to be solved iteratively. In this work only linear and quadratic interpolation schemes are used.

3.3.3

Two level return map algorithm

For complex multi-surface elasto-plasticity models, in particular in the presence of loading surfaces with large curvatures, the closest point projection P can lead to numerical difficulties. Therefore, instead of simultaneously solving for pn+1 , rn+1 , θn+1 , and ∆γn+1 within each (i) closest point projection P(σ tr n+1 , αn+1 ), a staggered approach is employed in the present implementation (Weihe, 1989) (Etse and Willam, 1996) (Macari et al., 1997). In the first phase of the two-level scheme, the Lode angle θn+1 is held fixed. In this phase, for a given value of θn+1 , i.e., θn+1 = θ¯n+1 , the system of equations ∂Q = 0, ∂p

(3.46)

∂Q tr tr = 0, Rr = rn+1 − rn+1 cos(θn+1 − θ¯n+1 ) + ∆γn+1 2G ∂r

(3.47)

Rp = pn+1 − ptr n+1 + ∆γn+1 K

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

f (pn+1 , rn+1 , θ¯n+1 ; qi ) = 0,

56

(3.48)

is solved for (pn+1 , rn+1 , ∆γn+1 ) by means of Newton’s method (see Equation (3.26)). The solution vector and the residual vector are    

pn+1 X= rn+1    ∆γ n+1

      

,

   

Rp R = Rr    f

whereas the Jacobian is given as 

J=

         

   

,

(3.49)

  

2 ∂Q ∂2Q 1 + ∆γn+1 K ∂ Q ∆γn+1 K ∂p∂r K 2 ∂p ∂p 2 2 ∂Q ∂ Q 1 + ∆γn+1 2G ∂ Q 2G ∆γn+1 2G ∂p∂r 2 ∂r ∂r ∂f ∂f 0 ∂p ∂r



     .    

(3.50)

∗ The values for pn+1 , rn+1 , and ∆γn+1 obtained from the first phase are denoted as p∗n+1 , rn+1 , ∗ ∗ ∆γ n+1 . With these values, a new value of θn+1 , i.e., θn+1 is computed in the second phase of the staggered algorithm from Equation (3.31). This is done by Newton’s method, solving the nonlinear scalar equation

(i+1)

(i)

θn+1 = θn+1 − (

∂Rθ (−1) θ (i) ) (R ) , ∂θ

(3.51)

with ∂Rθ ∂2Q ∗ tr tr ∗ = −rn+1 rn+1 cos(θn+1 − θn+1 ) − ∆γn+1 2G 2 . ∂θ ∂θ

(3.52)

∗ ∗ The values obtained from both iterations, i.e, p∗n+1 , rn+1 , and θn+1 , are inserted into the yield ∗ ∗ ∗ function for a fixed value of qi . For the case f (pn+1 , rn+1 , θn+1 ; qi ) ≤ TOL the iteration pro∗ ∗ cedure is terminated. For f (p∗n+1 , rn+1 , θn+1 ; qi ) > TOL a new iteration of the first algorithm is started. The two phases of the closest-point projection are illustrated in Figure 3.2.

3.3.4

Return map algorithm in the cone regions

The Extended Leon model was originally designed as a single-surface elasto-plasticity model. However, for the intersections of the loading surface with the hydrostatic axis apex points are obtained (see Figure 3.3). At the apex point in the tensile and compressive loading regime the direction of the plastic strains is not uniquely defined. Commonly this problem is bypassed by introducing a second yield function. The easiest yield function is the so-called

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

57

εpn , αn , σ tr n+1

f (σ tr n+1 , αn ) ≤ 0

σ n+1 = σ tr n+1 αn+1 = αn

yes

no elastic predictor - plastic corrector algorithm until convergence trial stress projection until convergence (2) p, r fixed

(1) θ = fixed r=

√

tr n+1

2J2

σ tr n+1

θ= (i) f (σ n+1 , q i ) = 0

π 3

θ=0

(i) n+1

(i)

σ n+1

yield surface update

p r=

√

2J2

σ tr n+1

(i)

f (σ n+1 , q i ) = 0 (i)

σ n+1

p

i σ n+1 = P(σ tr n+1 , αn )

Figure 3.2: Staggered iteration algorithm employed for the ELM cut-off function fCO , which is obtained from the original yield function given in Equation (2.23) by setting r=0 as q¯h fCO (p; qh , qs ) = 1 − fcu

!2

p fcu

!4

q¯h + fcu

!2

p q¯s m − fcu ftu

!

= 0.

(3.53)

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

58

r compressive meridian fLM

cone region tensile loading

cone region compressive loading p

pA apex point tensile loading tensile meridian

pB apex point compressive loading

Figure 3.3: Cone regions of the ELM for tensile and compressive loading regimes With this yield function at hand, the incremental plastic strains can be defined according to Koiter’s rule (Koiter, 1953) ∆εp = ∆γLM mLM + ∆γCO mCO ,

(3.54)

where the ∆γLM and ∆γCO denote the plastic multiplier of ELM and the cut-off function, respectively. mLM and mCO are the directions of the plastic flow given as mLM =

∂QLM ∂σ

and mCO =

∂fCO . ∂σ

(3.55)

For the algorithmic treatment of the cone region and hence when multi-surface plasticity theory must be applied, a criterion when this region becomes active must be formulated. The simplest criterion when the cone regions may become active is fLM > 0 and fCO > 0.

(3.56)

Figure 3.4 illustrates the yield surface for ideally plastic behavior. Trial stress points located in the dark grey region must be treated within single-surface theory. Whereas Equation (3.56) states that multi-surface theory must be applied for trial stress points located in the light grey region. However, Figure 3.4 clearly shows that a large set of trial stress points located in the light grey region must be treated within single-surface theory. Hence, using criterion (3.56) overestimates the activeness of trial stress states located in the cone region. In order to reduce the region where multi-surface theory must be applied a modified criterion will be employed. It is derived from the gradient of the yield potential QLM given in Equation (2.27) evaluated at the apex point pA (or pB ) of the yield surface leading the following cone

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

fLM > 0 fCO > 0

trial stress points σ tr n+1 treated within multi-surface theory

fLM > 0 fCO > 0

r fLM > 0

trial stress points σ tr n+1 treated within single-surface theory

fCO = 0

fCO = 0

σ n+1 fLM = 0

mLM

mLM mCO

σ n+1 mCO

59

pA

pB

p

Figure 3.4: Multi-surface regions of the ELM treated with cut-off function criterion condition fcone = fcone (p, r; pA ) = −(r − tan(φ))(p − pA ) = 0,

(3.57)

with the slope defined as tan(φ)=(2G ∂QLM /∂r)/(K ∂QLM /∂p) evaluated at the apex point. Figure 3.5 illustrates the regions where multi-surface theory is applied according to the cone condition (3.57). The staggered algorithm proposed in the previous section (see Equations (3.58), (3.59), and (3.60)) can be formulated within multi-surface theory as p

R = pn+1 −

ptr n+1

∂QLM ∂fCO + K ∆γLM + ∆γCO ∂p ∂p

!

= 0,

(3.58)

tr ¯ + 2G∆γLM ∂QLM = 0, Rr = −rn+1 cos(θ tr − θ) ∂r

(3.59)

fLM (pn+1 ; qi ) = 0.

(3.60)

r fLM > 0 fcone = 0 fLM > 0 fcone > 0

mLM

mCO

fcone = 0

fLM = 0

pA

mLM pB

fLM > 0 fcone > 0 mCO

p

Figure 3.5: Multi-surface regions of the ELM treated with cone function criterion

Numerical Integration of Material Laws

3.3: Return map algorithms (RMA)

60

In the following, it will be assumed that the gradient of the ELM in the cone region remains constant within each increment leading ∂QLM ∂r ∂QLM ∂p |t n + |t , ∂p ∂σ ∂r n ∂σ

mLM =

(3.61)

where the evaluation of (3.61) is made at the apex point. Performing the derivative of the cut-off function given in Equation (3.53) yields mCO =

∂fCO ∂p ∂p = sign(ptr ) , ∂p ∂σ ∂σ

(3.62)

The RMA for multi-surface plasticity in the case of a constant projection direction, employed in this work is shown in Figure 3.6(a). The general case of a variable projection direction is illustrated in Figure 3.6(b). This assumption allows a very efficient formulation of the r αn

r mn

αn+1

mn+1 σ tr n+1

αn

mn

αn+1

p pb

mn+1 σ tr n+1 p

σ n+1

pb

σ n+1 (b)

(a)

Figure 3.6: Projection directions for multi-surface regions of the ELM: (a) constant direction and (b) variable direction return map algorithm and of the consistent tangent moduli for stress points located in the cone region. Using the same update scheme as for the single-surface case, the return map (i) scheme can be performed ideally plastic. For a given hardening/softening state αn+1 , the (i) hydrostatic pressure pn+1 can be computed from Equation (3.53) with a standard Newton iteration. From Equation (3.59), the consistency parameter ∆γLM of the ELM can explicitly be computed as (i)

∆γLM =

tr ¯ rn+1 cos(θ tr − θ) . ∂QLM |t 2G ∂r n

(3.63)

Rearranging Equation (3.58) finally leads an expression for the plastic consistency parameter for the cut-off function as (i)

(i)

∆γCO =

(i)

ptr n+1 − pn+1 − ∆γLM K ∂fCO K ∂p

∂QLM ∂p

.

(3.64)

Numerical Integration of Material Laws (i)

(i)

3.4: Consistent tangent moduli 61 (i)

With the so obtained values pn+1 , ∆γLM and ∆γCO the update scheme described in Subsection 3.3.2 can be applied to compute the value of the hardening/softening variable αn+1 . The Picard function for the cone region must be formulated in terms of the plastic strain increments giving

fP I

 

αnh + xh1(p) k∆εp k for hardening, = s αn + xs1(p) kh∆εp ik for softening.

(3.65)

Remark 3.3.4: The evaluation of active yield surfaces in multi-surface plasticity is performed in an iterative manner (Simo and Hughes, 1998). Commonly, the trial state σ tr n+1 determines which yield surface is active.

3.4

Consistent tangent moduli

Use of tangent moduli, which are consistent with the integration algorithm, is essential for preserving the quadratic rate of convergence characterizing Newton methods (Simo and Taylor, 1986). In contrast to continuum elasto-plastic tangent moduli, which are obtained by enforcing the consistency condition on the continuum problem, consistent algorithmic tangent moduli are obtained by enforcing this condition on the discrete algorithmic problem. The aim of a derivation contained in Appendix A is to obtain an expression for the consistent tangent moduli which can formally be written as Cep T |n+1 =

3 X 3 3 X X dσ n+1 B A = aep m ⊗ m + σ ˆA CA,tr , AB dεn+1 A=1 B=1 A=1

(3.66)

where aep AB are the local tangent moduli formulated for principal axes. A detailed derivation ep of aAB is given in Appendices B.3 and B.4.

3.5

Numerical analyses

This section contains a numerical investigation concerning various properties of the proposed algorithms, such as convergence, robustness, and efficiency. The accuracy of the BE method will be studied in different regions of the ELM. Several examples will show the step-size dependency of the obtained numerical results. At the end of this section, results from a comparative study of the consistent tangent formulation with other tangent formulations will be presented.

Numerical Integration of Material Laws

3.5.1

3.5: Numerical analyses 62

Convergence properties of the algorithms used

Yield surfaces depending on the Lode angle θ often cause numerical problems because of inaccurate parameterization (see Appendix D). The deviatoric shape function g(θ, e) has a strong influence on the convergence properties of the CPPA. The ELM uses the elliptic function of Willam and Warnke (1975). Hence, the yield surface has a strong curvature near the compression meridian depending on the value of e. For an eccentricity value of e=0.5, the curvature at this meridian becomes infinite. Thus, numerical problems mainly result from iterative determination of the Lode angle θ. In order to study the iteration behavior in the deviatoric plane only, the material parameters m(qs ), q¯h , and q¯s of the ELM were taken as m(qs ) = 0, q¯h = fcu , q¯s = ftu , leading to 3 f (r, θ) = 2

rg(θ, e) fcu

!2

− 1 = 0.

(3.67)

Equation (3.67) describes a von Mises cylinder with an elliptic deviatoric meridian, for ideally-plastic material behavior. The algorithms developed in the previous section will be investigated. In the following they are denoted as • ALGO1: staggered algorithm, • ALGO2: standard Newton algorithm with LU-decomposition, • ALGO3: damped Newton algorithm with LU-decomposition. The damping strategy is chosen according to Deuflhard and Hohmann (1991), where the daming factor λk satisfies the condition λk ∈ {1, 1/2, 1/4, ..., λmin}. The damped Newton algorithm is started with λ1 = 1. If the monotony criterion of the Newton method is not satisfied a new damping factor is computed according to λ2 = λ1 /2 (for details, see (Deuflhard and Hohmann, 1991)). For evaluating the convergence properties of these algorithms, iteration maps have been investigated. The chosen trial state regions are represented in Table 3.1. The iteration definition of trial stress region deviatoric region Ω

tr tr tr σ tr n+1 = (p , r , θ ) 0, [9.00, 29.00], [0, π/3]

Table 3.1: Definition of the trial state region Ω for iteration maps maps for selected fixed values of the eccentricity parameter e employing the three different algorithms are depicted in Figures 3.7, 3.8 and 3.9. The isolines in these Figures show the number of iterations which were necessary to reach a prescribed tolerance value of TOL = 10−12 . For these investigations, iterations with more than 50 iteration steps were assumed to be non-converged iterations.

Numerical Integration of Material Laws

3.5: Numerical analyses 63

The iteration map for ALGO1 depicted in Figure 3.7 shows a strong increase of the number of iteration steps with decreasing e. For e=0.55 and 0.6 islands of no convergence are detected (gray region). Figure 3.8 shows the iteration map on the basis of ALGO2. Using the second algorithm leads to a moderate increase of the number of iteration steps with decreasing e. Similar to the first algorithm, an island of no convergence near the compressive meridian is observed for e=0.55. However, the location of this island differs from the one obtained on the basis of ALGO1. Finally the iteration map for the damped newton algorithm, ALGO3, is represented in Figure 3.9. For this algorithm, the number of iteration steps only slowly increases with decreasing e. For the chosen trial state region a solution was found irrespective of the value of e, indicating a very stable iteration behavior of ALGO3. From the iteration maps it can be concluded that the number of iteration steps increases with decreasing value of the eccentricity. Remark 3.5.2: According to the definition of the eccentricity parameter, i.e, e:=rt /rc =e(p; qh , qs ) (Pramono, 1988), e is a function of the hydrostatic pressure and the hardening/softening state (see Appendix B.1). Thus, for Newton-type algorithms also the derivatives with respect to p and the hardening/softening parameter have to be incorporated. From investigations concerning the robustness of the algorithms it was found that an implicit computation of e often leads to no convergence within the RMA. Hence, an explicit update scheme is proposed (see Appendix B.1). The information gained from the above iteration maps was incorporated in the update scheme for the eccentricity using an upper and lower bound for e, i.e., 0.665≤e≤1.

Numerical Integration of Material Laws

3.5: Numerical analyses 64

ITMAP_1: exz=1.0

ITMAP_1: exz=0.9 "ITMAP_1_1.DAT" 0

e=1.0

"ITMAP_1_2.DAT" 0.06 0.05 0.04 0.03 0.02 29 0.01 0

e=0.9

29

29 29

5

24 24

23 4

19 19

r-axis

0.26

0.52

0.26

0

0.78

0.52 θ th-axis [rad]

0.78

99 1.04

1.04

00

0.26 0.26

ITMAP_1: exz=0.8

0.52

0.52 θ th-axis [rad]

0.78

0.78

e=0.7

29

6

8

7

19 19

29

24

8

6

r-axis

10 12 14 16 18

14 14

0.26 0.26

00

0.52

0.78 0.78

0.52 th-axis θ [rad]

99 1.04 1.04

00

0.26 0.26

0.52

0.52 θ th-axis [rad]

0.78 0.78

10

14

e=0.55

24

19

2

4

6

18 8 10 14

24

30 40 50

0.26 0.26

0.52 0.52 th-axis θ [rad]

0.78 0.78

9 1.04

19 14

14

00

"ITMAP_1_6.DAT" 0.48 0.46 0.44 0.42 0.4 29 0.38 0.36 0.34 0.32 24 0.3 0.28 0.26 0.24 0.22 19 r-axis 0.2 0.18 0.16 0.14 14 0.12 0.1 0.08 0.06 9 0.04 0.02 1.04

29

r [M P a]

18 24 32 42 50

4

r-axis

99 1.04 1.04

00

0.26 0.26

Figure 3.7: Iteration map for ALGO1

0.52 0.52 th-axis θ [rad]

0.78 0.78

9 1.04

r [M P a]

"ITMAP_1_5.DAT" 0.48 0.46 0.44 0.42 0.4 29 0.38 0.36 0.34 0.32 24 0.3 0.28 0.26 0.24 0.22 19 r-axis 0.2 0.18 0.16 0.14 14 0.12 0.1 0.08 0.06 9 0.04 0.02 1.04

29

8

19 19

ITMAP_1: exz=0.55

e=0.6

6

24

14 14

ITMAP_1: exz=0.6

2

"ITMAP_1_4.DAT" 0.18 0.16 0.14 0.12 0.1 29 0.08 0.06 0.04 0.02 24 0

r [M P a]

24

r [M P a]

4

5

99 1.04 1.04

ITMAP_1: exz=0.7 "ITMAP_1_3.DAT" 0.08 0.07 0.06 0.05 0.04 29 0.03 0.02 0.01 0 24

e=0.8 2

r-axis

14 14

14 14

0

r [M P a]

19 19

r [M P a]

24 24

6

Numerical Integration of Material Laws

3.5: Numerical analyses 65

ITMAP_2: exz=1.0

ITMAP_2: exz=0.9 "ITMAP_2_1.DAT" 0.05 0.04 0.02 0 29

e=1.0

"ITMAP_2_2.DAT" 0.05 0.04 0.03 0.02 0.01 29 0

e=0.9

29

5

19 19

24 24

5 r-axis

19 19

4

1 2 3

14 14

0

0

0.26

0.26

0.52 0.52 th-axis θ [rad]

00

0.26 0.26

ITMAP_2: exz=0.8

0.52 0.52 th-axis θ [rad]

0.78 0.78

"ITMAP_2_3.DAT" 0.06 0.05 0.04 0.03 0.02 29 0.01 0

29

3 2

0.26

0.26

19 19

19 19

r-axis

14 14

0.78

0.52 0.52 th-axis θ [rad]

0.78

99 1.04 1.04

00

0.26 0.26

4

e=0.55 10 8

1.04 1.04

99

0.78 0.78

1.04 1.04

99

"ITMAP_2_6.DAT" 0.48 0.46 0.44 0.42 0.4 29 0.38 0.36 0.34 0.32 24 0.3 0.28 0.26 0.24 0.22 19 r-axis 0.2 0.18 0.16 0.14 14 0.12 0.1 0.08 0.06 9 0.04 0.02 1.04 0

50

29

24

6

19

r-axis

14 14

2 0.52 0.52 th-axis θ [rad]

0.78 0.78

14

4 2 0

0

0.26

0.26

Figure 3.8: Iteration map for ALGO2

0.52 0.52 th-axis θ [rad]

0.78 0.78

1.04

9

r [M P a]

18 29 8 12 24 19 19

0.26

14 14

0.52 0.52 th-axis θ [rad]

"ITMAP_2_5.DAT" 0.18 0.16 0.14 0.12 0.1 29 0.08 0.06 0.04 0.02 24 0

r [M P a]

6

0.26

r-axis

ITMAP_2: exz=0.55

e=0.6

0

4

2

ITMAP_2: exz=0.6

0

24 24

6

r [M P a]

4

r [M P a]

24 24

0

99

"ITMAP_2_4.DAT" 0.06 0.04 0.02 0 29

e=0.7

29

5

0

1.04 1.04

ITMAP_2: exz=0.7

e=0.8

1

r-axis

14 14

99 1.04 1.04

0.78

0.78

r [M P a]

4

2

r [M P a]

24 24

29

Numerical Integration of Material Laws

3.5: Numerical analyses 66

ITMAP_3: exz=1.0

ITMAP_3: exz=0.9 "ITMAP_3_1.DAT" 0.05 0.04 0.02 0 29

e=1.0

"ITMAP_3_2.DAT" 0.05 0.04 0.02 0.01 0 29

e=0.9

29

5

1919

24 24

5 r-axis

1

19 19

4

2

14 14

00

0.26 0.26

0.52

99 0

0

0.26

0.26

ITMAP_3: exz=0.8

e=0.8

0.52

0.78

1.04

0.52 0.78 th-axis θ [rad] ITMAP_3: exz=0.7

"ITMAP_3_3.DAT" 0.06 0.05 0.04 0.03 0.02 29 0.01 0

1.04

"ITMAP_3_4.DAT" 0.06 0.04 0.02 0 29

e=0.7

29

29

6

3

1 2 00

0.26 0.26

19 19

19 19

r-axis

14 14

0.52 0.52 th-axis θ [rad]

0

0

0.26

0.26

ITMAP_3: exz=0.6

8

0

0.26

0.26

0.52 0.78 th-axis θ [rad] ITMAP_3: exz=0.55

0.52

0.52 th-axis θ [rad]

0.78

0.78

"ITMAP_3_6.DAT" 0.08 0.06 0.04 0.02 0 29

e=0.55

29

99 1.04 1.04

8

6

2424 1919

r-axis

14 14

4

1.04

2 0

0

0.26

0.26

Figure 3.9: Iteration map for ALGO3

14 14

4

0.52 0.52 th-axis θ [rad]

0.78 0.78

99 1.04 1.04

r [M P a]

0

r [M P a]

19 19

2

99 1.04

0.78

29

24 24

6

r-axis

14 14

0.52

"ITMAP_3_5.DAT" 0.08 0.06 0.04 0.02 0 29

e=0.6

4

2

99 1.04 1.04

0.78 0.78

24 24

r [M P a]

5 4

6 r [M P a]

24 24

r-axis

14 14

99 1.04 1.04

0.78

0.78

0.52 th-axis θ [rad]

r [M P a]

4

2

r [M P a]

2424

29

r-axis

Numerical Integration of Material Laws kRk∞ e=0.9

10−4

3.5: Numerical analyses 67

ALGO1 ALGO2 ALGO3

10−8

kRk∞ e=0.55

10−4 10−8

TOL

10−12

TOL

10−12 10−16

10−16 iter 0

4

8

12

16

20

(a)

0

4

8 12 (b)

iter 20

16

Figure 3.10: Convergence behavior of investigated algorithms: kRk∞ vs. number of iteration steps for two converged stress states using (a) e=0.9 and (b) e=0.55 The convergence properties of the algorithms can further be investigated by looking at the residuum of the iteration procedure. Figure 3.10 shows the residuum measured in the infinity norm versus the number of iteration steps obtained for one selected converged stress state at e=0.9 and e=0.55. From Figure 3.10(a) it can be seen that for the chosen trial state the staggered algorithm converges faster than the Newton-type algorithms. For this case damping is not activated. Hence, the same results are obtained for both Newton algorithms. Figure 3.10(b) shows that for small values of e the Newton-type algorithms converge faster than the staggered algorithm. For the selected case ALGO3 converges faster than ALGO2, indicating activation of the damping procedure. The same investigation was performed for two trial stress states located within the region of the islands of no convergence (see Figure 3.11). In Figure 3.11(a) ALGO1 fails to reach the prescribed tolerance within the prespecified maximum number of iteration steps. The reason for this behavior is the slow convergence ALGO1 kRk∞ kRk∞ ALGO2 0 100 ALGO3 10 10−4

10−4

kRk∞ TOL

10−8

100 iter

TOL

10−12 10−16

iter 0

4

8 (a)

12

16

20

kRk∞ TOL 100 iter

10−8

TOL

10−12 10−16

iter 0

4

8

12

16

20

(b)

Figure 3.11: Convergence behavior of investigated algorithms: kRk∞ vs. number of iteration steps for two trial stress states leading to no convergence of (a) ALGO1 and (b) ALGO2 for e=0.55

Numerical Integration of Material Laws

3.5: Numerical analyses 68

order (p=1). However, increasing the maximum number of iteration steps from 50 to 100 leads to convergence of the staggered algorithm. It should be noted that the performance of such iterations within FE-simulations can be very time-consuming. The Newton-type algorithms have the same convergence properties. The residuum curve shown in Figure 3.11(b) indicates no convergence of ALGO2. It was found that in contrast to the previous case the tolerance limit could not be reached because of oscillating behavior of ALGO2. Thus, increasing the maximum number of iteration steps does not improve the convergence behavior of ALGO2. Computation of the condition number of the Jacobi matrix for ALGO2 has clearly shown that the problem is well posed in the sense of having a small condition number3 .

3.5.2

Accuracy analysis

The error measure for the BE integration algorithm was chosen as the relative error in the l2 -norm between a single load step ||σ n+1 − σ ∗n+1 ||2 , δ= ||σ ∗n+1 ||2

(3.68)

where σ n+1 is the result obtained when using this algorithm, whereas σ ∗n+1 is the exact solution corresponding to the specified strain increment. The exact solution for a given strain increment is obtained by repeated application of the algorithm by increasing the number of sub-increments. The value for which further sub-incrementation produces no change in the numerical result can be regarded as the exact solution. From numerical investigations of the considered regions the ’optimal’ sub-increment size was found to be 50. Therefore, Equation (3.68) can be reformulated as

δ=

kσ n+1 − σ ∗,50 n+1 k2 . 50 kσ n+1 k2

(3.69)

The accuracy of the CPPA has been assessed in different regions of the yield surface. Trial stress regions in the meridian plane together with the actual stress point are shown in Figure 3.12(a), whereas the trial stress region in the deviatoric plane is shown in Figure 3.12(b). Detailed information of the coordinates of the error regions can be found in Appendix B.5. In the following, a detailed description of the iso-error maps shown in Figure 3.13 and 3.14 will be given. 3

From this study it can be concluded that the Newton algorithm with damping is most reliable for the case of ideal plasticity within the framework of the von Mises criterion. However, for the case of I 1 -dependence of the yield condition and a non-associative flow rule, situations where there was no convergence were also found. For these cases a combination of ALGO3 and ALGO1 seems most promising. The strong influence of the eccentricity on the robustness of the algorithms leads to an explicit update scheme of the eccentricity e.

Numerical Integration of Material Laws

3.5: Numerical analyses 69 2

r fcu

θ=0

q¯h = 0.60 fcu 3

σ 3n

2

4

σ 4n -3 pA

θ=

-1

5-9

σ 5−9 n

σ 2n 1

0 -2

π 3

σ 1n

1

p fcu

0 pB

(a)

(b)

Figure 3.12: Trial stress regions chosen for the accuracy analysis: (a) meridian plane and (b) deviatoric plane • region 1: tension cone The p-axis in Figure 3.13 represents the trace of the yield surface in the meridian plane, and the dashed line represents the cone function given in Equation (3.57) separating the single-surface region of the ELM and the cone region. This iso-error map gives a good overview of the general trend, i.e., trial stress points located in the single-surface region, far away from the cone function, can be treated within single-surface theory. Hence, indicating a regular error distribution. On the other hand trial stress points located in the cone region, near to the hydrostatic axis, are treated within multi-surface plasticity theory. They are projected into the apex point. For this region only small errors are obtained. Trial stress points located near the cone function can be either projected in the apex point or onto the yield surface of the ELM depending on the chosen step size of the BE method. Choosing a large step size (one step only) results in projections of the trial stress point onto the yield surface located near the apex point. On the other hand small step sizes result in trial stress projections into the apex point. This switch between single- and multi-surface projections leads to pronounced errors around the cone function (see Figure 3.13). • region 2: low-medium compression regime The iso-error map for region 2 is equally distributed over the trial stress region (see Figure 3.13). The error is more pronounced in the low compressive region which follows from the location of the stress point σ n . • region 3: transition zone This zone is characterized by the transition from the ascending to the descending branch of the yield surface. The iso-error map of the transition from the ascending to the descending branch of the yield surface indicates a rather regular distribution of the error.

Numerical Integration of Material Laws

0.10 0.05 0.3 5.3 -0.30 5.30 pp-axis [MPa]

0.35

1.45

10 5

0 10.30

isoerror map

-21

-11 -16 -11 p p-axis [MPa]

-21

-16

17.45 17.45 -25 -25

0.025

25 20 20

1515"er_re4.dat" 0.025 -6-6 0.02 30 30 25

0.015

20 20

0.006

15 10 10

0.002

0.015 0.01 0.008 0.006 0.004 0.002 0

r [MPa]

-30 -30 pp-axis [MPa]

0.09 0.045 0.015

r [MPa]

22.45

-35 -35

0.18

32.45

0.028 0.014 27.45 0.002 0.006

-40 -40

30

r [MPa]

0.15

"er_re2.dat" 0.21 0.195 0.18 0.165 0.15 30 0.135 0.12 0.105 0.09 0.075 0.06 25 0.045 0.03 0.015 r-axis 0

r [MPa]

"er_re1.dat" 1.45 1.4 1.35 1.3 1.25 15 1.2 1.15 1.1 1.05 1 0.95 10 0.9 0.85 0.8 r-axis 0.75 0.7 5 0.65 0.6 0.55 0.5 0.45 0.4 0 0.35 0.3 10.3 "er_re3.dat" 0.036 0.25 0.034 0.2 0.032 0.15 0.03 0.1 0.028 0.05 0.026 0.024 0.022 0.02 0.018 27.45 0.016 0.014 0.012 0.01 r-axis 0.008 0.006 22.45 0.004 0.002 0

15

0.25

-4.70

3.5: Numerical analyses 70

r-axis

5

00 -63.42 -58.42 -53.42 -73.42-68.42 -63.42 -53.42-48.42 -43.42 p [MPa] p-axis

-73.42

iso-error map region 1

iso-error map region 2

iso-error map region 3

iso-error map region 4

Figure 3.13: Iso-error maps region 1-4: meridian plane • region 4: compression cone Contrary to the iso-error map of region 1, the iso-error map of region 4 is characterized by a uniform distribution of the error over the trial stress region. This follows from the fact that the gradient of the yield potential in the apex point encloses an acute angle with the hydrostatic axis. Only in the vicinity of the cone function near the multisurface region and the single-surface region the error distribution becomes irregular. For yield potentials characterized by perpendicular intersections with the hydrostatic axis no irregularities in the iso-error map should be detected, as there exists no cone region. • region 5-9: deviatoric regions for different hydrostatic pressures p For small values of the eccentricity e, i.e., for yield surfaces with a strong curvature, the iso-error maps have a kink near the compression meridian (θ=π/3). Trial stress

Numerical Integration of Material Laws

3.5: Numerical analyses 71

isoerror map

isoerror map

0.2

0.52 0.52 th-axis θ [rad]

0.1 0.08 0.06 0.04 0.02

r [M P a]

0.26 0.26

0.2

18.05 18.05

r-axis 13.05 13.05 8.05 8.05

0.02 00

23.05

0.78 0.78

3.05 1.04 1.04

0

0

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"er_re7.dat" 0.1 0.08 0.06 0.04 0.02 0

42

30 20

18.05

0.78 0.78

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0

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15 1.04 1.04 "er_re8.dat" 0.1 0.08 0.06 0.04 0.02 0

40

r-axis

35 30 25

27

r-axis

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22 1.04 1.04

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r-axis

23.05

r [M P a]

32

35

0.1 0.08 0.06 0.04 0.02

r [M P a]

37

40

th-axis θisoerror [rad] map

isoerror map

0.10 0.08 0.06 0.04 0.02

"er_re6.dat" 0.2 0.1 0.08 0.06 0.04 0.02 0

r [M P a]

0.1 0.08 0.06 0.04

"er_re5.dat" 0.2 0.1 0.08 0.06 0.04 23.05 0.02 0

0

0

0.26

0.26

0.52

0.78

0.52 th-axis θ [rad]

0.78

20 1.04 1.04

isoerror map

0.045

"er_re9.dat" 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

40 35 30

0.005

25 23.05

0

0

0.26 0.26

0.52 0.52 th-axis θ [rad]

0.78 0.78

r [M P a]

0.035 0.025 0.015

r-axis

region 5

region 6

region 7

region 8

region 9

20 1.04 1.04

Figure 3.14: Iso-error maps region 5-9: deviatoric planes points located on the compression meridian show only small errors. For e≈1, the error distribution becomes more regular. For the case of a von Mises yield surface with a circular deviatoric shape function no error in the CPPA is obtained. The dependence on the step size of the employed integration scheme will be studied next, both for low and high confined triaxial compression tests. Figure 3.15 shows the numerical results obtained with the BE integration scheme. Figures 3.15(a) and (b) clearly show that the numerical solution is not very sensitive with respect to the step size. However, the rather complex format of the yield function and hardening/softening law leads to a reduction of

Numerical Integration of Material Laws σz [N/mm2 ] -36

-24

-12

3.5: Numerical analyses 72

number of -80 increments: inc=5 inc=10 -60 inc=20 inc=40 inc=80 inc=160 -40

confinement: p=0.69 N/mm2

σz [N/mm2 ]

confinement: p=13.69 N/mm2

-20

εz [10−3 ]

εz [10−3 ] 0

0

-4

-8 (a)

-12

0

0

-4

-8 (b)

-12

Figure 3.15: Accuracy analysis of the implicit update scheme for a triaxial compression test with (a) low confinement and (b) high confinement the robustness of the global Newton iteration. In order to increase the robustness of the integration scheme, a similar approach as used for the update scheme of the hardening/softening parameter was introduced for the equilibrium iteration. In the following, this algorithm is denoted as semi-implicit update of the hardening/softening variable. For this integration algorithm the return map is performed ideally plastic. Once the stress point is obtained at the end of the time step the hardening/softening variable is updated. Investigations concerning the influence of the semi-implicit algorithms on the numerical results are also performed. Using this integration scheme improves the robustness of the global equilibrium iteration. However, Figures 3.16(a) and (b) indicate a rather pronounced step size dependence of the numerical results. Thus, using this algorithm should generally be accompanied by a control of the step size. σz [N/mm2 ] -36

-24

-12

0

0

number of -80 increments: inc=5 inc=10 inc=20 -60 inc=40 inc=80 inc=160 -40

-20 confinement: p=0.69 N/mm2 εz [10−3 ] 0 0 -4 -8 -12 (a)

σz [N/mm2 ]

confinement: p=13.69 N/mm2 -4

-8 (b)

εz [10−3 ]

-12

Figure 3.16: Accuracy analysis of the semi-implicit update scheme for a triaxial compression test with (a) low confinement and (b) high confinement

Numerical Integration of Material Laws

3.5.3

3.5: Numerical analyses 73

FEM efficiency analysis

The global convergence properties of the tangent operator are investigated for a triaxial compression test under low confinement (p=0.69 N/mm2 ) using small and large step sizes. Table 3.2 shows that for small increments the number of iterations for reaching a prespecified tolerance (TOL=1.0E-06) is equal for the continuum tangent and the consistent tangent, whereas a large number of iterations are required by using the elastic tangent. The convergence behavior obtained when using large increment sizes (see Table 3.2) indicates a rapid increase of the number of iterations for larger step sizes, using the elastic or the continuum tangent operator. Quadratic convergence was still maintained using the consistent tangent operator. Thus keeping the number of iterations very small. iter

elastic tangent

continuum tangent

consistent tangent

1 2 3 4 5 .. .

.104E-01 .816E-02 .649E-02 .519E-02 .417E-02 .. .

.823E-05 .909E-06

0.146E-04 0.221E-08

45 46 47 maxit 1 2 3 4 5 6 7 8 9 .. .

.125E-05 .102E-05 .835E-06 47 .330E-01 .238E-01 .181E-01 .142E-01 .113E-01 .916E-02 .747E-02 . . .. .

2 .180E-02 .661E-03 .242E-03 .882E-04 .323E-04 .120E-04 .452E-05 .178E-05 .758E-06

2 0.171E-02 0.348E-04 0.179E-08

57 58 59 maxit

.139E-05 .118E-05 .993E-06 59

9

3

Table 3.2: Global Newton-Raphson residuum for a traxial compression test under low confinement for small (∆u=0.000075 mm) and relatively large (∆u= 0.0003 mm) step sizes using various tangent formulations

Chapter

4

Finite Element Method 4.1

Introduction

The finite element method (FEM) emerged from the engineering literature of the 1950’s as one of the most powerful methods for the approximate solution of boundary/initial value problems such as field problems in structural mechanics, heat conduction and fluid dynamics, to name only a few of them. In general, the starting point of the FEM is a weak formulation of the field equations and boundary conditions. In case of time-dependent problems also initial conditions must be considered. The field variables are approximated by assumed shape functions φ over subdomains, referred to as finite elements and discrete values of the unknown field which are computed in the cause of the analysis. Generally these functions are polynomials. Summing up the contributions of the individual elements to the quantity which is approximated such as, e.g., the virtual work, finally results in an algebraic system of equations In the mid-fifties, engineering formulations of the FEM appeared in the open literature, see, e.g., Zienkiewicz and Taylor (1994) and Bathe (1996). In the seventies, the mathematical theory of the FEM emerged (see, e.g., Oden and Reddy (1976)). It is based on concepts used in functional analysis. Questions concerning the existence and uniqueness of the solution and convergence properties of the FE-approximations can be answered in an elegant manner within the framework of this theory. Especially for adaptive solution strategies, where the FE-meshes are adapted according to estimates of the error, this theory provides a powerful tool. In the following, a classical form of the FEM will be briefly described. Each shape function φ is weighted by the respective nodal quantity q. In structural mechanics, the aforementioned field variable may be the displacement vector u, with uT =bu1 , u2 , u3 c. Its approximation by

Finite Element Method

4.2: Continuum mechanics 75

the FE shape functions is obtained from uh (x, t) = Nu (x)qu (t) ,

(4.1)

where Nu represents a matrix containing the shape functions with Nu (x)1=1. qu is the vector of nodal displacements. The superscript “h” refers to quantities belonging to the FE approximation.

4.2

Continuum mechanics

4.2.1

Elastoplastic boundary value problem

The reference configuration of the considered body is defined by V ⊂ IR 3 , where V is the volume of the body. The continuum body is subjected to body forces b(x, t) in V , surface ¯ (x, t) on the complementary part Su tractions ¯t(x, t) on Sσ and prescribed displacements u of its boundary S: ¯ on Su u=u

and σ · n = ¯t on Sσ

(boundary conditions),

(4.2)

with Su ∩Sσ =0 and Su ∪Sσ =S and n denoting the normal vector. This vector is a unit vector which is directed to the outside of the body. The stress tensor is denoted as σ. The strain tensor is computed from the displacement field by ε = Du

(strain-displacement relation),

(4.3)

where D represents a differential operator matrix. The stress field is computed from ε as σ=

∂Ψ ∂ε

(constitutive law),

(4.4)

where Ψ denotes the free Helmholtz energy introduced in 2.2.1. For static problems the stresses have to fulfill the equilibrium conditions, given by div [σ] + b = 0

(equilibrium equations).

(4.5)

Finite Element Method

4.2.2

4.2: Continuum mechanics 76

Displacement formulation

Considering the equilibrium state expressed in the weak form, see, e.g., (Bathe, 1996), one gets1 GD (u; δu) = −

Z

V

δεT σ dV +

Z

V

δuT b dV +

Z

S

δuT ¯t dS = 0 ,

(4.6)

where δu is a kinematically admissible vector of infinitesimal displacements, with δui =0 on Sui , i=1,...3. The finite element approximation for the displacement field is given by me

h

u = N u qu

with Nu =

A [N

u,k ]

.

(4.7)

k=1

Nu,k denotes the matrix of shape functions of the finite element k and qu is the vector of nodal displacements. A is the assembly operator and me is the number of finite elements. Inserting (4.7) into (4.6) and noting that the resulting relation must hold for arbitrary variations δqu 6= 0, the algebraic form of the equilibrium equations is obtained as Fext − Fint = 0 .

(4.8)

In Equation (4.8), the vectors of the external and the internal forces are given by me

Fext =

A k=1 me

Fint =

A k=1

Z Z

Vk

Vk

NTu,k bk dV + BTk σ hk dV



Z

∂Vk

NTu,k ¯tk dS

with σ hk =



∂Ψ , ∂εhk

(4.9) (4.10)

where ∂Vk is the boundary of element k. Bk relates δεk to the vector of virtual nodal displacements, δqu,k : δεhk =Bk δqu,k , Bk =DNu,k does not depend on qu,k .

4.2.3

Hu-Washizu formulation

Since this formulation will be used in the next chapter, a short description of this method will be given. There are several alternatives to the displacement formulation of the FEM. Among them are mixed formulations. They were introduced to relax some or all of the conditions which the solution variables must fulfill. Herein, only the Hu-Washizu formulation, representing a three-field principle will be considered. In addition to the equilibrium equations, for the case of the Hu-Washizu formulation, the strain-displacement relation and the constitutive law are satisfied weakly. It is characterized by treating displacements, strains 1

It should be noted that Equation (4.6) represents the Eulerian form of the principle of virtual work.

Finite Element Method

4.2: Continuum mechanics 77

and stresses as independent variables. Hence, Equation (4.6) is extended to GHW (u, ε, σ; δu, δε, δσ) = − +

Z

T

V

δε

Z

∂Ψ −σ ∂ε

V

!

δεT σ dV + dV +

Z

V

Z

V

δuT b dV +

Z

S

δuT ¯t dS

δσ T (ε − Du) dV = 0 ,

(4.11)

where δu, δε and δσ are independent infinitesimal quantities. The approximations for the local, i.e., element-oriented strain and stress fields are given by εhk = Nε,k qε,k

and σ hk = Nσ,k qσ,k ,

(4.12)

with qε,k =0, qσ,k =0 for x6∈Vk . The approximation of the displacement field is the same as for the displacement formulation: me

u h = N u qu

with Nu =

A [N

u,k ]

.

(4.13)

k=1

Similar to the displacement formulation the condition Fext − Fint = 0 must hold. Details concerning the derivative of Fext and Fint employed for the Hu-Washizu formulation can be found, i.e, in (Zienkiewicz and Taylor, 1994)

4.2.4

The Newton-Raphson scheme

The solution of the algebraic equilibrium equation Fext − Fint = 0

(4.14)

is accomplished by the Newton scheme, in which a sequence of linearized equations is solved for the unknown increment of the displacement vector qu , h

i

Fext − Fint − KT ∆qu = 0 ,

(4.15)

where KT denotes the tangent stiffness matrix. In the geometrically linear theory, i.e., for Bk is constant, the tangent stiffness matrix of the displacement formulation takes the form (b and ¯t are assumed to be independent of the deformations) KD T =

i d h int F − Fext dqu

d = dqu

me

A k=1

"Z

Vk

BTk

∂Ψ dV − ∂εhk

Z

Vk

NTu,k bk dV

−

Z

∂Vk

NTu,k¯tk dS

#

Finite Element Method me

=

A k=1

"Z

Vk

4.3: Localization analyses 78 BTk

d dεhk

∂Ψ ∂εhk

!

dεhk dV dqu,k

#

.

(4.16)

Introducing the elastoplastic tangent, Cep T

∂Ψ ∂εh

d = h dε

!

,

(4.17)

the element stiffness matrix can be written as KD T,k

=

Z

Vk

BTk Cep T,k Bk dV .

(4.18)

It should be noted that the tangent stiffness matrix for the Hu-Washizu formulation can be also expressed as a function of Cep T . In order to guarantee the convergence of the Newton scheme, the starting value for the Newton algorithm should be close to the solution. Therefore, the load is divided into increments. The computed approximation of the equilibrium state of the preceding increment is used for starting the Newton iteration for the given load increment. Rewriting Equation (4.15) for the (i+1)-st iteration step within the Newton scheme of the (n+1)-st load increment yields h

(i)

i

(i)

(i+1)

int Fext n+1 − F (qu,n+1 ) − KT (qu,n+1 )∆qu,n+1 = 0 ,

(4.19)

with the update of the displacement vector (i+1)

(i)

(i+1)

qu,n+1 = qu,n+1 + ∆qu,n+1 .

(4.20)

The Newton iteration is terminated if a user-prescribed tolerance R is reached, i.e., for |Fext − Fint |/|Fext | 0, which confirms convexity of a functional and assures ellipticity of the governing differential equations and hence well-posedness of the solution. For rate-independent elasto-plasticity, with dσ=Cep :dε the loss of stability condition can be expressed as d2 W = dε : dσ = dε : Cep : dε = 0.

(4.21)

A zero value of the second order work density also gives a necessary condition for the stationarity condition, i.e., det(Cep )=0, where a singularity of the tangent operator is detected and the lowest eigenvalue of Cep becomes zero. • Stability postulate by Drucker: The second condition is the stability postulate by Drucker (1959), formulated for small strains, in the form of positive values of the second order plastic work density d2 W p >0. Hence, the condition of loss of stability, according to Drucker, can be expressed as d2 W p = dεp : dσ = 0.

(4.22)

The stability postulate by Drucker leads to the conditions of normality and convexity. An additive split of the second-order work density in an elastic and plastic part, i.e., d2 W =d2 W e + d2 W p , indicates that, because of d2 W e >0, the condition d2 W =0 implies d2 W p 0. Thus, there is no localization during the degradation of strength. Figure 4.9(b) shows that for small confinement the potential for localization is higher than for larger confinement. The minimum values of the normalized determinant of the localization tensor det(Qep ) shows the tendency to mixed mode failure. The angle of the slip planes decreases with increasing confinement. The numerical results obtained from the multi-surface model are shown in Figure 4.10. The stress-strain curves show an increase of strength with increase of confinement (see Figure

Finite Element Method (a) u ¯

4.3: Localization analyses 88

u ¯

σz [N/mm ]

p

zy x

(b) 1.0

2

0.8 0.6 0.4 0.2 0 -0.2

i=5

-30 -20

16.0

-10 i=1 i=2

εx 9.6

p=0.5 N/mm2

45

90

135

i=1 i=3 i=5 φ 180

ep

1.0 0.8 0.6 0.4 0.2 0 -0.2

i=3 i=4

p=2.0 N/mm2 p=1.0 p=0.5 p=0.0

det(Qep ) det(Qel )

εz [10−3 ] 0 3.2 0 -3.2 -6.4 -9.6

det(Q ) det(Qel )

p=2.0 N/mm2

45

i=1 i=3 i=5 90 135

φ 180

Figure 4.9: Numerical results obtained from single-surface model for triaxial confinement tests using a non-associative flow rule: (a) stress-strain curves and (b) localization analysis for e=rt /rc σz [N/mm2 ]

(a) u ¯

u ¯

(b)

-30 p

zy x

-20

1.0 0.8 0.6 0.4 0.2 0 -0.2

det(Qep ) det(Qel )

p=0.5 N/mm2

45

90

135

i=1 i=3 i=5 φ 180

3.2

0

0

i=2

εx

-3.2

i=5

i=4

-10 i=1

p=2.0 N/mm2 p=1.0 p=0.5 p=0.0

i=3

ep

εz [10−3 ] -6.4

1.0 0.8 0.6 0.4 0.2 0 -0.2

det(Q ) det(Qel )

p=2.0 N/mm2

45

90

135

i=1 i=3 i=5 φ 180

Figure 4.10: Numerical results obtained from multi-surface model for triaxial confinement tests using a non-associative flow rule: (a) stress-strain curves and (b) localization analysis 4.10(a)). The results from localization analysis, illustrated in Figure 4.10(b), indicate the tendency to shear-type failure for all levels of confinement. However, there is no localization for the entire softening regime. The inclination angle φ of potential slip planes remains almost equal for all levels of confinement, i.e., φ≈45◦ and 135◦ . 4.3.2.4

Plane strain test

Comprehensive studies concerning the failure mode obtained for plane strain experiments are reported in Van Geel (1995). From these experiments it can be concluded that because of the constraint condition, i.e., εy =0, pronounced shear-type failure is obtained. The numerical results obtained from the single-surface model for e=rt /rc and e=1, using a

Finite Element Method

4.3: Localization analyses 89

non-associative flow rule are shown in Figure 4.11. The formulation based on e=1 results in an increase of strength as compared to the one based on e=rt /rc and in ideally plastic behavior in the post-peak regime (see Figure 4.11(a)). This is caused by the increase of hydrostatic pressure at peak load leading to ppeak >pT P (see Subsection 2.3.1.4). The stressstrain curve obtained for the formulation based on e=rt /rc shows softening behavior in the post-peak regime characterized by ppeak 0. All other strain components remain zero. These kinematic constraints result in the so-called Reynolds effect of frictional materials (Willam, 1997). This effect is characterized by the development of normal stresses under monotonically increasing shear deformations. These stresses are a consequence of increasing inelastic dilatation and the supression of volumetric deformations by the boundary conditions of the simple shear test. Figure 4.19 shows the numerical results obtained from the ELM employing e=rt /rc and e=1. The increase of the eccentricity results in a moderate increase of the shear strength (see Figure 4.19(a)). The localization analysis indicates the development of a localized shear failure mode for both formulations (see Figure 4.19(b)). For e=rt /rc , the angle φ of localization is approximately 17◦ and 73◦ . For e=1, this angle is approximately 13◦ and 77◦ .

(a)

u ¯ τxz [N/mm2 ]

30

e=1

e=rt /rc

zy x

(b)

1.0 0.8 0.6 0.4 0.2 0 -0.2

0

0

0.2

0.4

0.6

i=4

i=3

i=2

i=1

10

0.8

i=5

20

γxz [10−2 ] 1.0

1.0 0.8 0.6 0.4 0.2 0 -0.2

det(Qep ) det(Qel )

φ 45

90

135

180

det(Qep ) det(Qel )

φ 45

90

135

180

Figure 4.19: Numerical results obtained from single-surface model for the simple shear test using a non-associative flow rule: (a) stress-strain curves and (b) localization analysis for e=rt /rc and e=1 Figure 4.20(a) shows the numerical results obtained from the ELM using an associative flow rule together with e=rt /rc . Whereas the non-associative flow rule is characterized by a significant reduction of shear stress because of the volumetric correction term in the plastic potential (see Equation (2.27)), the associative flow rule leads to a continuous increase of inelastic dilatation and, hence, of shear stress (see Figure 4.20(a)). Results from localization analysis based on the ELM using an associative flow rule strongly differ from the ones obtained when using a non-associative flow rule. A localized shear failure mode is obtained for the non-associative flow rule (see Figure 4.19(b)), whereas a distributed failure is obtained for the associative flow rule (see Figure 4.20(b)). The numerical results obtained from the Drucker-Prager criterion and the multi-surface model are shown in Figure 4.21.

Finite Element Method

4.3: Localization analyses 95 u ¯

(a)

det(Qep ) det(Qel )

τxz [N/mm2 ]

0.8 0.6

non-associative flow rule

0

i=4

i=3

i=2

i=1

10

0.4 0.2

i=5

20

0

1.0

zy x

associative flow rule

30

(b)

φ

0

γxz [10−2 ]

-0.2

45

90

135

180

0.2 0.4 0.6 0.8 1.0

Figure 4.20: Numerical results obtained from single-surface model for the simple shear test using a non-associative flow rule: (a) stress-strain curve and (b) localization analysis for e=rt /rc (a) 15

(b)

u ¯ τxz [N/mm2 ] R-DP DP

zy x

10

0

0

0.2

0.4

0.6

i=4

i=3

i=2

i=1

i=5

5

0.8

γxz [10−2 ] 1.0

1.0 0.8 0.6 0.4 0.2 00 -0.2 -0.4

1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4

det(Qep ) det(Qel )

φ 45

90

135

180

det(Qep ) det(Qel )

φ 45

90

135

180

Figure 4.21: Numerical results obtained from Drucker-Prager criterion and multisurface model for the simple shear test using a non-associative flow rule: (a) stress-strain curves and (b) localization analysis The use of the multi-surface model results in an increase of the shear strength under simple shear loading conditions compared to the Drucker-Prager criterion (see Figure 4.21(a)). Comparison of the localization behavior indicates the development of a shear-type failure mode for both formulations (see Figure 4.21(b)).

4.3.3

Conclusions

The information gained from the localization analyses can be summarized as follows • As regards comparison of the numerical and the experimental results: The experimental data provides the basis for the numerical modeling of the cracking

Finite Element Method

4.3: Localization analyses 96

behavior of concrete. Localization analyses gives the possibility to check whether the proposed material model is capable to predict the cracking behavior under a certain loading path accurately. Concrete experiments under compressive and shear loading paths are generally very sensitve with respect to the boundary conditions of the test set up. Hence, failure modes (crack patterns) for these tests can only be defined within a certain range. Although the cracking behavior of the models is rather different, the obtained cracking direction are within the accuracy of the experimental results. • As regards comparison of the material models: Comparison of the ELM with the multi-surface model showed that depending on the loading path rather different cracking behavior was obtained. For compressive loading paths similar response behavior was obtained. One the other hand shear loading paths indicate rather different response behavior. Stress paths reaching the corner point of the Drucker-Prager - Rankine criterion overestimates the shear loading capacity of concrete. • As regards the influence of material and model parameters: For the ELM the eccentricity parameter and the non-associative flow rule formulation had a strong influence on the cracking behavior of concrete. It was found that for tensile loading paths low values of e are essential for a realistic description of the cracking behavior of concrete. For the multi-surface model only small influence of the flow rule formulation on the numerical results was observed. The reason for this is that the plastic potential and the yield surface of the Drucker-Prager criterion are very similar. From the numerical modeling point of view informations gained from localization analyses can be used for model improvements such as the development of a plastic potential or a function controlling the deviatoric shape of the yield surface. For adaptive FE analyses the localization directions are used for mesh alignment.

Chapter

5

Numerical Simulations In order to discuss different aspects of the proposed concrete models on the structural level, altogether five applications, will be dealt with in this chapter. First, the capability for treating localization on the finite element level will be demonstrated by means of a plane strain panel test. In this simulation the influence of the previously described FE-formulation on the numerical results will be investigated. The second example is a cylinder splitting test. A mesh refinement study will demonstrate the convergence properties of the FE-simulation for nonlinear problems. The last three examples will deal with fastening systems. Starting with two headed anchor tests, with different assumptions for the embedment depth and the geometric dimensions of the anchor head, the last example is concerned with the investigation of an under-cut anchor characterized by large compressive stress states within the vicinity of the anchor head. For this simulation yielding of steel together with contact between steel and concrete will be considered. A substantial part of the results reported in this chapter was published in the open literature (Pivonka and Mang, 1999a) (Pivonka et al., 2001a) (Pivonka et al., 2001b) (Pivonka et al., 2001c) (Pivonka et al., 2001d). Table 5.1 refers to the examples solved in this chapter. Details of the finite elements from the MARC FE-package (MARC, 1996), employed for the numerical simulations, are given in Table 5.2. The employed computer hardware was a Hewlett-Packard workstation (HP 9000 L2000). Representative computing times for the largest application problem (high strength undercut anchor) were approximately 360 minutes using a HP PA-RISC 8500 440 MHz processor.

Numerical Simulations

4.3: Localization analyses 98

no

problem name

type of calculation:

1

plane strain test

2

cylinder splitting test

3

pull-out test I

4

pull-out test II

5

high strength undercut anchor

benchmark problem: • finite element formulation • reduced integration • FE-convergence study • material/model parameter study application: • FE-convergence study • influence of material/model parameters application: • material model study • influence of boundary conditions • influence of material/model parameters application: • a/d-ratio study • mesh study • material model study application: • description of steel: orthotropic material • setting of the anchor: contact problem • material model study • comparison with experimental data

Table 5.1: List of examples dealt with in this chapter: short description of the features of the performed numerical simulations

element type three-, four node, plane strain elements eight node 3D elements three-, four node, axisymmetric elements

problem no: 1 2 3, 4, and 5

Table 5.2: Types of finite elements used for the investigated problems

Numerical Simulations

5.1

5.1: Plane strain test

99

Plane strain test

As explained in detail in the previous chapter, localization of deformations plays a central role in computational inelasticity. The formation of spatial discontinuity surfaces is normally accompanied by the loss of ellipticity (static case), which in turn is responsible for the ill-posedness of the underlying initial boundary value problem. This results in extreme sensitivity of the numerical solutions with regard to mesh size and mesh orientation. In order to mitigate the problem of loss of objectivity, the use of the fracture energy concept (see Subsection 2.3.1.5 for details) should be generally accompanied by the following additional requirements: • the finite element formulation must be capable of reproducing the singularity on the constitutive level, also on the element level (enhanced element formulations (Steinmann and Willam, 1991)) • the FE-mesh should be oriented towards the discontinuity (mesh alignment). In order to analyze the influence of different element formulations and boundary conditions, single element compression tests will be investigated. For these tests only the ELM is considered. However, the results are also valid for the multi-surface model. The loaddisplacement curves obtained from the standard element formulation (Q4-element) and the assumed strain formulation (Q1E4-element) using different integration schemes are shown in Figure 5.1. Furthermore, the underlying boundary conditions together with the deformed configuration of the element are given in this figure. uy

(a)

(b)

uy

(c)

(d)

P /2 P /2 Q4-element Q4-element using reduced integration P [kN]

Q1E4-element Q1E4-element using reduced integration P [kN]

3.2 P [kN]

3.2

2.4

2.4

2.4

2.4

1.6

1.6

1.6

1.6

0.8

uy [mm]

0 0 0.12 0.36 0.60

0.8

3.2

3.2

P [kN]

0.8 uy [mm] uy [mm] 0 0 0 0 0.12 0.36 0.60 00.12 0.36 0.60 0 0.12 0.36 0.60 uy [mm]

0.8

Figure 5.1: Plane strain test under displacement control: investigation of the element formulation and integration scheme (a)-(d)

Numerical Simulations

5.1: Plane strain test

100

For the first set of test problems, see Figures 5.1(a) and (b), quadrilateral elements are employed. The load-displacement diagrams are based on analyses under displacement control. The vertical displacements of the two upper node points is the specified quantity. Figure 5.1(a) refers to a simulation with a support constraining the horizontal displacement. This simulation results in artificial stiffening. This element locking effect can be circumvented by using the reduced integration procedure incorporated in the MARC FE-package, leading to ideally plastic response behavior for both element formulations. The ideally plastic behavior is caused by the fact that the loading path intersects the failure surface of the ELM behind the transition point of brittle to ductile failure (p>pT P , see Subsection 2.3.1.4). Independent of the element formulation, the same results are obtained for the unconstrained problem shown in Figure 5.1(b). For this situation, the hydrostatic pressure at peak load is smaller than the one obtained at the transition point T P , i.e., ppT P . Based on the material parameters of the calibration test, pT P =18 N/mm2 . In fact, the hydrostatic pressure for the considered compression test was found to be very close to σ1

P [N]

2500

e=1.0 e=0.9 e=0.8 e=0.7 e=0.665

2000 1500 1000

σ2

σ3

500 0

0

0.1

0.2

0.3

0.4

u ¯ [mm] 0.5

Figure 5.7: Plane strain compression test: load-displacement curves obtained for selected values of e for Q4-element formulation using a non-associative flow rule

αs ∈ [0, 0.01]

αs ∈ [0, 0.05]

(c) e=0.8

(d) e=0.9

αs ∈ [0, 2.5 10−3 ]

(b) e=0.7

αs ∈ [0, 0.03]

(a) e=0.665

5.1: Plane strain test

104

(e) e=1.0

αs ∈ [0, 3 10−4 ]

Numerical Simulations

Figure 5.8: Plane strain compression test: distribution of internal variable αs at u ¯=0.5 mm obtained for different values of the eccentricity e (Q4-element) using a non-associative flow rule (a)-(e) the transition point. Since larger values of e resulted in an increase of confinement, ideallyplastic response was obtained. For e=0.665, the smallest peak load was obtained. The corresponding pressure in the specimen was lower than pT P resulting in softening behavior (see Figure 5.7). Figure 5.8 shows the influence of the eccentricity e on the failure mode by means of the distribution of the softening variable αs . For small values of e, i.e., e=0.665 and e=0.70, a similar inclination angle as for the analysis with variable e was obtained (φ≈52◦ ). As the value of e increases, the inclination angle of the shear band is decreasing. For e=1, an inclination angle of approximately 45◦ is obtained. Comparison with inclination angles of the localization analyses (see Subsection 4.3.2.4) confirms the similarity of the inclination angles obtained at the structural level and on the integration point level. The next investigation deals with the influence of the flow-rule formulation on the numerical results (see Figure 5.9). For the case of e=rt /rc shown in Figure 5.9(a) the load-displacement curve indicates softening behavior. [× 10−1 ] (a)

(b)

P [N]

3000

1.50

1.000e-03

e=1.0

1.20

7.800e-04

0.90

5.600e-04

2000

e=rt /rc

0.60

3.400e-04

0.30

1.200e-04

1000

0.00

-1.000e-04

0 0

u ¯ [mm] 0.1 0.2 0.3 0.4 0.5

Figure 5.9: Plane strain compression test: (a) load-displacement curves obtained for selected values of e and (b) distribution of αs at u ¯=0.5 mm (e=rt /rc ) for Q4-element formulation using an associative flow rule

Numerical Simulations

5.1: Plane strain test

105

[× 10−2 ] (a) 1500

(b)

P [N]

3.50

1.000e-03

2.80

7.800e-04

2.10

5.600e-04

1000 mesh4 mesh5 mesh6

500

1.40

3.400e-04

0.70

1.200e-04

0.00

-1.000e-04

0

u ¯ [mm] 0

0.1 0.2 0.3 0.4 0.5

Figure 5.10: Plane strain compression test: (a) load-displacement curves and (b) distribution of internal variable αs at u ¯=0.5 mm for triangular elements using a non-associative flow rule Independent of the flow-rule formulation similar peak loads are obtained (see Figure 5.5(a)). For e=1, however, a strong increase of the load is obtained. This increase is more pronounced when using an associative flow rule. The angle of the shear band for e=rt /rc is approximately 67◦ . The numerical results confirm that the inclination angle of the shear band strongly depends on the flow rule formulation. The final investigation deals with mesh alignment. The numerical results obtained from triangular meshes aligned towards the expected direction of the shear band are shown in Figure 5.10. Softening in the post-peak regime for aligned meshes is more pronounced than for non-aligned meshes (see Figure 5.10(a) and 5.5(a). Looking at the distribution of the internal variable αs indicates the development of a localized shear band (see Figure 5.10(b)). The width of this band is much smaller than the width obtained on the basis of the nonaligned meshes (see Figure 5.5(b)). In the following, the numerical behavior of the ELM under plane-strain tensile loading will be investigated Figure 5.11 shows the load-displacement curves together with the distribution [× 10−2 ] (a)

(b) P [N] 200

Q4-elements Q1E4-elements

1.5

1.000e-03

1.2

7.800e-04

0.9

5.600e-04

0.6

3.400e-04

100

0.3

1.200e-04

0.0

-1.000e-04

0 0

u ¯ [mm] 0.02

0.04

Figure 5.11: Plane strain tension test: (a) load-displacement curves obtained for different element formulations and (b) distribution of internal variable αs at u ¯=0.035 mm (Q4-element) using a non-associative flow rule

Numerical Simulations

5.1: Plane strain test

106

P [N] 250

σ1

e=1.0 e=0.9 e=0.8

200 150

σ2

100

σ3

e=0.7 e=0.665 u ¯ [mm]

50 0 0

0.02

0.04

Figure 5.12: Plane strain tension test: load-displacement curves obtained for selected values of e for Q4-element formulation using a non-associative flow rule of the internal variable αs for both element formulations. Independent of the element formulation, the load-displacement curves indicate mesh objectivity (see Figure 5.11(a)). This behavior can be explained best by inspecting the distribution of the internal variable αs shown in Figure 5.11(b). Contrary to compressive loading, where the localization band spreads over several rows of elements, the localization band for tensile loading is concentrated within one element row. Therefore, failure under tensile loading does not require an enhanced element formulation. The influence of the eccentricity e on the load-displacement curves for uniaxial tensile loading is shown in Figure 5.12. Similar as for the compressive case, an increasing of the eccentricity e leads to an increase of the peak load and in a different post-peak behavior. In Figure 5.13 the distribution of αs is used as a vehicle to illustrate the dependence of the failure mode on the value of e. This figure shows the transition of the failure mode from localized failure for small values of e (e) e=1.0 αs ∈ [0.6, 1.2] 10−3

(d) e=0.9 αs ∈ [0.6, 1.2] 10−3

(c) e=0.8 αs ∈ [0.6, 1.2] 10−3

(b) e=0.7

αs ∈ [0, 0.015]

αs ∈ [0, 0.015]

(a) e=0.655

Figure 5.13: Plane strain tension test: distribution of internal variable αs for selected values of e at u ¯=0.035 mm for Q4-element formulation using a nonassociative flow rule

Numerical Simulations

5.1: Plane strain test

107

(e=0.665 and 0.7) to a rather distributed failure characterized by spreading of the localization zone towards neighboring elements for larger values of e.

5.1.3

Numerical results for the multi-surface model

In the following, the mechanical behavior of the multi-surface model will be investigated under plane strain conditions. Because of the well known behavior of the Rankine criterion for the description of tensile loading, only the uniaxial plane strain compression test will be considered. For the analyses only mesh3 and mesh6 are employed. Figure 5.14 shows the load-displacement curves obtained from the multi-surface model for the Q4-element formulation using a non-associative flow rule. (a)

(b) P [N]

2000

[× 10−2 ] 9.0

5.000e-03

7.2

4.500e-03

1500

5.4 3.6

4.000e-03

1000

3.500e-03

1.8

3.000e-03

500

0.0

2.500e-03

u ¯ [mm]

0 0

0.5

1.0

1.5

2.0

Figure 5.14: Plane strain compression test: (a) load-displacement curves and (b) distribution of the internal variable αDP at u ¯=1.2 mm for Q4-element formulation using a non-associative flow rule A Comparison with the numerical result on the basis of the ELM (see Figure 5.5(a)) indicates that although almost the same peak load was predicted by both material models, the structural response in the post-peak regime differs significantly. The softening branch of the ELM is characterized by a sharp decrease of the load, for values of u ¯ for which there is yet no such decrease for the multi-surface model. For these values of u ¯ the residual force resulting of the ELM is approximately 1000 N. The multi-surface model, on the other hand, exhibits a pronounced plateau at the peak load, followed by a softening branch. The residual load of the multi-surface model is considerable lower than the one obtained from the ELM. The failure mode of the specimen is depicted in Figure 5.14(b) by means of the internal variable αDP of the multi-surface model. Similar to the ELM, a shear band starting from the weakened finite element in the lower left corner has developed. The inclination angle of the shear band obtained on the basis of the multi-surface model is approximately 43◦ . Figure 5.15 shows the load-displacement curve obtained from the multi-surface model for the Q1E4-element formulation, using a non-associative flow rule.

Numerical Simulations (a)

5.1: Plane strain test (b)

P [N]

2000

108

[× 10−2 ] 9.0 7.2

5.089e-03

4.546e-03

1500

5.4

4.003e-03

1000

3.6

3.460e-03

1.8

2.917e-03

500

0.0

2.374e-03

u ¯ [mm]

0 0

0.5

1.0

Figure 5.15: Plane strain compression test: (a) load-displacement curve and (b) distribution of the internal variable αDP at u ¯=1.2 mm for Q1E4-element formulation using a non-associative flow rule Comparison of the load-displacement curves obtained for this formulation with the one for the Q4-element formulation (see Figure 5.14(a)) indicates that the pre-peak regime is almost the same for the two formulations. However, a more brittle behavior in the post-peak regime is obtained with the Q1E4-element formulation. Comparison of the distribution of the internal variable αDP obtained on the basis of the Q4-element and the Q1E4-element formulation confirms that the width of the shear band is strongly affected by the element formulation. For the Q4-element formulation the shear band is spread over several rows of elements. For the Q1E4-element formulation, however, a more local shear band is observed. It is interesting to note that apart from the dependence of the width of the shear band on the element formulation also the inclination angle is slightly affected. The inclination angle of the shear band for the Q1E4-element formulation is approximately 47◦ . The final investigation for the multi-surface model deals with the use of triangular finite elements. Figure 5.16 shows the load-displacement curve and the distribution of the internal variable αDP obtained with the multi-surface model for the standard element formulation using a non-associative flow rule.

5.1.4

Conclusions

In this section the localization properties of the proposed models under plane strain conditions were investigated. The following conclusions concerning the structural response can be drawn from the numerical simulations: • Independent of the loading path nearly the same peak loads were obtained for both material models using the standard parameters. The peak load was practically not affected by the underlying finite element mesh and the chosen element formulation. • As regards the failure mode,

Numerical Simulations

5.1: Plane strain test

(a)

(b) P [N]

2000

109

[× 10−2 ] 12.0

3.738e-03

9.68

3.605e-03

1500

7.36 5.04

3.472e-03

1000

3.339e-03

2.72

3.206e-03

500

0.40

3.073e-03

u ¯ [mm]

0 0

0.5

1.0

Figure 5.16: Plane strain compression test: (a) load-displacement curve and (b) distribution of the internal variable αDP at u ¯=1.2 mm for triangular elements using a non-associative flow rule - for compressive loading a shear-type failure mode developed for both material models. However, the load-displacement curve in the post-peak regime was very sensitive with respect to the material model, the finite element mesh, and the element formulation. Mesh alignment of the employed finite element mesh resulted in more brittle failure and, hence, in pronounced softening. - for tensile loading a crack band perpendicular to the loading direction developed. The numerical results in the post-peak regime were independent of the finite element formulation. • As regards the dependence of the width and the inclination angle of the crack band on the element formulation: - for compressive loading a strong influence of the element formulation on the width of the shear band was observed. For the standard formulation (Q4-element) the shear band was distributed over a larger region than for the Q1E4-element formulation. However, the inclination angle of the shear band was hardly affected by the element formulation. - for tensile loading the width of the crack band was not affected by the element formulation. Localization was restricted to one row of elements. Furthermore, no change of orientation of the crack band was observed for the different element formulations. • As regards the shape of the loading surface in the deviatoric plane, a strong influence of the mechanical response on e was observed. Beside the peak load, the orientation of the failure mode changed for different values of e. Comparison of the inclination angles of the shear/crack band with the one obtained from localization analyses (see Section 4.3) indicates almost equal values of the orientation at the integration point level and the structural level.

Numerical Simulations

5.2

5.2: Cylinder splitting test

110

Cylinder splitting test

This section contains numerical results from the analysis of a cylinder splitting test. The test is also denoted as Brazilian test. It is commonly used to determine the uniaxial tensile strength ftu of concrete by means of the respective splitting-tensile strength ftsu of concrete (see Figure 5.17). Theoretical investigations concerning cylinder splitting tests were reported in Bonzel (1964a). P

P b

D

σt

(a)

(b)

ftsu ≈ftu

(c)

Figure 5.17: Cylinder splitting test: schematic illustration of (a) load configuration, (b) stress distribution, and (c) ideal rupture mode

5.2.1

Geometric dimensions and material parameters

The geometric dimensions of the concrete cylinder and the plywood loading platens as well as the material parameters are shown in Figure 5.18. According to Bonzel (1964b), the ultimate load for the considered geometric dimensions is estimated as Pu =292kN (see Lackner (2000)). Extensive numerical investigations based on the multi-surface model together with the use of adaptive mesh refinement strategies were material properties for concrete: concrete cylinder

304.8

152.4

3.2

25.4

loading platens

Ec =26200 N/mm2 νc =0.2 fcu =30.3 N/mm2 ftu =3.0 N/mm2 GIf =0.1 Nmm/mm2 GfII =5.25 Nmm/mm2 material properties for plywood: Ew =11030 N/mm2 νw =0.2

Figure 5.18: Cylinder splitting test: geometric dimensions of concrete specimen and plywood loading platens (in [mm]) together with material parameters

Numerical Simulations

5.2: Cylinder splitting test

111

made by Lackner (2000). Thus, in the following only the numerical results obtained on the basis of the ELM are reported.

5.2.2

Numerical results for the ELM: convergence study

Because of symmetry of the geometric properties and the boundary conditions, only a quarter of the specimen is discretized1 . The first study deals with objectivity of the numerical results with respect to mesh refinement. For this purpose, three finite element meshes were chosen for verification (see Figure 5.19). The second and third mesh (mesh2 and mesh3) were obtained by consistent refinement of the first mesh. The numerical simulations were performed displacement driven. The prescribed displacements u ¯ are shown in Figure 5.19.

u ¯

u ¯

u ¯

mesh1

n=35

mesh2

n=138

mesh3

n=548

Figure 5.19: Cylinder splitting test: finite element meshes employed (n: number of elements) Figure 5.20 shows the numerical results obtained from the mesh study. The chosen mesh refinement results in convergence of the peak load Ppeak . The values of Ppeak and the values of the corresponding displacement are given in Table 5.4. Comparison Ppeak [kN] mesh1

convergence properties

50 0

200 0

0

mesh2

mesh2

mesh3

100

mesh1

400

150

peak load according to Bonzel (1964b) mesh3

P [kN]

200

200

400

n 600

u ¯ [mm] 0

0.05

0.10

0.15

0.20

Figure 5.20: Cylinder splitting test: load-displacement curves obtained from the mesh study 1

Because of symmetry conditions, only one half of the load is applied. Hence, in order to obtain the ultimate load of the entire specimen the numerically obtained loads must be multiplied by 2.

Numerical Simulations

5.2: Cylinder splitting test

112

of the numerical results with the ultimate load obtained by Bonzel indicates convergence towards this peak load. no. mesh

lc [mm]

1 2 3

12.8 6.4 3.2

Ppeak [kN]

u ¯(P = Ppeak ) [mm]

2 × 191=382 2 × 179=358 2 × 161=322

0.16 0.14 0.11

Table 5.4: Cylinder splitting test: Ultimate loads obtained from mesh refinement study In order to obtain insight into the failure mechanism of the cylinder splitting test, the distribution of the internal variables of the ELM are considered. Figure 5.21 shows the distribution of the internal hardening variable αh at different loading states. [× 10−2 ]

[× 10−1 ] 5.0

2.0

1.84

5.00

1.65

4.50

4.0

1.6

1.47

4.00

1.28

3.50

3.0

1.2

1.10

3.00

0.92

2.50

2.0

0.8

0.73

2.00

0.55

1.50

1.0

0.4

0.37

1.00

0.18

0.50

0.0

0.0

0.00

0.00

(a)

(b)

Figure 5.21: Cylinder splitting test: distribution of internal hardening variable αh at (a) 30% and (b) 80% of peak load (10-fold magnification of displacements)

[× 10−2 ]

[× 10−3 ] 4.0

12.5

3.95

13.28

3.55

11.93

3.2

10.0

3.16

10.58

2.76

9.24

2.4

7.5

2.37

7.89

1.97

6.54

1.6

5.0

1.58

5.19

1.18

3.85

0.8

2.5

0.79

2.50

0.39

1.15

0.0

0.0

0.00

0.00

(a)

(b)

Figure 5.22: Cylinder splitting test: distribution of internal softening variable αs at (a) u=0.096 mm and (b) u=0.1 mm

Numerical Simulations

5.2: Cylinder splitting test

113

It indicates the development of damage in the vicinity of the loading platen. The plastic zone propagates towards the axis of symmetry. Inspecting the internal softening variable α s provides insight in the cracking behavior of the concrete specimen. Cracking starts at the center of the specimen finally propagating along the axis of symmetry towards the concrete surface. Investigations with the same geometry and material parameters on the basis of the multisurface model were made by Lackner (2000). The numerical results obtained from these investigations correspond very well with the results obtained from the ELM. In these simulations the development of the failure mode in the brazilian test was further investigated. It was found that for the considered geometry and material parameters a shear type failure mode with wedge formation was obtained (see Figure 5.23(a)). It should be noted that depending on the geometry and the material parameters different failure modes can be observed (see Figure 5.23).

(a)

(b)

(c)

Figure 5.23: Cylinder splitting test: schematic illustration of failure modes characterized by (a) ideal tensile failure, (b) tensile failure with development of secondary cracks, and (c) shear failure with wedge formation Failure of mortar and granite specimens by means of the formation of secondary cracks, as shown in Figure 5.23(b), were extensively investigated by Rocco et al. (1999). In these tests it was found that in addition to the development of a primary crack in the center of the concrete cylinder also secondary cracks form at the sides of the wooden bearing strips (see Figure 5.24(b)). These cracks start to open, when the primary crack reaches the bearing strips. The schematic load-displacement curve shown in Figure 5.24(a) indicates that after reaching the first peak (point A) unloading corresponding to the extension of the principal crack is obtained (point B). When the principal crack approches the bearing strips, the unloading stops and the load increases again. During this reloading, secondary cracks start to grow (points C and D). An increasing load with increasing dispacement is obtained until the second peak load (point E), the load decreases while the secondary crack continues to open (point F). This secondary crack opening proceeds until the total fragmentation of the specimen. Numerical investigations concerning these experiments were made by Rodriguez-Ferran and Huerta (2000). It should be noted that because of the snap back behavior (unloading) of the load displacement curve an arc length algorithm must be applied for stable computations.

Numerical Simulations

5.2: Cylinder splitting test

(a) 100

unloading E D

A

80 B

60

(b)

∆w

P [kN]

114

F A

B

C

D

E

F

C

40 20

∆w 0.5

1.0

1.5

2.0

2.5

Figure 5.24: Cylinder splitting test: schematic illustration of (a) load-displacement curve and (b) development of cracks according to (Rocco et al., 1999)

5.2.3

Numerical results for the ELM: model parameter study

In the following, the influence of selected model parameters of the ELM on the numerical results will be investigated. For this purpose, the second mesh is employed. The first study deals with the influence of the eccentricity parameter e on the numerical results. As in the previous section, different fixed values of e are considered. Figure 5.25 shows the loaddisplacement curves obtained from the eccentricity study. Expectedly, the value of e has a strong influence on the load-carrying behavior. Increasing the eccentricity leads to an increase of the peak load. Since the failure mode is mainly dominated by cracking because of tensile stresses (see Figure 5.22(b)), the influence of e on the failure mode is similar to the one for the case of tensile loading described in detail in the previous section. Hence, increasing the value of e leads to a spreading of the tensile crack over several element rows, finally causing distributed failure.

300

e=0.9

250 200

e=0.7

e=0.8

150

σ2

e=0.665

100 50 0

σ1

e=1.0

P [kN]

σ3

u ¯ [mm] 0

0.2

0.4

0.6

Figure 5.25: Cylinder splitting test: load-displacement curves obtained for different values of the eccentricity parameter

Numerical Simulations

5.2: Cylinder splitting test

115

The second investigation dealt with in this subsection is concerned with the influence of the formulation of the flow rule on the numerical results. Figure 5.26 shows the load-displacement curve obtained on the basis of the non-associative and the associative flow rule, respectively. associative flow rule

P [kN] 250

non-associative flow rule

200 150 100 50 0

u ¯ [mm] 0

0.2

0.4

Figure 5.26: Cylinder splitting test: load-displacement curves obtained for different flow rule formulations parameter eccentricity e: 0.665 0.7 0.8 0.9 1.0 flow rule: non-associative associative

Ppeak [kN]

u ¯(P = Ppeak ) [mm]

2 × 179=358 2 × 199=398 2 × 254=508 2 × 280=560 2 × 295=590

0.14 0.16 0.28 0.35 0.35

2 × 179=358 2 × 233=466

0.14 0.22

Table 5.5: Cylinder splitting test: Values of the ultimate load obtained from ELM

Numerical Simulations

5.3 5.3.1

5.3: Fastening systems

116

Fastening systems Introduction

The connection of steel and concrete structures in structural engineering is generally accomplished by means of fastening systems such as anchor bolts, undercut anchors, etc.. While the development of anchor devices in the past was mainly performed by means of experiments, todays developments are intensively supported by numerical simulations. Such simulations provide insight into the load transfer from the anchor to the surrounding concrete. They allow to monitor the development of the failure mode and to estimate the peak load of the anchor device. Both failure mode and peak load strongly depend on the concrete strength, the geometric dimensions of the anchor, and the steel strength. Depending on the installation process, a subdivision into cast-in-place systems (placed in formwork before casting of concrete) and post-installed systems (installed in hardened structural concrete) can be made. The load is transferred into the base material by mechanical interlock, friction, bond and combinations of these mechanisms. Many systems are further characterized by highly concentrated loads, where very large compressive stresses are induced within a relatively small area. Among the large number of different anchor systems, in the following only headed anchors will be considered. For an extensive description of anchor systems see e.g. CEB (1991). Headed anchors are characterized by a load-transfer mechanism consisting mainly of mechanical interlock (bearing) at the anchor head. In Figure 5.27 various failure mechanisms of headed anchors are illustrated. • concrete-cone failure: characterized by the formation of a roughly conical fracture surface radiating from the top of the anchor head. The primary factors determining the load at which the concrete cone will fail are the anchor embedment depth and the concrete tensile strength. • pull-out failure: characterized by ongoing crushing of concrete above the head of the anchor, followed by the formation of a concrete failure cone as the head of the anchor approaches the concrete surface. Pull-out failure generally occurs when the bearing stresses in the concrete surrounding the anchor head are extraordinarily large because of small head/shaft diameter ratios coupled with large embedment depths. • bursting failure: this may preclude a concrete-cone failure in cases where the head of the anchor is positioned close to the free edge. Failure occurs when the transverse (bursting) stresses around the head of the anchor exceed the tensile capacity of the concrete mass between the head and the adjacent free surface. • steel failure: Failure occurs by yielding and rupture of the anchor shaft or by thread stripping.

Numerical Simulations (a)

5.3: Fastening systems P

(b)

P

(c)

P (d)

117

P

Figure 5.27: Failure mechanism of headed anchors: (a) concrete-cone failure, (b) pull-out failure, (c) bursting failure and (d) steel failure Factors influencing the load at which steel failure will occur include the strength of steel and the cross-sectional area of the shaft.

5.3.2

Reformulation of fictitious crack concept for axisymmetric problems

In case of axisymmetric problems characterized by the development of both radial and circumferential cracks, special attention must be paid to the definition of the characteristic length `t used in the context of the fictitious crack concept. Experimental results confirm the development of cracks in different directions of the concrete block. Figure 5.28 shows a typical mode of concrete failure reported by Lehmann (1992). Figure 5.28(a) indicates the development of radial cracks, caused by circumferential stresses. Figure 5.28(b) shows a cone-shaped failure mode. In the following, an anisotropic formulation for the determination of `t accounting for the different directions of cracking will be proposed. It is based on a constant number of radial cracks. According to this formulation, `t related

(a)

(b)

Figure 5.28: Experimental results obtained by Lehmann (1992): (a) development of radial cracks caused by circumferential stresses and (b) obtained mode of concrete failure

Numerical Simulations

5.3: Fastening systems

118

to radial cracks, referred to as `t,circ , increases with increasing distance from the axis of symmetry. It is computed as (see Figure 5.29) P z S radial cracks

`t,circ axisymmetric plane RSe

Figure 5.29: Anisotropic characteristic length formulation: illustration of `t related to radial cracks, `t,circ

`t,circ

Ue = n

with U e = 2 RSe π.

(5.1)

U e denotes the circumferential length related to the barycenter S of the finite element and n is the number of radial cracks. The distance between S and the axis of symmetry is RSe (see Figure 5.29). In Figure 5.29 and in the standard analysis, four radial cracks are considered, i.e., n=4. For cracks leading to a√ cone-shaped failure mode (Figure 5.28), the characteristic length is computed as `t,max = Ae , where Ae represents the area of the finite element. The maximum plastic strains in the axisymmetric plane, εpmax , and the plastic strains in the circumferential direction, εpcirc , are used for determination of the characteristic length `t from `t,max and `t,circ . For this purpose, the angle β (see Figure 5.30) is introduced as εp + εprr εpcirc + β = arctan( p ) with εpmax = zz εmax 2

s

(εpzz − εprr )2 + (εprz )2 , 4

(5.2)

where r and z represent the coordinate in the axisymmetric plane. Assuming an elliptical interpolation function, the characteristic length `t can be computed as `t =

q

x2s + ys2

(5.3)

Numerical Simulations

5.3: Fastening systems `t related to opening of radial cracks

`t,circ

tan β =

119

εpcirc εpmax

elliptical interpolation

ys

`t related to development of cone-shaped failure mode

`t β xs

`t,max

Figure 5.30: Anisotropic characteristic length: elliptic interpolation of `t with xs =

5.3.3

s

((1/`t,max

)2

1 + tan2 β(1/`t,circ )2 )

and ys = xs tanβ.

(5.4)

Pull-out test I

The first pull-out test to be investigated is based on a round robin analysis summarized in Elfgren (1990). Three different investigations will be performed: • The first investigation deals with the influence of the used material model for concrete on the peak load and the failure mode. Special attention is paid to consideration of confinement used for the modified Drucker-Prager criterion. • The influence of the interface between concrete and steel on the numerical results will be investigated in the second study. • Finally, the influence of selected material and model parameters of the ELM on the peak load and the failure mode will be investigated in the third study. 5.3.3.1

Geometric dimensions and material parameters

In Figure 5.31, the geometric properties of the experimental set-up according to Elfgren (1990) are shown together with the material parameters. The steel bolt is modeled as an elastic member. 5.3.3.2

Numerical study I: influence of material model

Because of axial symmetry of the geometric dimensions and the loading conditions, the problem can be solved by means of axisymmetric analyses. Figure 5.32 shows the used FE mesh consisting of 677 four-node finite elements. As regards the mechanical model of the

Numerical Simulations

5.3: Fastening systems

120

a+c

a+c Ør 2c

2c dimensions: d=150 mm t=15 mm Ø=45 mm d

a=150 mm c=22.5 mm Ør =24 mm

concrete: Young’s modulus: Ec =30000 N/mm2 Poisson’s ratio: νc =0.2 uniaxial compressive strength: fcu =40 N/mm2 uniaxial tensile strength: ftu =3 N/mm2 fracture energy: GIf =0.1 N mm/mm2 I fracture energy: GII f =50Gf

t Ø 6d

steel: Young’s modulus: Es =210000 N/mm2 Poisson’s ratio: νs =0.3 a+d

a+d

Figure 5.31: Pull-out test I: geometric dimensions and material properties

Redrawing Done

Z

Y

X

1

anchor bolt, only the anchor head is discretized. The slip at the contact line between the anchor head and the concrete is disregarded. The analyses are performed displacementdriven. The displacement at the nodes of the anchor head located at the axis of symmetry is prescribed (see Figure 5.32). Softening functions appearing in the formulations of both models were calibrated according to the fictitious crack concept (Hillerborg et al., 1976). The application of the fictitious crack concept to the simulation of radial cracks in axisupport ring

no slip considered between concrete and steel u ¯

axis of symmetry

zoom zoom

Figure 5.32: Pull-out test I (study I): FE discretization

u ¯

Numerical Simulations

5.3: Fastening systems

121

symmetric analyses requires the input of the expected number of radial cracks (for details, see the preceding subsection). In the present analyses, four radial cracks are assumed to develop. Figure 5.33 contains load-displacement curves obtained from the multi-surface model. The peak load obtained from the Drucker-Prager criterion characterized by consideration of confinement (modified model) was found to be 340 kN. Disregard of the influence of confinement within the Drucker-Prager criterion (original model), i.e., assuming fcu (σ1 )=const., fcr (σ1 )=const., and αDP,m =0.0022, led to a reduction of the peak load by 47% (see Figure 5.33(a)). The difference between the displacement at the anchor head and the concrete sur400 P [kN]

ub

us











ub us

300

P

P [kN]

300 200

200 100 0

400

ub , us [mm] 0

0.5

(a)

1.0

ub us

100 0 0

0.5

(b)

ub , us [mm] 1.0

Figure 5.33: Pull-out test I (study I): (a) load-displacement curves obtained from multi-surface model with (a) original and (b) modified Drucker-Pager criterion face, ub − us , is an indicator for compressive failure of concrete over the anchor head. An almost constant evolution of ub − us indicates a rigid body motion in consequence of formation of a cone-like failure mode (see, e.g., Figure 5.36(a)). Small values of us together with continuously increasing values of ub − us indicate local failure of concrete over the anchor head. This failure mode was obtained from the analysis based on the original multi-surface model (see Figure 5.33(a)). The underestimation of the compressive strength and the ductility because of neglecting the influence of confinement resulted in compressive failure of concrete over the anchor head. The respective P − ub curve shows similar characteristics as the underlying hardening/softening curve used for the Drucker-Prager model (see Figure 2.15(b)). Moreover, failure of concrete over the anchor head resulted in unloading of the remaining part of the structure. This is reflected by the decrease of us in the post-peak regime. Consideration of confinement by the modified multi-surface model led to an increase of the compressive strength over the anchor head. The almost constant evolution of u b − us in the post-peak regime indicates the development of a cone-like failure mode (see Figure 5.36(a)). In Figure 5.34 the distribution of αDP at peak load is shown for both models. The compressive failure over the anchor head obtained from the original model is reflected by softening material behavior (see Figure 5.34(a)). Softening is characterized by αDP > αDP,m =0.0022. On the other hand, confinement considered in the simulation based on the modified multisurface model resulted in an increase of αDP,m over the anchor head (see Figure 5.34(c)).

Numerical Simulations

5.3: Fastening systems

28.0 22.4 16.8 11.2 5.6 0

5.000e+00

0.000e+00

(a) 103 · αDP

20.0 16.0 12.0 8.0 4.0 0

12.5 10.0 7.5 5.0 2.5 0

5.000e+00

5.000e+00

0.000e+00

-5.000e+00

-5.000e+00

-1.000e+01

-1.000e+01

-1.500e+01

-1.500e+01

-2.000e+01

-2.000e+01

(b) 103 · αDP

122

0.000e+00

-5.000e+00

-1.000e+01

-1.500e+01

-2.000e+01

(c) 103 · αDP.m

Figure 5.34: Pull-out test I (study I): distribution of the internal hardening/softening variable of the Drucker-Prager criterion, αDP , at peak load obtained from (a) original and (b) modified multi-surface model; (c) distribution of αDP,m at peak load obtained from modified multi-surface model (10-fold magnification of displacements) The distribution of αDP is shown in Figure 5.34(b). The values of αDP are smaller than the respective values of αDP,m , indicating that the compressive strength in this area did not reach the ultimate strength fcu (hardening regime). The strong influence of confinement on the obtained numerical results is a consequence of large compressive stresses over the anchor head. For the multi-surface model, the major principal stress is used as a measure for confinement. If the major principal stress is positive, no confinement is considered. Figure 5.35 shows the distribution of the two principal stresses in the r − z plane, denoted as σmin and σmax , and of the circumferential stress σcirc at peak load, obtained from the analysis based on the modified multi-surface model. Over the anchor head, negative stresses are observed for all three principal stresses. Hence, σ1 < 0, resulting in an increase of the compressive strength and the ductility (see Figure 2.17). The distribution of the minimum in-plane principal stress σmin (see Figure 5.35(a)) provides

5 0 -5 -10 -15 -20

5.000e+00

0.000e+00

-5.000e+00

-1.000e+01

-1.500e+01

-2.000e+01

(a) σmin

(b) σmax

(c) σcirc

Figure 5.35: Pull-out test I (study I): distribution of principal stresses in the r − z plane, σmin and σmax and the stress component in the circumferential direction, σcirc at peak load obtained from modified multi-surface model (in [N/mm2 ])

Numerical Simulations

5.3: Fastening systems

0.005 0.004 0.003 0.002 0.001 0

123

0.0009 0.0007 0.0005 0.0003 0.0001 -0.0001

5.000e+00

5.000e+00

0.000e+00

0.000e+00

-5.000e+00

-5.000e+00

-1.000e+01

-1.000e+01

-1.500e+01

-1.500e+01

-2.000e+01

-2.000e+01

(b) εpcirc

(a) εpmax

Figure 5.36: Pull-out test I (study I): distribution of the maximum plastic strain in r − z plane, εpmax and of the circumferential plastic strain εpcirc at peak load obtained from modified multi-surface model insight into the load-carrying behavior of the concrete specimen. The applied load at the anchor head is transferred by a compressive strut from the confined area over the anchor head to the support ring (see Etse (1998) for similar results). The respective maximum principal stress in this strut resulted in the development of a circumferential crack. This crack started to open at the anchor head, propagating towards the support ring, and finally causing coneshaped failure of the specimen. The crack pattern obtained at peak load on the basis of the modified multi-surface model is shown in Figure 5.36(a) by means of the distribution of the maximum plastic strain in the r − z plane, εpmax . In addition to the circumferential crack, radial cracks developed, starting from the corner at the concrete surface and propagating in the interior of the concrete block. As mentioned previously, four radial cracks were assumed to open in the context of the fictitious crack concept. In Figure 5.36(b), the location of these radial cracks is shown by means of the respective circumferential plastic strain, εpcirc . The load displacement curve obtained from the analysis based on the single-surface model is shown in Figure 5.37. The peak load was computed as 312 kN. The continuously increasing value of ub − us in the pre-peak regime indicates plastic material response above the anchor head. However, the actual failure occurs in consequence of circumferential cracks resulting in a cone-like concrete P [kN]

500

ub

us

P













400 300 200

ub us

100 0

0

0.5

ub , us [mm] 1.0

Figure 5.37: Pull-out test I (study I): load displacement curve obtained from single-surface model

Numerical Simulations

5.3: Fastening systems

124

failure surface. Similar to the analysis based on the modified multi-surface model, an almost constant evolution of ub − us is observed in the post-peak regime indicating this mode of failure. The distribution of αh at peak load is shown in Figure 5.38(a). αh describes the state of the yield surface in the hardening regime. αh = 0 refers to the initial yield surface defined by q¯h = fcy (see Figure 2.34(a)). If αh = 1, the failure surface is reached (¯ qh = fcu ) and softening or ideally-plastic behavior is initiated. Whether or not softening will take place depends on the level of confinement. For the single-surface model, confinement is represented by the hydrostatic pressure. For a hydrostatic pressure p = −(σ1 + σ2 + σ3 )/3 lower than the hydrostatic pressure pT P related to the so-called transition point T P , softening occurs. For stress states characterized by p > pT P , ideally-plastic material behavior is assumed. Based on the material parameters given in Figure 5.31, the hydrostatic pressure at the T P is computed as pT P =44.3 N/mm2 . 0 15 30 45 60 75

1.0 0.8 0.6 0.4 0.2 0

5.000e+00

5.000e+00

0.000e+00

0.000e+00

-5.000e+00

-5.000e+00

-1.000e+01

-1.000e+01

-1.500e+01

-1.500e+01

-2.000e+01

-2.000e+01

(a)

(b)

Figure 5.38: Pull-out test I (study I): distribution of (a) the internal hardening variable αh and (b) the hydrostatic pressure p, with p = −(σ1 + σ2 + σ3 )/3, (in [N/mm2 ]) at peak load obtained from single-surface model (5-fold magnification of displacements) The distribution of the hardening variable αh at peak load shown in Figure 5.38(a) is characterized by several zones with αh = 1. As regards the zone over the anchor head, the respective hydrostatic pressure is greater than the hydrostatic pressure at the transition point, i.e., p > pT P . This resulted in ideally-plastic material response and, hence, in plastic deformations already observed by the form of the evolution of ub − us depicted in Figure 5.37. Because of this ideally-plastic material response, no local failure mode over the anchor head developed. The remaining regions characterized by αh = 1 show small values for the hydrostatic pressure, resulting in softening material behavior. Similar to the crack pattern obtained from the multi-surface model (Figure 5.36), these regions refer to two circumferential cracks starting at the anchor head, and radial cracks propagating from the concrete surface into the interior of the concrete block.

Numerical Simulations 5.3.3.3

5.3: Fastening systems

125

Numerical study II: influence of boundary condition

As regards the contact condition at the steel-concrete interface, two different boundary conditions are considered in the second numerical study (see Figure 5.39). gap friction elements

u ¯

no slip considered between concrete and steel u ¯

u ¯

u ¯

h∆

(a)

(b)

Figure 5.39: Pull-out test I (study II): boundary conditions (BC) used for modeling of the steel-concrete interface: (a) use of ”gap-friction” elements accounting for slip between steel and concrete (BC1) and (b) fixed connection of the steel-concrete interface together with constrained lateral displacements near the anchor head (BC2) The first boundary condition, depicted in Figure 5.39(a), accounts for friction between the anchor head and the concrete specimen. The friction behavior is modeled by means of a Coulomb’s friction law employing ”gap-friction” elements (MARC, 1996). For these elements, the gap closure distance ∆ (see Figure 5.39(a)) and the friction coefficient µ serve as input parameters. In the present study, ∆ is set equal to 1.0E-05. The second boundary condition shown in Figure 5.39(b) is characterized by a fixed connection between the anchor head and the concrete at the steel-concrete interface. In addition, the lateral displacements of the concrete near the anchor head are constrained. Figure 5.40 contains the load-displacement curves obtained from the single-surface model using the first set of boundary conditions together with selected values of the friction coefficient µ. Figure 5.40 clearly shows that the simulation of the load-carrying behavior of the investigated anchor bolt depends on the friction coefficient. For the case of µ=1, the response is similar to the one based on the assumption of a fixed connection which was made in the preceding subsection (see Figure 5.37). For the value of µ=0, however, a significant drop of the peak load, followed by almost ideally plastic behavior in the post-peak regime, is observed. This strong deviation of the load-displacement curves might stem from the development of different failure modes on the structural level. The dependence of the failure mode on the value of the friction coefficient is elucidated by the distribution of the hydrostatic pressure at αh = 1 at the deformed configuration (see Figure 5.41).

Numerical Simulations

5.3: Fastening systems

P [kN]

400

u ¯ µ=1

300

126

P















µ=0

200 100 0

u ¯ [mm] 0

0.5

1.0

1.5

2.0

Figure 5.40: Pull-out test I (study II): load displacement curves obtained from ELM applying BC1 for friction coefficients µ=1 and µ=0 The decrease of the friction coefficient resulted in an increase of the hydrostatic pressure at peak load, above the anchor head. Inspection of the deformation plots clearly shows that for µ=1 the part of the contour of the model denoted as ”concave” (see Figure 5.41(a)) has the opposite curvature of the part of the contour of the model referred to as ”convex” (see Figure 5.41(b)). This model is based on disregard of friction, i.e., on µ=0. The load-displacement curves obtained using BC2 are shown in Figure 5.42. Comparison with the numerical results obtained when using BC1 (see Figure 5.40) shows an increase of the peak load by approximately 15 % together with an increase of stiffness of the loaddisplacement curve. This behavior can be explained by inspecting the distribution of the hardening variable αh and of the hydrostatic pressure at peak load (see Figure 5.43). A comparison with the distribution of the hydrostatic pressure reported in the preceding subsection (see Figure (a)

(b) 0

5.000e+00

convex 15 0.000e+00

concave

30

-5.000e+00

45

-1.000e+01

-1.500e+01

60

75

-2.000e+01

Figure 5.41: Pull-out test I (study II): distribution of the hydrostatic pressure p = −(σ1 + σ2 + σ3 )/3 (in [N/mm2 ]) obtained from the ELM at u ¯=1.2 mm for (a) µ=1 and (b) µ=0 (5-fold magnification of displacements)

Numerical Simulations

5.3: Fastening systems

127

5.38(b)) indicates that because of constraining of lateral deformations used for BC2, the hydrostatic pressure above the anchor head is strongly increasing. This led to the development of distributed cracking around the anchor head and, hence, to an increase of the peak load. ub P [kN]

400

us

P













300 200 ub us

100 0

ub , us [mm] 0

0.5

1.0

Figure 5.42: Pull-out test I (study II): load-displacement curves obtained from ELM applying BC2 [× 10−4 ] 4.50

5.000e+00

3.40

0.000e+00

2.30

-5.000e+00

1.20

-1.000e+01

0.10

-1.500e+01

0.00

-2.000e+01

5.000e+00

0

0.000e+00

15

-5.000e+00

30

-1.000e+01

45

-1.500e+01

60

75

-2.000e+01

(a)

(b)

Figure 5.43: Pull-out test I (study II): distribution of (a) the internal hardening variable αh and (b) the hydrostatic pressure p=−(σ1 +σ2 +σ3 )/3 (in [N/mm2 ]) at peak load Table 5.6 contains the numerical results obtained from investigation of influence of the boundary conditions. 5.3.3.4

Numerical study III: parameters of the ELM

Because of the rather complex format of the ELM, in the following a study of the influence of selected parameters on the numerical results was performed. The first investigation was

Numerical Simulations

5.3: Fastening systems

boundary condition BC1

friction ultimatecoefficient µ load 1 0 —

BC2

313 kN (≈ 210 kN) 362 kN

128

failuremode cone-failure ductile cone-failure

Table 5.6: Ultimate loads obtained from ELM for boundary condition study concerned with the influence of the anisotropic characteristic-length formulation on the numerical results described in the preceding section. Figure 5.44 shows the load-displacement curves obtained using 4, 6, and 8 radial cracks. For the numerical simulations performed in this study the same boundary conditions as in Subsection 5.3.3.2 are applied, i.e., a fixed connection of the steel-concrete interface. For the considered anchor bolt test, irrespective of the number of radial cracks n almost the same load-displacement curves are observed. u ¯

P [kN] 400 n=4

P













 n=8

300

n=6 200 100 0

u ¯ [mm] 0

0.5

1.0

1.5

Figure 5.44: Pull-out test I (study III): load-displacement curves obtained from ELM using different numbers of radial cracks used for the anisotropic characteristic length formulation The next investigation deals with the influence of the initial compressive yield strength parameter fcy on the numerical results. Figure 5.45 shows the load-displacement curves obtained for the standard formulation of the ELM, i.e., fcy =0.2fcu and fcy =fcu . The choice of fcy =fcu leads to a simplified version of the ELM, because of the fact that the entire hardening regime is neglected. Figure 5.45 indicates rather strong deviations of the loadcarrying behavior depending on the value of fcy . Using fcy = fcu leads to a reduction of the peak load by approximately 19 %. The load-displacement curve obtained on the basis of fcy =fcu strongly differs from the standard one. It is characterized by a first peak at approximately 200 kN (P1 ), followed by

Numerical Simulations

5.3: Fastening systems u ¯

P [kN] 400

129

P











fcy = 0.2 fcu 300

Pmax P1

200

P2

100 0

fcy = fcu

u ¯ [mm] 0

0.5

1.0

1.5

Figure 5.45: Pull-out test I (study III): load-displacement curves obtained from ELM using fcy =0.2fcu and fcy = fcu descending part. At a load level of 165 kN (P2 ) the load starts increasing again until the second and final peak (Pmax ) is reached. In order to elucidate the failure behavior of the pull-out test with this initial yield strength parameter, the distribution of the maximum plastic strain in the r −z plane, εpmax is shown in Figure 5.46 at different loading states. Figure 5.46(a) indicates that, similar to the standard simulation (fcy =0.2fcu ), two circumferential cracks start propagating from the anchor head towards the support ring. The first crack starts at the corner of the anchor head and propagates towards the support. This direction coincides with the direction of the compressive strut transferring the load to the support. The second circumferential crack originates from the geometric properties of the anchor bolt. However, the first crack directed towards the support ring opens much faster than the second one. At the loading state P2 this crack has already reached the support ring (see Figure 5.46(b)). Because of the fact that the finite element mesh is aligned towards the support and the dominating circumferential crack is located within the element row directed towards the support, a cone-shaped concrete failure surface can only develop by a change of crack direction. This change of crack direction is shown in Figure 5.46(c). The final investigation for the ELM deals with the formulation of the flow rule. The loaddisplacement curves obtained from the ELM based on the standard formulation, i.e, employing a non-associative flow rule, and the formulation based on an associative flow rule are shown in Figure 5.47. Comparison of the load-displacement curves clearly shows that the use of an associative flow rule leads to a strong overestimation of the ultimate load. This behavior is the consequence of large volumetric plastic deformations which in turn lead to highly confined stress states (see also the Reynolds effect described in Subsection 4.3.2.7). Hence, for the considered anchor bolt, characterized by confined stress states near the anchor head, the use of an associative flow rule does not lead to reasonable numerical results. Table 5.7 contains a list of values for the peak load obtained from the investigation of the influence

Numerical Simulations

5.3: Fastening systems

130

[× 10−3 ] 5.00 4.00 3.00 2.00 1.00 0.00

5.000e+00

0.000e+00

-5.000e+00

-1.000e+01

-1.500e+01

-2.000e+01

(b)

(a)

(c)

Figure 5.46: Pull-out test I (study III): distribution of the maximum plastic strain εpmax in the r − z plane at different load levels: (a) load level P1 , (b) load level P2 , and (c) peak load Pmax of model parameters on the mechanical behavior of the structure. 5.3.3.5

Conclusions

In this subsection the dependence of the structural failure of anchor bolts in the concrete on the material model and on the boundary conditions of the steel-concrete interface was investigated. For this purpose, two material models for plain concrete were used. One of them is a single-surface and the other one a multi-surface model. The latter was originally proposed by Meschke (1996) (it is referred to as the original model). This model was reformulated in order to account for the influence of confinement. The reformulated model is referred to as the modified multi-surface model. From the numerical simulations, the following conclusions concerning the structural response of the concrete specimen can be drawn: • the concrete located between the anchor head and the support ring is subjected to strong non-uniform triaxial stress states, characterized by P u ¯ P [kN] associative




600


flow rule 









 non-associative flow rule

400 200 0 0

0.5

1.0

u ¯ [mm] 1.5

Figure 5.47: Pull-out test I (study III): load-displacement curves obtained from the ELM using different flow rules

Numerical Simulations

5.3: Fastening systems parameters radial cracks: n=4 n=6 n=8 initial compressive yield strength fcy =0.2fcu fcy =fcu flow rule: associative non-associative

Pmax [kN]

failure mode

314 314 314

cone-failure cone-failure cone-failure

314 256

cone-failure cone-failure

545 314

cone-failure cone-failure

131

Table 5.7: Numerical results obtained from parameter study on the basis of the ELM - hydrostatic pressure over the anchor head, - a compressive strut from the anchor head to the support ring, and - circumferential cracking caused by tensile loading perpendicular to this strut; • the underlying material model has a crucial influence on the predicted failure mode characterized either - by local compressive failure over the anchor head, or - by a cone-shaped failure surface in consequence of the development of circumferential cracks • the underderlying boundary condition of the steel-concrete interface led to either - local compressive failure over the anchor head, or - to the development of a cone-shaped failure surface • the used model parameters of the ELM, e.g., the initial compressive yield strength parameter fcy and the flow rule formulation have a significant influence on the numerical results. The cone-shaped failure surface, which presumably represents the correct failure mode, was obtained by the modified multi-surface model and the single-surface model using the standard parameters. The respective peak loads deviated by 8%. Disregard of the influence of confinement on the material strength and ductility in the context of the original multi-surface model resulted in local compressive failure of concrete over the anchor head. The peak load related to this presumably incorrect failure mode was found to be significantly lower than the peak loads obtained from the analyses characterized by cone-shaped failure.

Numerical Simulations

5.3.4

5.3: Fastening systems

132

Pull-out test II

The second pull-out test investigated in this work is based on data given by the Hilti AG. The main difference from the previous example is the shape of the anchor head and the rather small embedment depth of the steel rod. Three different studies were performed: I The focus of the first study is on the influence of the geometric properties on the load-carrying behavior. II Different FE meshes are used in the second study in order to assess the influence of the underlying discretization on the numerical results. III The influence of the material model for concrete on the numerical results is investigated in the third study. 5.3.4.1

Geometric dimensions and material parameters

The headed stud considered in the numerical studies is characterized by an inclined shoulder (see Figure 5.48). The material properties of concrete and steel used in the numerical analyses are given in Table 5.8. In the numerical studies, the material behavior of the steel bolt was assumed to be linear elastic. At the cone-shaped contact surface between the anchor head and the concrete (see Figure 5.48), no slip is considered. The analyses are performed displacement-driven. The displacements are prescribed at the top of the steel rod. Because of axisymmetry of the geometric properties and the loading conditions, the problem is solved by means of axisymmetric analyses. a=200 a=200 14

50

coneshaped contact surface

5

4

d=50.8

100

260 203.65 5.55 50.8

40

50

ø1 = 12

ø2 = 22

334

32

334

Figure 5.48: Pull-out test II: geometric dimensions (in [mm])

Numerical Simulations

5.3: Fastening systems

133

concrete Young’s modulus Poisson’s ratio uniaxial compressive strength uniaxial tensile strength fracture energy: GIf fracture energy: GII f steel Young’s modulus Poisson’s ratio

30000 N/mm2 0.2 40 N/mm2 3 N/mm2 0.1 Nmm/mm2 50 GIf 210000 N/mm2 0.3

Table 5.8: Material parameters of concrete and steel 5.3.4.2

Numerical study I: variation of a/d-ratio

In order to investigate the influence of the a/d-ratio on the numerical results, the a/d-ratio of 4, depicted in Figure 5.48, was reduced to 3, 2, and 1. The different geometric properties considered in this study are shown in Figure 5.49.

(a) a/d=4

(b) a/d=3

(c) a/d=2

(d) a/d=1

Figure 5.49: Pull-out test II (study I): geometric properties of the headed stud considered in the numerical investigation Figure 5.50 shows the used FE mesh consisting of 2695 four-node elements. A relatively fine discretization was generated in the area where failure of concrete is expected. For the remaining part, a coarser mesh was designed.

mesh1

n=2695

axis of symmetry

zoom

Figure 5.50: Pull-out test II (study I): FE mesh (n: number of elements)

Numerical Simulations

5.3: Fastening systems

134

As regards failure of headed studs caused by concrete failure, there are two different types of failure modes. On the one hand, high compressive loading of concrete at the anchor head may cause local shear failure. On the other hand, a circumferential crack initiating at the anchor head and propagating towards the support may develop, finally resulting in a cone-shaped failure surface. As already pointed out in Pivonka et al. (2001d), the relative displacement between the anchor head and the concrete surface, ub − us (see Figure 5.51(a)), allows to distinguish between the two aforementioned failure modes. Whereas an almost constant value of ub − us is typical for cone-shaped failure, local shear failure at the anchor head is characterized by a continuously increasing value of ub − us . Figure 5.51(a) shows the histories of us and ub obtained from numerical analyses with a/d=4 and 1, respectively. For a/d=4, almost identical histories are observed for us and ub . This indicates rather small deformations and, hence, rather low compressive loading of concrete over the anchor head. On the other hand, a continuously increasing value of ub − us is obtained for the analysis based on a/d=1, indicating compressive failure of concrete over the anchor head. However, the value of ub − us in the post-peak regime is almost constant. Hence, similar to the analysis for a/d=4, a cone-shaped failure surface finally develops, causing failure of the headed stud. Figure 5.51(b) shows the history of the obtained load as a function of the prescribed displacement u ¯ for the considered values of the a/d-ratio. For decreasing values of the a/d-ratio, the value of the peak load is increasing. P 150

ub

P [kN]

us











100

150

a/d=4 a/d=3 a/d=2 a/d=1

P [kN]

100

50 0 0

u ¯

ub us 0.5

1.0 (a)

ub , us [mm] 1.5

50 0 0

u ¯ [mm] 0.5

1.0 (b)

1.5

2.0

Figure 5.51: Pull-out test II (study I): load-displacement curves obtained from multi-surface model (a) for a/d=4 and 1 and (b) for different values of the a/d-ratio The distribution of the internal variable of the Drucker-Pager criterion, αDP , in the vicinity of the anchor head is given in Figure 5.52 for a/d=4 and 1 at the respective peak loads. Figure 5.52(a) indicates an almost elastic material response of concrete over the anchor head for the headed stud with a/d=4. Peak values of αDP are observed in the region left of the upper outer corner of the anchor head. Figure 5.52(b) shows the distribution of αDP for an a/d-ratio equal to 1. In contrast to the distribution given in Figure 5.52(a), peak values of αDP are observed over the anchor head.

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135

[× 10−2 ] 1.0 0.8

5.000e+00

0.000e+00

0.6 0.4 0.2

-5.000e+00

-1.000e+01

-1.500e+01

0.0

-2.000e+01

(a)

(b)

Figure 5.52: Pull-out test II (study I): distribution of the internal hardening/softening variable of the Drucker-Prager criterion, αDP , in the vicinity of the anchor head obtained for (a) a/d=4 and (b) a/d=1 at the respective peak loads The respective plastic deformations are responsible for the increasing value of the relative displacement ub − us in Figure 5.51(a). The distribution of the minimum in-plane principal stress, σmin , provides insight into the load-carrying behavior of the headed stud (see Figure 5.53). In general, the load is transferred from the anchor head to the support ring by means of a compressive strut. For a/d-ratios ranging from 3 to 1, this strut can clearly be identified (see

(a) a/d=4

3.0

5.000e+00

0.000e+00 -13.6

(b) a/d=3

-5.000e+00 -30.2

-1.000e+01 -46.8

(c) a/d=2

-1.500e+01 -63.4

-2.000e+01 -80.0

(d) a/d=1

Figure 5.53: Pull-out test II (study I): distribution of minimum principal stress σmin (in [N/mm2 ]) in the axisymmetric plane

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Figures 5.53(b) to 5.53(d)). The previously observed increase of the peak load for decreasing values of the a/d-ratio is reflected by increasing compressive stresses σmin in the strut. The stress distribution shown in Figure 5.53(a) refers to a/d=4. The compressive strut between the anchor head and the support is divided into two parts. The lower part of the strut is similar to the strut obtained from the analysis with a/d=3. It starts at the anchor head and is aligned towards the support. However, for a/d=4 it does not reach the support. It becomes almost horizontal at a distance of approximately d/2 from the concrete surface. The upper part of the strut is parallel to the concrete surface, starting at the support ring. The load transfer between the two horizontal parts of the strut from a depth of d/2 to the concrete surface is accomplished by mixed compressive-tensile loading. Insight into the failure mechanism is gained from the distribution of the maximum plastic strain εpmax in the axisymmetric plane, depicted in Figure 5.54.

[× 10−3 ]

(a) a/d=4

1.00

5.000e+00

0.78

0.000e+00

(b) a/d=3

-5.000e+00

0.56 0.34

-1.000e+01

(c) a/d=2

0.12

-1.500e+01

-2.000e+01 -0.10

(d) a/d=1

Figure 5.54: Pull-out test II (study I): distribution of maximum plastic strain in the axisymmetric plane, εpmax εpmax represents the opening of circumferential cracks which finally cause cone-shaped failure of concrete. For the headed stud considered, two different circumferential cracks are observed. One of them is located in the previously mentioned compressive strut between the anchor head and the support ring. It is a consequence of high compressive loading in this strut. The second crack originates from the geometric properties of the anchor head. It starts from the upper outer corner of the anchor head. Depending on the geometric properties of the headed stud, described by the a/d-ratio, the two cracks are developing differently as regards both crack width and orientation. For the analysis with a/d=4, characterized by the compressive strut consisting of two parts, the second crack governs the structural failure. The first crack,

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i.e., the one located within the compressive strut, has only slightly developed. The opposite situation is found for a/d=1. Here, the crack located in the compressive strut dominates the failure of the headed stud. The second crack is almost horizontal and, hence, has no influence on the failure of the stud. The analyses based on a/d=2 and 3 represent transition states between the previously described two extreme cases a/d=4 and 1. The peak loads and the displacements at peak load obtained from this study are summarized in Table 5.9. 5.3.4.3

Numerical study II: variation of discretization

mesh2

n=994

axis of symmetry

axis of symmetry

The second investigation focuses on the influence of the FE mesh on the numerical results. For this purpose, three FE meshes consisting of 2695, 994, and 2676 four-node elements are considered (see Figures 5.50 and 5.55). Mesh1 was used in the numerical investigation reported in the previous subsection. For the two remaining meshes, mesh alignment was considered. Mesh alignment is the orientation of element edges in the direction of opening cracks (see Lackner and Mang (2001)). For mesh2, the element edges are aligned from the anchor head towards the support ring (such meshes are commonly used in anchor bolt analyses Elfgren (1990)). The element edges of mesh3 are adapted to the expected orientation of the circumferential crack, see Figure 5.55.

mesh3

n=2676

Figure 5.55: Pull-out test II (study II): FE meshes characterized by mesh alignment (n: number of elements) The load-displacement curves obtained from the multi-surface model for a/d=4 are depicted in Figure 5.56. Remarkably, the influence of the underlying FE mesh on the peak load is rather small. The values obtained for the peak loads vary only by 4.4%. Moreover, almost the same load-displacement response is observed for the aligned FE meshes, i.e., for mesh2 and mesh3. The orientation of the circumferential crack obtained by means of the structured mesh, i.e., mesh1 (see Figure 5.56(a)), does not coincide with the element edges of this mesh. This resulted in an artificial stress transfer in consequence of element locking (see Jir`asek and Zimmermann (1998), Huemer et al. (1999)) and, hence, in additional cracking in neighboring elements. An increase of the crack band width because of cracking of more than one row of finite elements leads to an increase of the released energy and, hence, to an overestimation of the load-carrying behavior (see pre-peak response obtained from mesh1 in Figure 5.56).

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5.3: Fastening systems











50

0

P u ¯

P [kN]

100

138

mesh1 mesh2 mesh3 0

0.5

1.0

u ¯ [mm] 1.5

2.0

Figure 5.56: Pull-out test II (study II): load-displacement curves Figure 5.57 shows the crack pattern at the respective peak loads by means of the distribution of εpmax . The distribution of εpmax obtained on the basis of mesh1 (see Figure 5.57(a)) indicates representation of the crack by at least three rows of elements. Mesh alignment towards the support ring (mesh2, see Figure 5.57(b)) provides the desired representation of the crack in the context of the fictitious crack concept (Hillerborg et al., 1976), i.e., the representation of the crack in one element row. However, crack propagation is strongly affected by the orientation of the element edges. Alignment towards the support resulted in a straight crack extending from the anchor head to the support. This crack pattern does not coincide with the crack pattern obtained by means of mesh1. Mesh1 and mesh3 (see Figures 5.57(a) and (c)) gave the presumably correct crack pattern. In contrast to mesh1, mesh3 provides the correct representation of the crack within one row of elements.

[× 10−3 ]

(a) mesh1

1.00

5.000e+00

0.78

0.000e+00

0.56 0.34

-5.000e+00

-1.000e+01

(b) mesh2

0.12 -0.10

-1.500e+01

-2.000e+01

(c) mesh3

Figure 5.57: Pull-out test II (study II): distribution of maximum plastic strain εpmax in the axisymmetric plane The numerical results obtained from the second study are summarized in Table 5.9.

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139

Numerical study III: variation of material model

The last part of the numerical investigations deals with the influence of the used material model on the numerical results for an a/d-ratio of 4. Mesh3 is used for the numerical simulations (see Figure 5.55). In the mesh study performed in the previous subsection this mesh was found to give best numerical results as regards the crack pattern and the representation of cracks by the FEM. For the analysis based on the ELM, a non-associative flow rule is considered. Moreover the dependence of the strength on the Lode angle is accounted for by means of e=rt /rc ≤1. Figure 5.58 shows the load-displacement curves obtained from the ELM and the multi-surface model and the value of the peak load obtained from the experiments. A strong influence of the used material model on the peak load is observed. The peak load obtained from the ELM is only 55% of the peak load predicted by the multi-surface model. P

P [kN] peak load obtained from experiment

100

50

u ¯











 multi-surface model Extended Leon Model u ¯ [mm]

0

0

0.5

1.0

1.5

Figure 5.58: Pull-out test II (study III): load-displacement curves obtained from the multi-surface model, the ELM, and the mean value of the peak load obtained from the experiment The reason for the large deviations between the two model answers probably stems from different behavior • on the constitutive level and/or • on the structural level caused by different modes of cracking. As regards the latter, similar crack patterns were obtained from both analyses. The dominating circumferential crack developed exactly in the row of finite elements which was prespecified for cracking by means of mesh alignment. Hence, the reason for the observed large deviations must be found at the constitutive level. Figure 5.59 shows the distribution of the internal variables of the multi-surface model, i.e., αDP and αRK for the Drucker-Prager and the Rankine criterion, respectively. A shear failure mode characterized by an active Drucker-Pager criterion and an active Rankine criterion is observed.

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[× 10−3 ] (a)

2.5

5.000e+00

2.0

0.000e+00

1.5

-5.000e+00

1.0

-1.000e+01

0.5

-1.500e+01

(b)

0.0

-2.000e+01

Figure 5.59: Pull-out test II (study III): distribution of internal variable of (a) the Drucker-Prager criterion, αDP , and (b) the Rankine criterion, αRK In order to investigate the performance of the used material models when applied to the simulation of such shear failure modes, a plane-stress benchmark problem is considered (see Figure 5.60(a)). The model depicted in Figure 5.60(a) is loaded by means of a vertical pressure p. Thereafter, a horizontal displacement u ¯ is prescribed at the top of the model. The respective stress path in the σ1 -σ2 stress space is depicted in Figure 5.60(b) for different values of p. According to the shapes of the initial and failure surfaces shown in Figure 5.61, this type of loading will result in the desired shear failure mode. (a)

p

(b) u ¯

application of confinement, p

σ2 σ1 increase of u ¯

Figure 5.60: Pull-out test II (study III) – benchmark problem: (a) loading conditions for the considered plane-stress benchmark problem and (b) stress path in the σ1 -σ2 stress space Figure 5.62 shows the stress paths obtained from the analysis of the benchmark problem. As regards the multi-surface model (see Figure 5.62(a)), σ1 increases with increasing value of u ¯ until the stress path reaches the Rankine loading surface. It is noteworthy that the contribution of the Rankine criterion to the plastic strain rate tensor is controlled by means of an associative flow rule. Consequently, softening, which is characterized by the decrease of σ1 , is accompanied by an increase of the compressive stress σ2 . The stress path drifts towards the Drucker-Prager loading surface, finally reaching the intersection point of the Rankine surface with the Drucker-Prager loading surface. Consideration of hardening within the Drucker-Prager criterion results in a continuous increase of the compressive stress σ 2 . The shape of the loading surface of the ELM and the used plastic potential (see Pivonka et al. (2000)) results in by far smaller values of the compressive stress σ2 (see Figure 5.62(b)).

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5.3: Fastening systems σ2 fcu

(a) -1.0

σ1 fcu

-1.0

initial surface

σ1 fcu

initial surface -1.0

failure surface

σ2 fcu

(b)

141

-1.0

failure surface

Figure 5.61: Pull-out test II (study III) – benchmark problem: initial and failure surface in σ1 -σ2 stress space of (a) multi-surface model and (b) ELM For the analyses with p=0, 1, and 2 N/mm2 , the stress path turns towards the origin of the stress space. The maximum values of the compressive stress σ2 are obtained as max|σ2 |=6.7, 9.3, and 11.8 N/mm2 for p=0, 1, and 2 N/mm2 , respectively. For the multi-surface model, the compressive stress σ2 increases until the uniaxial compressive strength is reached and, hence, softening of the Drucker-Prager criterion is initiated. The stress-strain curves corresponding to the considered benchmark problem are depicted in Figure 5.63. Whereas a reduction of the shear stress τ is obtained for the ELM for p=0, 1, and 2 N/mm2 , the multi-surface model shows a continuous increase of the shear stress for confined loading conditions, i.e., for p>0. The latter seems to be responsible for the restiffening observed in the load-displacement curve of the headed stud obtained from the multi-surface model (see Figure 5.58). Similar to the benchmark problem, restiffening is (a)

4

σ2 [N/mm2 ]

(b) 4

σ2 [N/mm2 ]

σ1 [N/mm2 ] -2 -4 -8

2 4 p=0 p=1 p=2 p=5

σ1 [N/mm2 ] -2

2 4

-4

p=0

-8

p=1

-12

-12

-16

-16

-20

-20

p=2

p=5

Figure 5.62: Pull-out test II (study III) – benchmark problem: stress paths in the σ1 -σ2 stress space obtained from (a) the multi-surface model and (b) the ELM

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observed until the uniaxial compressive strength is reached. Softening in the context of the Drucker-Prager criterion, characterized by αDP >αDP,m (see Figure 5.59(a)), finally leads to failure of the headed stud in the form of the expected cone-shaped failure surface 2 . The same failure mode was detected by the ELM at a by far smaller load level (see Figure 5.58). τ [N/mm2 ]

τ [N/mm2 ] 12

12

9 6 3 0 0

p= 5

9

6 p= 2 p= 1 3 p= 0 N/mm2 γ [10−3 ] 0 0.4 0.8 1.2 1.6 2.0 0 (a)

p= 5 p= 2 p= 0 N/mm2 p= 1 γ [10−3 ] 0.4 0.8 1.2 1.6 2.0 (b)

Figure 5.63: Pull-out test II (study III) – benchmark problem: stress-strain curves obtained from (a) multi-surface model and (b) ELM The numerical results obtained from the material model study are summarized in Table 5.9. study I: span/depth-ratio

II: FE mesh

III: material models

a/d

model

mesh

Pmax [kN]

u ¯ (P =Pmax ) [mm]

4 3 2 1 4 4 4 4 4

DP-R DP-R DP-R DP-R DP-R DP-R DP-R DP-R ELM

1 1 1 1 1 2 3 3 3

91.7 109.0 115.0 142.5 91.7 87.7 91.2 91.2 50.6

1.56 1.43 1.06 1.03 1.56 1.25 1.30 1.30 0.31

Table 5.9: Peak load Pmax and prescribed displacement u¯ at peak load obtained from numerical studies (DP-R: multi-surface model consisting of DruckerPrager and Rankine criterion, ELM: Extended Leon Model) 2

At αDP = αDP,m , the compressive strength is equal to the uniaxial compressive strength fcu . In the context of the used Drucker-Prager criterion, αDP,m depends on the level of confinement (see 2.3.2.4 in Section 2.3). For the case of no confinement, αDP,m =0.0022.

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5.3: Fastening systems

143

Conclusions

In this subsection, the dependence of the failure behavior of headed studs with inclined shoulder on the a/d-ratio, the discretization, and the material model was investigated. From the numerical simulations of a headed stud with inclined shoulder the following conclusions can be drawn: • As for the analyses based on different geometric properties, a strong influence of the a/d-ratio on the peak load and the load-carrying behavior was observed. For a/d-ratios ranging from 1 to 3, a compressive strut from the anchor head to the support ring was identified. From the analysis with a/d=4, a strut consisting of two separated parts was obtained. • As for the analyses based on different FE discretizations, almost the same value of the peak load was obtained for all considered FE meshes. In contrast to the used aligned FE meshes, the structured mesh resulted in element locking and, hence, in an overestimation of the load-carrying behavior in the pre-peak regime. • As for the analyses based on different material models for concrete, a crucial influence on the predicted value of the peak load was observed. Both the shape of the loading surface and the plastic potential were identified as the main reasons for the obtained deviations. In all analyses performed, the development of a cone-shaped failure surface was the reason for failure of the headed stud.

5.3.5

High strength undercut anchor

The final numerical simulation reported in this chapter is the investigation of an undercut anchor. This high strength undercut anchor was investigated by the Hilti AG for the purpose of strengthening of structural members, such as columns of bridges, to avoid damage as may result, e.g., from earthquake loading. Very high loads (up to 800-1000 kN) measured in the experiments and damage in the zone of the anchor head indicate the development of highly confined stress states in the region where the load transfer between the anchor and the concrete takes place. The loading frame and a detail of the concrete plate of the experimental set up are shown in Figure 5.64. From the mechanical point of view, there are many challenging aspects of this simulation, such as the combination of non-linear material models with the contact formulation incorporated in the finite element package MARC (MARC, 1996). The in situ installation of this type of anchor incorporates several loading states which will be described in detail in Subsection 5.3.5.2. A report of the investigation of the load-carrying behavior of the concrete plate by the ELM is contained in Subsection 5.3.5.3.

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Figure 5.64: High strength undercut anchor: experimental set up 5.3.5.1

Geometric dimensions and material properties

Figure 5.65 shows the geometric dimensions of the undercut anchor. The whole anchor system can be subdivided into four construction parts: (1) threaded rod: steel rod where the load is applied (2) nut: screwed on the threaded rod; connection member between rod and cone (3) cone: establishes the transfer of the load from the threaded rod, connected with the nut, to the expansion sleeve (4) expansion sleeve: provides the right positioning of the anchor system together with the transfer of the applied load to the concrete specimen. The deformable parts of the expansion sleeve are the expansion tongues. They are notched in order to act as a plastic hinge during the deformation process. The material properties of the concrete specimen and the different parts of the steel anchor are listed in Table 5.10. 5.3.5.2

Installation of the undercut anchor

The in situ installation of this type of anchor incorporates several loading states. In order to carry out a proper numerical simulation of these states, the respective construction states must be well understood:

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145

128.75 128.75 21

Ø1 = 32

195

Ø= 37 Ø= 59.5 Ø= 50.5

support ring Ø= 400

4

Ø= 105

1 275

17.5 3.5

61

60

26

21

3

25

400

Ø= 41 Ø= 59.5

30

195

100

30◦

2 Ø2 = 65

199 250

102

199 250

1

threaded rod

3

cone

2

nut

4

expansion sleeve

Figure 5.65: High strength undercut anchor: geometric dimensions (in [mm]) of test set up Material parameters concrete Young’s modulus 30303 N/mm2 Poisson’s ratio 0.2 uniaxial compressive strength 19.60 N/mm2 uniaxial tensile strength 2.22 N/mm2 fracture energy GIf 0.08 Nmm/mm2 threaded rod, nut, cone Young’s modulus 210000 N/mm2 Poisson’s ratio 0.28 yield strength 1450 N/mm2 expansion sleeve Young’s modulus 210000 N/mm2 Poisson’s ratio 0.28 yield strength 965 N/mm2

Table 5.10: High strength undercut anchor: material parameters 1. drilling: preparation of the drilling hole in the concrete plate

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146

2. prestressing: assembly of treaded rod, nut, cone, and expansion sleeve; prestressing of the expansion sleeve is achieved by applying a predefined vertical displacement on the expansion sleeve with a setting tool; thus, the cone gets positioned and deforms the expansion tongues in a predefined way 3. setting and pull-out: removal of the setting tool ⇒ relaxation of the expansion sleeve; application of the load on the threaded rod Figure 5.66 schematically shows the installation of the anchor device. prestressing performed with setting tool

u ¯2

fixing of threaded rod

drilling

P

final position

removal of setting tool ⇒ relaxation

 
  


positioning steelconcrete








prestressing

setting and pull-out

Figure 5.66: High strength undercut anchor: installation of the anchor device Because of symmetry of the geometric properties and the boundary conditions, the problem can be solved by means of axisymmetric analyses. The used finite element mesh consisting of 1961 three node axisymmetric elements (MARC, 1996) is shown in Figure 5.67. The plastic behavior of the steel parts of the anchor was considered by means of a von Mises plasticity model assuming linearly elastic - ideally plastic material behavior. This model is part of the MARC FEM package (MARC, 1996). The connection between the threaded rod and the nut was assumed to be fixed. Hence, these two construction members are modeled as one part. The in situ setting process of the anchor suggests subdivision of the numerical simulation into several load cases (LC). Depending on the construction state of the anchor these load cases are: LC1: prestressing of the expansion sleeve in oder to obtain the correct slope of the expansion tongue; for this load case the threaded rod is kept fixed LC2: rigid body motion of the prestressed expansion sleeve together with the threaded rod until contact with the concrete is reached and the contact force has attained a certain level

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147

axis of symmetry

support ring

1

enoD gniwardeR

enoD gniwardeR

zoom

1

Figure 5.67: High strength undercut anchor: finite element mesh with detail of the expansion sleeve and the cone LC3: removal of the boundary condition for the expansion sleeve resulting in a redistribution of contact forces; establishing contact before removal of the boundary conditions was found to be necessary for a stable computation; LC4: pull-out of the threaded rod, which results in a strong increase of the load; Figure 5.68 shows the load cases described. The setting of the anchor is analyzed by means of the single-surface model. The material parameters of the ELM are chosen such that fcy = 0.2 fcu . An associative flow rule was considered. The reason for this choice is stabilization of the algorithmic properties of the model. Results from a material/model parameter study will be presented in the next subsection. The first investigation dealt with in this subsection is numerical modeling of the expansion sleeve. In Figure 5.69 a photo of the expansion sleeve with eight tongues is shown together with a scheme for the axisymmetric analysis. During the installation process, where the expansion sleeve is moved towards the steel cone, the tongues of the expansion sleeve are moved until a prescribed inclination is reached. For this process a predefined plastic hinge is used in order to assure the right deformations.

Numerical Simulations

5.3: Fastening systems pull-out

prestressing u ¯2

[×103 ]

u2

u2

u ¯2

148

9.0

9.000e+03

relaxation first contact

loose of contact

strong increase of force

6.8

6.800e+03

4.8 2.4

4.600e+03

2.400e+03

0.2

2.000e+02

-2.0

-2.000e+03

u ¯1 = 0 LC1

u ¯1

u ¯1 = 0

u ¯1 LC2

LC3

LC4

Figure 5.68: High strength undercut anchor: distribution of external force (in [N]) for applied load cases and description of prescribed displacements u ¯ 1 and u ¯2 For axisymmetric analyses it must be assured that the circumferential stresses σcirc between the tongues vanish. This condition can be accomplished by using an orthotropic material description for the tongue of the expansion sleeve. The remaining steel parts are modeled by means of isotropic material laws.

axis of symmetry

tongues of expansion sleeve orthotropic material

y z

x

isotropic material Figure 5.69: High strength undercut anchor: modeling of the expansion sleeve

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149

The flexibility matrix for orthotropic materials can be expressed as 

C−1 =

           

− νEyxy − νEzxz 1 − νEzyz Ey − νEyzy − E1z 0 0 0 0 0 0

1 Ex − νExyx − νExzx

0 0 0

0 0 0 1 Gxy

0 0

0 0 0 0 1 Gyz

0 0 0 0 0

0

1 Gzx



      .     

(5.5)

Setting of Ex , Ey , νxy , νyx , Gxy 6= 0 and the remaining terms approximately zero results in a good approximation of the given plane stress condition of the tongue of the expansion sleeve. Figure 5.70 contains the load-displacement response for the installation of the high strength undercut anchor, i.e., LC1-LC4. Figure 5.70(a) shows the load-displacement curves obtained for the first load case (LC1) using various friction coefficients µ used for the Mohr-Coulomb friction law. Starting with a rigid body motion characterized by increasing values of u ¯2, with P remaining zero, the load begins to increase when the tongue of the expansion sleeve first gets in contact with the steel cone. At a displacement u ¯ 2 of approximately 7 mm the load remains almost constant with increasing displacement (see also Figure 5.71(a) and (b)). This indicates sliding of the expansion sleeve on the steel cone. A strong increase of the load is obtained at a displacement u ¯2 of approximately 21 mm, indicating self contact of the expansion sleeve (see Figure 5.71(c)). This state defines the end of LC1. non-converged equilibrium states

P [kN]

20

P [kN]

40

initial contact

friction coefficients 30 15 µ = 1.0 20 10 direction µ = 0.3 LC1 µ = 0.0 10 5 u ¯2 [mm] 0 0 0 5 10 15 20 0 (a) P [kN]

40

direction LC3

30

relaxation ⇒ decrease of load

20 10 0

0

1

2

3 (c)

4

end LC2

400

rigid body motion 5

10 (b)

direction end LC2 LC1 u ¯2 [mm] 20

15

strong increase of load

P [kN]

300

direction LC4

200 100 u ¯2 , u2 [mm] 0 0 5 6

end of LC3 u2 [mm] 1

2 (d)

3

4

5

Figure 5.70: High strength undercut anchor: load-displacement curves for load cases LC1-LC4

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Irrespective of the friction coefficient, similar load-displacement curves are obtained. In the following, only the case of no friction is considered3 . When the tongue of the expansion sleeve is positioned, the second load case (LC2) starts (see Figure 5.70(b)). For this load case, displacements u ¯1 at the bottom of the steel rod - nut connection and displacements u ¯2 on the top of the expansion sleeve are prescribed such that the relative distance between the steel members is kept constant. This guarantees that the state of prestressing is preserved. The figure shows a strong increase of the load when the expansion tongue gets in contact with the concrete specimen. It was found that the contact algorithm is very sensitive with respect to the geometric properties of the mesh. The position of the nodes of the contact bodies has a crucial influence on the convergence properties of the numerical simulation. Situations such as sticking of nodes where encountered during some computations. In general, this results in oscillations of the solution of the equilibrium iteration. As regards the algorithms used for the material model, the inclination angle of the tongue of the expansion sleeve has a strong influence on the rate of convergence. When only one node of the tongue penetrates the concrete, elements with very large contact forces are developing. In such cases the local return map algorithms of the material model fail to converge. Removal of the boundary conditions, u ¯2 , for the expansion sleeve results in relaxation of this 4 sleeve . This is reflected by the drop of the load for LC3 shown in Figure 5.70(c). At contact of the expansion sleeve and the concrete specimen encountered in LC2, the first damage of the concrete block occurs (see also Figure 5.68).

[× 102 ]

u ¯2

9.5

9.500e+02

7.6

7.600e+02

selfcontact

development of plastic hinge

5.7

5.700e+02

3.8

final position expansion tongue

3.800e+02

1.9

1.900e+02

0.0

0.000e+00

(a)

(b)

(c)

Figure 5.71: High strength undercut anchor: distribution of equivalent von Mises stress (in [N/mm2 ]) in the expansion sleeve at a displacement of u ¯2 (a) 6.9 mm, (b) 13.8 mm, and (c) 21 mm 3

The points marked with a square in Figure 5.70(a) indicate non-converged equilibrium states obtained using the contact algorithm in MARC. 4 It should be noted that u ¯2 denotes a prescribed displacement. Once this boundary condition is removed u2 denotes the measurement displacement.

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Figure 5.70(d) refers to the pull-out phase (LC4) of the anchor. It is characterized by a strong increase of the load with increasing displacement. From the numerical simulations of the setting process of the undercut anchor the following conclusions can be drawn: • the convergence behavior of the numerical simulation is very sensitive with respect to the applied step-sizes; large step sizes either lead to - divergence of the MARC contact algorithm or - divergence of the local return map algorithms of the material model. However, for an investigation of the failure behavior of the concrete plate the following structural assumptions can be made: • for LC4 the relative displacements between the cone and the tongues of the expansion sleeve are very small; hence, a fixed connection between these members can be assumed, • contact in the form of gap friction elements is assumed between the concrete and the tongues of the expansion sleeve, • the initial stress state in the expansion sleeve (see Figure 5.71) can be neglected, 5.3.5.3

Numerical results single-surface model

Based on the structural assumptions made in the previous subsection, in the following, only LC4 will be considered. The first simulation is performed using the strength parameters given in Table 5.10. The material parameters of the ELM were chosen such that fcy =0.2fcu . A non-associative flow rule was assumed. For the second simulation a uniaxial compressive strength of fcu =40.00 N/mm2 was used. The obtained load-displacement curves and the value of the maximum load5 obtained from the experiment are shown in Figure 5.72. Figure 5.72 indicates a strong influence of the uniaxial compressive strength fcu both on the peak load and the failure mode. Whereas a compressive strength of fcu =40.00 N/mm2 led to a peak load of approximately 750 kN followed by softening behavior, use of f cu =19.60 N/mm2 resulted in ideally plastic behavior at approximately 300 kN. Comparison of the peak load obtained for fcu =40.00 N/mm2 with the experimental data indicates rather good agreement. However, with respect to the ductility behavior rather strong deviations are observed. Possible reasons for this deviation are inaccurate numerical modeling and/or possible measurement errors of the experiment. With respect to numerical modeling the contact conditions between the construction members and the ductility function xh may 5

The experiment was terminated at a load level of approximately 800 kN. The tangent of the loaddisplacement curve for this load level was almost horizontal. Hence, in the following a comparison of the peak load obtained from the numerical simulations and the maximum load based on the experimental data will be made.

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u2 P [kN]

maximum value of load obtained from experiment

900

u ¯1 600 fcu =40.00 N/mm2 fcu =19.60 N/mm2

u ¯1 . . . prescribed u2

300 initital displacement obtained from experiment

u ¯ 1 , u2 [mm]

0 0

2

4

6

Figure 5.72: High strength undercut anchor: load-displacement curves for different uniaxial compressive strength parameters fcu using a non-associative flow rule and maximum value of load obtained from the experiment be described inaccurately. In order to analyze the influence of the uniaxial compressive strength fcu on the numerical results, a comparison of the stress distribution, the plastic strain distribution and the deformed configuration of the concrete plate will be made. The distribution of the minimum in-plane principal stress, σmin , (see Figures 5.73(a) and 5.75(a)) provides insight into the load-carrying behavior of the concrete plate. The applied load is transferred from the tongues of the expansion sleeve to the concrete plate and further by means of a compressive strut from the confined area over the anchor head to the support ring. The distribution of σmin is more pronounced for fcu =40.00 N/mm2 . The respective maximum principal stress σmax in this strut resulted in the development of circumferential cracks (see Figures 5.73(b) and 5.75(b)). These cracks started to open at the tongues of the expansion sleeve, propagating towards the the support ring. For the case of fcu =19.60 N/mm2 these cracks remained bounded in the region around the anchor head indicating local compressive failure (see Figure 5.74(a)). This behavior is also confirmed by looking at the deformed configuration of the plate, where large deformations were obtained at the tongues of the expansion sleeve (see 5.74(b)). However, for the case of fcu =40.00 N/mm2 the circumferential cracks propagated towards the support ring finally causing cone-shaped failure (see Figure 5.76(a)). From the deformation plot it can be concluded that no local failure above the tongues of the expansion sleeve was obtained (see 5.76(b)). In Figures 5.73(c) and 5.75(c) the distribution of the circumferential stress σcirc is shown, indicating the development of radial cracks. The different failure modes can be further analyzed by looking at the distribution of αh and of the hydrostatic pressure at peak load (see Figures 5.77 and 5.78). αh =0 refers to the initial yield surface defined by q¯h =fcy . If αh =1, the failure surface is reached and softening or

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1.000e+00

2.00

8.000e-01

-1.80

6.000e-01

-5.60

4.000e-01

-9.40 -13.2

2.000e-01

-17.0

0.000e+00

(a) σmin

(b) σmax

(c) σcirc

Figure 5.73: High strength undercut anchor: distribution of principal stresses in the r − z plane, σmin and σmax , and the stress component in the circumferential direction, σcirc (in [N/mm2 ]), at u ¯1 =2 mm for fcu =19.60 N/mm2 [× 10−4 ] 1.00

1.000e+00

0.80

8.000e-01

0.60

6.000e-01

zoom

0.40

4.000e-01

large local deformations

0.20

2.000e-01

0.00

0.000e+00

(a) εpmax

(b)

Figure 5.74: High strength undercut anchor: (a) distribution of the maximum plastic strain in the r −z plane, εpmax , and (b) the deformation of the concrete plate in the vicinity of the anchor at u ¯1 =2 mm for fcu =19.60 N/mm2 ideally-plastic behavior is initiated. Whether or not softening will take place depends on the level of confinement. For the ELM, confinement is represented by the hydrostatic pressure p = −(σ1 + σ2 + σ3 )/3. For a hydrostatic pressure smaller than the hydrostatic pressure related to the transition point T P , pT P , softening occurs. For stress states characterized by p > pT P , ideally-plastic material behavior is assumed. Based on the different values of the compressive strength fcu the hydrostatic pressure at T P is computed as pT P =16.80 N/mm2 (fcu =19.60 N/mm2 ) and pT P =55.44 N/mm2 (fcu =40.00 N/mm2 ). The distribution of the hardening variable αh at peak load for fcu =19.60 N/mm2 shown in Figure 5.77(a)

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1.000e+00

2.00

8.000e-01

-1.80

6.000e-01

-5.60

4.000e-01

-9.40 -13.2

2.000e-01

-17.0

0.000e+00

(a) σmin

(b) σmax

(c) σcirc

Figure 5.75: High strength undercut anchor: distribution of principal stresses in the r − z plane, σmin and σmax , and the stress component in the circumferential direction, σcirc , at peak load (in [N/mm2 ]) for fcu =40.00 N/mm2 [× 10−5 ] 8.00

1.000e+00

6.40

8.000e-01

4.80

6.000e-01

zoom

3.20

4.000e-01

1.60

2.000e-01

0.00

0.000e+00

(a) εpmax

(b)

Figure 5.76: High strength undercut anchor: (a) distribution of the maximum plastic strain in the r −z plane, εpmax , and (b) the deformation of the concrete plate in the vicinity of the anchor at peak load for fcu =40.00 N/mm2 is characterized by a zone with αh =1 around the anchor head. The hydrostatic pressure in this zone is greater than the hydrostatic pressure at the transition point, i.e., p > pT P . This resulted in ideally-plastic material response and, hence, in plastic deformations in that region already observed by the evolution of the maximum plastic strain εpmax depicted in Figure 5.74(a). For the case of fcu =40.00 N/mm2 , the distribution of the hardening variable αh shown in Figure 5.78(a) is characterized by a large region with αh =1. Only for the small zone above the anchor head the hydrostatic pressure is greater than the one at the transition point. Thus, ideally-plastic behavior is restricted to this small zone. Small values for the

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hydrostatic pressure in the remaining parts are characterized by αh =1, indicating softening.

1.00

1.000e+00

0

1.000e+00

0.80

8.000e-01

0.60

6.000e-01

0.40

4.000e-01

0.20

2.000e-01

0.00

0.000e+00

5

8.000e-01

15

6.000e-01

25

4.000e-01

2.000e-01

35 45

0.000e+00

(a)

(b)

Figure 5.77: High strength undercut anchor: distribution of (a) the internal hardening variable αh and (b) the hydrostatic pressure p = −(σ1 + σ2 + σ3 )/3 (in [N/mm2 ]) at u ¯1 =2 mm for fcu =19.60 N/mm2

0

1.00

1.000e+00

0.80

8.000e-01

0.60

6.000e-01

0.40

4.000e-01

2.000e-01

0.20

2.000e-01

0.000e+00

0.000e+00

1.000e+00

14

8.000e-01

28

6.000e-01

42

4.000e-01

56 70

0.00

(a)

(b)

Figure 5.78: High strength undercut anchor: distribution of (a) the internal hardening variable αh and (b) the hydrostatic pressure p = −(σ1 + σ2 + σ3 )/3 (in [N/mm2 ]) at peak load for fcu =40.00 N/mm2

5.3.5.4

Conclusions

In this subsection the structural behavior of a high strength undercut anchor was investigated. From the obtained numerical results the following conclusions can be drawn: • As regards the setting process, several load cases (LC1-LC4) were considered for the numerical simulation. • Convergence problems related to the interaction between the contact algorithms (incorporated in the FE-code MARC) and the material model (user subroutine) led to a special assumption on the structural level (consideration of LC4 only).

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• As regards the influence of the uniaxial compressive strength on the numerical results, a strong influence on the peak load and the failure mode was shown. Small values of fcu led to ductile material behavior over the anchor head. Large values of fcu resulted in cone-failure of the concrete plate.

Chapter

6

Summary and Conclusions Numerical methods such as the FEM together with sophisticated material models provide a powerful tool for the analyses of concrete structures. Among the large number of constitutive models for concrete, plasticity models are commonly used for large-scale application problems. In this work, two constitutive models for plain concrete formulated in the framework of plasticity theory were developed: • The Extended Leon Model, a single surface plasticity model previously formulated by Etse and Willam (1994), was reformulated with respect to the ductility function for description of ductile concrete behavior in the regime of large compressive stresses. Confinement was controlled by the hydrostatic pressure. This model accounts for the dependence of the concrete strength on the Lode angle by means of an elliptic deviatoric shape function. For an appropriate description of the dilatational behavior of concrete a non-associative flow rule was used. • The second model was a multi-surface model consisting of three Rankine surfaces for description of cracking of concrete and of a Drucker-Prager yield surface for description of compressive failure. This model was reformulated to account for the dependence of both strength and ductility of concrete on the confinement. Confinement was controlled by the major principal stress. For the Rankine criterion an associative flow rule was chosen. For the Drucker-Prager criterion a non-associative flow rule was used. The main steps of the development and verification of the models on the constitutive level were: • The development of a robust and efficient algorithmic formulation of the material models. For the numerical integration of the evolution equations the unconditionally stable Backward Euler integration scheme was used. The non-linear system of equations following from the numerical integration were solved by means of Newton-type

Summary and Conclusions

158

algorithms. For the multi-surface model the standard Newton algorithm was used. Because of the complexity of the ELM modified algorithms were used. For both models consistent tangent operators were derived. Use of such operators provide a quadratic rate of convergence of the global Newton-Raphson equilibrium iteration. • The performance of the models on the constitutive level was investigated by re-analyses of experimental data. For this purpose, a constitutive driver routine was used (see Pramono (1988) for details), allowing the prescription of complex loading paths. From these experiments the influence of material and model parameters on the numerical results was investigated. It was found that both models are capable to describe concrete behavior for a broad range of loading conditions. • The development of localized or diffuse failure was assessed by means of localization analyses. For this purpose, several loading paths were chosen for verification. From these tests it was found that the localization properties of the models depend strongly on the formulation of the flow rule and the shape of the loading surface in the deviatoric plane. One of the most important tasks in structural engineering is the estimation of the loadcarrying capacity of a structure together with an assessment of the failure mode. A lot of research was performed in the area of developing constitutive models for concrete and of verifying these models on the constitutive level. However, only few of these models were used for large-scale structural simulations. The application of the proposed material models to the analyses of anchor devices characterized by nonuniform triaxial stress states in the vicinity of the anchor head was one of the main goals of this thesis. From the structural simulations performed, the following conclusions can be drawn: • the material behavior of steel and concrete has a crucial influence on the failure behavior of the anchor device. As regards the influence of parameters of the constitutive models, a strong influence on the peak load and the failure mode was detected. For the ELM, the flow rule formulation and the shape of the loading surface in the deviatoric plane were found to have a significant influence on an appropriate description of concrete failure. On the other hand, the extension of the Drucker-Prager criterion to the description of confined compressive stress states was found to be indispensable for an accurate modeling of concrete behavior in the context of pull-out simulations. • As regards the geometric properties, characterized by the so-called span/depth ratio (a/d-ratio, see Subsection 5.3.4), a strong influence on the failure mode was encountered. Failure for large a/d-ratios is characterized by the development of a cone-shaped concrete failure surface in consequence of tensile stresses. Whereas small a/d-ratios lead to states of increasing triaxial compressive stresses above the anchor head and, hence, to an increase of the ductile behavior of concrete. From the above conclusions the following recommendations for future research can be made:

Summary and Conclusions

159

• As regards constitutive modeling: - extension of the multi-surface model to hydrostatic and oedometric loading by means of an additional cap loading surface (see Hofstetter et al. (1993)) should be made. Investigation of the model behavior under compressive and mixed loading states should be further addressed. - extension to dynamical loading: dynamical applications such as impact loading are characterized by highly concentrated loads acting over a short period of time. In the context of dynamical loading strength parameters must be formulated in a time dependent manner (see Sercombe et al. (1998) for details). - use of a parameter optimization program would provide an appropriate and efficient calibration of the models. With the aid of such a program suitable intervals of the used material and model parameters could be determined. • As regards structural simulations: - application and verification of the models in analyses of concrete structures together with a comparison with experimental data. - use of adaptive calculation schemes (Lackner, 2000) in order to guarantee objectivity of the numerical results.

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Appendix

A

Transformation to Principal Axes This section contains the transformation of the constitutive relations to the principal stress space (see also Meschke (1996)). For the case of isotropic constitutive laws the stress tensor σ is an isotropic function of ε e . Therefore, the principal axes nA of σ and of εe coincide. Hence, the spectral decomposition of the elastic strain tensor εe and the stress tensor σ take the form ε

e

=

3 X

εˆeA nA ⊗ nA ,

(A.1)

3 X

σ ˆ A nA ⊗ n A .

(A.2)

A=1

σ =

A=1

Principal values are indicated by the symbol ” ˆ “. For isotropic yield functions, f (σ; q) = fˆ(σ1 , σ2 , σ3 ; q), it follows that 3 X ∂f ∂ fˆ A = n ⊗ nA . ∂σ A=1 ∂ σˆA

(A.3)

The relation for determination of σ n+1 and the one between the elastic trial- stress tensor and strain tensor are given as σ n+1 = σ tr n+1 − ∆γn+1 C : e,tr σ tr n+1 = C : εn+1 p εe,tr n+1 = εn+1 − εn .

with

∂f , ∂σ

(A.4) (A.5) (A.6)

Inserting Equations (A.2) and (A.3) into Equation (A.4) shows that the principal directions

Transformation to Principal Axes

171

of the elastic trial strain tensor εe,tr and the stress tensor σ coincide: εe,tr =

3 X

A=1

A A εˆe,tr A n ⊗n .

(A.7)

Hence, the elastic predictor determines the principal directions nA of the final stress state σ already at the beginning of the CPPA. With the help of the Cardano formula the principal values of εe,tr can be obtained in closed form as the roots of the characteristic polynomial. Expressions for nA ⊗ nA may also be computed from a closed formula as (Meschke, 1996) (Miehe, 1998) e,tr mA = nA ⊗ nA = (εe,tr − εe,tr − εe,tr B 1I)(ε C 1I)/DA

(A.8)

where e,tr e,tr e,tr DA = (εe,tr B − εA )(εC − εA )

(A.9)

for A = 1, 2, 3, B = 1 + mod(A, 3) and C = 1 + mod(B, 3). For the case of two equal roots, e,tr |εe,tr − εe,tr − εe,tr 1 2 | < TOL or |ε2 3 | < TOL,

(A.10)

the easiest way of implementation is accomplished by perturbations of the respective eigenvalues. With the above definitions at hand, the algorithmic tangent material stiffness matrix is obtained by differentiation of Equation (A.2) with respect to ε as Cep T =

∂σ ∂εe,tr ∂σ = e,tr , e,tr ∂ε ∂ε ∂ε

(A.11)

where the relation ∂εe,tr /∂ε=I is used together with 3 X 3 3 X X ∂σ ∂mA ∂ σˆA ∂ εˆe,tr B A = m + σ ˆ A e,tr ∂εe,tr ∂εe,tr ˆe,tr B ∂ε A=1 B=1 ∂ ε A=1

=

3 X 3 X

A=1 B=1

aep AB

3 X ∂mA ∂ εˆe,tr B A m + σ ˆ . A ∂εe,tr ∂εe,tr A=1

(A.12)

The total differential of εe,tr is obtained from Equation (A.7) as dεe,tr =

3 h X

B=1

i

B B B B B B dˆ εe,tr ˆe,tr B (n ⊗ n ) + ε B (dn ⊗ n + n ⊗ dn ) .

(A.13)

Transformation to Principal Axes

172

Contracting (A.13) with nA ⊗ nA and consideration of knA k = 1, nA dnA = 0, and of B

B

A

A

A B

B

A

(n ⊗ n ) (n ⊗ n ) = (n n ) (n n ) =

(

1 for A = B 0 for A 6= B

(A.14)

gives e,tr ∂ εˆe,tr = mA . A /∂ε

(A.15)

Inserting (A.15) into Equation (A.12), finally yields Cep T =

3 X 3 X

A=1 B=1

B A aep AB m ⊗ m +

3 X

σ ˆA CA,tr .

(A.16)

A=1

Equation (A.16) consists of two parts. The first part contains the term, aep AB , denoting the A,tr A local tangent moduli referred to principal axes. The moduli C = ∂m /∂εe,tr , occurring in the second term of Equation (A.16) are independent of the chosen plastic model. Hence, for the global formulation the only ingredient needed from the model under consideration are the local plastic tangent moduli aep AB (see Appendix B.3) and the final principal stresses σ ˆA referred to principal axes. It should be noted that unsymmetries of Cep T can only stem from the first term in Equation (A.16). A,tr e,tr The moduli Cijkl = ∂mA ij /∂εkl are obtained by taking the derivative of (A.8) with respect to εe,tr as

A,tr Cijkl

=

e,tr ∂mA ij /∂εkl

3  1 X e,tr = (∂aim /∂εe,tr kl bmj + aim ∂bmj /∂εkl ) DA m=1

−

(A.17)

3 X 1 e,tr (∂D /∂ε (aim bmj )), A kl 2 DA m=1

where aim = εe,tr ˆe,tr im − ε B δim ,

bmj = εe,tr ˆe,tr mj − ε C δmj .

(A.18)

Taking the derivative of (A.18) with respect to εe,tr kl and using equations (A.15) and (A.1) gives B ∂aim /∂εe,tr = δik δml − δim nB kl k nl C ∂bmj /∂εe,tr = δmk δjl − δmj nC k nl kl A C A ∂DA /∂εe,tr = (mB εe,tr ˆe,tr εe,tr ˆe,tr C −ε A ) + (mkl − mkl ) (ˆ B −ε A ), kl kl − mkl ) (ˆ

(A.19)

Transformation to Principal Axes

173

with mA kl defined in Equation (A.8). Inserting (A.19) into (A.17) and considering mA ij

3 1 X aim bmj , = DA m=1

(A.20)

A,tr finally gives the coefficients Cijkl as

A,tr Cijkl

3 1 nX B C C ((δik δml − nB = k nl δim )bmj + (δmk δjl − nk nl δmj )aim ) DA m=1

o

B B A A C C A A −mA εe,tr ˆe,tr εe,tr ˆe,tr A )) . C −ε A ) + (nk nl − nk nl ) (ˆ B −ε ij ((nk nl − nk nl ) (ˆ

(A.21)

Appendix

B

Algorithmic Aspects In the following, details concerning the algorithmic formulation of the ELM, such as the update of the eccentricity (B.1), the Picard iteration (B.2) and the formulation of the local tangent moduli (B.3, B.4) will be given.

B.1

Update eccentricity

The deviatoric shape of the loading surface of the ELM is described by the elliptic function g(θ, e) (see Equation (2.25)) according to Willam and Warnke (1975). The parameter e is referred to as eccentricity. e controls the deviatoric shape of the loading surface. The value of e is ranging from 0.5 (triangular form of the yield surface) to 1 (circular form of the yield surface). The elliptic function g(θ, e) provides a proper representation of failure of concrete. The quantities rt and rc represent the distance from the tensile and compressive meridian, respectively, to the hydrostatic axis. They are computed from the Mohr-Coloumb-type model proposed by Pramono and Willam (1989b). The loading surface of the PramonoWillam model, fP W , reads  

fP W (σ1 , σ3 ; qh , qs ) =  1 − q¯h + fcu

!2

q¯h fcu

!

σ1 fcu

!2

+

σ1 q¯h m(qs ) − fcu fcy

2 

σ1 − σ 3 fcu 

!2

q¯s = 0. ftu

(B.1)

Algorithmic Aspects

175

Expressing the principal stresses σ1 and σ3 in (B.1) by means of the first and the third one of the following relations (Chen, 1982),    

σ1 σ2    σ 3

      

= p1I +

s

   

cos(θ) 2 r cos(θ − 2π ) 3 3    cos(θ + 2π ) 3

   

,

(B.2)

  

the loading surface (B.1) can be evaluated for the tensile and compressive meridian, i.e., for θ=0 and θ = π/3, giving   

fPt R (p; qh , qs ) =  1 − 

q¯h + fcu

!2

q¯h fcu

q ! p + 2/3 

fcu

m(qs )

p+

q

2/3 rt

fcu

rt

2 

+

q¯h − fcu

q

3/2 fcu

!2

2  rt   

q¯s = 0, ftu

(B.3)

and  

q¯h fPc R (p; qh , qs ) = 1−  fcu q¯h + fcu

!2

!

2 √ !2 q 3/2 rc  p + rc / 6 + fcu fcu 

√ !2 p + rc / 6 q¯h q¯s m(qs ) − = 0. fcu fcu ftu

(B.4)

Equations (B.3) and (B.4) are nonlinear equations for the determination of rt and rc as a function of the pressure p and the hardening/softening forces qh and qs . Accordingly, rt and rc and, hence, the eccentricity e depend on the actual pressure p as well as on the actual values of the hardening/softening forces qh and qs . In case of an implicit time integration scheme, the evaluation of e depends on the actual values of p, qh , and qs . In general, this causes a significant decrease of the robustness of the algorithmic formulation of the ELM. Hence, an explicit update of e is employed. Hereby, e is computed from the pressure p and the values of the hardening/softening force qh and qs at the end of the previous load increment. Consequently, e is constant during the actual load increment1 . 1

It should be noted that e is a material property rather than a model parameter. This is also confirmed by Men´etrey and Willam (1995). A simplification of the aforementioned update scheme could be achieved by assuming e to be a linear function depending only on the hydrostatic pressure p. This function could be calibrated to the uniaxial tensile strength ftu and the biaxial compressive strength fbc .

Algorithmic Aspects

176 σ1

rt

rc

σ1

circle

ellipse

rt σ3

σ2

rc

σ3

σ2

rt rt (b) e = 0. The next iteration value αn+1 is computed by means of the Picard iteration. (2)

2. Assume that f (σ tr n+1 , αn+1 ) ≤ 0. In this case the upper limit, i.e., αmax of αn+1 is computed from the consistency condition leading (2) (2) f (σ tr n+1 , αn+1 ) = 0, with αn+1 = αmax , which implies that (2) (2) (2) fP I (σ tr → fAOII (αn+1 ) = αn − αn+1 < 0. n+1 , αn+1 ) = αn (2) A lower bound for αn+1 is then found by reducing αn+1 such that (3) (3) (3) (3) (3) tr αn+1 = {α|fP I (σ tr n+1 , αn+1 ) ≥ αn+1 }, and f (σ n+1 , fP I (αn+1 ) > 0, resulting in fAOII (αn+1 ) ≥ 0. (2) (3) With the so obtained function values αn+1 and αn+1 the AOII-algorithm can be ap(4) (i) plied. If αn+1 ∈ [αn , αmax ], the next value αn+1 is obtained by means of a Picard (i) iteration. For i > 4, subsequent values of αn+1 are computed either by quadratic or linear inverse interpolation. In order to ensure that the cone regions do not become active, the consistency parameters are checked after every Picard iteration. For the particular case of the ELM, the transition from hardening to softening requires special considerations because the general strain hardening/softening parameter αn+1 is entirely different in the pre-peak and in the post-peak regime, both with regard to the evolution law and the physical meaning.

Algorithmic Aspects

177 h,(i)

If the update value of the strain hardening parameter is αn+1 > 1, which indicates that h,(i) the hardening regime has been exceeded, the limit value αn+1 = 1 is adopted for the next (i) projection of the trial stress. Therefore, the new stress state σ n+1 lies on the failure surface of the ELM. The Picard iteration can lead the following two cases h,(i+1)

1. αn+1 ≤ 1, i.e., in the increment only hardening is activated. The next values of h,(i+1) αn+1 are computed by the update scheme described above. h,(i+1)

2. αn+1 > 1, in the increment softening is activated. Softening is assumed to be active only if the trial state lies within the region defined by the transition point (ptr ≥ pT P ). The ductility function value xs in the softening regime is computed from the trial stress (i+2) (i+3) projection onto the failure surface. αn+1 , αn+1 are computed from Picard iterations. (i+k) For k > 3, subsequent values of αn+1 are computed either by linear or quadratic inverse interpolation.

B.3

Local tangent moduli: standard regions

This subsection contains a detailed derivation of the local algorithmic tangent moduli aep needed for the global formulation of the consistent algorithmic material stiffness matrix Cep T . In the following, all evaluations of functions must be performed at tn+1 and all derivatives must be carried out with respect to principal stresses. The total differential of Equations (2.34) and (2.43) is obtained as j j , = β j dαn+1 dqn+1

(B.6)

where the index j = h, s stands for the hardening and softening regime, respectively. In Equation (B.6) the following abbreviations are employed βh = β

s

(fcy − fcu )2 h (1 − αn+1 ) and h qn+1

n ftu = αu



s αn+1 αu

n−1

exp[−(

s αn+1 )n ]. αu

(B.7)

The internal strain-like hardening/softening parameters can be expressed as (see Equation (3.16)) j = αnj + ∆γn+1 Hj : m, αn+1

(B.8)

where Hh =

1 m xh kmk

and Hs =

1 hmi xs khmik

(B.9)

Algorithmic Aspects

178

are tensors of rank 2. The total differential of Equation (B.8) is obtained as j j dαn+1 = d(∆γn+1 )Hj : m + ∆γn+1 Zj : dσ n+1 + ∆γn+1 S j dqn+1

(B.10)

where Zj = ∂σ Hj : m + Hj : M with M = ∂σ m,

(B.11)

S j = ∂ q j Hj : m + H j : ∂ q j m ,

(B.12)

are tensors of rank 2 and scalars, respectively. Performing the derivatives of H j and S j with respect to σ and q j in Equations (B.11) and (B.12) and making use of k(m/kmk)k = 1, m : dm = 0 results in expressions employed for the numerical implementation: Zj = −

1 ∂xj ( ⊗ Hj ) : m + Hj : M, xj ∂σ

S j = Hj : ∂qj m.

(B.13) (B.14)

Since xs (ppeak ) is a constant value controlling the softening behavior (see Equation (2.42)), the first term in Equation (B.13) vanishes for the softening regime. Inserting Equation (B.6) into (B.10) and rearranging terms yields j = dαn+1

1 j 1 H : m d(∆γn+1 ) + j ∆γn+1 Zj : dσ n+1 , j A A

(B.15)

where Aj is a scalar given as Aj = 1 − ∆γn+1 S j β j .

(B.16)

With the Equation (B.16) at hand, Equation (B.6) can be expressed as j dqn+1 =

 βj  j j H : m d(∆γ ) + ∆γ Z : dσ . n+1 n+1 n+1 Aj

(B.17)

Inserting (B.17) into the expression for σ˙ given in Equation (2.18), leads to 

dσ n+1 = Ξjn+1 : dεn+1 − d(∆γn+1 )bj



(B.18)

Algorithmic Aspects

179

where Ξjn+1

= C

−1

+ ∆γn+1 M +

2 ∆γn+1

βj ∂q j m ⊗ Z j Aj

!−1

(B.19)

and bj = m + ∆γn+1

βj ∂q j m · Hj · m Aj

(B.20)

are tensors of rank 4 and 2, respectively. To obtain an expression for a differential of the consistency parameter, d(∆γn+1 ), a differential of the yield function has to be considered (see Equation (2.23)): j df |n+1 = n : dσ n+1 + ∂qj f dqn+1 .

(B.21)

Inserting (B.17) and (B.18) into (B.21) yields, after some algebraic manipulations, j d(∆γn+1 )=

(n : Ξjn+1 + ∆γn+1 r j (β j /Aj )Zj : Ξjn+1 ) : dεn+1 ˆpj + Eˆnj + ∆γn+1 E ˆ1j E

(B.22)

where rj =

∂f , ∂q j

(B.23)

Eˆpj = −r j /Aj Hj : m,

(B.24)

Eˆnj = n : Ξjn+1 : m,

(B.25)

j

β Eˆ1j = r j j Zj : Ξjn+1 : bj . A

(B.26)

Inserting (B.22) into (B.18) finally gives dσ n+1 = aep : dεn+1 ,

(B.27)

where aep = Ξjn+1 −

j Nj1 ⊗ Mj1 + ∆γn+1 r j (β j /Aj )Nj1 ⊗ Mj2 j ˆpj + Eˆnj + ∆γn+1 ˆ1j E E

(B.28)

Algorithmic Aspects

180

with Nj1 = Ξjn+1 : bj ,

B.4

Mj1 = n : Ξjn+1 ,

Mj2 = Zj : Ξjn+1 .

(B.29)

Local tangent moduli: cone regions

This subsection contains the derivation of the local algorithmic tangent moduli aep AB for the cone regions of the ELM. The starting point for this derivation is the following substitution: str s := tr . ksk ks k

(B.30)

This substitution is necessary because the derivative of the deviatoric radius r with respect to the stress tensor σ, i.e. ∂r/∂σ = s/ksk becomes singular for the intersection points of the yield surface with the hydrostatic axis. The same problem occurs at the intersection point of the Drucker-Prager yield function with the hydrostatic axis (see Hofstetter and Taylor (1991)). In the following the index 1 is used for the ELM, whereas the index 2 denotes the cut-off function. The plastic strain increment can be expressed as2 ∆εp = ∆γ1 m1 + ∆γ2 m2 .

(B.31)

Inserting Equation (B.31) into the stress-strain law (2.11) leads to σ n+1 = C : (εn+1 − εpn − ∆γ1 m1 − ∆γ2 m2 ).

(B.32)

Differentiation of Equation (B.32) with respect to the strain tensor leads to dσ n+1 = aep : dεn+1 ,

(B.33)

with aep = C − (C : m1 ) ⊗

d∆γ1 d∆γ2 dm1 − (C : m2 ) ⊗ − ∆γ1 (C : ), dε dε dε

(B.34)

where the relation dm2 /dε=0 has been employed. The expressions for d∆γ1 /dε, d∆γ2 /dε, and dm1 /dε are obtained as follows: 1. d∆γ1 /dε: 2

In the following the abbreviations ∆γ1 = ∆γ(n+1),1 and ∆γ2 = ∆γ(n+1),2 are used.

Algorithmic Aspects

181

Setting θ¯ = θ tr , the derivative of Equation (3.63) with respect to the strain tensor yields d∆γ1 = dε

dkstr k str 1 1 = . ∂f1 kstr k ∂f1 dε 2G |t |t ∂r n ∂r n

(B.35)

2. dm1 /dε: Because of the assumption of a constant projection direction in Subsection (3.3.4), the derivatives of the gradient of the yield potential of the ELM with respect to the strain tensor can be expressed as (see Equation (3.61)) dm1 d str 2G ∂Q1 ∂Q1 str str dev = |t = |t I − tr ⊗ tr . dε ∂r n dε kstr k ∂r n kstr k ks k ks k !

(B.36)

3. d∆γ2 /dε: Rewriting the Equation (B.8) requested for the hardening/softening law, in terms of plastic strains yields j αn+1 = αnj + Hj : ∆εp ,

(B.37)

where Hh =

1 ∆εp xh k∆εk

and Hs =

1 h∆εp i . xs kh∆εik

(B.38)

The total differential of Equation (B.37) is obtained as j dαn+1

dHj d∆εp = ∆ε : + Hj : dε dε p

!

: dε.

(B.39)

Using the chain rule together with Equation (B.34) yields the following expression for the first term in Equation (B.39) ∆εp :

  d∆γ dHj dHj dσ 1 = ∆εp : : = V j : C − V j : C : m1 − dε dσ dε dε 

j

V : C : m2



d∆γ2 dm1 − ∆γ1 Vj : C : dε dε

!

(B.40)

with dxj 1 Vj = − ∆εp : Hj ⊗ xj dσ

!

.

(B.41)

Algorithmic Aspects

182

The second term in Equation (B.39) can be expressed as d∆γ1 d∆γ2 dm1 d∆εp = H j : m1 ⊗ + m2 ⊗ + ∆γ1 H : . dε dε dε dε !

j

(B.42)

The expression for the derivative of the consistency parameter ∆γ2 for the cut-off function with respect to ε is obtained from the consistency condition df2 = 0 as n2 :

df2 dq dσ : dε + dα = 0 dε dq dα

(B.43)

with n2 = ∂f2 /∂σ. Inserting Equation (B.39) into Equation (B.43) using (B.40) finally leads d∆γ2 1 = dε Ep + E 1 + E 2 − ∆γ1

"

!

d∆γ1 df2 dq j V : C − c1 n2 : C + dq dα dε

dm1 dm1 dm1 df2 dq j + (V : C : − Hj : ) n2 : C : dε dq dα dε dε

!#

(B.44)

with Ep = n 2 : C : m2 , E1 =

(B.45)

df2 dq j V : C : m2 , dq dα

(B.46)

df2 dq j H : m2 , dq dα

(B.47)

E2 = −

c1 = n 2 : C : m1 +

df2 dq j (V : C : m1 − Hj : m1 ). dq dα

(B.48)

With these expressions the local tangent moduli for the cone regions can be determined.

B.5

Definition of error regions

In Table B.1 the regions for the accuracy analyses described in Subsection 3.5.2 are listed. The values of the stress tensor σ n are given in terms of invariants (p, r, θ). For the accuracy analyses in the meridian plane a constant Lode angle, θ=0.96 rad, was chosen. The trial stress tensor σ tr n+1 is given in form of intervals for p and r, together with the respective Lode angle. For the accuracy analyses in the deviatoric plane, fixed values of p were chosen. The intervals r and θ for the trial stress tensor are given in Table B.1. For region 9, the value of the eccentricity was set equal to 1. Hence, a circular loading surface in the deviatoric plane was chosen.

Algorithmic Aspects

183

definition of trial stress regions meridian plane region 1 region 2 region 3 region 4 deviatoric plane region 5 region 6 region 7 region 8 region 9

σ n dummy σ tr n+1 dummydumm (2.80,0.00,0.96) [ − 4.70, 10.30], [ 0.00, 15.00], 0.96 (-13.50,18.30,0.96) [−21.00, −6.00], [15.00, 30.00], 0.96 (-32.50,24.95,0.96) [−40.00, −25.00], [17.45, 32.45], 0.96 (-58.42, 0.00,0.96) [−73.42, −43.42], [ 0.00, 30.00], 0.96 σ n dummy (0.0,5.05,0.96) (-15.00,19.28,0.96) (-30.00,24.83,0.96) (-45.00,20.74,0.96) (-45.00,20.74,π/6)

σ tr n+1 dummydumm 0.00, [3.05, 23.05], [0, π/3] -15.00, [15.00, 40.00], [0, π/3] -30.00, [22.00, 42.00], [0, π/3] -45.00, [20.00, 40.00], [0, π/3] -45.00, [20.00, 40.00], [0, π/3]

Table B.1: intervals of the meridian and deviatoric planes employed for accuracy analyses

Appendix

C

Derivatives of the ELM In the following, all derivatives employed for the implementation of the ELM are given. It is noted that for complex loading surfaces computer algebra systems, e.g., MATHEMATICA or MAPLE, where derivatives of functions can be expressed analytically provide an indispensable tool.

C.1

Invariants of the stress tensor σ 1 p = I1 , 3

r=

q

2J2 ,

√ 3 3 J3 , cos(3θ) = 2 J23/2

(C.1)

referred to global axes: I1 = σii ,

1 J2 = sij sij , 2

1 J3 = sik skj sji , 3

sij = σij − pδij ,

(C.2)

referred to principal axes: σ1 ≥ σ2 ≥ σ3 1 s1 I1 = σ1 + σ2 + σ3 , J2 = [(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ], cos(θ) = √ 6 12J2 ,

(C.3)

where s1 = 2σ1 − σ2 − σ3 . For the formulation in the direction of the principal axes, relations for σ1 , σ2 , σ3 in terms of invariants p, r, θ are employed (see Equation (B.2)).

Derivatives of the ELM

C.2

185

Derivatives of the invariants

The first derivative of p, r and θ with respect to the stress tensor σ yields the following expressions for 2nd -order tensors: ∂p 1 = 1I, ∂σ 3

(C.4)

1 ∂r = s, ∂σ r

(C.5)

∂θ ∂J2 ∂θ ∂J3 ∂θ = + , ∂σ ∂J2 ∂σ ∂J3 ∂σ

(C.6)

where √ ∂θ 3 3 J3 = ∂J2 4 sin(3θ) J25/2

and

√ ∂θ 3 1 , = 3/2 ∂J3 2 sin(3θ) J2

(C.7)

∂J2 = s, ∂σ

(C.8)

∂J3 2 = s · s − J2 1I. ∂σ 3

(C.9)

The second derivative of p, r and θ with respect to the stress tensor σ yields the following expressions for 4th -order tensors: ∂2p = 0, ∂σkl ∂σij

(C.10)

∂2r 1 1 skl sij = (δik δjl − δij δkl ) − 3 , ∂σkl ∂σij r 3 r √ ∂2θ 1 cos(3θ) 3 1 1 1 = 2 sij − sjr sri + √ δij . 3 ∂σkl ∂σij 2r sin(3θ) 2 r sin(3θ) 3 rsin(3θ)

(C.11) (C.12)

It should be noted that for an efficient numerical implementation vector and matrix notation should be used. Hence, the previously derived tensors of second and fourth order must be transformed to vectors and matrices, respectively (for details see Helnwein (2001)).

C.3

Yield function and yield potential f = A2L + k 2 ms BL2 − k 2 c = 0, Q = A2L +

with ms = m(q),

k2 rg(θ, e) (mq + ms √ )k 2 c = 0, fcu 6

(C.13) (C.14)

Derivatives of the ELM

186

with the abbreviations k = q¯h /fcu AL

"

rg(θ, e) p = (1 − k) + √ fcu 6fcu

BL =

C.4

and c = q¯s /ftu ,

"

#2

(C.15) ,

#

p rg(θ, e) . + √ fcu 6fcu

(C.16) (C.17)

First derivatives of f and Q

The gradients of the yield function and of the yield potential are given as follows: n = m =

∂f ∂f ∂p ∂f ∂r ∂f ∂θ = + + , ∂σ ∂p ∂σ ∂r ∂σ ∂θ ∂σ

(C.18)

∂Q ∂Q ∂p ∂Q ∂r ∂Q ∂θ = + + . ∂σ ∂p ∂σ ∂r ∂σ ∂θ ∂σ

(C.19)

The derivatives of f and Q with respect to the stress invariants are computed as1 4(1 − k) k 2 ms ∂f = AL BL + , ∂p fcu fcu

(C.20)

k 2 ∂mQ ∂Q 4(1 − k) AL BL + = , ∂p fcu fcu ∂p

(C.21)

∂Q g(θ, e) g(θ, e)ms k 2 ∂f √ = = AL [4(1 − k)BL + 6] + √ , ∂r ∂r 6fcu 6fcu

(C.22)

∂Q rg 0 (θ, e) ∂f ∂f = = , ∂θ ∂θ g(θ, e) ∂r

(C.23)

g(θ, e) =

a(θ, e) , b(θ, e)

(C.24)

g 0 (θ, e) =

a0 (θ, e) a(θ, e) 0 − b (θ, e), b(θ, e) b(θ, e)

(C.25)

where

with a(θ, e) = 4(1 − e2 ) cos θ 2 + (2e − 1)2 , 1

(C.26)

As described in Section B.1, an explicit update scheme for the eccentricity parameter e is employed. Hence, no derivatives of e need to be performed. For the case of implicit consideration of e, derivatives with respect to the hydrostatic pressure p and the hard-/softening variables must be accounted for.

Derivatives of the ELM

187 q

b(θ, e) = 2(1 − e2 ) cos θ + (2e − 1) 4(1 − e2 ) cos θ 2 + 5e2 − 4e, a0 (θ, e) = −8(1 − e2 ) cos θ sin θ,

(C.27) (C.28)

4(2e − 1)(1 − e2 ) cos θ sin θ . b0 (θ, e) = −2(1 − e2 ) sin θ − q 4(1 − e2 ) cos θ 2 + 5e2 − 4e

C.5

(C.29)

Second derivatives of f and Q

∂2Q ∂p ∂ 2 Q ∂p ∂ 2 Q ∂r ∂ 2 Q ∂θ ∂Q ∂ 2 p = ( 2 + + )+ + ∂σkl ∂σij ∂σkl ∂p ∂σij ∂p∂r ∂σij ∂p∂θ ∂σij ∂p ∂σkl ∂σij ∂ 2 Q ∂r ∂ 2 Q ∂θ ∂Q ∂ 2 r ∂r ∂ 2 Q ∂p ( + + ) + + ∂σkl ∂r∂p ∂σij ∂r ∂σkl ∂σij ∂r 2 ∂σij ∂r∂θ ∂σij ∂θ ∂ 2 Q ∂p ∂ 2 Q ∂r ∂ 2 Q ∂θ ∂Q ∂ 2 θ ( + + ) + . ∂σkl ∂θ∂p ∂σij ∂θ∂r ∂σij ∂θ ∂σkl ∂σij ∂θ 2 ∂σij

(C.30)

The second derivative of f is obtained by replacing Q for f in Equation (C.30). The derivatives of f with respect to the stress invariants are computed as ∂2f ∂p2

"

#

4(1 − k) ∂AL ∂BL , = BL + A L fcu ∂p ∂p

(C.31)

∂2Q ∂BL 4(1 − k) ∂AL k 2 ∂ 2 mQ B + A + = , L L fcu ∂p ∂p fcu ∂p2 ∂p2 "

∂2f ∂r 2

∂2Q g(θ, e) √ = 2 = ∂r 6fcu

#

(

)

∂BL ∂AL [4(1 − k)BL + 6] + 4(1 − k)AL , ∂r ∂r

∂2Q g 00 (θ, e) ∂f r 2 g 02 (θ, e) ∂ 2 f = , + g 0 (θ, e) ∂θ g 2 (θ, e) ∂r 2 ∂θ 2 √ # " ∂2f ∂2Q 2 6g(θ, e)(1 − k) rg(θ, e) 2 = = , (1 − k)BL + BL + √ 2 ∂p∂r ∂p∂r fcu 6fcu ∂2f ∂θ 2

=

∂2f ∂2Q rg 0 (θ, e) ∂ 2 f = = , ∂p∂θ ∂p∂θ g(θ, e) ∂p∂r ∂2f ∂2Q g 0 (θ, e) = = ∂r∂θ ∂r∂θ g(θ, e)

(C.32)

∂f ∂2f +r 2 ∂r ∂r

(C.33) (C.34) (C.35) (C.36)

!

,

(C.37)

with ∂AL BL = 2(1 − k) ∂p fcu

and

∂BL 1 = , ∂p fcu

(C.38)

Derivatives of the ELM

188

∂AL g(θ, e) = 2(1 − k)BL √ + ∂r 6fcu

s

3 g , 2 fcu

and

∂BL g(θ, e) = √ , ∂r 6fcu

(C.39)

and g 00 (θ, e) =

a0 (θ, e)b0 (θ, e) a(θ, e)b02 (θ, e) a00 (θ, e) −2 + 2 , b(θ, e) b2 (θ, e) b3 (θ, e)

(C.40)

where a00 (θ, e) = −8(1 − e2 )(cos θ 2 − sin θ 2 ), b00 (θ, e) = −2(1 − e2 ) cos θ − − sin θ 2 +

4(2e − 1)(1 − e2 ) h cos θ 2 N 1/2

(C.41) (C.42)

4(1 − e2 ) cos θ 2 sin θ 2 i , N

N = 4(1 − e2 ) cos θ 2 + 5e2 − 4e.

(C.43)

Appendix

D

Parametrization of Deviatoric Shape Function This section contains numerical investigations concerning parameterization of the deviatoric shape function g(θ, e). The starting point is the derivative of an isotropic yield function f (p, r, θ; q) with respect to the stress tensor leading to the gradient n of the yield function (see Equation (C.18)). In the following, only the third term of Equation (C.18) is considered. Taking the parameterization for the Lode angle θ of Equation (C.1)3 and making use of the chain rule leads to √ ! ∂f ∂θ 3 ∂f 1 ∂J3 3 J3 ∂J2 q = − 3/2 = ∂θ ∂σ ∂θ 2 1 − cos2 (3θ) 2 J25/2 ∂σ J2 ∂σ √ ∂f 3 ∂θ 2sin(3θ)

!

3 J3 ∂J2 1 ∂J3 − 3/2 . 5/2 2 J2 ∂σ J2 ∂σ

(D.1)

Equation (D.1) becomes infinite for values of the Lode angle θ= 0 and π/3. The remaining term ∂f /∂θ defined in Equation (C.23) contains only one term, i.e., g 0 (θ, e) which can become zero. Thus, the following expression must be investigated: g 0 (θ, e) 1 b(θ, e)a0 (θ, e) − a(θ, e)b0 (θ, e) I = lim = 2 lim . θ →0,π/3 sin(3θ) b (θ, e) θ →0,π/3 sin(3θ)

(D.2)

Using L’Hopital’s rule leads to Iθ=0 =

2 1 − e2 1 00 00 lim (b(θ, e)a (θ, e) − a(θ, e)b (θ, e)) = − , 3b2 (θ, e) θ →0 3 e(2 − e)

(D.3)

Parametrization of Deviatoric Shape Function Iθ=π/3 = −

190

1 1 − e2 00 00 lim (b(θ, e)a (θ, e) − a(θ, e)b (θ, e)) = − . 3b2 (θ, e) θ →π/3 (2e − 1)2

(D.4)

Hence, for 0.5