Contact methods integrating plasticity models with application to soil mechanics

Von der Fakultat fur Maschinenbau

der Gottfried Wilhelm Leibniz Universitat Hannover

zur Erlangung des akademischen Grades Doktor-lngenieur

genehmigte Dissertation

von

Dipl.-lng. Christian WeiRenfels

geboren am 30.01.1979 in Rosenheim

2013 Contents

1 Introduction 1 1.1 State of the art 3 1.2 Structure of this work 4

2 Continuum mechanics 7 2.1 Kinematics 8 2.1.1 Motion 8

2.1.2 Deformation measures 9 2.2 Balance principles 10 2.2.1 Continuity equation 10

2.2.2 Momentum, mechanical energy arid angular momentum ..... 10 2.2.3 Energy and entropy 11 2.3 Constitutive models 13 2.3.1 Elasticity 14 2.3.2 Plasticity 15 2.4 Variational form 18 2.4.1 Reduction to surface description 18 2.4.2 Reduction to contact description 21

3 Finite elements 27 3.1 Concept of finite elements 28 3.2 Space discretization 31 3.3 Newton iteration for nonlinear equations 33 3.4 Assembling and solver 35

4 Contact discretization 37 4.1 Contact solution methods 38 4.1.1 Lagrange multiplier method 40 4.1.2 Penalty method 40 4.1.3 Augmented method 41 4.2 Node to surface 44 4.2.1 Global search of contact elements 44 4.2.2 Projection point and base vectors 46 4.2.3 Contact kinematical relations 47 4.2.4 Integration domain 48

vii viii

4.2.5 Linearized quantities 49 4.2.6 Residual vector and tangent matrix 50 4.3 Mortar method 53 4.3.1 Setup of contact element 53 4.3.2 Integration point and base vectors 56 4.3.3 Integration 57 4.3.4 Kinematical contact relations 57 4.3.5 Linearized quantities 60 4.3.5.1 Contact element quantities 61 4.3.5.2 Kinematical quantities 65 4.3.6 Residual vector and tangent matrix 69 4.3.6.1 Lagrange multiplier method 70 4.3.6.2 Penalty method 73 4.3.6.3 Augmented Lagrange multiplier method 75 4.3.6.4 Mixed method 77 4.4 Numerical solutions 80 4.4.1 Hertzian contact 80 4.4.2 Rotating blocks 81 4.4.3 Ironing 83

5 Soil mechanics 87 5.1 Classification of the soil 87 5.2 Behavior under loading 89 5.2.1 Plastic behavior 89 5.2.2 Change of volume 89 5.2.3 Dependency on pressure and porosity 90 5.2.4 Localization 91 5.2.5 Shakedown and runaway ratcheting 91 5.2.6 Liquefaction and consolidation 92 5.3 Modeling strategies 92 5.4 Elders soil models 94 5.4.1 Stress strain relation 94 5.4.2 Yield criterion and evolution equation 95 5.4.3 Hardening concept 97 5.4.4 Derivatives 97 5.4.5 Algorithmic implementation 98 5.4.6 Substepping scheme 101 5.4.7 Viscoplastic regularization 102 5.4.8 Numerical triaxial and footing test 102

6 Theory of porous media 107 6.1 Concept of volume fractions 107 6.2 Kinematics of the mixture 108

6.3 Balance of mass and momentum of the mixture 108 6.4 Incompressible biphasic model 109 ix

6.5 Weak form of porous media HI 6.6 Discretization of porous media 112 6.7 Consolidation test 113

7 Projection strategies 115 7.1 Projection over the coefficient of friction 115 7.1.1 Link between contact and continuum 116 7.1.2 Linearization 118 7.2 Direct link between yield and slip criterion 119 7.2.1 Projection of Ehlers yield criterion 121 7.2.2 Linearization 124

7.3 Connection to the theory of porous media 125 7.4 Numerical tests 125 7.4.1 Direct shear test 125

7.4.2 Pull out of a wall 128

8 3D contact element 133 8.1 3D contact kinematical relations 133 8.2 Linearized quantities 136 8.3 Residual vector and tangent matrix 140 8.4 Connection to the theory of porous media 142 8.5 Numerical direct shear test 143

9 XFEM contact element 145 9.1 Description of elements with interfaces 145 9.2 Linear embedded element 147 9.2.1 Weak form 148 9.2.2 Discretization 149 9.2.3 Solution algorithm 152 9.3 Linear embedded contact element 152 9.3.1 Weak form 153 9.3.2 Segmentation 153 9.3.3 Discretization 154 9.3.4 Solution algorithm 156 9.4 Numerical examples 157

10 Conclusion and Outlook 161

A Principle stresses 165

B Voigt notation 167 B.l B matrices 167 B.2 Elastic strain tensor 168 B.3 Invariants and its derivations 168 B.4 Shell transformation tensor 169 X

C Contact vectors and storage arrays 171 C.l Node to surface vectors 171 C,2 Mortar tensors 172 C.3 Mortar storage arrays 173

D Derivation of the Reynolds equation 175