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Alexander Konyukhov has been working at Karlsruhe Institute of Technology in Germany since 2002, where he has made his Habilitation in 2010 in the field of Geometrically exact theory of contact interaction. His main areas of research are computational and theoretical mechanics. He is recognized expert in the field of Computational Contact Mechanics, authors of other books in Computational Contact Mechanics and has numerous publications in international journals. His special teaching interest in Computational Contact Mechanics has led to this book.
Ridvan Izi is a member of the academic staff in the Institute of Mechanics at Karlsruhe Institute of Technology. His work focuses on computational contact problems.Preface viii
Acknowledgements xi
Part One Theory 1
1 Introduction with a spring-mass frictionless contact system 2
1.1 Structural part - deflection of spring-mass system 3
1.2 Contact part – non-penetration into rigid plane 3
1.3 Contact formulations 4
2 General formulation of a contact problem 11
2.1 Structural part – formulation of a problem in linear elasticity 11
2.2 Formulation of the contact part (Signorini’s problem) 14
3 Differential geometry 19
3.1 Curve and its properties 19
3.2 Frenet formulas in 2D 23
3.3 Description of surfaces by Gauss coordinates 24
3.4 Differential properties of surfaces 32
4 Geometry and kinematics for arbitrary two body contact problem 40
4.1 Local coordinate system 41
4.2 Closest Point Projection (CPP) procedure – Analysis 43
4.3 Contact kinematics. 50
5 Abstract form of formulations in computational mechanics 54
5.1 Operator necessary for the abstract formulation 54
5.2 Abstract form of iterative method 55
5.3 Fixed point theorem (Banach) 56
5.4 Newton iterative solution method 58
5.5 Abstract form for contact formulations 61
6 Weak formulation and consistent linearization 65
6.1 Weak formulation in the local coordinate system 66
6.2 Regularization with penalty method 67
6.3 Consistent linearization 67
6.4 Application to the Lagrange multipliers and to the following forces 71
6.5 Linearization of the convective variation δξ 73
6.6 Nitsche method 73
7 Finite element discretization 76
7.1 Computation of the contact integral for various contact approaches 76
7.2 Node-To-Node (NTN) contact element 78
7.3 Nitsche Node-To-Node (NTN) contact element 80
7.4 Node-To-Segment (NTS) contact element 81
7.5 Segment-To-Analytical-Surface (STAS) approach 88
7.6 Segment-To-Segment (STS) Mortar approach 94
8 Verification with analytical solution 99
8.1 Hertz problem 99
8.2 Rigid flat punch problem 104
8.3 Impact on moving pendulum – center of percussion 106
8.4 Generalized Euler-Eytelwein problem 108
9 Frictional contact problems 111
9.1 Measures of contact interactions – sticking and sliding case. Friction law. 111
9.2 Regularization of tangential force and return mapping algorithm 112
9.3 Weak form and its consistent linearization 118
9.4 Frictional Node-To-Node (NTN) contact element 119
9.5 Frictional Node-To-Segment (NTS) contact element 123
9.6 NTS frictional contact element 125
Part Two Programming and Verification Tasks 127
10 Introduction into programming and verification tasks 128
11 Lesson 1 Nonlinear structural truss – elmt1.f 132
11.1 Implementation 134
11.2 Examples 138
12 Lesson 2 Nonlinear structural plane – elmt2.f 144
12.1 Implementation 145
12.2 Examples 150
13 Lesson 3 Penalty Node-To-Node (NTN) – elmt100.f 154
13.1 Implementation 156
13.2 Examples 158
14 Lesson 4 Lagrange multiplier Node-To-Node (NTN) – elmt101.f 161
14.1 Implementation 163
14.2 Examples 165
15 Lesson 5 Nitsche Node-To-Node (NTN) – elmt102.f 167
15.1 Implementation 169
15.2 Examples 171
16 Lesson 6 Node-To-Segment (NTS) – elmt103.f 173
16.1 Implementation 175
16.2 Examples 178
16.3 Inverted contact algorithm – following force 182
17 Lesson 7 Segment-To-Analytical-Segment (STAS) – elmt104.f 186
17.1 Implementation 188
17.2 Examples 191
17.3 Inverted contact algorithm – general case of following forces 194
18 Lesson 8 Mortar / Segment-To-Segment (STS) – elmt105.f 202
18.1 Implementation 204
18.2 Examples 207
18.3 Inverted contact algorithm – following force 209
19 Lesson 9 Higher order Mortar / STS – elmt106.f 213
19.1 Implementation 215
19.2 Examples 219
20 Lesson 10 3D Node-To-Segment (NTS) elmt107.f 223
20.1 Implementation 225
20.2 Examples 229
21 Lesson 11 Frictional Node-To-Node (NTN) – elmt108.f 233
21.1 Implementation 235
21.2 Examples 237
22 Lesson 12 Frictional Node-To-Segment (NTS) – elmt109.f 239
22.1 Implementation 241
22.2 Examples 245
23 Lesson 13 Frictional higher order NTS – elmt110.f 250
23.1 Implementation 251
23.2 Examples 256
24 Lesson 14 Transient contact problems 259
24.1 Implementation 260
24.2 Examples 262
A Numerical integration 264
A.1 Gauss quadrature 266
B Higher order shape functions of different classes 268
B.1 General 268
B.2 Lobatto class 268
B.3 Bezier class 271
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