# Introduction to Computational Contact Mechanics

## A Geometrical Approach

Wiley Series in Computational Mechanics

1. Edition June 2015

304 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-118-77065-8**

Introduction to Computational Contact Mechanics: A Geometrical Approach covers the fundamentals of computational contact mechanics and focuses on its practical implementation. Part one of this textbook focuses on the underlying theory and covers essential information about differential geometry and mathematical methods which are necessary to build the computational algorithm independently from other courses in mechanics. The geometrically exact theory for the computational contact mechanics is described in step-by-step manner, using examples of strict derivation from a mathematical point of view. The final goal of the theory is to construct in the independent approximation form /so-called covariant form, including application to high-order and isogeometric finite elements.

The second part of a book is a practical guide for programming of contact elements and is written in such a way that makes it easy for a programmer to implement using any programming language. All programming examples are accompanied by a set of verification examples allowing the user to learn the research verification technique, essential for the computational contact analysis.

Key features:

* Covers the fundamentals of computational contact mechanics

* Covers practical programming, verification and analysis of contact problems

* Presents the geometrically exact theory for computational contact mechanics

* Describes algorithms used in well-known finite element software packages

* Describes modeling of forces as an inverse contact algorithm

* Includes practical exercises

* Contains unique verification examples such as the generalized Euler formula for a rope on a surface, and the impact problem and verification of thå percussion center

* Accompanied by a website hosting software

Introduction to Computational Contact Mechanics: A Geometrical Approach is an ideal textbook for graduates and senior undergraduates, and is also a useful reference for researchers and practitioners working in computational mechanics.

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 deltaxi 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