Lecture Notes on Solid Mechanics

Lecture Notes on Solid Mechanics
Prof. Dr. Ing. Martin Schanz
Institute of Applied Mechanics
Graz University of Technology
Dr. Ing. Jens Uwe B¨ohrnsen
Institute of Applied Mechanics
Spielmann Str. 11
38104 Braunschweig
Contents
1 Introduction and mathematical preliminaries 3
1.1 Vectors and matrices 3
1.2 Indical Notation . 5
1.3 Rules for matrices and vectors . 6
1.4 Coordinate transformation . 8
1.5 Tensors . 9
1.6 Scalar, vector and tensor fields . 10
1.7 Divergence theorem . 11
1.8 Summary of chapter 1 . 11
1.9 Exercise . 15
2 Traction, stress and equilibrium 17
2.1 State of stress 17
2.1.1 Traction and couple{stress vectors . 17
2.1.2 Components of stress 18
2.1.3 Stress at a point 19
2.1.4 Stress on a normal plane 21
2.2 Equilibrium . 21
2.2.1 Physical principles . 21
2.2.2 Linear momentum . 22
2.2.3 Angular momentum 23
I2.3 Principal stress . 24
2.3.1 Maximum normal stress 24
2.3.2 Stress invariants and special stress tensors 27
2.4 Summary of chapter 2 . 28
2.5 Exercise . 31
3 Deformation 33
3.1 Position vector and displacement vector 33
3.2 Strain tensor 34
3.3 Stretch ratio{finite strains . 36
3.4 Linear theory 37
3.5 Properties of the strain tensor . 38
3.5.1 Principal strain . 38
3.5.2 Volume and shape changes . 38
3.6 Compatibility equations for linear strain . 40
3.7 Summary of chapter 3 . 41
3.8 Exercise . 42
4 Material behavior 44
4.1 Uniaxial behavior 44
4.2 Generalized Hooke’s law 45
4.2.1 General anisotropic case 45
4.2.2 Planes of symmetry . 46
4.2.3 Isotropic elastic constitutive law 48
4.2.4 Thermal strains . 49
4.3 Elastostatic/elastodynamic problems . 50
4.3.1 Displacement formulation . 50
4.3.2 Stress formulation 51
4.4 Summary of chapter 4 . 51
4.5 Exercise . 53
II5 Two{dimensional elasticity 55
5.1 Plane stress . 56
5.2 Plane strain . 57
5.3 Airy’s stress function 59
5.4 Summary of chapter 5 . 60
5.5 Exercise . 62
6 Energy principles 63
6.1 Work theorem 63
6.2 Principles of virtual work 65
6.2.1 Statement of the problem . 65
6.2.2 Principle of virtual displacements . 66
6.2.3 Principle of virtual forces 68
6.3 Approximative solutions 70
6.3.1 Application: FEM for beam 72
6.4 Summary of chapter 6 . 77
6.5 Exercise . 79
A Solutions 80
A.1 Chapter 1 80
A.2 Chapter 2 84
A.3 Chapter 3 91
A.4 Chapter 4 95
A.5 Chapter 5 100
A.6 Chapter 6
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