Elasticity with Mathematica: An introduction to continuum mechanics and linear elasticity

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Acknowledgments......Page 11
MOTIVATION......Page 13
WHAT WILL AND WILL NOT BE FOUND IN THIS BOOK......Page 16
Lagrangian description......Page 20
Particle path......Page 21
Eulerian description......Page 23
Streamline......Page 24
Streakline......Page 25
Ideal inviscid potential flow......Page 26
Deformation gradient......Page 31
The rotation and stretch tensors......Page 32
Geometrical interpretation of the stretch tensors......Page 34
Computation of strain, stretch, and rotation tensors......Page 35
Trigonometric representation of strain, stretch, and rotation tensors......Page 37
Plotting stretch and rotation of material vectors......Page 39
1.3 SMALL STRAIN TENSOR......Page 40
1.4 COMPATIBILITY EQUATIONS AND INTEGRATION OF SMALL STRAINS......Page 41
Compatibility conditions for small strains......Page 42
Equivalent compatibility conditions for small strains......Page 46
EXERCISES......Page 47
2.1 FORCES AND MOMENTA......Page 53
2.2 VIRTUAL POWER AND THE CONCEPT OF STRESS......Page 54
Ideal fluid......Page 55
Continuum solid......Page 56
Bending of thin plates......Page 57
2.3 THE STRESS TENSOR ACCORDING TO CAUCHY......Page 58
Example stress states......Page 59
2.4 POTENTIAL REPRESENTATIONS OF SELF-EQUILIBRATED STRESS TENSORS......Page 60
1. Balance of linear and angular momentum......Page 62
4. Stress balance on infinitesimal volume elements (I)......Page 63
7. Stress tensors in a spherical shell......Page 64
9. Local stress representation using Mohr’s circles (Salencon, 2001; Timoshenko, 1951)......Page 65
10. Stress balance and the Beltrami potential......Page 67
3.1 LINEAR ELASTICITY......Page 68
3.2 MATRIX REPRESENTATION OF ELASTIC COEFFICIENTS......Page 70
Mathematica Programmaing......Page 73
3.3 MATERIAL SYMMETRY......Page 77
Monoclinic elastic materials......Page 78
Tetragonal elastic materials......Page 79
Orthotropic elastic materials......Page 80
Isotropic elastic materials......Page 81
Mathematica programming......Page 82
Displaying the symmetry planes......Page 83
3.4 THE EXTENSION EXPERIMENT......Page 84
3.5 FURTHER PROPERTIES OF ISOTROPIC ELASTICITY......Page 87
Thermal expansion......Page 88
Residual stresses......Page 89
3.6 LIMITS OF LINEAR ELASTICITY......Page 90
1. Extension of cylindrical bar......Page 92
3. Uniform compression......Page 93
5. Bending of a plate......Page 94
8. Elastic moduli in isotropic elasticity......Page 95
10. Special strain–stress states in isotropic elasticity......Page 96
4.1 THE COMPLETE ELASTICITY PROBLEM......Page 98
4.2 DISPLACEMENT FORMULATION......Page 100
4.3 STRESS FORMULATION......Page 101
4.4 EXAMPLE: SPHERICAL SHELL UNDER PRESSURE......Page 103
4.5 SUPERPOSITION PRINCIPLE......Page 106
4.7 UNIQUENESS OF SOLUTION......Page 107
Existence of the strain energy potential......Page 108
Existence of the complementary energy potential......Page 110
4.9 RECIPROCITY THEOREMS......Page 111
4.10 THE SAINT VENANT PRINCIPLE......Page 113
Counter-examples proposed by Hoff......Page 120
2. Isotropic elastic spherical shell subjected to pressure and temperature loading......Page 121
6. Transversely isotropic elastic sphere under pressure (Lehnitski, 1981)......Page 122
8. Finite cylindrical tube subjected to thermal loading......Page 123
11. Thermal assembly (‘frettage’) of cylindrical tubes (Ballard, in Polytechnique 1990– 2005).......Page 124
13. Cylindrical rod of arbitrary cross section under torsion......Page 126
5.1 PLANE STRESS......Page 128
Harmonic Airy stress function:  A0(x, y) = 0......Page 131
Reducibility of differential operators......Page 132
5.3 AIRY STRESS FUNCTION WITH A CORRECTIVE TERM: A0 (x y) – z2A1(x y)......Page 134
5.4 PLANE STRAIN......Page 136
5.6 BIHARMONIC FUNCTIONS......Page 138
5.7 THE DISCLINATION, DISLOCATIONS, AND ASSOCIATED SOLUTIONS......Page 142
5.8 A WEDGE LOADED BY A CONCENTRATED FORCE APPLIED AT THE APEX......Page 145
Axial force Fx......Page 146
Transverse force Fy......Page 147
5.9 THE KELVIN PROBLEM......Page 149
5.10 THE WILLIAMS EIGENFUNCTION ANALYSIS......Page 151
MATHEMATICA programming......Page 153
5.11 THE KIRSCH PROBLEM: STRESS CONCENTRATION AROUND\rA CIRCULAR HOLE......Page 157
5.12 THE INGLIS PROBLEM: STRESS CONCENTRATION AROUND\rAN ELLIPTICAL HOLE......Page 159
1. Displacement field around a disclination......Page 164
4. Diametrical compression of a cylinder by equal and opposite concentrated forces......Page 165
6. Concentrated and distributed loading at the surface of a half-plane......Page 166
7. Curved beam under shearing force at one end (Barber, 2002)......Page 167
OUTLINE......Page 169
6.1 PAPKOVICH–NEUBER POTENTIALS......Page 170
Uniform deformation and stress states......Page 172
Concentrated force in infinite space – the Kelvin solution......Page 173
Momentless force dipoles......Page 175
Centre of dilatation in the infinite space – the Lam´e solution......Page 176
Force dipoles with moment......Page 178
Centre of rotation in an infinite elastic solid......Page 179
Concentrated load normal to the surface of a half space – The Boussinesq solution......Page 181
Concentrated load parallel to the surface of a half space – The Cerruti solution......Page 183
Concentrated loads at the tip of an elastic cone......Page 186
6.2 GALERKIN VECTOR......Page 194
6.3 LOVE STRAIN FUNCTION......Page 195
Integral transform methods......Page 197
SUMMARY......Page 198
4. Stresses around a spherical cavity......Page 199
7. Torsion of a cylindrical shaft with a spherical hole (Barber, 2002)......Page 200
Strain energy......Page 201
Complementary energy......Page 202
Equality of potentials......Page 203
7.2 EXTREMUM THEOREMS......Page 204
7.3 APPROXIMATE SOLUTIONS FOR PROBLEMS OF ELASTICITY......Page 208
7.4 THE RAYLEIGH–RITZ METHOD......Page 209
Example: compression of a cylinder between two fully adhered rigid platens......Page 211
Series solutions......Page 214
Calculus of variations......Page 215
Well-posed problem of elastodynamics......Page 216
The spectrum of free vibrations......Page 217
Approximate spectra......Page 220
Example: vibration of a cantelever beam......Page 221
2. Bending of a plate......Page 224
3. Torsional stiffness of a cylindrical rod......Page 225
5. A dam loaded by the water pressure and its own weight (Obala, 1997)......Page 227
6. A heated disk glued to a planar rigid substrate......Page 228
7. Torsion applied to an adhesive bond (Dumontet et al., 1998)......Page 229
8. Elastic deformation of a hemi spherical joint (Dumontet et al., 1998)......Page 230
Orthogonal curvilinear coordinate systems......Page 231
Length, surface, and volume elements......Page 233
Defining tensors in Mathematica......Page 234
Differential operators in curvilinear coordinate systems......Page 236
The gradient operator......Page 239
The divergence operator......Page 240
Laplacian, biharmonic, and inc operators......Page 241
Potential fields and Stokes’ theorem......Page 242
Solutions of the Poisson equation......Page 243
Gradient integration......Page 244
A.2.1 List manipulation......Page 247
Assigning element names in lists......Page 248
Other operations on lists: Map, Apply, Thread......Page 249
Vector-type operations on lists: Inner, Outer......Page 250
A.2.2 Functions......Page 251
A.2.3 Algebraic handling of expressions......Page 253
A.2.4 Graphics......Page 254
Building a 2D mesh......Page 255
Building a 2D surface in 3D......Page 257
Building a 3D box as a collection 2D surfaces......Page 258
The ParametricMesh package......Page 259
Bibliography......Page 261
Index......Page 263