Elasticity of Transversely Isotropic Materials (eBook)

Contents 6
Preface 12
1 BASIC EQUATIONS OF ANISOTROPIC ELASTICITY 14
1.1 TRANSFORMATION OF STRAINS AND STRESSES 14
1.2 BASIC EQUATIONS 17
1.2.1 Geometric Equations 17
1.2.2 Equations of Motion 20
1.2.3 Constitutive Equations 21
1.3 BOUNDARY AND INITIAL CONDITIONS 36
1.3.1 Boundary Conditions 36
1.3.2 Initial Conditions 37
1.4 THERMOELASTICITY 38
2 GENERAL SOLUTION FOR TRANSVERSELY ISOTROPIC PROBLEMS 42
2.1 GOVERNING EQUATIONS 42
2.1.1 Methods of Solution 42
2.1.2 Governing Equations for the Displacement Method 43
2.1.3 Equations for a Mixed Method – the State-Space Method 46
2.2 DISPLACEMENT METHOD 51
2.2.1 General Solution in Cartesian Coordinates 51
2.2.2 General Solution in Cylindrical Coordinates 64
2.3 STRESS METHOD FOR AXISYMMETRIC PROBLEMS 67
2.4 DISPLACEMENT METHOD FOR SPHERICALLY ISOTROPIC BODIES 75
2.4.2 Relationship between Transversely Isotropic and Spherically Isotropic Solutions 80
2.4.1 General Solution 75
3 PROBLEMS FOR INFINITE SOLIDS 84
3.1 THE UNIFIED POINT FORCE SOLUTION 84
3.1.1 A Point Force Perpendicular to the Isotropic Plane 84
3.1.2 A Point Force within the Isotropic Plane 88
3.2 THE POINT FORCE SOLUTION FOR AN INFINITE SOLID COMPOSED OF TWO HALF- SPACES 92
3.2.1 A Point Force Perpendicular to the Isotropic Plane 93
3.2.2 A Point Force within the Isotropic Plane 98
3.2.3 Some Remarks 106
3.3 AN INFINITE TRANSVERSELY ISOTROPIC SPACE WITH AN INCLUSION 106
3.4 SPHERICALLY ISOTROPIC MATERIALS 112
3.4.1 An Infinite Space Subjected to a Point Force 112
3.4.2 Stress Concentration near a Spherical Cavity 114
4 HALF-SPACE AND LAYERED MEDIA 120
4.1 UNIFIED SOLUTION FOR A HALF-SPACE SUJECTED TO A SURFACE POINT FORCE 120
4.1.1 A Point Force Normal to the Half-Space Surface 121
4.1.2 A Point Force Tangential to the Half-Space Surface 125
4.2 A HALF-SPACE SUJECTED TO AN INTERIOR POINT FORCE 131
4.2.1 A Point Force Normal to the Half-Space Surface 132
4.2.2 A Point Force Tangential to the Half-Space Surface 134
4.3 GENERAL SOLUTION BY FOURIER TRANSFORM 136
4.4 POINT FORCE SOLUTION OF AN ELASTIC LAYER 146
4.5 LAYERED ELASTIC MEDIA 154
5 EQUILIBRIUM OF BODIES OF REVOLUTION 160
5.1 SOME HARMONIC FUNCTIONS 160
5.1.1 Harmonic Polynomials 160
5.1.2 Harmonic Functions Containing 160
5.1.3 Harmonic Functions Containing 161
5.2 AN ANNULAR (CIRCULAR) PLATE SUBJECTED TO AXIAL TENSION AND RADIAL COMPRESSION 161
5.3 AN ANNULAR (CIRCULAR) PLATE SUBJECTED TO PURE BENDING 163
5.4 A SIMPLY-SUPPORTED ANNULAR (CIRCULAR) PLATE UNDER UNIFORM TRANSVERSE LOADING 165
5.5 A UNIFORMLY ROTATING ANNULAR (CIRCULAR) PLATE 167
5.6 TRANSVERSELY ISOTROPIC CONES 171
5.6.1 Compression of a Cone under an Axial Force 171
5.6.2 Bending of a Cone under a Transverse Force 175
5.7 SPHERICALLY ISOTROPIC CONES 178
5.7.1 Equilibrium and Boundary Conditions 178
5.7.2 A Cone under Tip Forces 181
5.7.3 A Cone under Concentrated Moments at Its Apex 186
5.7.4 Conical Shells 189
6 THERMAL STRESSES 196
6.1 TRANSVERSELY ISOTROPIC MATERIALS 196
6.2 A DIFFERENT GENERAL SOLUTION FOR TRANSVERSELY ISOTROPIC THERMOELASTICITY 202
6.2.1 General Solution for Dynamic Problems 202
6.2.2 General Solution for Static Problems 205
6.3 SPHERICALLY ISOTROPIC MATERIALS 212
7 FRICTIONAL CONTACT 218
7.1 TWO ELASTIC BODIES IN CONTACT 218
7.1.1 Mathematical Description of a Contact System 218
7.1.2 Deformation of Transversely Isotropic Bodies under Frictionless Contact 221
7.1.3 A Half-Space under Point Forces 227
7.2 CONTACT OF A SPHERE WITH A HALF-SPACE 230
7.2.1 Contact with Normal Loading 230
7.2.2 Contact with Tangential Loading 235
7.3 CONTACT OF A CYLINDRICAL PUNCH WITH A HALF-SPACE 237
7.3.1 Contact with Normal Loading 238
7.3.2 Contact with Tangential Loading 241
7.4 INDENTATION BY A CONE 243
7.4.1 Contact with Normal Loading 244
7.4.2 Contact with Tangential Loading 246
7.5 INCLINED CONTACT OF A CYLINDRICAL PUNCH WITH A HALF- SPACE 249
7.5.1 Contact with Normal Loading 250
8 BENDING, VIBRATION AND STABILITY OF PLATES 260
8.1 GENERAL SOLUTION METHOD 260
8.1.1 Rectangular Plates 260
8.1.2 Circular Plates 270
8.2 THE STATE-SPACE METHOD FOR LAMINATED PLATES 279
8.2.1 Laminated Rectangular Plates 279
9 VIBRATIONS OF CYLINDERS AND CYLINDRICAL SHELLS OF TRANSVERSELY ISOTROPIC MATERIALS 296
9.1 THREE SIMPLE MODES OF VIBRATION 296
9.1.1 Axisymmetric Torsional Vibration 298
9.1.2 Breathing Mode Vibration 305
9.1.3 Thickness-Shear Vibration 309
9.2 ASYMMETRIC VIBRATION 314
9.3 VIBRATION OF A LAYERED CYLINDRICAL SHELL 323
9.3.1 State-Space Formulations 323
9.3.2 Layerwise Method and State Vector Solution 326
9.3.3 Free Vibration Analysis and Numerical Results 327
9.4 VIBRATION OF A CYLINDRICAL SHELL COUPLED WITH FLUID 329
10 SPHERICAL SHELLS OF SPHERICALLY ISOTROPIC MATERIALS 340
10.1 FREE VIBRATION 340
10.1.1 Basic Equations and Solution 340
10.1.2 Free Vibration Analysis 344
10.2 FREQUENCY EQUATIONS AND NUMERICAL RESULTS 347
10.2.1 Frequency Equations of a Single-Layered Hollow Sphere 347
10.2.2 Some Special Cases 349
10.2.3 An Example 351
10.3 VIBRATION COUPLED WITH FLUID 358
10.3.1 Effect of Fluid 358
10.3.2 Frequency Equations 361
10.3.3 Numerical Results 363
10.4 VIBRATION COUPLED WITH THE SURROUNDING ELASTIC MEDIUM 369
10.4.1 Pasternak Model of Elastic Foundation 370
10.4.2 Frequency Equations 373
10.4.3 Numerical Results 374
10.5 LAMINATED SPHERICAL SHELLS 377
10.5.1 State-Space Formulations for Spherically Isotropic Elasticity 377
10.5.2 Layerwise Method and State Vector Solution 380
Appendix A ADDITIONAL NOTES AND BIBLIOGRAPHY TO CHAPTERS 386
Ch. 1 386
Ch. 2 386
Section 2.1 386
Section 2.2 386
Section 2.4 387
Ch. 3 387
Section 3.1 387
Section 3.2 388
Section 3.3 388
Ch. 4 389
Section 4.2 389
Ch. 5 390
Sections 5.2-5.5 390
Ch. 6 390
Section 6.1 390
Section 6.3 391
Ch. 7 391
Ch. 8 394
Section 8.1 394
Section 8.1 394
Section 8.2 394
Section 8.2. 395
Ch. 9 395
Ch. 10 396
Section 10.3 397
Section 10.4 397
Section 10.5 398
Appendix B SPECIAL FUNCTIONS 400
B. 1 HELMHOLTZ EQUATION AND SEPARATION OF VARIABLES 400
B. 2 LEGENDRE FUNCTION AND ASSOCIATED LEGENDRE FUNCTION 402
B. 3 SPHERICAL HARMONICS 403
B. 4 BESSEL FUNCTIONS 404
B. 5 BESSEL FUNCTIONS OF OTHER KINDS 406
B.5.1 Modified Bessel Functions 406
B.5.2 Hankel Functions 407
B.5.3 Sperical Bessel Functions 408
Appendix C NOMENCLATURE 410
REFERENCES 418
INDEX 444