Introduction to Rational Elasticity

Title......Page 2
Contents......Page 4
Dedication......Page 8
Acknowledgment......Page 9
Preface......Page 10
1. Finite-Dimensional Vector Spaces......Page 12
2. Inner-Product Spaces......Page 30
3. Euclidean Spaces......Page 36
4. Differentiable Manifolds......Page 51
5. Parallelisms on Manifolds......Page 63
6. Lie Groups......Page 77
1. Kinematics......Page 92
2. Frame-Indifference......Page 103
3. Axioms of Dynamics and Thermodynamics in Inertial Frames......Page 117
4. Frame-Indifferent Dynamical Principles......Page 127
5. Cauchy's Postulate and Fundamental Theorem......Page 131
6. Field Equations and Jump Conditions......Page 138
7. Constitutive Relations......Page 145
8. Representations for Constitutive Relations......Page 166
1. The Constitutive Equation of an Elastic Point......Page 176
2. The Symmetry of an Elastic Point......Page 185
3. Representation for the Response Function of an Elastic Point......Page 197
4. The Constitutive Equation of a Hyperelastic Point......Page 200
5. The Symmetry of a Hyperelastic Point......Page 207
6. Representation for the Stored-Energy Function of a Hyperelastic Point......Page 216
7. Inequalities in Elasticity, I. General Background......Page 221
A. The Generalized Coleman-Noll Inequality......Page 224
B. The Strong Ellipticity Condition and Infinitesimal Stability......Page 231
C. Hill's Inequalities......Page 236
A. Specialization of the GCN Condition......Page 249
B. Specialization of the S-E Condition......Page 256
C. Specialization of Hill's Inequalities......Page 259
D. Closing Remarks......Page 268
1. The Concept of an Elastic Body......Page 274
2. Static Universal Solutions......Page 283
3. A Central Example: Simple Shear of a Rectangular Block......Page 291
4. Static Universal Solutions for Some Incompressible Bodies......Page 302
5. Examples of Static Universal Solutions for Incompressible Isotropic Bodies......Page 315
6. Dynamic Universal Solutions for Some Incompressible Elastic Bodies......Page 326
7. Examples of Dynamic Universal Solutions for Incompressible Isotropic Bodies......Page 342
1. The Material Tangent Bundle of an Elastic Body......Page 353
2. The Bundle of Reference Frames......Page 361
3. The Concept of a Material Connection......Page 367
4. The Existence of Material Connections......Page 373
5. Material Connection in Component Form......Page 380
6. The Integrability Condition of a Material Connection......Page 387
7. The Concept of a Characteristic Field......Page 394
8. The Equations of Motion for an Elastic Body......Page 402
9. Static and Dynamic Universal Solutions for Some Incompressible Laminated Elastic Bodies......Page 408
10. Examples of Static and Dynamic Universal Solutions for Some Incompressible Laminated Bodies......Page 415
1. Conditions of Compatibility for Singular Surfaces......Page 428
2. Waves in Deformable Bodies......Page 438
3. The Propagation Condition......Page 445
4. Acceleration Waves in Isotropic Elastic Bodies, I. Gradients of the Response Function......Page 452
5. Acceleration Waves in Isotropic Elastic Bodies, II. The Propagation Condition......Page 460
6. Acceleration Waves in Isotropic Elastic Bodies, III. The Displacement Derivative of the Amplitude Vector......Page 465
7. Acceleration Waves in Homogeneous or Laminated Bodies......Page 471
1. Formulation of Boundary-Value Problems......Page 481
2. Signorini's Expansion and Theorems of Uniqueness and Compatibility......Page 487
3. A Central Example: Second-order Effects in Torsion and Extension of an Isotropic Circular-Cylindrical Tube......Page 498
4. Stoppelli's Theorems of Existence, Uniqueness, and Analyticity of the Solution of a Class of Traction Boundary-value Problems......Page 505
5. Van Buren's Analysis of Existence and Uniqueness of Solutions of Boundary-value Problems in Elasticity......Page 515
6. Beju's Analysis of Existence, Uniqueness, and Stability of Solutions of Boundary-value Problems of Place for Hyperelastic Bodies......Page 521
7. The Concept of Dynamic Stability in Elasticity......Page 529
8. Static Stability and the Energy Criterion......Page 541
Index of Persons Named or Cited......Page 556
Index of Matters Treated......Page 560