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An Introduction to Modern Anistropic Elasticity

1-56700-107-6 (Print)

An Introduction to Modern Anistropic Elasticity

K. F. Chernykh
St. Petersburg University, St. Petersburg, Russia


The author, a leading Russian authority in the field of elasticity, has obtained some fundamental results in the investigation of a number of general problems in the mechanics of solids, the theory of shells, biomechanics, and the mechanics of elastomers. Fundamental theoretical investigations were combined with the practical applications to real construction problems. This book deals with symmetry, linear anisotropic elasticity, constitutive equations, plane problems, anisotropic and reinforced shells, brittle fracture, Volterra's dislocations, etc.

248 pages, © 1998


To the Reader
Chapter I. Symmetry Considerations
1. Symmetry of finite bodies
2. Notion of a symmetry group
3. Symmetry point groups
4. Crystal classes
5. Symmetry of the physical properties of crystals. The Neumann principle
6. Curvilinear anisotropy
Chapter II. Linear Elasticity
1. Structure of Hooke’s law
2. Principal axes of anisotropy
3. Bounds on the variation of the components of a positive definite symmetric matrix
4. Bounds on the variation of the elastic constants. Volumetric and shearing strains
5. Orthotropic material
6. Transversally isotropic material
7. Isotropic material
8. Reduction of the elastic moduli
9. Orthotropic cylinder under pressure
Chapter III. Nonlinear Elasticity
1. Main equations of the nonlinear theory of elasticity
2. Law of elasticity
3. Nonlinear orthotropic material
4. Transversally isotropic material
5. Isotropic material
6. Construction of the strain-energy density function
7. Hollow cylinder of incompressible material
Chapter IV. Deformation Anisotropy
1. Perturbation of the equilibrium configuration of the body
2. Deformation anisotropy of an initially isotropic material
3. Elastomers reinforced with filaments
Chapter V. Thin Anisotropic Shells under Large Deformations
1. Deformation of a shell
2. Stress resultants and moments. Boundary force values
3. Equations of motion
4. Law of elasticity
5. Nonlinear orthotropic shells
6. Nonlinear transversally isotropic shells
7. Isotropic incompressible material
8. Construction of strain-energy density functions
9. Uniform membrane state of a plate of incompressible transversally isotropic material
10. Axially symmetric deformation of a shell of revolution
Chapter VI. Reinforced Shells
1. Reinforcement in the middle surface
2. Extension of a cylindrical plate
3. Reinforcement of a cylindrical shell in the middle surface
4. Hollow conic shock-absorber reinforced in the middle surface
5. Reinforcing a shell uniformly through the thickness by inextensible cords
6. Bending of a cylindrical plate reinforced uniformly through the thickness
Chapter VII. Problems in Plane Elasticity (A Linear Approach)
1. Generalized plane strain and plane stress
2. Orthotropic material
3. Stress function
4. Use of functions of a complex variable
5. Plane containing an elliptic hole. Rectilinear cut
Chapter VIII. Problems in Plane Elasticity (A Nonlinear Approach)
1. Basic equations
2. Laws of elasticity
3. Physically linear problem
4. Plane with an elliptic hole
5. Plane with a rectilinear cut
Chapter IX. Nonlinear Theory of Cracks (Brittle Fracture)
1. A geometrically and physically nonlinear theory of cracks in an isotropic elastic material
2. On the nonlinear theory of cracks in an anisotropic material
Chapter X. Nonlinear Volterra’s Dislocations
1. Volterra’s dislocations in the linear theory of elasticity
2. Nonlinear dislocations and disclinations
3. Wedge disclinations
4. Continuation (a weakly compressible material)
Chapter XI. Some Aspects of Theory
1. Group tensor basis
2. Semisymmetric tensor of rank four
3. The associated basis
4. Tensor bench mark
5. Single axis in common
6. Coaxial tensors
7. Tensor functions of vector type
8. Six-dimensional space
9. Work conjugates. Objective rates of change of stress
10. Stability of an anisotropic material
11. Stability of an isotropic material
12. Structural invariants of large deformations
13. “True” measures of stresses and strains
Appendix A
Appendix B
Subject Index