Nonlinear Continuum Mechanics of Solids

by Basar, Yavuz; Weichert, Dieter

• ISBN13: 9783540666011
• ISBN10: 354066601X
• Format: Hardcover
• Copyright: 2000-03-01
• Publisher: Springer Verlag

Summary

The aim of the book is the presentation of the fundamental mathematical and physical concepts of continuum mechanics of solids in a unified description so as to bring young researchers rapidly close to their research area. Accordingly, emphasis is given to concepts of permanent interest, and details of minor importance are omitted. The formulation is achieved systematically in absolute tensor notation, which is almost exclusively used in modern literature. This mathematical tool is presented such that study of the book is possible without permanent reference to other works.

Table of Contents

Mathematical fundamentals	1	(42)
Simple tensors	1	(2)
General tensors	3	(4)
Special tensors	7	(8)
Orthogonal tensors	15	(1)
Spherical tensor, deviatoric tensor	16	(1)
Differential operators	17	(2)
Differentiation rules	19	(3)
Invariants of a second-order tensor	22	(1)
The eigenvalue problem of a second-order tensor	23	(6)
Rotation tensor, rotation vector	29	(4)
Analytical solution of eigenvalue-problems	33	(2)
Tensor functions on the basis of power series	35	(1)
Exponential, skew-symmetric tensors	36	(2)
Summary of notations and formulae	38	(5)
Exercises	41	(2)
Deformation	43	(56)
General backgrounds	43	(4)
Deformation gradient	47	(7)
Deformation gradient in material and spatial coordinates	54	(4)
Polar decomposition	58	(8)
Green-Lagrange strain tensor, Almansi strain tensor	66	(5)
Eigenvectors and eigenvalues of deformation variable	71	(7)
Unified definitions of strain tensors	78	(3)
Isochoric and volumetric deformations	81	(1)
Rate of deformation tensor and spin tensor	82	(6)
Pull-back and push-forward operations	88	(4)
Isotropic tensor functions of C and b	92	(7)
Exercises	97	(2)
Stresses	99	(16)
Cauchy Stress tensor	99	(5)
Stress tensors	104	(5)
Energy conjugate stress and strain variables	109	(3)
Summary of important definitions	112	(3)
Exercises	113	(2)
Time derivative	115	(8)
Definitions	115	(2)
Velocity and acceleration	117	(1)
Examples for material time derivative	118	(5)
Exercises	122	(1)
Balance laws	123	(16)
Conservation of mass	123	(1)
Balance of momentum	124	(3)
Balance of moment of momentum	127	(2)
Balance of kinetic energy	129	(2)
Conservation of energy	131	(3)
Principle of virtual work	134	(5)
Exercises	137	(2)
Constitutive modelling	139	(36)
General principles	139	(3)
Objective tensors	142	(2)
Elastic material	144	(2)
Isotropic elastic material	146	(2)
Derivatives of scalar-valued functions	148	(4)
Hyperelastic material or Green-elastic materials	152	(4)
Isotropic hyperelastic material	156	(3)
Special constitutive models for isotropic hyperelasticity	159	(4)
St.Venant-Kirchhoff material	163	(3)
Hookean material	166	(4)
Linearization and comparison of various material models	170	(5)
Exercises	174	(1)
Appendix 1	175	(10)
A1.1 Index notation	175	(1)
A1.2 Metric tensor and geometrical properties	176	(3)
A1.3 Vector decompositions, tensor components of first order	179	(1)
A1.4 Definition of higher-order tensor components	179	(2)
A1.5 Permutation tensor	181	(1)
A1.6 Christoffel symbols, covariant differentiation	182	(3)
References	185	(6)
Index	191