ISBN: 3540637214
TITLE: Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies
AUTHOR: Guz, A.N.
TOC:

Preface to the Russian edition VII
Preface to the English translation X
Translator's preface XII
Chapter 1. Fundamentals of nonlinear solid mechanics 1
1. Essentials of tensor analysis 3
1.1 Notations and co-ordinate systems 3
1.2 Base vectors, metric tensor 5
1.3 Geometrical objects. Invariants of the second order tensor 10
1.4 Covariant differentiation. Physical components 17
1.5 Particular cases 21
2 Description of state of strain 23
2.1 Strain tensor and its invariants 23
2.2 Changes in geometrical objects 28
2.3 Definition of "tracking" loads 31
2.4 Simplifications for small deformation 33
2.5 Simplifications for small rotation angles 36
3 Description of state of stress 43
3.1 Stress tensors 43
3.2 Equations of motion. Boundary conditions 50
3.3 Virtual work of external forces 54
3.4 Simplification for small deformations 57
3.5 Feasible generalisation of the small deformations theory 58
4 Elastic solids 63
4.1 General elastic solid 63
4.2 Hyperelastic solid 65
4.3 Linear elastic anisotropic solid 73
4.4 On the development of the simplest nonlinear elasticity theory 81
5 Plastic solids 93
5.1 Deformation theory (theory of small elastoplastic deformations) 93
5.2 Flow theory (general equations) 99
5.3 Flow theory of strain-hardening solids (inversion of relationships, particular cases) 111
5.4 Theory of the perfectly plastic solid 124
6 Solids with rheological properties 129
6.1 The simplest relationships of linear theory 129
6.2 Linear theory of viscoelasticity (integral form) 135
6.3 Nonlinear theory of viscoelasticity (nonlinear creep) 140
6.4 Complex media. Constitutive equations for viscoelastoplastic solids 145
Chapter 2. Fundamentals of linearised solid mechanics 155
7 States of stress and strain 157
7.1 Principles of construction of linearised theory 157
7.2 Geometrical relationships 160
7.3 Equations of motion. Boundary and initial conditions 164
7.4 Simplifications for small deformations. Classification of linearised problem statements 166
7.5 Definition of "tracking" load 171
7.6 Uniform states 173
7.7 Feasible generalisations of the small initial deformations theory 184
8 Elastic solids 187
8.1 Compressible solids 187
8.2 Incompressible solids 194
8.3 On the analogy with the linear elasticity theory 200
8.4 Theorem on the uniqueness of solution in the linearised elasticity 204
8.5 Sufficient conditions of stability for compressible bodies 207
8.6 Sufficient conditions of stability for incompressible bodies 212
9 Nonelastic solids 217
9.1 Deformation theory (theory of small clastoplastic deformations) 217
9.2 Flow theory of strainhardening plastic solids 222
9.3 Theory of the perfectly plastic solids 231
9.4 Viscoelastic solids 235
9.5 Viscous solids. Inversion of constitutive equations 244
9.6 Viscoclastoplastic solids 247
Chapter 3. General issues of threedimensional linearised theory of deformable bodies stability (TLTDBS) 257
10 Stability criteria for deformable bodies 259
10.1 On surface and volume forces. Classification of problems 259
10.2 Stability criteria and statement of the problem for elastic bodies 261
10.3 Criterion of stability of the state of equilibrium and problem statement for plastic solids. Generalised concept of continuing loading 263
10.4 Criterion of stability of deformation. Comparative analysis of two criteria of stability 266
10.5 Stability criterion and problem statement for bodies with rheological properties 273
11 General statement of stability problem for deformable bodies 275
11.1 Representation of linearised constitutive equations for solids with rheological properties. Nonuniform initial state 275
11.2 Representation of linearised constitutive equations for solids with rheological properties. Uniform precritical state 283
11.3 General statement of stability problem 287
11.4 Application of Galerkin's method 291
12 Sufficient conditions of applicability of the static method 295
12.1 Compressible bodies. General case 295
12.2 Incompressible bodies. General case 300
12.3 "Tracking" loads applied to the whole surface of a body 303
12.4 "Tracking" loads applied to the part of body surface 305
13 Variational principles 309
13.1 Compressible bodies under "dead" loads. Particular case 310
13.2 Compressible bodies under "dead" loads. General case 312
13.3 Incompressible bodies under "dead" loads. Particular case 314
13.4 Incompressible bodies under "dead" loads. General case 315
13.5 "Tracking" loads applied to the whole surface of a body 317
13.6 "Tracking" loads applied to the part of body surface 319
14 General solutions for uniform precritical states 329
14.1 Compressible bodies. General relationships 329
14.2 Compressible bodies. Plane and anti-plane problems 332
14.3 Compressible bodies. Threedimensional problems 336
14.4 Incompressible bodies. General relationships 347
14.5 Incompressible bodies. Plane and antiplane problems 350
14.6 Incompressible bodies. Threedimensional problems 354
14.7 On the representations of solutions for one feasible generalisation of the small initial deformations theory 361
14.8 Determination of coefficients for particular types of solids 366
15 Approximate approach in threedimensional theory of stability 381
Chapter 4. Analysis of the simplest problems 387
16 Allround compression of isotropic simply connected body. Application of the integral stability criteria 389
16.1 Application of sufficient conditions of stability in the integral form to compressible bodies. Analysis of results for various types of bodies 389
16.2 Implementation of the problem statement in the differential form to compressible bodies. Analysis of results for various types of bodies 399
16.3 Application of sufficient conditions of stability in the integral form to incompressible bodies. Analysis of results for various types of bodies 404
16.4 Implementation of the problem statement in the differential form to incompressible bodies. Analysis of results for various types of bodies 408
17 Internal (structural) instability. Propertics of the basic system of simultaneous equations 413
17.1 Compressible bodies 416
17.2 Incompressible bodies 424
17.3 Bodies with rheological properties 430
17.4 Implementation of the approximate approach 433
18 Nearthesurface instability. Problems for semirestricted regions 435
18.1 Halfplane. Analysis of results for various compressible and incompressible bodies 436
18.2 Halfspace. Analysis of results for various compressible and incompressible bodies 449
19 Compression of a strip (plane strain problem) 471
19.1 Incompressible strips 474
19.2 Compressible strips 480
19.3 Asymptotic analysis of thinwalled strip 483
19.4 Approximate approach. Analysis of results for isotropic and composite strips 486
19.5 Analysis of results for a thickwalled strip 493
20 Compression of highelastic noncircular cylindrical body. Implementation of variational principles 501
20.1 Basic equations. Implementation of variational principles 502
20.2 Bar mode of stability loss. Numerical examples 508
Supplement. Exact solutions of mixed plane problems of linearised solid mechanics 517
References 547
References supplement 553
Biography 557
END
