Foundations of the Non-Linear Mechanics of Continua (eBook)

Front Cover 1
Foundations of the Non-Linear Mechanics of Continua 4
Copyright Page 5
Table of Contents 6
EDITOR'S PREFACE 8
FOREWORD 10
INTRODUCTION 14
CHAPTER I. OUTLINE OF TENSOR CALCULUS 18
1. CURVILINEAR SYSTEMS OF COORDINATES 18
2. TRANSFORMATION OF COORDINATES 24
3. THE CONCEPT OF A TENSOR AND THE FUNDAMENTALS OF THE ALGEBRA OF TENSORS 27
4. TENSORS OF SECOND RANK 35
5. TENSOR FUNCTIONS 47
6. FUNCTIONS OF SEVERAL TENSORS 62
7. TENSOR POTENTIAL FUNCTIONS 66
8. DIFFERENTIATION OF TENSORS WITH RESPECT TO SPACE COORDINATES 70
9. THE RIEMANN-CHRISTOFFEL TENSOR 74
10. DIFFERENTIATION OF TENSORS WITH RESPECT TO PARAMETERS 78
CHAPTER II. KINEMATICS OF A DEFORMABLE MEDIUM 90
1. GENERAL FEATURES OF CONTINUOUS DEFORMATIONS 90
2. AFFINE DEFORMATIONS 93
3. REGULAR FINITE DISPLACEMENTS OF A CONTINUUM 98
4. THE DEFORMATION TENSORS 101
5. THE VECTOR OF ROTATION OF THE AXES OF DEFORMATION 110
6. KINEMATIC TENSOBS CHABACTEBIZING A DEFOBMATION 113
7. THE DIFFERENTIATION OF NON-LINEAR TENSOR FUNCTIONS 116
8. THE CONDITIONS OF COMPATIBILITY 122
CHAPTER III. DYNAMIC AND THERMODYNAMIC EXPOSITION 126
1. PHYSICAL CONSIDERATIONS 126
2. THE CONCEPT OF A MATERIAL CONTINUUM 129
3. THE EQUATIONS OF CONTINUITY 131
4. DYNAMICAL EQUATIONS 134
5. THE THEOREM OF KINETIC ENERGY AND THE WORK OF THE INTERNAL SURFACE FORCES 146
6. THERMODYNAMIC SYSTEMS AND CYCLES. THE EQUATION OF CONSERVATION OF ENERGY AND THE CONCEPT OF THE INTERNAL ENERGY OF A SYSTEM 148
7. THE BASIC CONCEPTS AND CONSEQUENCES OF THE SECOND LAW OF THERMODYNAMICS 156
8. IDEAL LIQUIDS AND GASES 163
9. THE SIMPLEST IDEAL PROCESSES 172
10. SPECIAL EXAMPLES OF IDEAL COMPRESSIBLE MEDIA 176
11. THE EQUATIONS OF MOTION OF GAS MIXTURES WITH PHYSICO-CHEMICAL INTERACTION BETWEEN THE COMPONENTS 181
12. VISCOUS LIQUIDS AND GASES 199
13. THE TURBULENT FLOW OF CONTINUOUS MEDIA 202
14. THE MODEL OF AN ELASTIC BODY 205
15. THERMOELASTIC ISOTROPIC MEDIA 221
16. SOME REMARKS ON MECHANICAL MODELS FOR IRREVERSIBLE PROCESSES 228
17. PLASTIC, ELASTIC AND TOTAL DEFORMATIONS 232
18. THE ELASTIC RANGE AND THE LOADING SURFACE 239
19. THE BASIC LAWS IN THE THEORIES OF PLASTIC BODIES 244
20. MODELS OF PLASTIC MEDIA FOR WHICH THE LOADING SURFACE POSSESSES SINGULAR POINTS 255
REFERENCES 261
INDEX 266