Foundations of the Non-Linear Mechanics of Continua

International Series of Monographs on Interdisciplinary and Advanced Topics in Science and Engineering, Volume 1: Foundations of the Non-Linear Mechanics of Continua deals with the theoretical apparatus, principal concepts, and principles used in the construction of models of material bodies that fill space continuously. This book consists of three chapters. Chapters 1 and 2 are devoted to the theory of tensors and kinematic applications, focusing on the little-known theory of non-linear tensor functions. The laws of dynamics and thermodynamics are covered in Chapter 3.

This volume is suitable for persons who intend to do research on the development of the theory of dynamics and thermodynamics or solve specific theoretical problems on the motion of a continuous medium with finite deformations.

Contenu

Editor's Preface
Foreword
Introduction
Chapter I. Outline of Tensor Calculus

1. Curvilinear Systems of Coordinates
2. Transformation of Coordinates
3. The Concept of a Tensor and the Fundamentals of the Algebra of Tensors
4. Tensors of Second Rank
5. Tensor Functions
6. Functions of Several Tensors
7. Tensor Potential Functions
8. Differentiation of Tensors with Respect to Space Coordinates
9. The Riemann-Christoffel Tensor
10. Differentiation of Tensors with Respect to Parameters
    Chapter II. Kinematics of a Deformable Medium
11. General Features of Continuous Deformations
12. Affine Deformations
13. Regular Finite Displacements of a Continuum
14. The Deformation Tensors
15. The Vector of Rotation of the Axes of Deformation
16. Kinematic Tensors Characterizing a Deformation
17. The Differentiation of Non-Linear Tensor Functions
18. The Conditions of Compatibility
    Chapter III. Dynamic and Thermodynamic Exposition
19. Physical Considerations
20. The Concept of a Material Continuum
21. The Equations of Continuity
22. Dynamical Equations
23. The Theorem of Kinetic Energy and the Work of the Internal Surface Forces
24. Thermodynamic Systems and Cycles. The Equation of Conservation of Energy and the Concept of the Internal Energy of a System
25. The Basic Concepts and Consequences of the Second Law of Thermodynamics
26. Ideal Liquids and Gases
27. The Simplest Ideal Processes
28. Special Examples of Ideal Compressible Media
29. The Equations of Motion of Gas Mixtures with Physico-Chemical Interaction Between the Components
30. Viscous Liquids and Gases
31. The Turbulent Flow of Continuous Media
32. The Model of an Elastic Body
33. Thermoelastic Isotropic Media
34. Some Remarks on Mechanical Models for Irreversible Processes
35. Plastic, Elastic and Total Deformations
36. The Elastic Range and the Loading Surface
37. The Basic Laws in the Theories of Plastic Bodies
38. Models of Plastic Media for which the Loading Surface Possesses Singular Points
    References
    Index