Relativistic Theories of Materials

The theory of relativity was created in 1905 to solve a problem concerning electromagnetic fields. That solution was reached by means of profound changes in fundamental concepts and ideas that considerably affected the whole of physics. Moreover, when Einstein took gravitation into account, he was forced to develop radical changes also in our space-time concepts (1916). Relativistic works on heat, thermodynamics, and elasticity appeared as early as 1911. However, general theories having a thermodynamic basis, including heat conduction and constitutive equations, did not appear in general relativity until about 1955 for fluids and appeared only after 1960 for elastic or more general finitely deformed materials. These theories dealt with materials with memory, and in this connection some relativistic versions of the principle of material indifference were considered. Even more recently, relativistic theories incorporating finite deformations for polarizable and magnetizable materials and those in which couple stresses are considered have been formulated. A broader description of the development of these relativistic topics is contained in § 13. The purpose of this book is to describe the foundations of the general relativistic theories that include constitutive equations, and to present some applications, mainly to elastic waves, of these theories. This tract is divided into two parts. In the first part only the Eulerian point of view is considered; basic equations of general relativity, other than constitutive equations, are stated in full generality (except for couple stresses which are considered in part 2). Part 1 also thoroughly covers fluids, including constitutive equations.

Inhalt

1. Introduction.- § 1. On the Beginning of Relativity.- § 2. The Space-Time Structure of Special Relativity and First Basic Consequences.- § 3. On the Operational Aspect of Physical Concepts.- § 4. New Ideas on Mass and Energy, in Contrast with Classical Physics, Accepted on the Basis of Special Relativity Kinematics.- § 5. On Forces, Cauchy Equations of Continuous Media, and the First Principle of Thermodynamics in Special Relativity.- § 6. On Electromagnetism, Heat Conduction, and Constitutive Equations in Special Relativity.- § 7. Gravitation and Relativity.- § 8. On the Local Equivalence Principle and the Basic Local Laws of the Electromagnetic Field and Continuous Media, Other than the Poisson Equation, in General Relativity. A Criterion Connecting those Laws with Their Analogues in Classical Physics or Special Relativity.- § 9. On the Invariance of Physical Equations and on the Possible Physical Equivalence of the Frames in which these Equations have the Same Form. On a Privileged Absolute Concept of Event Point.- § 10. On Harmonic Coordinates and the Existence of General Frames not Physically Equivalent in General Relativity.- § 11. Some Distinctive Properties of General Relativity. On the Equivalence of General Frames in General Relativity.- § 12. What We Mean by General Theory of Relativity.- § 13. On the Development of General Relativity. Inclusion of Elasticity, Electromagnitostriction, Couple Stresses, and Hereditary Phenomena.- § 14. Scope and Plan of the Present Tract.- Footnotes to Chapter 1.- I. Basic Equations of Gravitation, Thermodynamics and Electromagnetism, and Constitutive Equations from the Eulerian Point of View.- 2. Space-Time Kinematics Including Masses.- 3. Gravitation and Conservation Equations. Fluids and Elastic Waves.- 4.Electromagnetism from the Eulerian Point of View. Polarizable Fluids.- 5. On Media Capable of Electromagnetic Phenomena from the Eulerian Point of View. Magneto-Elastic Waves in Ideal Conductors.- II. Materials from the Lagrangian Point of View.- 6. Kinematics and Stresses from the Lagrangian Point of View.- 7. Elasticity, Acceleration Waves, and Variational Principles for Simple Materials.- 8. Piezo-Elasticity and Magnetoelastic Waves from the Lagrangian Point of View.- 9. Materials with Memory and Axiomatic Foundations.- 10. Couple Stresses and More General Stresses.- Appendix A. Double Tensors.- §A1. Definition of Double Tensors Related to Two Topological Spaces.- §A2. Partial Covariant Derivative and Total Covariant Derivative Based on a Mapping.- §A3. On Differentiation of Double Tensors, Functions of Double Tensors.- Case of Arguments Fulfilling Typical Regular Conditions.- Appendix C. On the Divergence of Spatial Vectors in Space-Time.- References.