A Geometric Approach to Thermomechanics of Dissipating Continua: (31 Progress in Mathematical Physics)

Across the centuries, the development and growth of mathematical concepts have been strongly stimulated by the needs of mechanics. Vector algebra was developed to describe the equilibrium of force systems and originated from Stevin's experiments (1548-1620). Vector analysis was then introduced to study velocity fields and force fields. Classical dynamics required the differential calculus developed by Newton (1687). Nevertheless, the concept of particle acceleration was the starting point for introducing a structured spacetime. Instantaneous velocity involved the set of particle positions in space. Vector algebra theory was not sufficient to compare the different velocities of a particle in the course of time. There was a need to (parallel) transport these velocities at a single point before any vector algebraic operation. The appropriate mathematical structure for this transport was the connection. I The Euclidean connection derived from the metric tensor of the referential body was the only connection used in mechanics for over two centuries. Then, major steps in the evolution of spacetime concepts were made by Einstein in 1905 (special relativity) and 1915 (general relativity) by using Riemannian connection. Slightly later, nonrelativistic spacetime which includes the main features of general relativity I It took about one and a half centuries for connection theory to be accepted as an independent theory in mathematics. Major steps for the connection concept are attributed to a series of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.

Table of Contents:
Geometry and Kinematics.- 2.1 Introduction to continuum motion.- 2.2 Geometry of continuum.- 2.3 Discontinuity of fields on continuum.- 2.4 Deformation of continuum.- 2.5 Kinematics of continuum.- Conservation Laws.- 3.1 Introduction.- 3.2 Boundary actions and Cauchy’s theorem.- 3.3 Conservation laws.- Continuum with Singularity.- 4.1 Introduction.- 4.2 Continuum with singularity of the rate type.- 4.3 Operators on continuum with singularity.- 4.4 General equations of continuum.- Thermoviscous Fluids.- 5.1 Fluids without singularity distribution.- 5.2 Fluids with singularity distribution.- 5.3 Overview of fluid-like models.- Thermoviscous Solids.- 6.1 Solids without singularity distribution.- 6.2 Solids with singularity distribution.- 6.3 Intermediate configurations.- 6.4 Overview of solid-like models.- 6.5 Elastic waves in nonclassical solids.- Solids with Dry Microcracks.- 7.1 Geometry.- 7.2 Kinematics.- 7.3 Conservation laws.- 7.4 Constitutive laws at the crack interface.- 7.5 Concluding remarks.- Conclusion.- A Mathematical Preliminaries.- A.1 Vectors and tensors.- A.1.1 Vector, space, basis.- A.1.2 Linear maps and dual vector spaces.- A.1.3 Tensors, tensor product.- A.2 Topological spaces.- A.2.1 Topological spaces.- A.2.2 Continuous maps.- A.2.3 Compactness.- A.2.4 Connectedness.- A.2.5 Homeomorphisms and topological invariance.- A.3 Manifolds.- A.3.1 Definition of manifold.- A.3.2 Tangent vector.- A.3.3 Tangent dual vector.- A.3.5 Mappings between manifolds.- B Invariance Group and Physical Laws.- B.1 Conservation laws and invariance group.- B.1.1 Newton spacetime.- B.1.2 Leibniz spacetime.- B.1.3 Galilean spacetime.- B.1.4 Physical roots of conservation laws.- B.2 Constitutive laws and invariance group.- B.2.1 Spacetime of Cartan.- B.2.2 Objectivity (frameindifference) of constitutive laws.- C Affinely Connected Manifolds.- C.1 Riemannian manifolds.- C.1.1 Metric tensor.- C.2 Affine connection.- C.2.1 Metric connection, Levi-Civita connection.- C.2.2 Affine connections.- C.2.3 Covariant derivative of tensor fields.- C.3 Curvature and torsion.- C.3.1 Lie-Jacobi bracket of two vector fields.- C.3.2 Exterior derivative.- C.3.3 Poincaré Lemma.- C.3.4 Torsion and curvature.- C.3.5 Holonomy group.- C.4.1 Orientation on connected manifolds.- C.4.4 Stokes’ theorem.- C.5 Brief history of connection.- D Bianchi Identities.- D.1 Skew symmetry.- D.2 First identities of Bianchi.- D.3 Second identities of Bianchi.- E Theorem of Cauchy-Weyl.- E.1 Theorem of Cauchy (1850).- E.2 Theorem of Cauchy-Weyl (1939).- References.

Review :
"In my opinion, the book is excellent: it is well structured, well focused on an interesting topic, and it is clearly developed. It combines mathematical rigor with deeply physical motivations, many of them of much current interest in material sciences or in the fundamentals of thermodynamics. The proposals of the author seem to be a worthwhile contribution to a mathematically sound and physically fruitful description of many intresting phenomena." - Zentralblatt MATH "By its very nature and subject matter the book will have a specialized audience.  To those happy few this unique book is warmly recommended as it will certainly initiate discussions and further extensions."(Mathematical Reviews, Maugin, Gérard A.)