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

This monograph focuses on the modeling of continua that exhibit the property of singularities in fields. A rational theory of continuum thermomechanics that takes into account the existence and distribution of scalar and vector discontinuities is presented. Guiding the work is the notion that discontinuities should be modeled by the torsion and curvature tensors (Cartan), thus forming an attractive alternative way of representing dissipating continua with singularities. Also emphasized in this study is the relationship between singularities and certain irreversible properties that occur in solid and fluid models. The required mathematical background is given in several appendices, although some basic understanding of differential geometry and tensor analysis would be helpful. The book is largely self-contained and should serve a readership of graduate students, researchers, and engineers in applied mathematics, mechanics, and physics.

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.)