Kružík M., Roubíček T. Mathematical Methods in Continuum Mechanics of Solids

Springer, 2019. — 624 p. — (Interaction of Mechanics and Mathematics). — ISBN: 3030020649.This book primarily focuses on rigorous mathematical formulation and treatment of static problems arising in continuum mechanics of solids at large or small strains, as well as their various evolutionary variants, including thermodynamics. As such, the theory of boundary- or initial-boundary-value problems for linear or quasilinear elliptic, parabolic or hyperbolic partial differential equations is the main underlying mathematical tool, along with the calculus of variations. Modern concepts of these disciplines as weak solutions, polyconvexity, quasiconvexity, nonsimple materials, materials with various rheologies or with internal variables are exploited.
This book is accompanied by exercises with solutions, and appendices briefly presenting the basic mathematical concepts and results needed. It serves as an advanced resource and introductory scientific monograph for undergraduate or PhD students in programs such as mathematical modeling, applied mathematics, computational continuum physics and engineering, as well as for professionals working in these fields.Description of Deformable Stressed Bodies
Elastic Materials
Polyconvex Materials: Existence of Energy-Minimizing Deformations
General Hyperelastic Materials: Existence/Nonexistence Results
Linearized Elasticity
Linear Rheological Models at Small Strains
Nonlinear Materials with Internal Variables at Small Strains
Thermodynamics of Selected Materials and Processes
Evolution at Finite Strains
Suggestions for further reading and some open problems
Appendixes
Elements of abstract functional analysis
Lebesgue and Sobolev function spaces
Appendix: Elements of modern theory of differential equations
Parametrized measures
Solutions or hints for selected exercises