×

Nonlocal continuum field theories. (English) Zbl 1023.74003

New York, NY: Springer. xvi, 376 p. (2002).
This interesting book covers a gap in the literature on continuum field theories. It contains and updates the own research of the author and his collaborators for almost the last forty years. Nonlocal field theories are concerned with physics of material bodies whose behaviour at any interior point depends on the state of all other points in the body. They are natural extensions of classical field theories, constructing constitutive equations which describe the responses of points within the medium. The purpose of the book is to present a unified approach to the formulations of nonlocal field theories for memory-dependent elastic solids, viscous fluids, electromagnetic solids, fluids that conduct heat, nonlocal microcontinua and liquid crystals.
The book is divided into fithteen chapters. It also includes an appendix containing a derivation of Riemann-Christoffel curvature tensor, and the Laplacian of stress tensor in coordinates. In the first chapter, the author introduces some basic notions on motion and deformation. Using the axiom of Galilean invariance, nonlocal balance laws are obtained in chapter 2. Chapter 3 discusses constitutive equations constructed using eight axioms as was postulated by the author in a previous work. In chapter 4, the interaction of electromagnetism with deformable bodies is studied, and the corresponding nonlocal balance laws are obtained. The constitutive equations for electromagnetic thermoelastic solids and electromagnetic thermoviscous fluids are constructed in chapter 5. Chapter 6 is probably one of the most interesting chapters of the book. The author discusses nonlocal linear elasticity, dislocations, interactions between defects and dislocations, etc. The theory of nonlocal fluid dynamics (including constitutive equations, field equations, problems of lubrication) is considered extensively in chapter 7. Chapter 8 come backs again to nonlocal electromagentic theory, including a general theory of superconductivity. The next three chapters are devoted to the study of memory-dependent nonlocal thermoelastic solids, thermofluids, electromagnetic elastic solids, and electromagnetic thermofluids, respectively. Chapter 13 is concerned with the author’s theory of nonlocal microcontinua, including variational principles. The field equations of memory-dependent nonlocal micropolar electromagnetic elastic solids are obtained in chapter 14, and, finally, the nonlocal electromagnetic continuum theory of liquid crystals is presented in chapter 15. Every chapter ends with an interesting collection of problems.
The reviewer would like to quote some important references which are not included in the bibliography, and that could the reader to get additional information: [G. Capriz, Continua with microstructure, Springer Tracts in Natural Philosophy, 35. New York etc.: Springer-Verlag (1989; Zbl 0676.73001), and G. A. Maugin, Material inhomogeneities in elasticity. Applied Mathematics and Mathematical Computation. 3. London: Chapman & Hall. xii (1993; Zbl 0797.73001)].

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
74Axx Generalities, axiomatics, foundations of continuum mechanics of solids
76A02 Foundations of fluid mechanics
82Dxx Applications of statistical mechanics to specific types of physical systems
PDFBibTeX XMLCite
Full Text: DOI