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An introduction to the mathematical theory of dynamic materials. (English) Zbl 1125.74001

Advances in Mechanics and Mathematics 15. New York, NY: Springer (ISBN 978-0-387-38278-4/hbk). xviii, 181 p. (2007).
The monograph includes six chapters, four appendices and a subject index, and deals with some topics in the physical response of dynamic materials. Its contents are essentially a collection of the author’s papers in the field of dynamic materials. The author introduces the concept of dynamic or time-dependent materials in mechanical and electromagnetic context in order to study varying environmental conditions and changing properties, structures or compositions of materials both in space and time.
Chapter 1 (pp. 1–16): “A general concept of dynamic materials” introduces dynamic materials with their definition, applications, and implementations in electronics and optics. Chapter 2 (pp. 17–49): “An activated elastic bar: effective properties”. The longitudinal vibrations of a thin elastic bar are discussed, and effective material parameters of a laminate are defined in one spatial dimension and time, in order to illustrate the concept of activated dynamic material. Chapter 3 (pp. 51–89 ): “Dynamic materials in electrodynamics of moving dielectrics”. A brief account of the fundamentals of Maxwell’s theory is given, and the one-dimensional wave propagation in an activated dielectric laminate is studied. Chapter 4 (pp. 91–107): “\(G\)-closures of a set of isotropic dielectrics with respect to one-dimensional wave propagation”. A set of effective properties of all mixtures that are produced when the original materials become intermingled on a microscale is defined as a \(G\)-closure, and the \(G\)-closures of isotropic dielectrics are investigated with respect to one-dimensional wave propagation. Chapter 5 (pp. 109–139): “Rectangular microstructures in space-time”. In this chapter, a general rectangular microstructure is examined in one spatial dimension and time. Chapter 6 (pp. 141–161): “Some applications of dynamic materials in electrical engineering and optimal design”. A plane electromagnetic wave travelling in three-dimensional space through an activated laminate structure is studied, and a related material control problem is considered.
In total, the monograph seems to be a pioneering step in establishing the mathematical theory of dynamic materials, and without chapter problems, computer programs and design projects, which is desirable for normal classroom use, it may basically be of interest to researchers and engineers.

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74Hxx Dynamical problems in solid mechanics
74Q15 Effective constitutive equations in solid mechanics
74F15 Electromagnetic effects in solid mechanics
74Jxx Waves in solid mechanics
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