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Elastic media with microstructure II. Three-dimensional models. Transl. from the Russian. (English) Zbl 0536.73003
Springer Series in Solid-State Sciences, 44. Berlin etc.: Springer- Verlag. VIII, 272 p., 20 figs. DM 98.00; $ 28.90 (1983).
This book is the second part of a revised and updated edition of the author’s monograph which was originally published in Russian in 1975. While the first part (1982; Zbl 0527.73002) presented a self contained theory of one-dimensional models for nonlinear dispersive models of elastic continua (composite materials, polycrystals, grids and multibar systems, Cosserat continua, ionic crystals), this second volume considers the more realistic three-dimensional theory.
Like in the first volume a special attention is paid to the effects of microstructure, internal degrees of freedom, nonlocality, defects, deterministic and stochastic properties, approximate models and transition to classical elasticity in the limit. In contrast, however, nonlinear wavelike phenomena are not studied, in reason of their complexity, for the three-dimensional theory. The emphasis is rather placed on local defects and dislocations which are considered in detail.
More specifically, the present volume is made of seven chapters and four appendices which develop as follows. Chapter one is the same introduction as in the first volume. Chapter two is devoted to media with simple (i.e., monoatomic) structure. Here again the notions of quasicontinuum, elastic-energy operator and approximations of the nonlocal behavior are introduced, but in the three-dimensional framework. Media with complex structure (i.e., polyatomic models in the jargon of lattice dynamics) are the object of chapter three with a nice transition to the case of Cosserat continua.
Chapter four, devoted to local defects, is the most important one in the volume for it introduces the systematic use of the Green function technique and the fact that boundary-value problems in elasticity with inhomogeneities can be formulated in terms of a special class of global pseudo-differential operators (projection operators and Green’s operators for stress and strain). This is applied to the case of ellipsoidal inhomogeneities and cracks. A similar technique is used in the remainder of the book. Indeed, chapter five deals with the case of internal stresses and point defects (vacancies, interstitial atoms, etc.) and a general outline of the theory of internal stresses in nonlocal elasticity is proposed. Screw and edge dislocations are the object of the short chapter six.
Of great interest is the last chapter where elastic media with random fields of inhomogeneities are considered. There the method of effective fields is applied to solve problems for composites and cracked solids. In particular, formulas for first and second moments of random stress-strain fields are presented. The first moments are needed to evaluate the effective elastic moduli tensor while the second statistical moment helps one to find the numerical characteristics of fluctuations of the elastic field, and this is needed for a description of phenomena, such as fracture and plastic threshold, which depend on the fine structure of the material. Here the wellknown self-consistent method (which is the Hartree-Fock approach in the quantum theory of the atom) which owes much to E. Kröner and R. Hill for its application to continuum mechanics, is developed and applied to evaluate the random fields of ellipsoidal inhomogeneities, elliptic cracks and point defects. Appendices on fourth- order tensors, Green operators and calculations of certain conditional means and a rich and updated bibliography complete the volume. Several sections, chapters and appendices were written in collaboration with S. K. Kanaun.
As a whole this second volume complements harmoniously the first one. It will be a very useful tool to scientists working in a variety of branches of applied sciences (continuum mechanics, solid state physics, lattice dynamics, science of materials). One word of caution to mechanicians, however, since, like the first volume, this one borrows much to the formalism and language of quantum mechanics, a fact which may be discouraging at the start, but which proves to be rewarding in the end.
Reviewer: G.A.Maugin

MSC:
74A35 Polar materials
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74A40 Random materials and composite materials
74A60 Micromechanical theories
74M25 Micromechanics of solids
74J20 Wave scattering in solid mechanics
82D25 Statistical mechanics of crystals
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
74R05 Brittle damage
74R99 Fracture and damage
74E05 Inhomogeneity in solid mechanics
74E30 Composite and mixture properties