Anisotropic Elasticity: Theory and Applications

Abstract

The author states in the preface to this book that, in the early 1980s, motivated by the upsurge in research on composite materials, he embarked on anisotropic elasticity research "with little background on isotropic elasticity" and "reluctant and apprehensive in venturing into anisotropic elasticity." He need not have worried. The book under review is a masterly account of the fundamental theory of linear anisotropic elasticity and its applications, with emphasis on the two-dimensional theory. The book consists of 15 chapters. Following a brief 30-page introductory chapter on a summary of relevant results from Matrix Algebra, Chapter 2 presents the basic stress-strain laws for general anisotropic elastic materials, including classification of materials according to the number of symmetry planes. Chapter 3 is concerned with the basic theory and applications of antiplane shear deformations. It is refreshing to see this topic treated in a linear elasticity book before embarking on the considerably more-complicated plane problems. Chapter 3 discusses some very recent developments from the research literature on the anti-plane shear theory. The remainder of the book, except for the final Chapter 15, is concerned with the two-dimensional plane theory of elasticity. The well-known Lekhniskii formulation, involving a fourth-order partial differential equation for an Airy stress function, is briefly summarized in Chapter 4 (15 PP). The remaining chapters form the core of this book. The author is one of the pioneers in the use of the Stroh formalism as an alternative to the Lekhnitskii approach, and this method is described in detail in Chapters 5-7 (108 pp). As the author points out in the preface, this algebraic method was first developed by A.N. Stroh in 1958 and 1962; it has been widely used by the physics, materials science, and applied mathematics communities. The present account is the first to appear in book form, and the author clearly hopes to persuade solid mechanics researchers of its utility. A nice personal touch is provided at the end of Chapter 5, where a brief historical account, including a biography of Stroh (1926-1962), is given. Applications of the Stroh formalism to special subjects are presented in Chapters 8-12, whose contents may be surmised from the chapter headings. Topics covered include Green's functions for infinite space, half-space, and composite space; particular solutions, stress singularities, and stress decay; anisotropic materials with an elliptic boundary; anisotropic media

Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

“…where K is a transformation matrix [58]. In 3D, when we rotate the material about the third coordinate vector by an angle ‚, the transformation matrix reduces to…”

Interaction of a screw dislocation with a nano-sized, arbitrarily shaped inhomogeneity with interface stresses under anti-plane deformations

“…Let t 3 be the only non-zero traction component along the x 3 -direction on a boundary L. If s is the arc-length measured along L such that, when facing the direction of increasing s, the material is on the left-hand side, it is readily shown that [18] …”

“…For the anti-plane shear deformation of an isotropic elastic material, the two shear stress components σ 31 and σ 32 , out-of-plane displacement w = u 3 and stress function φ can be expressed in terms of a single analytic function f (z) of the complex variable z = x 1 + ix 2 as [18] σ 32 + iσ 31 = μf (z), φ + iμw = μf (z), (2.4) where f (z) = df (z)/dz. Let t 3 be the only non-zero traction component along the x 3 -direction on a boundary L. If s is the arc-length measured along L such that, when facing the direction of increasing s, the material is on the left-hand side, it is readily shown that [18] …”

A micromechanical approach of crack-induced damage in orthotropic media: Application to a brittle matrix composite

“…Despite the progresses concerning the mathematical formulation of constitutive laws and the evaluation of overall properties of engineering materials ( [29,9,10,48,19,51,34,41,21,45] and many others, especially in the 2D context), the 3D modelling of interaction between initial and damage-induced anisotropies remains a difficult task, even in the context of phenomenological models (see, for example, [26,44,16,5]). In this paper, this coupling is addressed by means of Eshelby-type homogenization techniques.…”