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Deconstructing plane anisotropic elasticity. I: The latent structure of Lekhnitskii’s formalism. (English) Zbl 0988.74012

A linear homogeneous anisotropic elastic body is said to be in a generalized plane-strain state with respect to \((x_1,x_2)\) plane, if behavior of the body is represented by an ordered array of functions \([u,E,S]\) that depend on \((x_1,x_2)\) and satisfy the fundamental system of field equations: (i) \(E=\widehat \nabla u\), (ii) \(\text{div} S=0\), and (iii) \(E=K [S]\) on \((x_1,x_2)\) plane. Here, \(u,E\), and \(S\) denote the displacement, strain, and stress fields, respectively, such that \(E_{33}= S_{33}=0\), and \(K\) is a symmetric compliance matrix containing 15 independent material constants. By postulating \(u\) in the form \(u=af (x_1+px_2)\), where \(a\) is a vector and \(p\) is a scalar, and \(f(z)\) is a prescribed function, the author reduces the problem of finding a class of particular generalized plane-strain states to the study of an eigenvalue problem for a pair \((a,p)\). Depending on the nature of eigensolution \((a,p)\), five distinct types of anisotropic materials are defined: (i) normal material with three simple eigenvalues; (ii) normal material with one simple and one double eigenvalue; (iii) normal material with one triple eigenvalue; (iv) abnormal material with one simple and one double eigenvalue, and (v) abnormal material with one triple eigenvalue. Also, for each of the five materials the author derives generalized plane-strain state formulas. An application of the method to a concrete physically sound boundary value problem for anisotropic body subject to generalized plane-strain conditions is, however, not given.

MSC:

74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics
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