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Anisotropic ElasticityTheory and Applications$
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T. T. C. Ting

Print publication date: 1996

Print ISBN-13: 9780195074475

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195074475.001.0001

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The Lekhnitskii Formalism

The Lekhnitskii Formalism

Chapter:
(p.118) Chapter 4 The Lekhnitskii Formalism
Source:
Anisotropic Elasticity
Author(s):

T. C. T. Ting

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195074475.003.0007

A two-dimensional deformation means that the displacements ui (i = 1,2,3) or the stresses σij depend on x1 and x2 only. Among several formalisms for two-dimensional deformations of anisotropic elastic materials the Lekhnitskii (1950, 1957) formalism is the oldest, and has been extensively employed by the engineering community. The Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The formalism begins with the stresses and assumes that they depend on x1 and x2 only. The Stroh formalism, to be introduced in the next chapter, starts with the displacements and assumes that they depend on x1 and x2 only. Therefore the Lekhnitskii formalism is in terms of the reduced elastic compliances while the Stroh formalism is in terms of the elastic stiffnesses. It should be noted that Green and Zerna (1960) also proposed a formalism for two-dimensional deformations of anisotropic elastic materials. Their approach however is limited to materials that possess a symmetry plane at x3=0. The derivations presented below do not follow exactly those of Lekhnitskii.

Keywords:   Airy function, Lekhnitskii formalism, compatibility conditions, eigenvalues of elastic constants, generalized plane strain, invariants, isotropic materials, monoclinic materials, reduced elastic compliances, stress function, transversely isotropic

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