The general solution of second order effects in elastic half-space acted upon by a non-uniform shear load

Abstract

This paper is a continuation of [1]. Key words elastic half-space, shear load, second order elasticity effects, integral transform I, IntroductionIn finite elasticity theory the mathematical equations governing the deformation of an isotropic compressible elastic material are highly nonlinear. As a result, the exact solutions of the boundary value problems have been possible only in some restricted cases and, often approximate methods have been taken to obtain results. The method of successive approximations is a technique which has received considerable attention. The second order solutions include terms which are quadratic in the displacement gradients, and obtaining a particular integral in explicit form becomes a formidable task. Rivlin I21 and Green and Spratt t31 were among the first to formulate the second order theories and a comprehensive account of the method was given by Truesdell and Nollt~] and Green and Adkins tS]. Goodman and Naghdi trj have presented the use of displacement potentials for the solution of compressible second order elasticity problems and have applie d it to plane strain problems. For incompressible materials a variety of techniques for the second order theories have been formulated by Chan and Carlson tn, Selvadurai and Spencer ~8], Carroll and Mooney tg) and Lindsay ~176 Choi and Shield ttq have used the inverse deformation approach to study axisyrnmetric problems for a certain class of the second order elastic materials.By using the method of successive approximation, the displacements, stresses, etc. are expanded in a power series, in some suitable parameter, with non-zero radius of convergence. Signorini 021 and Stoppeli t~3" ~4] have discussed the results on existence and uniqueness of series solution under suitable differentiability conditions. Stoppeli, in particular, has shown that the displacement can be expanded as an absolutely convergent power series in some parameter, with

The illustration calculations of second order effects in elastic half-space acted upon by a non-uniform shear load

“…'-~) JL'--+/ jK,.,(-,, x) x' 0s 1o By substituting (1.2) into. (l.1) we can get r'z and r', , In the above B# can be found in [1], -P(r)is defined in (1.3), and T is given by (3.14) in paper [1].…”

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AbstractThis paper is a continuation of [1]. An Key words elastic half-space, shear load, second order elasticity effects, integral transform

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