Bounds on stress concentration factors in finite anti-plane shear

Abstract

This paper provides an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous, isotropic, incompressible materials. The problem of an infinite slab with traction-free circular cavity subject to a state of f'mite simple shear deformation is considered. Explicit estimates are obtained for the maximum shearing stress in terms of the applied stress at intrmity and constitutive parameters. The analysis is based on application of maximum principles for second-order quasilinear uniformly elliptic equations.

Stress concentration factors in finite anti-plane shear: numerical calculations and analytical estimates

“…Our purpose in this note is to assess the accuracy of the analytical estimates obtained in [1,2] by comparing these with the results of direct numerical computation. Stress concentration factors are computed using the APE computer program developed by one of the authors [3,6].…”

“…For the problem of an infinite slab, with a traction-free circular or elliptical cavity, subject to a state of finite simple shear deformation, explicit estimates for the maximum shearing stress were obtained in terms of the cavity geometry, applied stress at infinity and constitutive parameters. The analysis in [1,2] is based on application of maximum (or comparison) principles for the governing second-order uniformly elliptic partial differential equation (see equation (2.4) …”

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In two recent papers [1,2], an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous isotropic incompressible materials was presented. For the problem of an infinite slab, with a tractionfree circular or elliptical cavity, subject to a state of finite simple shear deformation, explicit estimates for the stress concentration factor were obtained in terms of the cavity geometry, applied stress at infinity and constitutive parameters.

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Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes

“…As discussed in [4], apart from their intrinsic interest, crack problems in anti-plane shear (mode III) provide valuable guidance for investigation of the more complicated plane problems (modes I, II). Stress concentration problems in finite anti-plane shear have also been investigated [5][6][7], A precise characterization of anti-plane shear deformations within the framework of three-dimensional finite elastostatics has been given by Knowles for both incompressible [8] and compressible [9] isotropic materials. As pointed out in [8,9], not all nonlinearly elastic materials can sustain finite anti-plane shear deformations in a cylinder of arbitrary cross section.…”

“…When r = R, the corresponding stresses follow from (2.15)-(2.19) as Trr = lo + fit+P-^ + l), (3)(4)(5) 1ee ~ Po + P\ + ^-i' (3-6) In what follows, it will be convenient for us to consider the implications of differentiating both sides of (3.4) with respect to R . Performing this operation and using the chain rule, we obtain Since W is assumed sufficiently smooth, it follows from (3.14) that + W2 does not change sign on A < R < B and so, without loss of generality, we take Wl + W2> 0 on A<R<B.…”