Near tip fields for a stationary mode III crack between a linear elastic and an elastic power law hardening material

The asymptotic stress field near the tip of an antiplane crack lying along a planar bimaterial interface between an elastic and an elastic power law hardening material is analysed. Deformation plasticity theory is assumed in the analysis. We show that the shear stress field near the tip is of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabes8a0naaDaaaleaacaWGPbaabaGaeyySaelaaOGaaeiiaiab% gYJi+jaabccacaWGYbWaaWbaaSqabeaadaWcgaqaaiabgkHiTiaaig% daaeaacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaaaaaaakiaabcca% caWGubWaa0baaSqaaiaaicdadaWgaaadbaGaeyySaelabeaaaSqaai% aacIcacaWGPbGaaiykaaaakiaabccacaGGOaGaeqiUdeNaaiykaiaa% bccacqGHRaWkcaqGGaGaamOCamaaCaaaleqabaGaamiDamaaBaaame% aacaaIXaaabeaaliabgkHiTiaaigdaaaGccaqGGaGaamivamaaDaaa% leaacaaIXaWaaSbaaWqaaiabgglaXcqabaaaleaacaGGOaGaamyAai% aacMcaaaGccaqGGaGaaiikaiabeI7aXjaacMcacaqGGaGaey4kaSIa% aeiiaiabl+UimjaabccacqGHRaWkcaqGGaGaamOCamaaCaaaleqaba% GaamiDamaaBaaameaacaWGRbaabeaaliabgkHiTiaaigdaaaGccaqG% GaGaamivamaaDaaaleaacaWGRbWaaSbaaWqaaiabgglaXcqabaaale% aacaGGOaGaamyAaiaacMcaaaGccaqGGaGaaiikaiabeI7aXjaacMca% caqGGaGaey4kaSIaaeiiaiabl+Uimbaa!809A!\[\tau _i^ \pm {\text{ }} \sim {\text{ }}r^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {(n + 1)}}} \right. \kern-\nulldelimiterspace} {(n + 1)}}} {\text{ }}T_{0_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }}r^{t_1 - 1} {\text{ }}T_{1_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }} \cdots {\text{ }} + {\text{ }}r^{t_k - 1} {\text{ }}T_{k_ \pm }^{(i)} {\text{ }}(\theta ){\text{ }} + {\text{ }} \cdots \]for. Here r is the radial distance from the crack tip, θ is the angle measured from the interface, n is the hardening exponent, and + and — indicate the plastic and elastic regions respectively. The exponents t k are uniquely determined by n, and for k⩾1,t k+1> t k , t 1. For k≤M, where M is the largest positive integer for which (n(M+1)-M)/(n+1) < 0.5 (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaakaaabaGaamOBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa% iodacaaI0aGaamOBaiabgUcaRiaaigdaaSqabaaaaa!431D!\[\sqrt {n^2 + 34n + 1} \] + 1 + n − 1)/(n + 1),t k = (n(k + 1) − k)/(n + 1). The corresponding angular functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadsfadaqhaaWcbaGaam4AamaaBaaameaacqGHXcqSaeqaaaWc% baGaaiikaiaadMgacaGGPaaaaOGaaiikaiabeI7aXjaacMcaaaa!45AB!\[T_{k_ \pm }^{(i)} (\theta )\] are determined by the J-integral and material parameters and can be obtained completely from the asymptotic analysis. Some of the terms of stresses with k≤M may be singular. For k>M, t k can be obtained numerically, and the corresponding % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadsfadaqhaaWcbaGaam4AamaaBaaameaacqGHXcqSaeqaaaWc% baGaaiikaiaadMgacaGGPaaaaOGaaiikaiabeI7aXjaacMcaaaa!45AB!\[T_{k_ \pm }^{(i)} (\theta )\] can be obtained completely or within multiplicative constants. All the terms of stresses with k>M vanish as r α, when r→0, where α>0, for all 1<n<∞. It is important to note that although the individual terms of the stress expansion is variable separable, the resultant stress field is non-separable. The values of t 1,...,t 5 for 1<n≤20 and the first three terms of stresses for various values of n and material parameters are computed explicitly in the paper. Our analysis shows that, in the series solution for stresses in the plastic domain, the effect of the linear elastic material appears in the second or higher order terms depending on the value of n. In spite of this effect of elasticity on the higher order terms, the region of dominance of the HRR field in the plastic zone % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadkhadaahaaWcbeqaaiabgkHiTmaalyaabaGaaiikaiaaigda% aeaacaWGUbGaey4kaSIaaGymaiaacMcaaaaaaOGaamivamaaDaaale% aacaaIWaWaaSbaaWqaaiabgUcaRaqabaaaleaacaGGOaGaamyAaiaa% cMcaaaGccaGGOaGaeqiUdeNaaiykaaaa!4B3E!\[r^{ - {{(1} \mathord{\left/ {\vphantom {{(1} {n + 1)}}} \right. \kern-\nulldelimiterspace} {n + 1)}}} T_{0_ + }^{(i)} (\theta )\] may be significantly reduced compared to the corresponding region of dominance when the crack is in a homogeneous elastic power law hardening material.