‘When can we apply LEFM principles to elastic softening materials?’

Abstract

The paper focusses on the determination of R, the size of the fully developed softening zone associated with a semi-infinite crack in a remotely loaded infinite elastic softening solid. R is a characteristic length for a material, and is important in that if R is less than an appropriate characteristic dimension of a structure, then LEFM principles can be used to describe the structure's failure. With p _c and δ_c being respectively the maximum stress and displacement within the softening zone, then provided the softening is not particularly pronounced, i.e. the area under the stress (p)-displacement (v) curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH+aGpaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB5!\[\widetilde > {\text{ 0}}{\text{.25}}pc\delta c\], it is shown that R ∼ 0.4E ₀δ_c/P _c and R is relatively insensitive to the precise p-v softening behaviour (E ₀ = E/(1 − v ²) where E is Young's modulus and ν is Poisson's ratio. However, when the area under the curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH8aapaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB1!\[\widetilde < {\text{ 0}}{\text{.25}}pc\delta c\], then R increases above this 0.4E ₀δ_c/P _c value. For this case, and provided most of the area under the p-v curve is not associated with the tail in the softening law, a more appropriate expression for R is R ∼ 0.1E ²₀ /δ ²₀ /K ²_∞ , with K ²_∞ /E ₀ being the area under the p-v curve and K _∞ being the stress intensity associated with the full development of a softening zone.