Quasi-statically steady crack growth in rate dependent solids-one-parameter field encompassing different constraints

A finite element analysis is presented for quasi-statically steady crack growth in an elastic-viscoplastic material under Mode I, plane strain and small scale yielding conditions. The effects of material rate-sensitivity on the fields in the vicinity of the moving crack tip are examined. Our analysis employs a modified boundary layer formulation whereby the remote tractions are given by the first two-terms of elastic asymptotic stress field, characterized by K _Iand T. When the physical coordinates are scaled by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Hcaca% WFlbWaaSbaaSqaaGqaaiaa+fdaaeqaaOGaai4laiabeo8aZnaaBaaa% leaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa!3ECA!\(K_1 /\sigma _0 )^2 \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa% aaleaacaaIWaaabeaaaaa!3A07!\\sigma _0 \] is the tensile yield stress, the near-tip fields over a wide range of stress triaxialities are members of a family of self-similar solutions parameterized by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-rfaca% WFVaGaeq4Wdm3aaSbaaSqaaiaa-bdaaeqaaaaa!3B8E!\T/\sigma _0 \]. Members of this family are found to collapse into a single near-tip distribution when the physical coordinates are normalized by a characteristic length L _g, which is a significant fraction of the plastic zone length directly ahead of the crack tip. This distribution depends only on the relative crack speed given by the dimensionless number % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\V/(L_g \dot \in _0 )\] where V is the crack speed and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGGaciqb-HGioB% aacaWaaSbaaSqaaiaaicdaaeqaaaaa!39D6!\\dot \in _0 \] is the material's viscoplastic strain rate at a reference stress. Near-tip field distributions are obtained for several values of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\V/(L_g \dot \in _0 )\] and material strain rate sensitivity, m. Our results show that strong material rate sensitivity and high crack speed elevate the stress level ahead of the moving crack tip.