Matched asymptotic expansions in an elastic-creeping material: a new view on the Hui-Riedel equation

摘要

The work deals with the asymptotic stress-strain field around a crack tip, steadily propagating in a viscous material for antiplane conditions. A solution of this problem has been offered by Hui and Riedel, but with some unexpected features. In particular, the solution generally leads to an autonomous crack growth (independent on the loading state). This problem is revisiting here, using a multiscale asympotic analysis. Small scale yielding and low crack velocity are assumed. A small parameter ε, proportional to the crack growth rate, is introduced to switch from the inner solution (close to the crack tip) to the outer one (far field), using an asymptotic expansion of the solution. The outer solution is equivalent to the non linear elastic HRR field at the first order for ε=0, while the viscosity appears at the second order. Close to the crack tip, the viscous effects arise at the first order and the corresponding asymptotic field is governed the elastic field associated to the crack velocity, while the non linear term, corresponding to the nonlinear elasticity emerges at the second order. This is a basic difference with the Hui-Riedel solution where the two scale orders are merged. The matching conditions allow to link the far and close fields, and to correct the paradox whereby the crack velocity should not depend to the far field governed by the loading (except for perfect plasticity (n→∞) where the solution remains autonomous).