13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Matched asymptotic expansions in an elastic-creeping material : a new view on the Hui-Riedel equation. Radhi ABDELMOULA1,* Gilles DEBRUYNE2 , Jia LI1 1 Paris XIII University, LSPM ,Avenue J.B. Clément 93430 Villetaneuse, France 2 LaMSID-EDF-CEA, 1 avenue du Gal de Gaulle 92141 Clamart, France * Corresponding author: , radhi.abdelmoula@univ-paris13.fr Abstract 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 , 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 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). Keywords Hui-Riedel solution, creep, steadily growing crack, singularity, matched asymptotic expansions. 1. Introduction An antiplane asymptotic solution for a steadily slowly growing crack in an elastic-non linear viscous medium has been suggested by Hui and Riedel [1]. A power law creep is considered. For uniaxial tension, the Norton law has the following form : n B (1) For 3 n some paradox events arise, in particular the solution is autonomous, independent on the remote loading (this phenomenon is described by Bui as an analogy to the “soliton” in non linear waves problems [2]). Some authors have corrected this paradox, but with substantive changes to the law or introducing a threshold [3,4]. Keeping the original Norton law, a new antiplane shear analysis is offered here, using a matched asymptotic expansion method. Higher expansion terms of the stress function will allow to connect the « inner solution » (near the viscous crack tip) to the « outer solution » (corresponding to the far HRR field).
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