Bifurcation in Growth Patterns for Arrays of Parallel Griffith, Edge and Sliding Cracks

摘要

This paper presents the recent finding by Muhlhaus et al [1] that bifurcation of crack growth patterns exists for arrays of two-dimensional cracks. This bifucation is a result of the nonlinear effect due to crack interaction, which is, in the present analysis, approximated by the "dipole asymptotic" or "pseudo-traction" method. The nonlinear parameter for the problem is the crack length/ spacing ratio #lambda# = a/h. For parallel and edge crack arrays under far field tension, uniform crack growth patterns (all cracks having same size) yield to nonuniform crack growth patterns (i.e. bifurcation) if #lambda# is larger than a critical value #lambda#_(cr) (note that such bifurcation is not found for collinear crack arrays). For parallel and edge crack arrays respectively, the value of #lambda#_(cr) decreases monotonically from (2/9)~(1/2) and (2/15.96)~(1/2) for arrays of 2 cracks, to (2/3)~(1/2)#pi# and (2/5.032)~(1/2)/#pi# for infinite arrays of cracks. The critical parameter #lambda#_(cr) is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain the exact solution of #lambda#_(cr) for infinite crack arrays. For geomaterials, bifurcation can also occurs when array of sliding cracks are under compression.