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

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 bifurcation 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 λ = 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 λ is larger than a critical value λcr (note that such bifurcation is not found for collinear crack arrays). For parallel and edge crack arrays respectively, the value of λcr decreases monotonically from (2/9)1/2 and (2/15.096)1/2 for arrays of 2 cracks, to (2/3)1/2/π and (2/5.032)1/2/π for infinite arrays of cracks. The critical parameter λcr is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain the exact solution of λcr for infinite crack arrays. For geomaterials, bifurcation can also occurs when array of sliding cracks are under compression.