Bifurcation of crack pattern in arrays of two-dimensional cracks

Abstract

Theoretical calculations based on simple arrays of two-dimensional cracks demonstrate that bifurcation of crack growth patterns may exist. The approximation used involves the ‘dipole asymptotic’ or ‘pseudo-traction’ method to estimate the local stress intensity factor. This leads to a crack interaction parametrized by 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 (bifurcation) if λ is larger than a critical value λcr. However, no such bifurcation is found for a collinear crack array under tension. 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 λcr for infinite crack arrays. For infinite parallel crack arrays under uniaxial compression, a simple shear-induced tensile crack model is formulated and compared to the modified Griffith theory. Based upon the model, λcr can be evaluated numerically depending on μ (the frictional coefficient) and c 0/a (c 0 and a are the sizes of the shear crack and tensile crack, respectively). As an iterative method is used, no closed form solution is presented. However, the numerical calculations do indicate that λcr decreases with the increase of both μ and c 0/a.