Bifurcation of crack pattern in arrays of two-dimensional cracks

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 λ_cr. However, no such bifurcation is found for a collinear crack array under tension. For parallel and edge crack arrays, respectively, the value of λ_cr 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 λ_cr is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain λ_cr 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, λ_cr can be evaluated numerically depending on μ (the fractional coefficient) and c₀/a (c₀ 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 λ_cr decreases with the increase of both μ and c₀/a.