Bifurcation in periodic arrays of growing cracks. 2-D analytical solutions

The existence of bifurcation in the process of stable growth of cracks arranged in collinear periodic arrays is shown for two types of loading: (a) the uniform tensile load and (b) pairs of concentrated forces of equal magnitudes applied at the centre of each crack. It is assumed that the cracks grow simultaneously in a controllable manner such that the condition of crack growth is always satisfied. The bifurcation analysis is conducted to check the sensitivity of the simultaneous crack growth with respect to small disturbances in crack lengths. The analysis leads to an infinite system of singular integral equations. The application of discrete Fourier transform reduces the infinite system to a single parametric singular integral equation with the same kernel as would be for the case of periodic crack array. The analysis shows that the bifurcation is only possible when the crack lengths are disturbed symmetrically (both crack ends shift at the same distance in opposite directions). It happens at the ratios between crack semi-lengths to the intercrack spacing equal to 0.38 for uniform loading and 0.44 for concentrated forces. Thus bifurcation occurs at the crack lengths smaller than required for crack coalescence. The results are applied to modelling of a macrocrack process zone associated with microcrack growth, fault instability caused by development and interaction of sliding zones and brittle fracture in uniaxial compression.