Alternative conditions for the bifurcation of critical equilibrium in simple arrays of cracks.

Recent approximate analyses suggest that the critical equilibrium solution for an equally spaced array of cracks undergoes a bifurcation at some critical ratio of the crack length over the spacing. These results pertain to periodic, infinitesimal departures from the uniform solution of all cracks having equal length. Similar results for finite departures from the uniform solution are less easily obtained. Indeed, demonstration of such solutions is likely to involve numerical methods. By considering solutions that are periodic, it is possible to reduce the infinite crack array problem to a finite dimensional problem, thereby facilitating the numerical solution of the full non-linear set of equations. Moreover, a close look at the situation shows that the initial bifurcation has infinite wave length, and corresponds to the point separating stable and unstable regimes of crack growth. Thus, for more accurate descriptions of the crack array, the existence of non-uniform solutions might be inferred from the existence of such a point. Finally, the mathematical techniques involved in performing some of the infinite sums of influences are useful for extending the results to more accurate asymptotic descriptions of the crack stress field.