Stability and post-critical growth of a system of cooling or shrinkage cracks

When a system of parallel equidistant cooling cracks propagates into an elastic halfspace, it reaches at a certain depth of the cracks a critical point of instability, and the equilibrium path of the system bifurcates. Further extension of equally long cracks is unstable and impossible. The stable post-critical path consists of extension of every other crack upon further cooling, initially with a crack jump at constant temperature, while the intermediate cracks stop growing and gradually diminish their stress intensity factor until it becomes zero. This represents a second critical state at which these intermediate cracks suddenly close over a finite length at no change in temperature and at constant length of the leading cracks. Subsequently, as the cooling front further advances, the leading cracks grow at equal length until they again reach a critical state, at which every other crack stops growing, and the process in which the crack spacing doubles is repeated. In this manner, the spacing of the opened cooling cracks fluctuates around roughly the one-half value of the cooling penetration depth. The instability is determined by the sign of the second variation of the work needed to create the cracks, which leads to positive definiteness conditions for a matrix consisting of partial derivatives of the stress intensity factors with regard to crack lengths, subjected to admissibility conditions for the eigenvector of crack length increments. The first initial state of crack arrest is characterized by the vanishing of the diagonal element of the matrix, while the second critical state of crack closing is characterized by the vanishing of the determinant of this matrix. The critical states and the postcritical crack growth are calculated numerically by finite elements. The solution is applied to the cooling of a hot granite mass, the cracking of which is important for one recently proposed geothermal heat extraction scheme. The solution is also of interest for drying shrinkage cracks, especially in concrete.