On the active control of crack growth in elastic media

Let S be a 2-D elastic structure submitted to a fixed boundary load f and containing a crack γ . In the framework of the linear fracture theory, a common tool used to describe the smooth evolution of γ is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith’s criterion postulates the evolution of the crack if this rate – positive measure of the singularity which depends quadratically on the displacement field – reaches a critical value. In this work, we numerically investigate whether or not this rate may be reduced by applying an additional boundary load with a support disjoint from the support of the initial load f possibly responsible of the growth. We first introduce a well-posed relaxed formulation of this optimal location problem, and then compute explicitly the variation of the relaxed energy release rate with respect to the location of the additional force and also with respect to its intensity, taken into account the contact condition on the crack lips. Numerical simulations, based on a gradient descent method permit to optimize the support and amplitude of the extra load and so to reduce significantly the energy release rate. The optimal extra force highlights the balance between the opening and the in-plane shear modes. The case of a multi-crack structure is considered as well.