Optimizing microstructure for toughness: the model problem of peeling

We consider the problem of peeling an adhesive thin film from a substrate that has a non-uniform distribution of adhesive. When the length scale of the non-uniformity is small compared to the overall dimensions of the film being peeled, it is possible to describe the overall peeling behavior with an effective adhesive strength. In this paper, we seek to find the distributions of adhesive strength at the microscale that optimize various aspects of the effective adhesive strength at the macroscale. We do so using both analytic bounds and topology optimization. We formulate the problem of peeling as a free boundary problem, and the effective strength as a maximum principle over the trajectory. For topology optimization, we replace the maximum with an integral norm, and use an adjoint method for the sensitivity. The problem of peeling may be viewed as a model problem in fracture mechanics where the crack (peel) front is confined to a plane, and thus our analysis as a first step toward studying the more general problem of optimizing microstructure for toughness.