Inverse problems of inhomogeneous fracture toughness using phase-field models

We propose inverse problems of crack propagation using the phase-field models. First, we study the crack propagation in an inhomogeneous media in which fracture toughness varies in space. Using the two phase-field models based on different surface energy functionals, we perform simulations of the crack propagation and show that the $J$-integral reflects the effective inhomogeneous toughness. Then, we formulate regression problems to estimate space-dependent fracture toughness from the crack path. Our method successfully estimates the positions and magnitude of tougher regions. We also demonstrate that our method works for different geometry of inhomogeneity.