Boundary Integrals for Data Reconstruction on an Elastostatic Crack

The elastostatic Cauchy problem of fracture mechanics is studied in a two-dimensional bounded domain containing a crack. Given Cauchy data on the boundary of the domain, the displacement and normal stress (traction) are reconstructed on the crack. The reconstruction is done by reducing the original problem, via the elastostatic potential, to a system of integral equations to be solved for densities over the boundary of the domain and the crack. Discretization is carried out by the Nyström method using quadrature formulas adjusted for singularities manifesting at the endpoints of the crack. Tikhonov regularization is applied for the stable solution of the discretized system. The results of numerical experiments for different input data and parameters are given showing that relevant physical quantities on the crack can be stably reconstructed.