Regular perturbation methods for a region with a crack

The paper considers a model problem for Poisson's equation for a region containing a crack or a set of cracks under arbitrary linear perturbation. Variational formulation of the problem using smooth mapping of regions yields a complete asymptotic expansion of the solution in the perturbation parameter, which is a generalized shape derivative. This global asymptotic expansion of the solution was used to derive representations of arbitrary-order derivatives for the potential energy function, stress intensity factors, and invariant energy integrals in general form and for basis perturbations of the region (shear, tension, and rotation). The problem of the local growth of a branching crack for the Griffith fracture criterion and the linearized problem of optimal location of a rectilinear crack in a body with the energy function as a cost functional were formulated.