  A class of non-smooth and non-convex optimization problems with penalty
constraints linked to variational inequalities (VI) is studied with respect to
its shape differentiability. The specific problem stemming from quasi-brittle
fracture describes an elastic body with a Barenblatt cohesive crack under the
inequality condition of non-penetration at the crack faces. Based on the
Lagrange approach and using smooth penalization with the Lavrentiev
regularization, a formula for the shape derivative is derived. The explicit
formula contains both primal and adjoint states and is useful for finding
descent directions for a gradient algorithm to identify an optimal crack shape
from a boundary measurement. Numerical examples of destructive testing are
presented in 2D.
