Analysis of the derivative of the energy functional with respect to the length of a curvilinear crack in an elastic body with a possible crack-edge contact

A homogeneous two-dimensional body with a crack of variable length is considered. At the crack edges, conditions are formulated in the form of inequalities that describe mutual nonpenetration of the edges. The derivative of the elastic-energy functional with respect to the length of the curvilinear crack is analyzed. It is shown that the derivative is independent of the crack path, provided that the curve along which the crack propagates is reasonably smooth.