Crack nucleation due to bending of a variable-thickness band

A mathematical model of crack nucleation in a band (beam) bent in its plane by a given system of external loads (constant bending moments, uniformly distributed pressure, etc.) is constructed. Pre-fracture zones modeled as zones of attenuated interparticle bonds of the material are assumed to appear in the variable-thickness band in the course of its loading. The solution of the problem of equilibrium of an isotropic band of variable thickness with a nucleating crack is reduced to the solution of a nonlinear singular integrodifferential equation with a kernel of the Cauchy kernel type, which yields the forces in the zone of crack nucleation. The condition of crack nucleation in the variable-thickness band is formulated with allowance for the criterion of critical tension of the material bonds.