Gauge model of a crack

Abstract

The paper proposes a model of a crack system as local symmetry breaking for a group of 3D rotations compensated by fictitious fields that make Lagrangian equations of elastic energy density covariant in the effective Riemannian space. Equilibrium equations are solved using the perturbation theory with the number of notches being a small parameter, and exact expressions are derived for stress and gauge fields in a 2D problem by applying Hilbert transform to orthogonal Chebyshev polynomials. The model is generalized to nonlinear elasticity (deformation theory of plasticity) and statistical mesomechanics models. Also presented is a solution for stress concentration in a system of arbitrarily oriented cracks which takes into account their mutual influence at any order of the perturbation theory and reduces to a system of linear equations with explicit exact solutions.