On the possibility of reducing the problem of stress concentration in an elastic medium to the nonlinear problem for dummy gauge fields

System of cracks in a solid is considered as a violation of local symmetry group of three-dimensional rotations, which are introduced to compensate for the bogus field, making the gauge-invariant expression for the lagrangian density of elastic energy. Equilibrium equations are solved by perturbation theory in the scaling parameter of the gauge group, which is the concentration of notches.

For the two-dimensional case managed to obtain exact expressions for the voltage and the gauge field by using the Hilbert transform for orthogonal Chebyshev polynomials. The generalization of the model to the case of nonlinear elastic (deformation theory of plasticity) and statistical models mesomechanics.

We obtain a solution of the problem of stress concentration in the system of arbitrarily oriented cracks, taking into account their mutual influence in any order of perturbation theory. It reduces to a system of linear equations for which it is always possible to obtain an exact solution explicitly.