Usage of generalized functions formalism in modeling of defects by point singularity

Abstract

The paper proposes a new approach to the construction of point defect models, based on the solution of boundary value problems with non smooth coefficients. Heterogeneity is included in the determining equation of the boundary problem. This approach allows us to formalize defects at the stage of use of state equations, and thus automatically reconciles the defect with the hypotheses of diminution of dimension and does not break the energy closed. The solution is sought in the form of weakly convergent series of generalized functions. The proposed approach simplifies the mechanical interpretation of defect parameters and is demonstrated in several examples. In the first example, the Green function for harmonic oscillations of an elastic beam with a point defect is constructed. The defect model is a limiting state of elastic inclusion with weakening or strengthening. The second example considers the inclusion of an elliptical shape in the problem of harmonic oscillations of the elastic plate. The first approximation of the equivalent volumetric force is constructed and the path to the following approximations is indicated. In the third example, a model of a brittle crack with a known displacement jump is constructed for a static two-dimensional problem of elasticity theory.