On Integral Equations of Cracks of a New Type

• Published in: Vestnik, St. Petersburg University. Mathematics St. Petersburg University. Mathematics, 2022-09, Vol.55 (3), p.267-274
• Main Authors: Babeshko, V. A., Evdokimova, O. V., Babeshko, O. M.
• Format: Article
• Language: English
• Subjects: Analysis; Boundary value problems; Cracks; Formability; Fractures; Helmholtz equations; Integral equations; Mathematical models; Mathematics; Mathematics and Statistics; Modelling; Partial differential equations; Plates; Rheological properties; Rheology
• ISSN: 1063-4541; 1934-7855
• DOI: 10.1134/S1063454122030049

For the first time, the paper develops a method for the modeling of cracks of a new type that allows their description in environments of complex rheologies. It is based on a new universal modeling method previously published by us and used in boundary-value problems for systems of partial differential equations. The advantage of the method is the possibility of avoiding the need to solve complex boundary-value problems for systems of partial differential equations by replacing them with separate differential equations, among which the Helmholtz equations are the simplest. Namely, with the help of combinations of solutions to boundary-value problems for this equation, it is possible to describe the behavior of complex solutions of multicomponent boundary-value problems. In this paper, for the first time, the method is applied to a mixed boundary-value problem for cracks of a new type. Cracks of a new type, complementing Griffiths cracks, were discovered during the study of fractures of lithospheric plates that converge at the ends when meeting along the Conrad discontinuity. In the course of the study, Kirchhoff plates were adopted as models of lithospheric plates. The method developed in the published article is aimed at the possibility of describing models of approaching objects similar to lithospheric plates in the form of deformable plates of more complex rheologies. In particular, it can be a thermoelectroelastic plate or other rheology. In the process of solving the problems using Kirchhoff models for lithospheric plates, there is the problem of calculating some functionals that needed to be determined. This method demonstrates an approach that eliminates this drawback. The derivation of integral equations of cracks of a new type, the method of their solution, and the approach to application in more complex rheologies are given.