Equilibrium Problem With Nonlinear Boundary Conditions For A Two-dimensional Body With A Cut

This work relates to the destruction of solid mechanics,specifically,to its mathematical modeling using the theory of cracks and the theory of elasticity.Alexander Khludnev,Novosibirsk State University professor,made a great contribution to the development of this research field.He developed a general method for proving the solvability of problems in crack theory by studying the energy functional and the set of admissible displacements.We repeatedly refer to the works of Alexander Khludnev,as well as other scientists,in the process of our work.The study of the bodies' behavior having certain inhomogeneities under the influence of external loads is a promising modern direction in the development of mathematical modeling of the deformation processes of various engineering structures.These include both examples of object's integrity violation(cracks,defects,cuts)and compositional features of the material(inclusions of a different nature).The relevance of our work is a previously unstudied crack shape,to be precise,a quadrangular cutout with parallel sides and a rigid inclusion extending from one of the corners.We suppose,that the width of the cutout is very small and comparable with the movements of the body points.The inclusion has a detachment from the elastic matrix forming a crack.Thus,the problem is considered in a domain with a quadrangular cutout and a cut coming out of it,in where a rigid inclusion is placed.The inclusion is modeled as a Bernoulli-Euler beam.Non-classical boundary conditions of the inequality type are specified at the both cutout and crack surfaces excluding mutual penetration of the crack faces into each other.This is the difference between this case and classical linear problems,which allow the crack faces to penetrate each other,leading to a contradiction from a practical point of view.The goal of the work is to prove the solvability of problem's variational formulation,the uniqueness of the solution,to derivate the full system of boundary conditions and to prove its equivalence to the variational inequality,as well as the limiting transition with respect to the stiffness parameter of the thin inclusion.In the investigating process of the formulated problem,all goals have been achieved.Solvability is proved by the method of studying the energy functional and the set of admissible displacements.The problem is defined as the problem of minimizing the energy functional,which,in turn,is equivalent to the problem of variational inequality.A system of boundary conditions is derived,the equivalence of differential and variational statements is proved.The fulfillment of these goals guarantees a great contribution to the scientific development in this field.The need for an accurate description of the deformation processes in two-dimensional bodies is an actual trend in mathematics in connection with the rapid development of engineering devices and equipment.