Variational method of investigation of three-dimensional mixed problems of a plane cut in an elastic medium in the presence of slip and adhesion of its surfaces
The mixed problem of the equilibrium of an elastic space is studied on the hypothesis that a plane cut, horizontal in the plane \(Z=0\), occupies a region in the elastic medium bounded by a known curve \(\Gamma\). The parallel surfaces of the cut are in contact or are separated on some parts of the region, unknown before-hand. The forces applied on the surfaces of the cut are anti-symmetric body-or surface forces. The author formulates the boundary value problem which is separated into two other ones, on the basis of the superposition principle. The two boundary value problems concern: 1. the pressure and the region of contact and 2. the shear stress and displacements. These problems are formulated in variational form by means of systems of equalities and inequalities for some nonquadratic functionals. The resultant statements of the study concern: 1. existence and uniqueness conditions for the solutions, 2. the qualitative behaviour of the solution and of its integral characteristics. The variational formulation of the problem allows to construct a numerical solution using methods of mathematical programming.
equivalent variational statement of input boundary value problem
scientific article; zbMATH DE number 3868722
scientific article