Transverse compression of a thin elastic disc (Q6565674)

The work is devoted to the asymptotic calculation of a thin elastic disk subjected to compression normal to its faces. Solving such a problem using analytical approximate methods is important for efficient multiple calculations of various similar elements such as gaskets made of soft materials or elements of nanostructures. All earlier studies by other authors did not consider boundary layers at the edges of the disk, but adapted the boundary conditions to solutions obtained for the interior of the disk. At the same time, the Saint-Venant principle is not applicable to problems of transverse compression with non-classical boundary conditions along the faces. In this paper, lateral compression is modeled using mixed boundary conditions along both faces. The main attention is paid to the axisymmetric deformation of a thin elastic disk, compressed by normal stresses, under the assumption of zero tangential displacements along the faces. The work considers compression due to a given stress rather than displacement, which provides a solution for the boundary layer in Fourier series. The asymptotic solution of the formulated problem is divided into an external component and a flat boundary layer localized near the edge. The two-term asymptotic expansion of the external solution is obtained from the canonical elasticity relations for displacements and stresses. This analytical solution is compared with the FEM solution. Thus, the work obtains a complete asymptotic solution to the axisymmetric problem of transverse compression of a thin elastic disk. Adapted mixed boundary conditions along the disk faces support an explicit internal solution corresponding to a flat boundary layer with exponential decay. The work demonstrates an effective technique for asymptotic analysis of elasticity theory equations, which per se can be a template for solving similar problems. The work is of undoubted scientific interest and can be practically applied, for example, in the development of refined mathematical models of nanostructure deformation.