Non-Axisymmetric Deformation of a Circular Three-Layer Plate in its Own Plane

Abstract

A statement is given for the boundary value problem of non-axisymmetric deformation of an elastic threelayer circular plate in its own plane. The plate contour is pinched. Physical equations of state in the plate layers are described using the linear theory of elasticity, taking into account temperature influence on the elastic characteristics of materials. Equilibrium equations are obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the boundary value problem is reduced to finding the radial and tangential displacements in the layers of the plate. These displacements satisfy an inhomogeneous system of ordinary linear differential equations. To solve it, the method of decomposition into trigonometric Fourier series is applied. After substituting the series into the original system of equilibrium equations and performing the corresponding transformations, a system of ordinary linear differential equations is obtained to determine the four radial functions in each term of the series. The analytical solution is written out in the final form in the case of cosine radial and sinusoidal circumferential loads that depend linearly on the radial coordinate. The load is applied in the middle plane of the filler. Numerical approbation of the solution is carried out. The dependence of radial and tangential displacements on polar coordinates and temperature is investigated. Graphs of changes in displacements along the radius of the plate for different values of the angular coordinate are given. The weak dependence of displacements on temperature is illustrated when the plate contour is fixed.

Keywords

• three-layer circular plate
• elasticity
• non-axisymmetric load
• displacements
• numerical results