Non-axisymmetric deformation of a freely supported three-layer plate in its own plane | Mekhanika | kompozitsionnykh | materialov i konstruktsii

Non-axisymmetric deformation of a freely supported three-layer plate in its own plane



Three-layer structural elements are widely used in aerospace and transport engineering, construction, production and transportation of hydrocarbons. The theory of bending of circular three-layer plates those are not symmetrical in thickness under external various kinds of force actions is currently developed quite fully. Here we present a statement of the boundary value problem of non-axisymmetric deformation of an elastic three-layer circular plate in its plane. The center of the plate is fixed, its contour is freely supported. The physical equations of state relate stresses and deformations to the relations of the linear theory of elasticity. The equilibrium equations are obtained by the Lagrange variational method. Force boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding two desired functions – radial and tangential displacements in the layers of the plate. These functions satisfy an inhomogeneous system of linear partial differential equations. To solve this problem, the method of decomposition into trigonometric Fourier series is applied. To determine the desired amplitude functions of each of the series members, a system of four ordinary linear differential equations is obtained. The analytical solution of this system is written out in the final form in the case of a non-axisymmetric cosine radial load with constant amplitude. The load is applied in the middle plane of the filler. The numerical approbation of the solution is carried out for a freely open plate contour. The dependence of radial and tangential displacements on polar coordinates is numerically investigated. Graphs of changes in displacements along the radius of the plate for different values of the angular coordinate are given. The dependence of the displacements on the thickness of the bearing layers and the aggregate is illustrated.