Variable Thermal-Force Bending of a Three-Layer Bar with a Compressible Filler

Abstract

Deformation of a three-layer elastoplastic bar with a compressible filler in a temperature field is considered. To describe the kinematics of a pack asymmetric across its thickness, the hypothesis of broken line is accepted, according to which the Bernoulli hypothesis is true in thin bearing layers, and the Timoshenko hypothesis is valid for a filler compressible across the its thickness, with a linear approximation of displacements across the layer thickness. The work of filler in the tangential direction is taken into account. The physical stress-strain relations correspond to the theory of small elastoplastic deformations. Temperature variations are calculated from a formula obtained by averaging the thermophysical properties of layer materials across the bar thickness. Using the variational method, a system of differential equilibrium equations is derived. On the boundary, the kinematic conditions of simply supported ends of the bar are assumed. The solution of the boundary problem is reduced to the search for four functions, namely, deflections and longitudinal displacements of median surfaces of the bearing layers. An analytical solution is derived by the method of elastic solutions with the use of the Moskvitin theorem on variable loadings. Its numerical analysis is performed for the cases of continuous and local loads.