Representation of the thermo-stress state of a plate based on the 3D elasticity theory

¹ Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NASU, 3-b, Naukova st., Lviv, 79060, Ukraine
² Institute of Applied Mechanics of Russian Academy of Sciences, 7, Leningradsky prospekt, Moscow, 125040, Russia

^* e-mail: victor.rev.12@gmail.com

Abstract

The general formulation of the 3D temperature static problem of the theory of elasticity is considered. A new representation of the general solution of static equations of thermoelasticity is proposed. The three-dimensional stress state of a thick or thin plate are divided into three parts: bending of the plate, symmetric compression of the plate on it’s flat end surfaces and symmetric temperature problem. Third thermoelastic stress state of the plate is studied in detail. It have been reduced to a two-dimensional state after integration the stresses components along the normal to the middle surface. The displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the flat surfaces of the plate. Only the assumption has been used that the normal stresses perpendicular to the middle surface are insignificant. The introduced harmonic functions are determined from the boundary conditions on the lateral surface of the plate. The formula for the experimental determination of the sum of normal stresses has been found through the measured deflections of the end surfaces of the plate.