Non-stationary coupled axisymmetric thermoelasticity problem for a rigidly fixed round plate
A new closed solution is constructed for the axisymmetric dynamic problem of the classical (CTE) theory of thermoelasticity for a rigidly fixed circular isotropic plate in the case of a temperature change on its face surfaces (boundary conditions of the first kind). The mathematical formulation of the problem under consideration includes linear equations of thermal conductivity and equilibrium in a spatial setting, assuming that their inertial elastic characteristics can be neglected in the structures under study. In constructing a general solution of related non-self-conjugate equations, we use the mathematical apparatus of separation of variables in the form of finite integral transformations i.e. Hankel along the radial coordinate and biorthogonal transformation (FIT) with respect to the axial variable. At each stage of the investigation, a procedure is performed to reduce the boundary conditions to a form that allows the corresponding transformation to be applied. A particular feature of this solution is the application of a FIT based on a multicomponent relation of the eigenvector functions of two homogeneous boundary value problems. An important point in the procedure of the structural algorithm is the separation of the adjoint operator, without which it is impossible to solve non-self-adjoint linear problems of mathematical physics. This transformation is the most effective method for studying similar boundary value problems. The calculated design relationships make it possible to determine the stress-strain state and the character of the distribution of the temperature field in a rigidly fixed circular isotropic plate for an arbitrary external temperature effect with respect to time. Numerical analysis of the strength characteristics of the concrete structure shows that during the period of the unsteady load the maximum values of mechanical stresses are observed. Later, at a constant temperature regime, as a result of heating the entire plate, the displacements increase and the stresses fall.