Solution of the coupled nonstationary problem of thermoelasticity for a rigidly fixed multilayer circular plate by the finite integral transformations method
Abstract:
A new closed solution of an axisymmetric non-stationary
problem is constructed for a rigidly fixed round layered plate in the case
of temperature changes on its upper front surface (boundary conditions of
the 1st kind) and a given convective heat exchange of the lower front
surface with the environment (boundary conditions of the 3rd kind).
The mathematical formulation of the problem under consideration includes
linear equations of equilibrium and thermal conductivity (classical theory)
in a spatial setting, under the assumption that their inertial elastic
characteristics can be ignored when analyzing the operation of the structure
under study.
When constructing a general solution to a non-stationary problem described
by a system of linear coupled non-self-adjoint partial differential
equations, the mathematical apparatus for separating variables in the form
of finite integral Fourier–Bessel transformations and generalized
biorthogonal transformation (CIP) is used. A special feature of the solution construction is
the use of a CIP based on a multicomponent relation of
eigenvector functions of two homogeneous boundary value problems, with the
use of a conjugate operator that allows solving non-self-adjoint linear
problems of mathematical physics. This transformation is the most effective
method for studying such boundary value problems.
The calculated relations make it possible to determine the stress-strain
state and the nature of the distribution of the temperature field in a rigid
round multilayer plate at an arbitrary time and radial coordinate of
external temperature influence. In addition, the numerical results of the
calculation allow us to analyze the coupling effect of thermoelastic fields,
which leads to a significant increase in normal stresses compared to solving
similar problems in an unrelated setting.
Keywords:round multilayer plate, classical theory of thermoelasticity, nonstationary temperature influence, biorthogonal finite integral transformations.
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