Thermoelastic non-stationary fields in a rigidly fixed plate

In this article, a new closed solution of the axisymmetric dynamic problem of the theory of thermoelasticity is constructed for a rigidly fixed circular isotropic plate in the case of temperature changes on its face surfaces. The mathematical formulation of the problem includes linear equations of motion and thermoelasticity in the spatial formulation with respect to the components of the displacement vector, as well as the function of temperature change. The study of non-self-adjoint equations is carried out in an unrelated statement. Initially, we consider the initial boundary value problem of thermoelasticity without taking into account the deformation of the plate, and at the next stage, we study the problem of elasticity theory under the action of a given (defined) temperature change function. To solve the problems, we use a mathematical apparatus for separating variables in the form of finite integral transformations: Fourier, Hankel, and generalized integral transformation (CIP). In this case, at each stage of the study, the procedure is performed to bring the boundary conditions to the form that allows you to apply the appropriate transformation.


Introduction
Uneven non-stationary heating of structures for various purposes leads to the thermal deformations and stresses occurence, which must be taken into account in the case of a comprehensive analysis of the strength characteristics of elastic systems of finite dimensions [1,2]. At present, various theories of thermoelasticity have been developed (CTE, GHI -GHIII, LS) [3,4], which solves this problem with varying accuracy degrees.
The mathematical formulation of the considered initial-boundary value problems in a linear threedimensional formulation includes coupled non-self-adjoint differential equations of motion and heat conduction. As a rule, this system of differential equations is considered in an unrelated setting [5][6][7][8][9][10]. In this case, when an external non-stationary heat load acts on the elastic system, the effect of the rate of change in the volume of the body on the temperature field is not taken into account.
In a related formulation, closed solutions of dynamic problems of thermoelasticity are presented in a few works.
In particular, studies [11][12][13] were carried out for a finite isotropic cylinder with membrane fixing of its end surfaces. In [11], using the generalized method of finite integral transformations [14], and [12,13] -the biorthogonal integral transform [15,16]. Research [17,18] was carried out using hyperbolic (GHII, GHIII) theories of thermoelasticity and helps to analyze the frequency equations, as well as the forms of harmonic waves in an infinite cylindrical waveguide.
In this work, a rigidly fixed round isotropic plate is investigated. The case of the action on the upper and lower surfaces of an unsteady axisymmetric temperature load (boundary conditions of the IOP Publishing doi:10.1088/1757-899X/1181/1/012026 2 1st kind) is considered. The numerical results of calculating this problem in an unconnected formulation [19] allow us to conclude that the inertial forces of an elastic system affect its stress-strain state only in very thin structures ( thickness and radius of the plate) under the action of a high-frequency load. Taking into account these results, the inertia forces are not taken into account when solving the system of non-self-adjoint differential equations of the classical (CTE) theory of thermoelasticity, i.e. the constraint is used for the considered constructions The constructed solution of the coupled problem in a three-dimensional formulation makes it possible to take into account the effect of the rate of change in its volume (rate of dilatation) on the nature of the distribution of the temperature field and the stress-strain.

Materials and methods
Let a round rigidly fixed plate occupy the region  : . On the upper and lower surfaces, the temperature is set, the value of which depends on the radial coordinate and initial conditions:

Results and discussion
The initial boundary value problem (1) -(3) is solved by the method of integral transformations, using successively the Hankel transform [20] with finite limits in the variable and the degenerate biorthogonal finite transformation [15] in the coordinate z . At each stage of the solution, the procedure of standardization of the corresponding boundary conditions is carried out [21]. Transformants and the inversion formulas of the corresponding transformations have the following form: where k F − matrix is a column of standardizing functions. The algorithm for solving the initialboundary value problem of thermoelasticity (1) -(3) is described in detail in [22].

Conclusions
As an example, we consider a rigidly fixed round plate( 1 b = m) made of steel, which has the following physical and mechanical characteristics of the material: where ( )− t H~ the single function of Heaviside ( ( ) ).
The temperature field and stress-strain state are analyzed for plates with a thickness 0. ( ) Analysis of the calculation results allows us to draw the following conclusions: • the connectivity of thermoelastic fields at a given temperature load (7) leads to a slower heating of the plate over time ( figure 2). In this case, the rate of change in the volume of the body, which is taken into account in the heat conduction equation (1) that are used in the initial differential equations of thermoelasticity (1). As a result, there is an increase in the numerical values of the axial component of the displacement vector ( figure 3, graphs 1, 2); • at a given temperature load, the coupling of thermoelastic fields decreases over time ( figure  3). In addition, as a result of warming up the structure, an increase in displacements is observed ( figure 3,4), and with a steady temperature regime on the lower surface there are no radial displacements (figure 4, graph 3); • the linear nature of the change in the radial component of the displacement vector along the height of the plate, allows us to conclude that when solving thermoelasticity problems for homogeneous elastic systems with the help of applied theories, it is possible to use the kinematic hypothesis of plane sections; • the numerical values of radial displacements ( )  Here you can draw a conclusion: • the greatest influence of the field coupling on the stress tensor component ( ) In conclusion, we can conclude that when calculating structures of finite dimensions in the case of a high-speed thermal load, the coupling of temperature and elastic fields has a significant effect on its stress-strain state. Moreover, this feature is more pronounced in thin plates.