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**Solution of the coupled nonstationary problem of thermoelasticity for a rigidly fixed multilayer circular plate by the finite integral transformations method.**
*(Russian.
English summary)*
Zbl 07380830

Summary: 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.

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.

### MSC:

74F15 | Electromagnetic effects in solid mechanics |

74S20 | Finite difference methods applied to problems in solid mechanics |

74K20 | Plates |

74H05 | Explicit solutions of dynamical problems in solid mechanics |

### Keywords:

round multilayer plate; classical theory of thermoelasticity; nonstationary temperature influence; biorthogonal finite integral transformations
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\textit{D. A. Shlyakhin} and \textit{Z. M. Kusaeva}, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 25, No. 2, 320--342 (2021; Zbl 07380830)

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