Dynamic Problem Of Thermoelasticity Research Articles

Study on coupled problems of thermoelasticity in Strains

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.

Coupled Problem on Thermo-Elasticity in Strains for an Isotropic Parallelepiped

This work is devoted to mathematical and numerical modeling of the coupled dynamic problem of thermoelasticity in deformations. A numerically related boundary value problem of thermoelasticity in deformations for a parallelepiped with the corresponding initial and boundary conditions is formulated and solved. Grid equations are constructed using the finite-difference method in the form of explicit and implicit schemes. In this case, the solution of the explicit scheme is reduced to recurrent relations with respect to deformations and temperature. In the case of implicit difference schemes, the equations are solved by sequential application of the sweep method. The validity of the formulated boundary value problems is justified by comparing the numerical results obtained with different methods based on two different models.

Accelerating the computational process for analyzing dynamic thermal behavior of an innovatively designed longitudinally-graded sandwich plate: Deep-learning approach

Employing the solutions through artificial intelligence (AI) to solve mechanical engineering problems can pave the way toward accelerating the computation process, especially for those engaged with the dynamic nature of the time-variable systems. In this regard, this article attempts to employ deep neural networks (DNN) for the coupled thermoelasticity assessment of the longitudinally sandwiched plates exposed to intensive thermal shock. The left and right face sheets of the sandwich plate are enriched with graphene platelets (GPLs) along the longitudinal direction. The thermal shock is considered as the instantaneous heat flux applied in the axial direction. Elasticity theory is employed for deriving the governing equations in an exact template and differential quadrature approach (DQA) in conjunction with the Laplace transform plays the role of solution approaches to attain the thermoelastic reaction of the sandwich system at training points. Then, the DNN utilizes these points to adjust a predictor mechanism for analyzing the behavior of the sandwich system in every load-exertion circumstances. Additionally, the DNN would be optimized by a brand-new metaheuristic algorithm according to some error estimation criteria to minimize the error of regression. The mentioned solution is validated by comparing its performance with the outcomes of the published literature. This study provides a novel platform to use the great potential of AI-based methods in solving dynamic thermoelastic problems in the world of solid structures.

The Role of the Coupling Coefficient in the Dynamic Problem of Thermoelasticity with Localized Inclusions

The Role of the Coupling Coefficient in the Dynamic Problem of Thermoelasticity with Localized Inclusions

ON THE ROLE OF COUPLING COEFFICIENT IN DYNAMIC PROBLEM OF THERMOELASTICITY WITH LOCALIZED INCLUSION

In the present paper paper the dynamic problem of thermoelasticity with a localized inclusion in the medium is considered. It is shown that in the resonant regime an important role is played by the coupling coefficient of the temperature and strain fields. The coupling of the problem and the presence of a discrete spectrum lead to the appearance of an additional term in the expression for temperature, which is localized and does not describe the diffusion process.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selected dynamic thermoelasticity problems for the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a new procedure. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel-Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

Numerical analysis of dynamic thermoelastic two-dimensional problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

Numerical analysis of dynamic thermoelastic two-dimensional problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

Numerical analysis of dynamic thermoelastic two-dimensional crack problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

In recent years, a variety of microscopes using heat sources, such as photoacoustic microscopes, have been developed. In this observation principle, reflection and diffraction phenomena of two waves with different properties, namely thermal wave and elastic wave, are utilized. However, the thermal wave cannot be reproduced using the classical Fourier-type heat conduction equation. In order to overcome this mathematical difficulty, we had derived a coupled nonlinear partial differential equation consisting of the heat conduction with a delay time and the dynamic thermoelastic equation, and had simulated numerically the propagation process of thermal and elastic waves for one-dimensional problem. In this study, thermal and elastic waves were simulated for a two-dimensional plane problem with and without defect using the finite-difference time-domain method. The results showed that the reflection and interference phenomena of two waves can employ for detecting small internal defects existing near the surface of target sample.

Implementation of a high-accuracy manifold element modelling scheme for dynamic fracture analysis under thermal-mechanical shock

In this study, a high-accuracy numerical manifold modelling scheme is proposed for simulating dynamic thermoelastic fracture problems of two-dimensional stationary cracks in linear elastic solids. The advantages of the numerical manifold method in discretizing the physical model with multiple discontinuities, and those of the precise time step integration method in time integral are integrated together in the proposed numerical scheme. The implementation details of the proposed modelling scheme are well presented. Dynamic stress intensity factor considering the inertial effect is computed. Through three dynamic fracture examples with pure thermal shock, heat flux shock and mechanical shock, respectively, the validity of the present scheme is first verified. Then the proposed scheme is applied for dynamic fracture analysis under mixed thermal-mechanical shock, in which different loading forms, such as step loading, linear loading and harmonic loading, are considered. The results show that the present modelling scheme is extremely stable and convergence. The advantages of the present scheme on accuracy, efficiency, time-step-independence over commonly used Newmark method are also fully demonstrated.

Peculiarities of formulation and solving the dynamic problems of thermoelasticity

The problem of propagation of spherical waves in a thermoelastic medium is considered. Two approaches to taking into account the mutual influence of dynamic fields of deformations and temperature are compared. A generalized model of coupled thermoelasticity is used for calculation in the first approach and the second one is based on the ratio of the theory of thermal stresses, which are neglecting the change of temperature distribution under mechanical loads action. The amplitude-frequency characteristics of radial displacements and normal tangential stresses at the boundary of a spherical cavity being under action of load, which changes according to the harmonic law in time, are obtained. The correspondence between the value of the coupling parameter and the results error caused by the use of the simplified model of field interaction is traced. Wave processes in solids of modern polymeric materials, such as polyvinyl butyral and polyvinyl butyralfurfural belonging to the family of polyvinyl acetals, which have a fairly high coefficient of field connectivity of 0.18 and 0.41, respectively, are considered. It is shown that the use of a simplified model of coupled thermoelasticity for the calculation of structure of such materials leads to unacceptably large differences in the results. Thus, the maximum values of the stress-strained state parameters obtained using the generalized model were 18% higher than in the case of the application of the theory of temperature stresses for polyvinyl butyral. The results difference obtained using this two approaches at some frequencies exceeded 30% for the polyvinyl butyralfurfural medium. It is concluded that the simplified model of the interaction of deformation and temperature fields can be a rough approximation in the analysis of the dynamic reaction of massive structural elements made of such materials.

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

Numerical analysis of dynamic thermoelastic problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

Numerical analysis of dynamic thermoelastic problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

Термическая реакция при тепловом ударе массивного тела с внутренней цилиндрической полостью

The paper examines mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to specific cases during intense heating of a solid boundary. We introduce a stress compatibility equation for dynamical problems, generalizing the well-known Beltrami --- Mitchell relation for quasi-static cases. It is convenient to use this relation when considering numerous special cases in the theory of heat shock in Cartesian coordinates both for bounded bodies of a canonical form, i.e., an infinite plate, and for partially bounded bodies, i.e., space bounded from the inside by a flat surface. In the latter case, the obtained analytical solutions of dynamic problems of thermoelasticity lead to visual and convenient for physical analysis functional structures describing the kinetics of thermal stresses. For cylindrical and spherical coordinate systems, we propose a compatibility equation in displacements, which is convenient for studying the problem of thermal shock in bodies with a radial heat flux and under conditions of central symmetry. In the study, we singled out a class of problems in which the consideration of the geometric dimensions of a structure investigated for a thermomechanical reaction under conditions of intense heating concerns mainly the near-surface layers. According to the experimental results, it is these layers that absorb the main amount of heat during a time close to the beginning of heating, which corresponds to the time of the microsecond duration of the inertial effects. We investigated the thermal reaction of a massive body with an inner cylindrical cavity within the framework of dynamic thermoelasticity under various modes of intense heating of the cavity surface. Finally, we carried out numerical experiments and described the wave character of thermal stresses with the corresponding quasi-static values, and established the role of inertial effects in mathematical models of the theory of thermal shock

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

Dynamic thermoelasticity problem for a finite hollow cylinder with a cut along the generatrix

We consider the steady state problem of incoherent thermoelasticity for a finite hollow elastic cylinder on the inner and outer cylindrical surfaces of which ideal contact conditions are given. Without limiting the generality of reasoning, we consider a dynamic normal load applying at the upper face of the cylinder, and there are no tangential stress there. The bottom face of the cylinder is fixed. It is required to determine the thermos-wave field of the cylinder under steady state oscillations.To solve this problem, it is proposed to reduce the equations of dynamic incoherent thermoelasticity relatively to the displacements to two jointly and one separately solved equations. The solution of the two separately solved problems is carried out by the method of integral transformations, which allows us to construct the exact solutions for the separately solved problems in the explicit forms. In the case of the problem formulated using a system of two partial differential equations, it is proposed to use the apparatus of a matrix differential calculus. The matrix Greens function is constructed and the solution of the problem is derived in explicit form.

Heat Transfer in the Formation of Thermoelastic and Thermoelectric Responses of Metals to Laser Pulse Action

We analyze the effect of heat transfer in metals on parameters of the thermoelastic and thermoelectric responses of metals to a pulsed laser action. It is shown that two-stage analysis of the dynamic problem of thermoelasticity and the thermoelectric effect makes it possible to obtain an adequate description of experimental results. The results described here demonstrate a considerable dependence of the response parameters on the microstructure, which indicates the possibility of development of a high-efficiency method of nondestructive control of metal quality.

Computer Simulation of the Coupled Dynamic Thermoelasticity Problem for a Two-Dimensional Isotropic Bodies

The two-dimensional coupled thermodynamic boundary problem for isotropic bodies is formulated. The explicit and implicit two-dimensional finite difference equations are constructed. The received explicit and implicit schemes are solved using the elimination method and recurrence formulas, respectively. The comparison of the obtained numerical results by the two methods shows a good coincidence.

Analysis of the generalized heat equation applied to the solution of the dynamic problem of thermoelasticity

Based on the approximate solution of the dispersion equation, the paper presents an analysis of the system of dynamic thermoelasticity equations taking into account the generalized heat equation. It is noted that during the wave process of heat transfer, a sufficiently intensive process of energy exchange between thermal and elastic fields is realized, while depending on the relations of the characteristic relaxation times, the direction of energy exchange can change.

Analysis of the Generalized Heat Equation for Solution of the Dynamic Thermoelasticity Problem

On the basis of the approximate solution of the dispersion equation, the set of equations of dynamic thermoelasticity is analyzed with taking into account the generalized heat equation. It is noted that a reasonably intense process of energy exchange between thermal and elastic fields is implemented during the heat-transfer wave process, while the energy-exchange direction can vary depending on the relation between the characteristic relaxation times.

NUMERICAL RESOLUTION OF AN EXACT HEAT CONDUCTION MODEL WITH A DELAY TERM

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.