Equations Of Thermoelasticity Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

Propagation of bending waves along the edge of a point-loaded piezoelectric plate on elastic foundation

An investigation on the propagation of bending waves along the edge of a piezoelectric thin plate resting on an elastic foundation (cf. Pasternak foundation) and under the action of a point-load is carried out in this paper. The displacement field of the plate is based on the classical plate theory. The Kelvin-Voigt type viscoelastic model is used to examine the viscous effect on the characteristics of the bending wave. The Moore-Gibson-Thompson (MGT) thermoelastic equation is solved for an external time variant harmonic heat source at the upper surface of the plate. The piezoelectric potential is obtained for an electrode-covered upper surface and an electrically shorted lower surface. The displacements of the propagating bending wave along the edge are examined under a time-harmonic point-load acting near the vicinity of the free edge of the plate. The well-known Green function technique and Fourier transform method are implemented to generate an analytical solution for the non-homogeneous boundary value problem. The examination of the derived explicit form of the dispersion relationship and the displacements of the bending edge wave under external loading are discussed and presented both analytically and graphically, and the results are summarized.

A study on solid-shell finite element formulations applied to nonlinear thermoelastic analysis of thin-walled structures

The solid-shell elements have shown great advantages in three-dimensional (3D) finite element (FE) simulation of thin-walled structures. To overcome the numerical lockings for 3D element with a large span-thickness ratio, the assumed natural strain (ANS) method combined by either the hybrid stress (HS) or enhanced assumed strain (EAS) formulations are commonly used. In this work, two types of finite element formulations of the solid-shell elements are developed for nonlinear thermoelastic analysis of thin-walled structures, which are termed as the “ANS+HS” and “ANS+EAS” formulations. Accordingly, two eight-node solid-shell elements (CSSH8 and CSSE8) are developed based on the “ANS+HS” and “ANS+EAS” formulations, respectively. The Green–Lagrange displacement–strain relation is applied to involve the geometrical nonlinearities, and both the thermal expansion and temperature-dependent material properties are considered. Nonlinear thermoelastic equilibrium equations are constructed in the framework of the two types of finite element formulations using the Hellinger–Reissner and conventional variational principles, respectively. The proposed method can easily realize three different coupling analyses for thermal–mechanical loads by modifying the parameters of nonlinear thermoelastic equilibrium equations. Numerical examples demonstrate that the proposed method using CSSH8 and CSSE8 elements traces the nonlinear thermoelastic response of the isogrid stiffened panel as accurately as the shell, solid-shell, and solid elements of ABAQUS. Furthermore, the path-following capability of the proposed method using the CSSH8 with “ANS+HS” formulation is slightly superior to that using the CSSE8 with “ANS+EAS” formulation, but both better than ABAQUS.

Coupled Thermo-Electric-Elastic Piezoelectric Vibration Energy Harvester With Axial Movement: Modeling, Verification, and Analysis

Abstract In the field of rail transport and aerospace field, vibration energy harvesting is inevitably subjected to coupled excitations, including train wheel–track interaction induced friction heat and forced vibration, periodic thermal radiation, and vibration excitation. This paper investigates a coupled thermo-electric-elastic piezoelectric vibration energy harvester with axial movement under external heat flux and mechanical force load. The coupled forced vibration equation, coupled electric equation, and coupled thermoelastic heat conduction equation are derived and solved by Green's function theory. To analyze the effect of excitations on the response characteristics, the decoupled method is utilized to solve the coupled multi-field equations and obtain the displacement, electric, and temperature distribution closed-form solutions. The displacement coupling effect induced temperature distribution and the thermo-electric coupling effect triggered displacement are respectively decoupled and analyzed. The obtained closed-form temperature distribution and displacement solutions are verified by the finite element method. To further verify the obtained solutions, a numerical method is conducted by decoupling the coupled multi-field equations and comparing them with prior solutions. Additionally, the different height-to-length ratios, axially moving speeds, and external force load are analyzed in detail. The results indicate that the displacement, temperature distribution, and output voltage vary with external conditions due to the coupled multi-field effect. Overall, this work investigates the thermo-electric-elastic coupling effect on the axially moving piezoelectric energy harvesting, which is beneficial to promote theoretical investigations of the coupled multi-field energy harvesting system and accelerate the practical applications in the aerospace field.

The equations coupled by Von Karman system with thermoelasticity

The aim of this paper is to study the model of the von Karman evolution coupled to the thermoelastic equation, with rotational inertia and clamped boundary conditions. We establish the existence and uniqueness of a weak solution related to the dynamic model. Towards the conclusion, we employ the finite difference method to approximate the solution to our problem.

Solution of the plane thermoelasticity problems based on a new set of equations

The interaction of thermal and mechanical fields in modern technologies for the synthesis and processing of materials attracts more and more attention. This is due to the need to predict the behavior and properties of new materials and products depending on the conditions of their creation. In the present work, attention is focused on the comparison of the solutions to the thermoelasticity plane problems with and without accounting for the temperature dependence of the material properties. It is assumed that an external heat source traveling over the surface with a specific velocity and following a given trajectory is responsible for a significant temperature increase. It may correspond to a laser source, for example. The new way of problem formulation in stresses is used. The temperature dependence of the characteristics leads to nonlinear thermoelasticity equations when the equilibrium equations, the requirement of strain compatibility, and the Duhamel-Neuman relation for the plane stress state case are combined. To determine the temperature field, a nonlinear thermal conductivity equation is used. To solve the problem numerically, the use of an implicit finite-difference scheme and coordinate splitting for the thermal conductivity equation and the finite-difference method of successive over-relaxation with Chebyshev acceleration for the equations for stresses was made. As a result, the difference in the solution of linear and nonlinear problems is demonstrated for the considered cases. Using the example of a mixture of titanium and aluminum powders, numerical modeling demonstrated that some stress components can reach values up to 3.5 times higher than those obtained in the case of temperature-independent material properties when there is a 500-degree local temperature increase.

Variational temporal convolutional networks for I-FENN thermoelasticity

Machine learning (ML) has been used to solve multiphysics problems like thermoelasticity through multi-layer perceptron (MLP) networks. However, MLPs have high computational costs and need to be trained for each prediction instance. To overcome these limitations, we introduced an integrated finite element neural network (I-FENN) framework to solve transient thermoelasticity problems in Abueidda and Mobasher (2024). This approach used a physics-informed temporal convolutional network (PI-TCN) within a finite element scheme for solving transient thermoelasticity problems. In this paper, we introduce an I-FENN framework using a new variational TCN model trained to minimize the thermoelastic variational form rather than the strong form of the energy balance. We mathematically prove that the I-FENN setup based on minimizing the variational form of transient thermoelasticity still leads to the same solution as the strong form. Introducing the variational form to the ML model brings the advantages of lower requirement for the differentiability of the basis function and, thus, lower memory requirement and higher computational efficiency. Also, it automatically satisfies zero Neumann boundary conditions, thus reducing the complexity of the loss function. The formulation based on the variational form complies with thermodynamic requirements. The proposed loss function reduces the difference between predicted and target data while minimizing the variational form of thermoelasticity equations, combining the benefits of both data-driven and variational methods. In addition, this study uses finite element shape functions for spatial gradient calculations and compares their performance against automatic differentiation. Our results reveal that models leveraging shape functions exhibit higher accuracy in capturing the behavior of the thermoelasticity problem and faster convergence. Adding the variational term and using shape functions for gradient calculations ensure better adherence to the underlying physics. We demonstrate the capabilities of this I-FENN framework through multiple numerical examples. Additionally, we discuss the convergence of the proposed variational TCN model and the impact of hyperparameters on its performance. The proposed approach offers a well-founded and flexible platform for solving fully coupled thermoelasticity problems while retaining computational efficiency, where the efficiency scales proportional to the model size.

A novel hybrid-stress method using an eight-node solid-shell element for nonlinear thermoelastic analysis of composite laminated thin-walled structures

Composite laminated thin-walled structures, widely used in high-speed aircrafts, undergo a complex thermal–mechanical coupling environment. Geometrical nonlinearities with a thermal effect bring significant challenge to finite element analysis of structures. In this paper, a novel hybrid-stress method based on the solid-shell element is proposed for nonlinear thermoelastic analysis. An eight-node solid-shell element (CSSH8) is developed based on the assumed natural strain method and hybrid-stress formulations to overcome various locking problems and achieve an effective 3D simulation for structures with a large span-thickness ratio. The Green–Lagrange displacement-strain relation is selected to take the geometrical nonlinearities into account. The modified generalized laminate constitutive model is extended to consider both the thermal expansion and temperature-dependent material properties. A temperature variation along the laminate thickness can also be assumed in the constitutive model. Nonlinear thermoelastic equilibrium equations are derived using the Hellinger–Reissner variational principle, in which five different coupling cases for thermal–mechanical loads can be fully involved. Numerical examples demonstrate that the proposed method with CSSH8 element is insensitive to various distorted meshes and numerically robust to pass the buckling point; meanwhile large step sizes can be achieved in the path-following nonlinear thermoelastic analysis.

The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Моделирование и анализ распределений температуры и термомеханических напряжений в многокристальном электронном модуле

A mathematical thermomechanical model of a multi-chip electronic module (MEM) containing three silicon dies of high-power transistors fixed with a conductive adhesive on a copper plate placed on a radiator is considered in the form of a system of equations of thermal conductivity and thermoelasticity with specified boundary conditions. As a result of computational studies of the model in the COMSOL Multiphysics software environment, distributions of temperature and thermomechanical stresses in the MEM structural elements were obtained depending on the heating power, the thickness of the adhesive and the size of the model defect in the contact connection of one of the MEM crystals with a copper plate in the form of voids in the adhesive layer. It is shown that voids in the adhesive layer lead to a sharply non-uniform temperature distribution over the crystal area and thermomechanical shear stresses in the contact layer that exceed the maximum permissible values for the adhesive. It has been established that thermomechanical stresses decrease with increasing thickness of the adhesive layer and increase with increasing size of the defect(voids) in this layer. The simulation results are in good agreement with the results of measuring the thermal impedance of power transistor crystals using the modulation method, which indicates the correctness and adequacy of the developed model.

Analytical model for laser-induced transient grating measurements of thermal diffusivity in non-opaque materials

The thermal transport and elastic properties of materials are often measured using the laser-induced transient grating spectroscopy (TGS) technique. The analysis of the TGS signal usually involves fitting well-known expressions, derived assuming the limiting cases of opaque or transparent materials, to the measured data. In this paper, the system of thermoelastic equations is analytically solved for an isotropic homogeneous material assuming finite laser penetration depth, which is an important consideration when the penetration depth is on the order of the acoustic wavelength. The need to use such a solution is discussed and compared to the expression for opaque material. The solution is benchmarked against TGS signal measured on single-crystal silicon with {100} surface orientation and is found to significantly improve the accuracy of the calculated thermal diffusivity as compared to using the expression for opaque material.

Thermoelastic analysis of axisymmetric conical shells: Investigating stress–strain response under uniform heat flow with semi-coupled approach

ABSTRACT This study addresses the issue of thermal stresses and strains in a thin-walled conical shell subjected to uniform heat flow along its side surfaces and at both ends of the thermal insulation shell. The governing equations are derived using a semi-coupled static thermoelastic equation and an energy equation. The energy equation ignores strain rate effects, but static behavior equations consider temperature variations in axisymmetric thin-walled conical shells. Semicoupled methods, due to their simpler analysis, are suitable for thermal stress analysis when structural and thermal disturbances differ significantly. To obtain solutions, the Galerkin finite element method is employed, yielding valuable insights. Using the Galerkin reduced weight residual method, the sum of the weight residuals on the shell thickness is set to zero. Through a comprehensive problem-solving approach, the effects of increasing the angle of the cone apex in different thermal boundary conditions are investigated in two one-ended and two-ended cases.

Calculations of Temperature and Thermo-Elastic Displacement Distributions Generated by Laser-Ultrasonic Inspection System

Calculations of temperature and thermo-elastic displacement distributions generated by laser-ultrasonic inspection system are performed. The goal is to determine temperature distributions created in the test objects by generation laser and the respective thermo-elastic displacement distributions causing the ultrasonic waves propagating within the test objects during laser-ultrasonic inspection of parts. The respective heat propagation equation and thermo-elastic equation of motion are solved analytically for 1D and 3D test objects with different geometries.

Thermal analysis for laminated plates with arbitrary supports under non-uniform temperature boundary conditions

This paper aims to investigate thermal behaviors of composite laminated plates with different supports under non-uniform temperature boundary conditions. The thermo-elastic solutions of temperature, displacements, and stresses for the laminated plate with arbitrary layer numbers and thickness are obtained based on the three-dimensional theory of thermoelasticity. By applying the Fourier law of heat conduction and the thermoelastic equations, the state-space equations are established by employing the temperature, heat flux, displacements, and stresses as state variables without the hypothesis of displacement form. The differential quadrature method is introduced to discretize the in-plane state variables. On the basis of the continuities of state variables at the interfaces in the laminated plate, the relationships of temperature, heat flux, displacements, and stresses between the top and bottom surfaces can be derived. By simultaneously considering temperature and mechanical boundary conditions applied to the surfaces of the laminated plate, the initial state variables for temperature, displacements, and stresses can be determined uniquely. The errors in the temperature, displacements and stresses resulting from the differential quadrature method can be eliminated by incrementally augmenting the number of sampling points in the convergence study. The accuracy of the present method is thoroughly verified by comparing the current results with both the finite element solutions and the findings reported in previous literature. Finally, the influences of surface temperature, length-to-thickness ratios, material properties, layer numbers, and support types for the distributions of the temperature field, displacements, and stresses in the laminated plate are discussed in detail.

Generalized frequency shift and attenuation in simply-supported micro/nano-beam resonators with thermoelastic dissipation

The low frequency shift, weak frequency attenuation, and large quality factor (Q) are strongly required for micro/nano-resonators with excellent operation performance. Therefore, the accurate predictions of frequency shift, frequency attenuation, and Q based on thermoelastic dissipation (QTED ) are significant when optimally designing micro/nano-resonators. In this work, employing the modified-couple-stress (MCS) and nonlocal-single-phase-lag (NSPL) models, the generalized analytical expressions of the frequency-shift ratio, frequency-attenuation ratio, and QTED are first proposed for the simply-supported micro/nano-beam resonator with two-dimensional (2D) heat conduction. In the modeling procedure, utilizing the Galerkin approach, the function of 2D fluctuation temperature is solved from the governing equation of NSPL thermoelasticity. For simplicity, the function of one-dimensional (1D) fluctuation temperature is presented directly. Various expressions of complex natural frequency are obtained from the governing equation of MCS vibration with thermoelastic coupling, and then the models of the frequency-shift ratio, frequency-attenuation ratio, and QTED are developed based on the relevant expressions of complex natural frequency. It is confirmed that the present developed models with generality can effectively degenerate into the classical models. In the simulation cases, silicon (Si) that is a representative semiconductor material is selected. Results reveal that the MCS size-dependence effect can affect the frequency shift more markedly than the NSPL and heat-conduction-dimension (HCD) effects. Moreover, for the beam with a small thickness and operated in a high-order mode, the NSPL and HCD effects will significantly determine behaviors of the frequency shift, frequency attenuation, and QTED .

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.

Analysis of pressure derivatives of bulk modulus for materials at high compressions

Pressure derivatives of bulk modulus play a central role in describing thermoelastic behaviour and equation of state for materials at high pressures. It is demonstrated here that the relationships between pressure derivative of bulk modulus and volume compression used by earlier workers are not consistent with the thermodynamic constraints based on the second-, and third order pressure derivatives of bulk modulus, and third order Grüneisen parameter in the limit of infinite pressure. It is pointed out that the compatibility of Keane’s equation and the Stacey reciprocal equation of state as considered by Sinha et al. does not hold at high pressures. The Keane equation does not satisfy the thermodynamic constraint according to which the third order Grüneisen parameter must be less than the pressure derivative of bulk modulus at extreme compression. Values of second-, and third order pressure derivatives of bulk modulus for the Earth lower mantle and outer core determined using the K-primed equations reported by Sinha et al. are found to deviate significantly from the results based on the seismic data at finite pressures obtained by fitting the Stacey reciprocal K-primed EOS.

Solutions of the theory of thermoelasticity and thermal conductivity in the cylindrical coordinate system for axisymmetric temperature

The paper uses the system of Navier equations in the stationary case. A cylindrical coordinate system is considered, when the temperature does not depend on the angular variable. A partial solution of the system of Navier equations, which does not contain elastic displacements, is called a purely temperature solution. It was established that for purely temperature solutions the sum of normal stresses is zero and the volume deformation is equal T e  = 3 . An analytical expression of purely temperature displacements and stresses in the cylindrical coordinate system in the axisymmetric case was found. The solution of the boundary value problem of thermal conductivity, when the cylinder is heated on one end, cooled by liquid on the other with known heat losses on the side surface, is proposed. The solution of the boundary value problem of thermal conductivity for such a cylinder is given in the form of the sum of the basic temperature, which describes the heat balance, and the perturbed temperature. The basic temperature has a polynomial form and integrally satisfies the boundary conditions. The perturbed temperature has an exponential decrease with distance from the heated end and does not carry out integral heat transfer. The found dependencies were used and a new solution to the heat conduction equation was written in a cylindrical coordinate system. Simple formulas for expressing temperature changes have been obtained. A new temperature solution to the system of thermoelasticity equations in a cylindrical coordinate system has been written, when the temperature does not depend on the angular variable.

Thermo-mechanical behavior of 2D functionally graded porous-auxetic metamaterial rotating disk with an auxetic foundation

In the present study, the thermo-mechanical behavior of two-directional functionally graded porous-auxetic metamaterial rotating disk (2D-FGPAMRD) has been analyzed. The rotating disk subjected to combination of transient thermal and steady mechanical loads and supported by an auxetic foundation. Haber-Schaim (H-S) frictionless elastic foundation model is considered to drive the structure-structure interaction. The governing equations are obtained based on the thermoelasticity theory. The response of disk to mechanical and thermal loads has been obtained by solving nonhomogeneous governing equations. To this end, the alternating direction implicit (ADI) finite difference scheme and combination of state-space and differential quadrature (DQ) methods have applied to solve the Fourier heat conduction and thermoelasticity equations, respectively. In the numerical examples, the effects of porosity and auxeticity, as well as inhomogeneity indices of the disk on its thermo-mechanical behavior have been investigated for the first time. The results reveal that, the elastic field components of rapid heating disk are significantly affected by the nature of temperature diffusion, inhomogeneity indices and the auxeticity of the disk material. Besides, the changes of thermal expansion gradient indices provide a relaxation in the normal transverse stress.

Transient responses of dielectrics during ultrashort pulsed laser irradiation using a generalized thermoelastic model considering temperature-dependent thermophysical properties

A novel model involving electron dynamics, electron-lattice interactions, and thermoelastic excitation for dielectrics exposed to ultrashort pulsed laser is proposed. The thermoelastic equations are modified based on the G-L generalized thermoelasticity and consider the temperature-dependent thermophysical properties. An axisymmetric problem of ultrashort pulsed laser irradiation on fused silica is presented. The fully-coupled equations are solved simultaneously using the finite element method to obtain the transient electron density, electron temperature, lattice temperature, and stresses. The evolution of transient responses and the effects of laser intensity, pulse duration, and temperature-dependent thermophysical properties are analyzed in detail. It was found that there are kinks in the distribution of transient responses, which is attributed to the rapid change of reflectivity. The axial stress has wave-like characteristics. The electron temperature is slightly lower than the lattice temperature after the thermal equilibrium is achieved. Maybe the temperature-dependence of lattice thermal conductivity could be neglected to reduce computational expenses.