Equations Of Thermoelasticity Research Articles

Existence of a weak solution to a nonlinear fluid-structure interaction problem with heat exchange

In this paper, we study a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear thermoelasticity equations and encompasses a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise to a novel nonlinear moving boundary fluid-structure interaction problem involving heat exchange. The existence of a weak solution is obtained by combining three approximation techniques – decoupling, penalization and domain extension. In particular, the penalization and the domain extension allow us to use the methods already developed for compressible fluids on moving domains. In such a way, the proof is more elegant and the analysis is drastically simplified. Let us stress that this is the first time the heat exchange in the context of fluid-structure interaction problems is considered.

Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticity

This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and C∞ Semigroups

We consider Fourier multiplier systems on Rn with components belonging to the standard Hörmander class S1,0mRn, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector Λ⊂C (introduced by Denk, Saal, and Seiler) we show the generation of both C∞ semigroups and analytic semigroups (in a particular case) on the Sobolev spaces WpkRn,Cq with k∈N0, 1≤p<∞ and q∈N. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces.

Thermal analysis of the laser-induced thermal deformation of a diffractive optical element in a single-aperture coherent beam combining system

Diffractive optical element (DOE) is a critical device for combining multiple laser beams into a single beam in a coherent beam combining (CBC) architecture. This study proposes a determination method for calculating the intrinsic absorption rate of the DOE, and the corresponding experimental system is established. We present a theoretical thermal deformation model of the laser-irradiated DOE based on the thermoelastic equation and thermal conduction theory. The temperature and thermal deformation of the DOE are simulated using different parameters, including the laser power density, substrate size, substrate material, laser incident time, and clamping method. The simulations indicated that the thermal deformation is directly proportional to substrate area and inversely proportional to substrate thickness. The thermal deformation of the DOE can also be decreased by using a two-surface fixing method, and the maximum decrease is 4.4%. The quantitative discussion and analysis of the DOE temperature field and thermal deformation are important for designing a DOE to increase the combining efficiency and improve the combined beam quality of a practical DOE-based CBC system.

Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model

In this work, the effect of the fractional time derivative on the piezo-thermo-elastic medium is studied, using the hybrid Laplace transform and finite element methods (LFEM). The generalized fractional piezoelectric–thermoelastic basic equations for piezo-thermo-elastic medium are formulated. The Laplace transforms are used for the time derivatives, and the finite element method is used to discretize for the space derivatives. The inversions process is performed using the Stehfest numerical technique. The finite element approach is used to obtain the solutions of complex coupled formulations of this problem. The effects of fractional time derivative and thermal relaxation time on piezoelectric–thermoelastic mediums are studied. It can be seen from the distribution that the thermal-induced displacement, the temperature and the stress of the medium increase at a high fractional parameter.

Schmidt method to study the disturbance of steady-state heat flows by an arbitrary oriented crack in bonded functionally graded strips

This article aims to investigate the impact of a partially insulated crack on vertical and horizontal components of heat flow. The crack is arbitrary oriented and its center is located at the interface of bonded finitely thick functionally graded strips. Employing the integral transformation, the thermoelastic equations are reduced to a set of singular integral equations. The unknown variables of these equations are the jump in temperature and displacements across the crack surfaces. Representing the unknowns as a series of Jacobi polynomials, the solutions of the singular integral equations are obtained by employing the Schmidt method which further helps to determine the analytical forms of crack tip heat flux intensity factors (HFIFs) and thermal stress intensity factors (TSIFs). The results of the present study are also validated for a particular case.To quantify the strength of heat flow components, the crack tip HFIFs and TSIFs as a function of crack angle, crack insulation parameter, strips’ width, and non-homogeneity parameters’ ratio are demonstrated pictorially.

Size-dependent thermoelastic damping analysis in nanobeam resonators based on Eringen’s nonlocal elasticity and modified couple stress theories

Thermoelastic damping has emerged as a critical issue in the modeling and design of micro and nanomechanical systems. Therefore, an extensive research interest is being devoted towards the reduction in thermoelastic damping for micro and nanomechanical systems. The present study intends to examine thermoelastic damping in nanobeam resonators using the modified couple stress theory and Eringen’s nonlocal elasticity theory, thus so-called modified nonlocal couple stress theory within the context of the recently proposed Moore–Gibson–Thompson thermoelasticity theory. In order to observe size effects, the size-dependent coupled thermoelastic equations are derived by combining modified couple stress theory and nonlocal elasticity theory in the frame of Moore–Gibson–Thompson thermoelasticity. The coupled governing equations for thermoelastic damping of nanobeam resonators are solved analytically in terms of the inverse quality factor. The size-dependent thermoelastic damping of nanobeams is presented graphically with the help of numerical results predicted by modified nonlocal couple stress theory. The results obtained under the combined effects of modified couple stress theory and nonlocal theory in the present context are further compared to those of the classical, modified couple stress, and nonlocal elasticity theories as special cases of the modified nonlocal couple stress theory. It is observed that thermoelastic damping weakens at the submicron scale under Eringen’s nonlocal elasticity theory, whereas it becomes greater at the submicron scale under modified couple stress theory. However, at the submicron scale, the modeling of nanobeam resonators using modified nonlocal couple stress theory reveals a smaller amount of thermoelastic damping than modified couple stress, nonlocal, and classical theories. In addition, as compared to the Green–Naghdi thermoelasticity theory of type III (GN-III), the Moore–Gibson–Thompson model predicts a higher thermoelastic damping value, while agreeing with the results predicted by the Lord–Shulman (LS) model under modified nonlocal couple stress theory. Some important points highlighted here are believed to be useful in the design of mechanical resonators at micron and submicron scales.

An accurate thermoelastic model and thermal output error analysis of a hemispherical resonator gyroscope under varying temperatures

This paper proposes an accurate thermoelastic model of a hemispherical shell resonator (TM-HSR) taking into consideration thermal stress and reveals the effect of its thermal deformation on the hemispherical resonator gyroscope (HRG) output error for the first time. First, a set of thermoelastic dynamic equations of the TM-HSR are derived by leveraging the thermoelastic shell theory, and the relationship between temperature and the equivalent thermal load is determined. Next, an approximate analytical solution of thermal deformation of HSRs is derived using Bubnov-Galerkin method and its accuracy is evaluated by comparing its result with ANSYS simulation results of HSRs with different structures. The approximate analytical solution is found to be very accurate, which is important for temperature testing of HRGs and application to various vibratory sensors with hemispherical shell structures. Subsequently, the effects of thermal deformations under uniform and nonuniform temperature distributions on the HRG output error are revealed by using a thermal output error model. Comparative simulation and experimental results demonstrate the effectiveness and practicability of the proposed model. The proposed model can be used for accurate error analysis and developing a compensation scheme for the thermal output error of HRGs.

Structural Stability on the Boundary Coefficient of the Thermoelastic Equations of Type III

This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-infinite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is obtained. Based on the results of decay, the continuous dependence and the convergence results on the boundary coefficient are established by using the differential inequality technique and the energy analysis method. The main work of this paper is to extend the study of continuous dependence to a semi-infinite cylinder, which can be used as a reference for the study of other types of partial differential equations.

Thermoelastic wave propagation in thin beams under thermal shock loading

An analytical solution is presented for the thermoelastic longitudinal and flexural wave propagation in transversely isotropic thin beams subjected to a thermal shock loading using the Lord–Shulman generalized thermoelasticity theory. The formulation considers a general through-thickness profile of the applied thermal loading, does not assume the across-thickness distribution of the temperature field and considers a general convective boundary condition at the beam surfaces. The thin beam is modeled using the Euler–Bernoulli beam theory. The governing equations of motion and the variationally consistent boundary conditions are derived from the extended Hamilton’s principle. The coupled thermoelastic equations are solved in the Laplace domain satisfying the thermal and mechanical boundary conditions using the homotopy perturbation method. The Laplace inversion to obtain the final solution in the time domain is performed numerically by using Durbin’s method. The formulation is validated by comparing the results for longitudinal wave propagation response under uniform thermal loading with those available in the literature. The developed solution is then employed to study the longitudinal and flexural wave propagation behaviour of a cantilever beam subject to uniform and non-uniform symmetric, and linear and non-linear anti-symmetric thermal shock loading applied at the free end. The effects of thermal boundary conditions at the beam surfaces and the relaxation time parameter on the response of the beam are illustrated.

MODELING THE EFFECT OF STOCHASTIC DEFECTS FORMED IN PRODUCTS DURING MACHINING ON THE LOSS OF THEIR FUNCTIONAL DEPENDENCIES

The article investigates the influence of hereditary defects formed in the surface layer of products from metals of heterogeneous structure on the quality of surfaces treated with finishing methods. The research is based on an integrated approach based on the results of the deterministic theory of defect development and methods of probability theory. The treated layer of the product is considered as a medium weakened by random defects that do not interact with each other, namely: structural changes, cracks, inclusions, the parameters of which are random variables with known laws of their probability distribution. The causes of structural changes, crack formation on the treated surface product depending on different types of probability distribution of dimensions are investigated: length, depth of defects, and their orientation. From these positions, technological possibilities of their elimination by definition of branch of combinations of the technological parameters providing necessary quality of the processed surfaces are considered. Modeling of thermomechanical processes in the treated surface containing hereditary defects is carried out based on thermoelastic equations with discontinuous boundary conditions in the places of their accumulation. The research used the apparatus of boundary value problems of mathematical physics equations, the method of singular integral equations for solving problems of fracture mechanics, Fourier-Laplace integral transformations for obtaining exact solutions, the method of constructing discontinuous functions. The dependences determining the intensity of stresses in the vertices of hereditary defects are obtained. A method for predicting the nature of crack formation depending on the probability distribution of defects, the values of heat flux entering the surface layer of the processed product has been developed. It is established that the increase in the homogeneity of the material leads to an increase in the value of heat flux, which corresponds to a fixed probability of failure.

Representation of the thermo-stress state of a plate based on the 3D elasticity theory

The general formulation of the 3D temperature static problem of the theory of elasticity is considered. A new representation of the general solution of static equations of thermoelasticity is proposed. The three-dimensional stress state of a thick or thin plate are divided into three parts: bending of the plate, symmetric compression of the plate on it’s flat end surfaces and symmetric temperature problem. Third thermoelastic stress state of the plate is studied in detail. It have been reduced to a two-dimensional state after integration the stresses components along the normal to the middle surface. The displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the flat surfaces of the plate. Only the assumption has been used that the normal stresses perpendicular to the middle surface are insignificant. The introduced harmonic functions are determined from the boundary conditions on the lateral surface of the plate. The formula for the experimental determination of the sum of normal stresses has been found through the measured deflections of the end surfaces of the plate.

Construction of static solutions of the equations of elasticity and thermoelasticity theory

New solutions to the theories of thermoelasticity and elasticity in the Cartesian coordinate system are found in this paper. New explicit partial solutions of thermoelasticity equations, when the temperature field is defined by 3D or 2D harmonic functions, are constructed. Displacements, deformations, and stresses determined by these partial solutions are called temperature functions. A simple formula for the expression of normal temperature stresses is obtained and it is shown that their sum is zero. Separate cases when the temperature depends on the product of harmonic functions of two variables on the degree of coordinate z are also considered. Partial and general solutions are derived for them. General solutions of thermoelasticity equations (Navier’s equations) through four harmonic functions, when the temperature field is given three-dimensional or two-dimensional harmonic functions, are constructed. The thermoelastic state of the body is divided into symmetric and asymmetric stress states. It is proposed to present the solutions of the theory of elasticity, which are expressed by the product of the harmonic function of two variables to the degree of the coordinate. Polynomial solutions that depend on three coordinate variables are recorded. An example of the application of the proposed solution is given.

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.

Influence of Second-Order Effects on Thermoelastic Behaviour in the Proximity of Crack Tips on Titanium

BackgroundThe Stress Intensity Factor (SIF) is used to describe the stress state and the mechanical behaviour of a material in the presence of cracks. SIF can be experimentally assessed using contactless techniques such as Thermoelastic Stress Analysis (TSA). The classic TSA theory concerns the relationship between temperature and stress variations and was successfully applied to fracture mechanics for SIF evaluation and crack tip location. This theory is no longer valid for some materials, such as titanium and aluminium, where the temperature variations also depend on the mean stress.ObjectiveThe objective of this work was to present a new thermoelastic equation that includes the mean stress dependence to investigate the thermoelastic effect in the proximity of crack tips on titanium.MethodsWestergaard’s equations and Williams’s series expansion were employed in order to express the thermoelastic signal, including the second-order effect. Tests have been carried out to investigate the differences in SIF evaluation between the proposed approach and the classical one.ResultsA first qualitative evaluation of the importance of considering second-order effects in the thermoelastic signal in proximity of the crack tip in two loading conditions at two different loading ratios, R = 0.1 and R = 0.5, consisted of comparing the experimental signal and synthetic TSA maps. Moreover, the SIF, evaluated with the proposed and classical approaches, was compared with values from the ASTM standard formulas.ConclusionsThe new formulation demonstrates its improved capability for describing the stress distribution in the proximity of the crack tip. The effect of the correction cannot be neglected in either Williams’s or Westergaard’s model.

Analytical solution for thermoelastic oscillations of nonlocal strain gradient nanobeams with dual-phase-lag heat conduction

In order to examine the impact of structural and thermal scale parameters on thermoelastic vibrations of Euler-Bernoulli nanobeams, this article intends to provide a size-dependent generalized thermoelasticity model with the help of nonlocal strain gradient theory (NSGT) in conjunction with dual-phase-lag (DPL) heat conduction model. To highlight the role of each scale parameter in size-dependent motion and heat conduction equations, normalized forms of these nonclassical coupled thermoelastic equations are extracted by introducing and exploiting some dimensionless parameters. By exploiting power series expansion as a general solution for arbitrary boundary conditions, system of partial differential equations reduces to system of ordinary differential equations in time domain. The Laplace transform is then utilized to attain the analytical solution of this set of differential equations. By conducting a case study on a hinged-hinged nanobeam, with the aim of illustrating the impact of nonclassical structural and thermal parameters on thermoelastic vibrations, size-dependent numerical results are compared with those estimated by classical elasticity theory and heat conduction model. Meaningful disparity between classical and nonclassical results reveals the pivotal role of scale parameters in realistic assessment of thermoelastic behavior of nanobeams. Outcomes also indicate that corresponding to the relationship between the magnitudes of two scale parameters of NSGT, the nonclassical model of nanobeam can exhibit either softening or hardening behavior compared to the classical one. This affirms the ability of NSGT to justify both stiffness softening and stiffness hardening phenomenon in nanostructures.

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.

AVERAGED CONTROLLABILITY OF THERMOELASTICITY EQUATIONS. AVERAGE STATE OF A RECTANGULAR PLATE

The concept of averaged controllability has been introduced relatively recently aiming to analyse the controllability of systems or processes containing some important parameters that may affect the controllability in usual sense. The averaged controllability of various specific and abstract equations has been studied so far. Relatively little attention has been paid to averaged controllability of coupled systems. The averaged state of a thermoelastic rectangular plate is studied in this paper using the well-known Green's function approach. The aim of the paper is to provide a theoretical background for further exact and approximate controllability analysis of fully coupled thermoelasticity equations which will appear elsewhere.

Size-dependent generalized thermoelasticity model for thermoelastic damping in circular nanoplates

This paper assesses thermoelastic damping (TED) in circular nanoplates by incorporation of the small-scale effect into structural and thermal domains. The nonlocal elasticity theory and dual-phase-lag (DPL) heat conduction model are exploited for achieving the size-dependent coupled thermoelastic equations. By choosing time-harmonic and asymmetric form for deflection and temperature change, and solving the size-dependent thermoelastic eigenvalue problem, the damped natural frequency of circular nanoplate is extracted. On the basis of the complex frequency approach, an analytical relation, for the description of TED in circular nanoplates, is derived. For different boundary conditions and vibration modes, a comparison study is performed between the size-dependent results and those provided by classical continuum mechanics and heat conduction theories. Outcomes show a discrepancy between results acquired by the size effect on the structure and heat conduction. It is elucidated that how nonlocal and DPL characteristic parameters can represent the size-dependent phenomenon and can affect the magnitude of TED. A comprehensive parametric study is conducted to specify the influence of boundary constraints, vibration mode and the type of material on the amount of energy loss from TED.

РОЗРОБКА МЕТОДУ ВИЗНАЧЕННЯ ТЕРМОПРУЖНІХ НАПРУГ У ПЛАСТИНАХ GAAS

Purpose. А study of the temperature field of the melt during the cultivation of GaAs single crystals from under a layer of liquid flux. Thermoplastic stresses were measured on plates cut from the upper, middle and lower part of the ingot of undoped GaAs.Methodology. Finite element method is used to calculate temperature profiles and internal thermoplastic stresses. The mechanism of theoretical and experimental researches which allow to predict thermoplastic stresses in the course of cultivation of ingots is offered. For the analysis and mathematical calculations of the stationary differential equation in partial derivatives and the equations of thermoelasticity, respectively, use the finite element method, the calculations were performed in the programs THERMIX and INCA. Temperature profiles and internal thermoplastic stresses were calculated. Thermoplastic stresses were measured on plates cut from the upper, middle and lower part of the ingot of undoped GaAs (cm-3) with a thickness of 1 mm with a resistivity of 108 Ohm x cm, diameter 50 mm, orientation (111).The axial temperature gradient is determined. Experimentally obtained values of ison voltage lines along the plane of GaAs plates cut from different parts of the ingot. To measure the internal (thermoplastic) stresses in the work used the method of photoelastic-guests in infrared polarized light. The integrated picture of thermoplastic stresses was obtained using the "Polaron" installation, and the point measurement with the construction of the iso-voltage line was obtained on the "Polaron-2" installation. Originality. As a result of the research it can be concluded that the mechanism of theoretical and experimental researches is offered in the work, which allows to predict thermoplastic stresses in the process of growing GaAs ingots and, finally, to develop a procedure for reducing dislocation density in GaAs ingots. The practical value. the proposed method will improve the technology of growing ingots of gallium arsenide with a more homogeneous technology, which will be a good indicator for the future creation of gas sensors from this material.