Boundary Value Problems 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.

Investigation of the Stress-Strain State of a Rectangular Plate after a Temperature Shock

In this paper, the temperature shock phenomenon is considered. This phenomenon occurs during the operation of engineering structures on Earth and in outer space. A rectangular plate has been selected as a structural element exposed to temperature shock. It has a rigidly sealed edge and three free edges. A one-dimensional third initial boundary value problem of thermal conductivity was posed and solved to study the stress–strain state of the plate. Fourier’s law was used to solve this problem, taking into account the inertial term, since the temperature shock is a fairly fast-dynamic phenomenon. It was believed that all the thermophysical properties of the plate are constant and do not depend on its temperature. As a result, the temperature field of the plate was obtained after the temperature shock. This temperature field generates temperature stresses inside the plate, which lead to temperature deformations. To determine these deformations, the initial boundary value problem of thermoelasticity was posed and solved in this work. The Sophie Germain equation was used while solving this problem. To describe the plate, the theory of flexible plates was used, taking into account the stresses in the middle surface of the plate. Next, the accuracy of analytical solutions for the points displacement of a homogeneous plate subjected to a temperature shock was investigated. The temperature field was constructed using a numerical simulation. Functions of the displacement vector components were obtained using approximate analytical solutions. The accuracy of approximate analytical solutions for the components of the plate points deformation vector was estimated. The obtained results allow us to describe the stress–strain state of the plate after the temperature shock. The results of this work can be used in the design of engineering structures for both terrestrial and space purposes in terms of stability calculations and the implementation of deformation constraints.

Temperature stresses in a rectangular two-layer plate under the action of a locally distributed temperature field

A rectangular isotropic two-layer plate of an irregular structure is considered, the edges of which are freely supported, and a constant temperature is maintained on them. Two-dimensional Kirchhoff-type thermoelasticity equations and two-dimensional heat equations written for an inhomogeneous material were used to study the temperature stresses in the plate. Using the method of double trigonometric series in spatial variables and the Laplace integral transformation over time, the general solutions of boundary value problems of thermoelasticity and heat conductivity for this plate under the action of a locally distributed temperature field specified at the initial moment of time are written down. The normal stresses in the layers of the plate are numerically analyzed depending on the geometric parameters, heat transfer coefficient, and time.

Plane Boundary Value Problems of Thermoelasticity and Their Generalized Solutions

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation by use general functions method (GFM). The generalized solutions of boundary value problems are constructed in the space of generalized vector functions. Fundamental Green tensor, stress tensor and their antiderivatives over time are used for construction their regular integral representations. By use analogue of Gauss formula for fundamental stress the singular boundary integral equations are constructed to determine the unknown boundary functions. The resulting formulas have important engineering application. They make possible to determine the thermal stressed state of the medium by the boundary values of stresses, displacements, temperature and heat flux, without solving singular boundary integral equations. Because for real engineering problems these physical characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stressstrain state. The last one is very important in designing structures made of thermoelastic materials.

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

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.

Analytical Solution for the Lifetime of a Spherical Shell of Arbitrary Thickness Under the Pressure of Corrosive Environments: The Effect of Thermal and Elastic Stresses

Abstract This article presents analytical solutions to initial boundary value problems of thermoelasticity with a priori unknown evolving boundaries. To be more precise, we consider a spherical shell of arbitrary thickness subjected to the internal and external pressures of corrosive media at generally different temperatures, with taking into account the mechanochemical effect and inhibition of corrosion process. Conditions under which the circumferential stress can serve as the equivalent stress are determined. It was found that the life of the shell was influenced by the competing effects of the pressures and temperatures on the corrosion rate and stress values, as well as by possible moving the location of the maximal stress. It was also concluded that the elastic stress component played a decisive role in the synergistic growth of the total thermoelastic stress and the corrosion rate.

ON THE SOLUTION OF THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY OF ISOTROPIC CONTINUOUSLY INHOMOGENEOUS BODIES BY THE METHOD OF BOUNDARY INTEGRAL EQUATIONS

ON THE SOLUTION OF THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY OF ISOTROPIC CONTINUOUSLY INHOMOGENEOUS BODIES BY THE METHOD OF BOUNDARY INTEGRAL EQUATIONS

Метод обобщенных функций в плоских краевых задачах несвязанной термоэластодинамики

Non-stationary boundary value problems of uncoupled thermoelasticity are considered.             A method of boundary integral equations (BIME) in the initial space-time has been developed for solving non-stationary boundary value problems of thermoelasticity under plane deformation. On the basis of the method of generalized functions, generalized solutions of boundary value problems are constructed using the Green's function for the heat equation and the Green's tensor of the Lame equations under the action of non-stationary power and heat sources of various types. Integral representations of the solution of boundary value problems are obtained. These solutions allow, based on known boundary values and initial conditions (displacements, temperature, stresses and heat flux), to determine the thermally stressed state of the medium under the influence of various power and thermal loads. Resolving boundary integral equations are constructed to determine the unknown boundary functions.

Three-dimensional nonlocal thermoelasticity in orthotropic medium based on Eringen’s nonlocal elasticity

ABSTRACT The present article deals with thermoelastic boundary value problem in three-dimensional orthotropic medium based on Eringen’s nonlocal elasticity. A time-dependent thermal shock is applied on the traction-free surface of homogeneous orthotropic solid and the problem is treated in the context of Green-Naghdi model type III. The governing equations of three-dimensional problem of generalized thermoelasticity are obtained as a vector matrix differential equation form by employing normal mode analysis to the considered equations which are then solved by eigenvalue approach. The effect of nonlocal parameter on the displacements and temperature are presented graphically.

Explicit solution of one boundary value problem of thermoelasticity for a circle with diffusion, microtemperatures, and microconcentrations

In the present paper, an explicit solution of the Dirichlet BVP for an isotropic circle with diffusion, microtemperatures, and microconcentrations is presented. The general solution of the system of equations for isotropic materials with diffusion, microtemperatures, and microconcentrations by means of elementary (harmonic, bi-harmonic, and meta-harmonic) functions is constructed. The obtained solution of the Dirichlet BVP is presented as absolutely and uniformly convergent series.

Development of a structurally similar element of a helicopter blade with an active twist system

A mathematical model is formulated for calculating the stress-strain state (SSS) of piezoelectric layered polymer composite materials (PCM) equipped with a control piezoactuator. The model implements the principle of thermo-piezoelectric analogy, which allows to move from the coupled boundary value problem of electroelasticity to the boundary value problem of thermoelasticity. The mathematical model was tested with a numerical calculation of the mechanical behavior of a sample structurally similar to a rotor blade, equipped with control piezo-actuators and made of equal strength and unidirectional fiberglass.

Three-dimensional thermoelastic problem in orthotropic medium

The present article is concerned with a thermoelastic boundary-value problem with a time-dependent thermal shock on traction-free half-space for a homogeneous orthotropic heat-conducting solid. The governing equations of the three-dimensional problem of generalized thermoelasticity in orthotropic medium are obtained as a vector-matrix differential equation form by employing normal mode analysis to the considered equations which is then solved by the eigenfunction expansion method. The distribution of thermal stress, displacements, and temperature are presented graphically and compared with other thermoelastic models.

Numerical calculation of the stress-strain state of piezoelectric layered polymer composite materials equipped with a control piezoactuator

A mathematical model is formulated for calculating the stress-strain state (SSS) of piezo-electric layered polymer composite materials (PCM) equipped with a control piezoactuator. The model implements the principle of thermo-piezoelectric analogy, which allows to move from the coupled boundary value problem ofelectroelasticityto the boundary value problem of thermoelasticity. A comparison of the deformation fields for model layered samples from PCM with a different arrangement of piezoactuatorswas carried out. Verification of the developed mathematical models was carried out. The main patterns and mechanisms are revealedfor controlling the geometry of a layered PCM using embedded piezo-actuators.

Thermoelastic influence functions and solution to a locally mixed BVP for a strip

ABSTRACTThis article presents in closed form new influence functions for displacements and stresses to a boundary value problems (BVP) of thermoelasticity within a strip, caused by a unit point heat source. We also obtain the respective new integration formula of Green’s type, which directly determines the thermal stresses in the form of integrals of the products between specified internal heat sources, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). The general Green’s type integral formula permits to derive new solution to one particular BVP of thermoelasticity for a strip in the form of elementary functions. Graphical representation of thermal stresses, created by an internal point heat source and by a boundary temperature, is included.

A method to derive thermoelastic Green’s functions for bounded domains (on examples of two-dimensional problems for parallelepipeds)

This article presents new constructive formulas to steady-state thermoelastic Green’s functions for a plane generalized boundary value problem of thermoelasticity for a generalized rectangle. The constructive formulas are expressed in terms of Green’s functions for Poisson’s equation. These results are formulated in a special theorem, which is proved using the author’s developed harmonic integral representations method. On the base of derived constructive formulas, it is possible to obtain many analytical expressions for Green’s functions for thermoelastic displacements and stresses to 28 concrete boundary value problems for: rectangle-16, half-strip-8, strip-4. An example of such kind is presented for a concrete plane boundary value problem for a rectangle, Green’s functions of which are presented in the form of a sum of elementary functions and ordinary series. These results are presented in another theorem, which is proved on the base of derived general constructive formulas. In the particular case for a half-strip and strip, ordinary series vanish and Green’s functions are presented in elementary functions. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle subjected to a constant boundary temperature gradient are also derived. Numerical investigation has shown that the infinite series are convergent. All solutions obtained for thermal stresses, caused by a constant temperature gradient and by a unit heat source are validated by checking the respective equilibrium equation and continuity of deformation equations (Beltrami–Michel equations), written in the terms of thermal stresses. The graphics for thermal stresses and their infinite series also are presented.

Steady-state Green's functions for thermal stresses within rectangular region under point heat source

ABSTRACTThis article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.

Explicit Solutions of Boundary-Value Problems of Thermoelasticity with Microtemperatures for a Half-Space

We consider the statics case of the theory of linear thermoelasticity with microtemperatures materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in this paper is expressed by means of four harmonic and three metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrare an application of these formulas to the III type boundary value problem for a half-space. Uniqueness theorems are proved. Solutions are obtained in quadratures.

Thermoelastic Analysis of Functionally Graded Hollow Circular Disc

Abstract A thermoelastic boundary value problem of a hollow circular disc made of functionally graded materials with arbitrary gradient is analysed. The steady-state temperature distribution is assumed to be the function of the radial coordinate with prescribed temperature at the inner and outer cylindrical boundary surfaces. The material properties are assumed to be arbitrary smooth functions of the radial coordinate. A coupled system of ordinary differential equations containing the radial displacement and stress function is derived and used to get the distribution of thermal stresses and radial displacements caused by axisymmetric mechanical and thermal loads. General analytical solutions of functionally graded disc with thermal loads are not available. The results obtained by the presented numerical method are verified by an analytical solution. The considered analytical solution is valid if the material properties, except the Poisson ratio, are expressed as power functions of the radial coordinate.