Thermoelastic interactions without energy dissipation due to a line heat source (original) (raw)

1998, Acta Mechanica

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Abstract

The linear theory of thermoelasticity without energy dissipation is employed to study thermoelastic interactions due to a continuous line source of heat in a homogeneous and isotropic unbounded solid. Laplace and Hankel transforms are employed to solve the problem. Exact expressions, in closed form, for the temperature and stress fields are obtained. Numerical results for a hypothetical, copper-like material are presented with the view of illustrating the theoretical results.

Mechanical and Thermal Sources in a Generalized Thermoelastic Half-Space

Journal of Thermal Stresses, 2001

A dynamical two-dimensional problem of thermoelasticity has been considered to investigate the disturbance due to mechanical (horizontal or vertical) and thermal source in a homogeneous, thermally conducting orthorhombic material. Laplace-Fourier transforms are applied to basic equations to form a vector matrix differential equation, which is then solved by eigenvalue approach. The displacements, stresses and temperature distribution so obtained in the physical domain are computed numerically and illustrated graphically. The numerical results of these quantities for zinc crystal-like material are illustrated to compare the results for different theories of generalised thermoelasticity for an insulated boundary and a temperature gradient boundary.

Generalized Thermoelastic Interaction in a Half-Space under a Nonlocal Thermoelastic Model

Mathematics

In the current article, the nonlocal thermoelastic theory is used to discuss the wave propagation in unbounded thermoelastic materials. Due to the inclusion of relaxation time in thermal conduction formulation and the equations of motion, this model was developed using Lord and Shulman’s generalized thermoelastic model. The theory of the nonlocal continuum proposed by Eringen is used to obtain this model. The integral transforms of the Laplace transform methods used to generate an analytical solution for physical variables are utilized to produce the analytical solutions for the thermal stress, displacement, and temperature distribution. The effects of nonlocal parameters and relaxation time on the wave propagation distributions of physical fields for material are visually shown and explored.

On Linear Theory of Thermoelasticity for an Anisotropic Medium Under a Recent Exact Heat Conduction Model

Communications in Computer and Information Science

The aim of this paper is to discuss about a new thermoelasticity theory for a homogeneous and anisotropic medium in the context of a recent heat conduction model proposed by Quintanilla (2011). The coupled thermoelasticity being the branch of science that deals with the mutual interactions between temperature and strain in an elastic medium had become the interest of researchers since 1956. Quintanilla (2011) have introduced a new model of heat conduction in order to reformulate the heat conduction law with three phase-lags and established mathematical consistency in this new model as compared to the three phase-lag model. This model has also been extended to thermoelasticity theory. Various Taylor's expansion of this model has gained the interest of many researchers in recent times. Hence, we considered the model's backward time expansion of Taylor's series upto second-order and establish some important theorems. Firstly, uniqueness theorem of a mixed type boundary and initial value problem is proved using specific internal energy function. Later, we give the alternative formulation of the problem using convolution which incorporates the initial conditions into the field equations. Using this formulation, the convolution type variational theorem is proved. Further, we establish a reciprocal relation for the model.

Thermoelastic Disturbances in Transversely Isotropic Half-Space

2008

In this article attempt is made to study the fracas due to a thermal line load in a homogeneous transversely isotropic half-space in linearized theory of generalized thermo elasticity. A combination of Fourier and Laplace transform technique is applied to obtain the solutions of governing equations. The transformed solutions are then inverted using Cagniard technique for small times. The results obtained theoretically, for temperature, thermal stresses are computed numerically for a crystal of zinc, and found that variations in stresses and temperature are more prominent at small times and decrease with passage of time. The

Thermoelastic Disturbances in a Transversely Isotropic Half-Space Due to Thermal Point Load

The objective of this paper is to study disturbances due to thermal point load in a homogeneous transversely isotropic half-space in genera-lized thermoelasticity. A combination of the Fourier and Hankel trans-form technique is applied to obtain the solutions to governing equations. Cagniared's technique is used to invert the transformed solutions for small times. Theoretically obtained results, for temperature, stresses are computed numerically for a zinc material. It is found that variations in stresses and temperature are more prominent at small times and decrease with passage of time. Theg results obtained theoretically are represented graphically at different values of thermal relaxation times.

Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc

European Journal of Mechanics - A/Solids, 2007

This paper deals with the problem of thermoelastic interactions in a homogeneous isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity with energy dissipation (TEWED). The governing equations are expressed in Laplace-Fourier transform domain. The inversion of Fourier transform is carried out analytically while that of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained and presented graphically. A comparison has been made with the results obtained earlier for thermoelasticity without energy dissipation (TEWOED).

A Generalized Thermoelasticity Problem for a Half-Space with Heat Sources and Body Forces

International Journal of Thermophysics, 2010

In this work, we solve a transient two dimensional problem for an infinite thermoelastic half space whose surface is traction free and subjected to a known axisymmetric temperature distribution. The problem is considered within the context of the theory of generalized thermoelasticity with one relaxation time. Axisymmetric heat sources permeate the medium. The problem is solved using the Laplace and Hankel transforms. The solution in the transformed domain is obtained by using a direct approach. The inverse transforms are obtained using a numerical technique. Numerical results for the temperature, stress and displacement distributions are obtained and represented graphically.

Integral representation for three-dimensional steady state size-dependent thermoelasticity

arXiv (Cornell University), 2017

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multiphysical response, especially in the linear case for which boundary-only formulations are possible. This paper presents the integral equation formulation for size-dependent linear thermoelastic response of solids under steady state conditions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couplestress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this general size-dependent thermoelastic theory. Then, an integral equation formulation for the three-dimensional isotropic case is derived, along with the corresponding singular infinite space fundamental solutions or kernel functions. Remarkably, for isotropic materials within this theory, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter l , while the thermal coupling is defined in terms of the thermal expansion coefficient  , as in the classical theory of steady state isotropic thermoelasticity. This simplification permits the development of the required kernel functions from previously defined fundamental solutions for isotropic media.

Exact Solution of Thermoelastic Problem for a One-Dimensional Bar without Energy Dissipation

ISRN Mechanical Engineering, 2014

We consider a homogeneous isotropic thermoelastic half-space in the context of the theory of thermoelasticity without energy dissipation. There are no body forces or heat source acting on the half-space. The surface of the half-space is affected by a time dependent thermal shock and is traction free. The Laplace transform with respect to time is used. The inverse transforms are obtained in an exact manner for the temperature, thermal stress, and displacement distributions. These solutions are represented graphically and discussed for several cases of the applied heating. Comparison is made between the predictions here and those of the theory of thermoelasticity with one relaxation time.

Spatial behavior for some non-standard problems in linear thermoelasticity without energy dissipation

Journal of Mathematical Analysis and Applications, 2010

In the present paper we consider a prismatic cylinder occupied by an anisotropic and homogeneous compressible linear thermoelastic material within the framework of the linear theory of thermoelasticity without energy dissipation. The cylinder is subject to zero body force and heat supply and zero lateral specific boundary conditions and the motion is induced by a time-dependent displacement and thermal displacement specified pointwise over the base. Further, the motion is constrained such that the displacement, thermal displacement, velocity and temperature variation at points in the cylinder and at a prescribed time are in given proportions to, but not identical with, their respective initial values. It is shown that certain integrals of the solution spatially evolve with respect to the axial variable. Conditions are derived that show the integrals exhibit alternative behavior and in particular for the semi-infinite cylinder that there is either at least exponential growth or at most exponential decay, provided the elasticity tensor is positive definite or strongly elliptic.

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Thermoelastic Processes by a Continuous Heat Source Line in an Infinite Solid via Moore–Gibson–Thompson Thermoelasticity

Materials

Many attempts have been made to investigate the classical heat transfer of Fourier, and a number of improvements have been implemented. In this work, we consider a novel thermoelasticity model based on the Moore–Gibson–Thompson equation in cases where some of these models fail to be positive. This thermomechanical model has been constructed in combination with a hyperbolic partial differential equation for the variation of the displacement field and a parabolic differential equation for the temperature increment. The presented model is applied to investigate the wave propagation in an isotropic and infinite body subjected to a continuous thermal line source. To solve this problem, together with Laplace and Hankel transform methods, the potential function approach has been used. Laplace and Hankel inverse transformations are used to find solutions to different physical fields in the space–time domain. The problem is validated by calculating the numerical calculations of the physical ...

Effect of two temperature and anisotropy in an axisymmetric problem in transversely isotropic thermoelastic solid without energy dissipation and with two temperature

The present study is concerned with the thermoelastic interactions in a two dimensional homogeneous, transversely isotropic thermoelastic solids without energy dissipation and with two temperatures in the context of Green -Naghdi model of type-II. The Laplace and Hankel transforms have been employed to find the general solution to the field equations. Concentrated normal force, normal force over the circular region and concentrated thermal source and thermal source over the circular region have been taken to illustrate the application of the approach. The components of displacements, stresses and conductive temperature distribution are obtained in the transformed domain. The resulting quantities are obtained in the physical domain by using numerical inversion technique. Numerically simulated results are depicted graphically to show the effect of two temperature and anisotropy on the components of normal stress, tangential stress and conductive temperature.

Thermoelastic interactions in a hollow cylinder due to a continuous heat source without energy dissipation

Materials Research Express

The linear generalized Green-Naghdi thermoelasticity theory without energy dissipation is employed. The study of thermoelastic interactions in a hollow cylinder under a continuous heat source is carried out. Firstly, Laplace and Hankel transforms are employed to solve the problem without the time domain. Then, the state space approach is employed to get the exact solution of the problem in the space domain. Once again, the inverse Laplace transforms is used to get the solutions in the time domain. Accurate terminologies for the temperature, thermoelastic potential, axial displacement, dilatation, and stresses are derived. Numerical outcomes for field variables are presented with the view of illustrating the theoretical results.

Reciprocal and variational principles in linear thermoelasticity without energy dissipation

Mechanics Research Communications, 2010

In the present paper we consider the equations which govern the behavior of an anisotropic and inhomogeneous centrosymmetric material within the framework of the linear theory of thermoelasticity without energy dissipation. We establish a reciprocal relation which is based on a characterization of the boundary-initial value problem in which the initial conditions are incorporated into the field equations. Further, a variational principle is presented too.

On a two-dimensional model of generalized thermoelasticity with application

Scientific Reports

A 2D first order linear system of partial differential equations of plane strain thermoelasticity within the frame of extended thermodynamics is presented and analyzed. The system is composed of the equations of classical thermoelasticity in which displacements are replaced with velocities, complemented with Cattaneo evolution equation for heat flux. For a particular choice of the characteristic quantities and for positive thermal conductivity, it is shown that this system may be cast in a form that is symmetric t-hyperbolic without further recurrence to entropy principle. While hyperbolicity means a finite speed of propagation of heat waves, it is known that symmetric hyperbolic systems have the desirable property of well-posedness of Cauchy problems. A study of the characteristics of this system is carried out, and an energy integral is derived, that can be used to prove uniqueness of solution under some boundary conditions. A numerical application for a finite slab is considered ...

Thermo-Mechanical Responses of an Annular Cylinder with Temperature Dependent Material Properties under Thermoelasticity without Energy Dissipation

2017

The present work is concerned with thermoelasticity without the energy dissipation theory for a problem of an infinitely long and isotropic annular cylinder of temperature dependent physical properties. We employ the thermoelasticity theory of GN-II and derive the basic governing equations with variable material properties. The formulation is then applied to solve a boundary value problem of an annular cylinder with its inner boundary assuming to be stress free and subjected to exponential decay in temperature and sinusoidal temperature distribution. The outer boundary is also assumed to be stress free and is maintained at reference temperature in both cases. We solve the non-linear coupled differential equations by applying the finite difference approach efficiently. We analyze the numerical results in a detailed way with the help of different graphs. The effects of temperature dependency of material properties on the thermo-mechanical responses for two different time dependent temperature distributions applied at the inner boundary are highlighted.

Thermoelastic stress due to a rectangular heat source in a semi-infinite medium. Presentation of an analytical solution

Engineering Geology, 1998

The thermoelastic response due to a time-dependent rectangular heat source in a semi-infinite medium is analyzed. The problem originates from studies of nuclear waste repositories in rock. Canisters containing heat-emitting nuclear waste are deposited over a large rectangular area deep below the ground surface. The solution for a time-dependent heat source is obtained from the corresponding instantaneous heat source by superposition. The thermoelastic problem for the instantaneous rectangular heat source in an infinite surrounding is solved exactly. An important step is the introduction of so-called quadrantal heat sources. The solution for the rectangle is obtained from four quadrantal solutions. The solution for the quadrantal heat source depends on the three dimelasionless coordinates only. Time occurs in the scale factors only. The condition of zero normal and shear stresses at the ground surface is fulfilled by using a mirror heat source and a boundary solution. The boundary solution accounts for the residual normal stress at the ground surface. Using a Hertzian potential, a surprisingly simple solution is obtained. The final analytical solution is quite tractable considering the complexity of the initial problem. The solution may be used to test numerical models for coupled thermoelastic processes. It may also be used in more detailed numerical simulations of the process near the heat sources as boundary conditions to account for the three-dimensional global process.

On Some Non-Standard Problems in Linear Thermoelasticity

2009

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A Problem on Thermoelastic Interactions in an Infinite Medium with a Cylindrical Hole in Generalized Thermoelasticity III

Journal of Thermal Stresses, 2008

The aim of the present article is to study the thermoelastic interactions in an infinite elastic medium with a cylindrical hole in the context of generalized thermoelasticity III, recently developed by Green and Nagdhi [1]. The boundary of the hole is assumed to be stress free and is subjected to a ramp type heating. In order to make a comparison between this thermoelastic model with other thermoelastic models, the problem is formulated on the basis of three different theories of thermoelasticity, namely: the extended thermoelasticity proposed by Lord and Shulman [2], the thermoelasticity without energy dissipation (Green and Nagdhi [3]) and thermoelasticity with energy dissipation (thermoelasticity type III [1]) in a unified way. The solutions for displacement, temperature and stresses are obtained with the help of Laplace transform procedure. Firstly the short time approximated solutions for three different theories have been obtained analytically. Then following a numerical method for the inversion of Laplace transforms, the numerical values of the physical quantities are also computed for the copper material and results are displayed in graphical forms to compare the results obtained from different models of generalized thermoelasticity.

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Effects of phase lags on wave propagation in an infinite solid due to a continuous line heat source

Acta Mechanica, 2011

The present work is concerned with the wave propagation in a homogeneous, isotropic and unbounded solid due to a continuous line heat source under the theory of thermoelasticity with three phase-lags (Roychoudhari in J Therm Stress 30: [231][232][233][234][235][236][237][238] 2007). For the solution of the problem, we employ a potential function approach together with Laplace and Hankel transform method. Analytical expressions for the distributions of different fields like temperature, displacement and stresses inside the medium are derived by inverting Laplace transforms in an approximate manner for small values of time. The problem is illustrated by computing numerical values of the field variables for a particular material. The theoretical as well as numerical results are compared with the corresponding results for other theories of thermoelasticity reported earlier.

Thermoelastic interactions in a hollow cylinder due to a continuous heat source without energy dissipation

Materials Research Express

Thermoelastic interactions in an isotropic unbounded medium due to moving heat source using GNIII model

Latin American Journal of Solids and Structures, 2015

In this work, we study a problem of thermoelastic interaction due to moving heat source in an isotropic infinite medium under Green and Naghdi model of type III (GNIII). The form of vector-matrix differential equation in the Laplace transform domain, the basic equations have been written, which is then solved by an eigenvalue technique. The analytical solution in the Laplace transforms domain with eigenvalue approach has been obtained. Numerical results for the displacement, temperature and the stress distributions are represented graphically. Some comparisons have been shown in figures to estimate the effect of heat source velocity and time in the physical quintets.