Lord-Shulman Theory Research Articles

Vibration analysis of thermoelastic double porous microbeam subjected to laser pulse

ABSTRACTThe present work deals with vibration phenomenon of a homogeneous, isotropic thermoelastic double porous microbeam induced by laser pulse heating, in context of Lord–Shulman theory of thermoelasticity with one relaxation time. Laplace transform technique has been used to obtain the expressions for lateral deflection, axial stress, axial displacement, volume fraction field and temperature distribution. The resulting quantities are obtained in the physical domain by applying a numerical inversion technique. Variations of axial displacement, axial stress, lateral deflection, volume fraction field and temperature distribution with axial distance are depicted graphically to show the effects of porosity and laser intensity parameter. Some particular cases are also deduced.

Eigenvalue formulation to micropolar porous thermoelastic circular plate using dual phase lag model

PurposeThe purpose of this paper is to study the axisymmetric problem in a micropolar porous thermoelastic circular plate with dual phase lag model by employing eigenvalue approach subjected to thermomechanical sources.Design/methodology/approachThe Laplace and Hankel transforms are employed to obtain the expressions for displacements, microrotation, volume fraction field, temperature distribution and stresses in the transformed domain. A numerical inversion technique has been carried out to obtain the resulting quantities in the physical domain. Effect of porosity and phase lag on the resulting quantities has been presented graphically. The results obtained for Lord Shulman theory (L-S, 1967) and coupled theory of thermoelasticity are presented as the particular cases.FindingsThe variation of temperature distribution is similar for micropolar thermoelastic with dual (MTD) phase lag model and coupled theory of thermoelasticity. The variation is also similar for tangential couple stress for MTD and L-S theory but opposite to couple theory. The behavior of volume fraction field and tangential couple stress for L-S theory and coupled theory are observed opposite. The values of all the resulting quantities are close to each other away from the sources. The variation in tangential stress, tangential couple stress and temperature distribution is more uniform.Originality/valueThe results are original and new because the authors presented an eigenvalue approach for two dimensional problem of micropolar porous thermoelastic circular plate with dual phase lag model. A comparison of porosity, L-S theory and coupled theory of micropolar thermoelasticity is made. Such problem has applications in material science, industries and earthquake problems.

Two-dimensional generalized thermo-elastic problem for anisotropic half-space

This paper concerns with the study of wave propagation in fibre reinforced anisotropic half space under the influence of temperature and hydrostatic initial stress. Lord-Shulman theory is applied to the heat conduction equation. The resulting equations are written in the form of vector matrix differential equation by using Normal Mode technique, finally which is solved by Eigen value approach.

Dispersion relations for the hyperbolic thermal conductivity, thermoelasticity and thermoviscoelasticity

The Maxwell–Cattaneo heat conduction theory, the Lord–Shulman theory of thermoelasticity and a hyperbolic theory of thermoviscoelasticity are studied. The dispersion relations are analyzed in the case when a solution is represented in the form of an exponential function decreasing in time. Simple formulas that quite accurately approximate the dispersion curves are obtained. Based on the results of analysis of the dispersion relations, an experimental method of determination of the heat flux relaxation time is suggested.

Response of fractional ordered micropolar thermoviscoelastic half-space with diffusion due to ramp type mechanical load

• Interactions caused by mechanical shock in micropolar diffusive medium are studied. • Problem has been modeled in context of generalized fractional thermoelasticity. • Laplace and Fourier transforms are used to solve the problem. • Effects of fractional parameter, micropolarity and diffusion are noticed. The current manuscript is devoted to scrutinize the effects of fractional parameter, micropolarity and diffusion on the interactions caused by a ramp type mechanical shock in a generalized thermo-viscoelastic solid half-space. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The problem has been modeled by employing the fractional generalization of the Lord–Shulman theory to carry the investigation. The governing equations in xz -plane are handled with an analytical–numerical technique based on Laplace–Fourier transforms. Expressions for displacement, stresses, temperature and mass concentration in the physical domain are obtained using a numerical inversion technique. The problem is illustrated by computing the numerical values of the field variables for a magnesium crystal like material. Comparisons of the physical quantities are shown in figures to study the effects of fractional parameter, micropolarity, diffusion and ramp parameter. Some particular cases of interest have also been inferred from the present problem.

Analytical Solutions of a Two-Dimensional Generalized Thermoelastic Diffusions Problem Due to Laser Pulse

In this paper, we apply the generalized thermoelastic theory with mass diffusion to a two-dimensional problem for a half-space. The surface of the half-space is taken to be traction-free and heated by laser pulse. The analytical solution is adopted for the temperature, the displacement components, concentration, the stress components and chemical potential. The nonhomogeneous basic equations have been written in the form of a vector–matrix differential equation, which is then solved using the eigenvalue approach. A comparison is made in the case of the absence and presence of a mass diffusion between the coupled and Lord–Shulman theories. The results obtained are presented graphically for the effects of the laser pulse and the mass diffusion to display the phenomena of physical significance.

Effect of the gravity on the photothermal waves in a semiconducting medium with an internal heat source and one relaxation time

The purpose of the present paper is to study the gravitational effect in a homogeneous isotropic semiconducting medium subjected to an internal heat source, based on Lord–Shulman theory. The problem is formulated in the dimensionless form and then solved analytically using the normal mode analysis method. The physical quantities are obtained and tested by a numerical study using the parameters of silicon as a target and presented graphically. The distribution of these quantities is represented graphically in the presence and absence of the internal heat source as well as gravity.

A two-temperature rotating-micropolar thermoelastic medium under influence of magnetic field

In the present paper, we introduced a general model of the equations of the formulation in the context of Lord–Shulman theory which includes one thermal relaxation time and Green–Lindsay theory with two thermal relaxation times as well as the classical dynamical coupled theory to study the effect of the rotation and the magnetic field on the total deformation of a micropolar thermoelastic medium with an internal heat source that is moving with a constant speed. The analytical method used to obtain the formula of the physical quantities is the normal mode analysis. Comparisons made with the results of the three theories in the presence and absence of the magnetic field as well as an internal heat source. A comparison is also made with the results of the three theories for different values of the rotation.

Statement of Retraction

This article refers to:On the completeness of a set of potential functions for the solution of generalized thermoelastic problems based on Lord–Shulman theory

Fractional order theory to an infinite thermo-viscoelastic body with a cylindrical cavity in the presence of an axial uniform magnetic field

The model of fractional time-derivative of generalized magneto–thermo–viscoelasticity equations for isotropic media in the presence of a constant magnetic field is considered. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

Effect of rotation on a micropolar magneto-thermoelastic solid in dual-phase-lag model under the gravitational field

The purpose of this paper is to study the effect of rotation and gravitational field on a micropolar magneto-thermoelastic solid in the context of the dual-phase-lag model (DPL). The normal mode analysis is used to obtain the exact expressions for displacement components, force stresses, the micro-rotation and temperature. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made in the presence and absence of rotation, gravity and magnetic field in the context of the dual-phase-lag model and the Lord-Shulman theory.

Effect of magnetic field on piezo-thermoelastic medium with three theories

Abstract In the present paper, we use the classical dynamical coupled theory, Lord-Shulman theory with one relaxation time, and Green-Lindsay with two relaxation times of thermoelasticity to study the influence of magnetic field on piezo-thermoelastic medium. The normal mode method is used to transfer partial differential equation to an ordinary differential equation. Numerical computations are performed for a specific material and the results obtained are represented graphically. A comparison is made between the three theories in the absence and presence of a magnetic field. The presence of magnetic field plays an important role on all physical quantities.

The Characteristic of Thermoelastic Waves in Transversely Isotropic Finite Cylinders

A theoretical as well as a numerical investigation of the propagation of thermoelastic waves and vibration of transversely isotropic cylinders of finite length is discussed. Lord-Shulman theory is adopted here to model the thermoelastic deformation of cylinders. A semi analytical finite element (SAFE) method is employed to study dispersion of thermoelastic waves and natural frequencies of vibration of finite cylinders with traction free curved surfaces having both ends insulated and constrained by frictionless rigid walls. Numerical results obtained by the SAFE method for the frequencies of vibration of a sapphire rod are found to be in excellent agreement with published results. Natural frequencies of vibration for first three axisymmetric and asymmetric modes are presented graphically for a silicon nitride thermoelastic cylinder. Also, numerical results showing dispersion of both propagating and evanescent circumferential waves in infinite and finite cylinders are presented also.

On (non-)exponential decay in generalized thermoelasticity with two temperatures

We study solutions for the one-dimensional problem of the Green–Lindsay and the Lord–Shulman theories with two temperatures. First, existence and uniqueness of weakly regular solutions are obtained. Second, we prove the exponential stability in the Green–Lindsay model, but the non-exponential stability for the Lord–Shulman model.

Photothermal waves for two temperature with a semiconducting medium under using a dual-phase-lag model and hydrostatic initial stress

The dual-phase-lag (DPL) model with two different time translations and Lord–Shulman (LS) theory with one relaxation time are applied to study the effect of hydrostatic initial stress on medium under the influence of two temperature parameter(a new model will be introduced using two temperature theory) and photothermal theory. We solved the thermal loading at the free surface in the semi-infinite semiconducting medium-coupled plasma waves with the effect of mechanical force during a photothermal process. The exact expressions of the considered variables are obtained using normal mode analysis also the two temperature coefficient ratios were obtained analytically. Numerical results for the field quantities are given in the physical domain and illustrated graphically under the effects of several parameters. Comparisons are made between the results of the two different models with and without two temperature parameter, and for two different values of the hydrostatic initial stress. A comparison is carried out between the considered variables as calculated from the generalized thermoelasticity based on the DPL model and the LS theory in the absence and presence of the thermoelastic and thermoelectric coupling parameters.

Analytical solution of magnetothermoelastic interaction in a fiber-reinforced anisotropic material

The present paper is concerned with the investigation of the analytical solution of a fiber-reinforced anisotropic material under generalized magnetothermoelastic theory using the eigenvalue approach. Based on the Lord-Shulman theory, the formulation is applied to generalized magnetothermoelasticity with one relaxation time. Based on eigenvalue approach, exponential Fourier transform and Laplace techniques, the analytical solutions has been obtained. The inverses of Fourier transforms are obtained analytically. Numerical computations for a fiber-reinforced-like material have been performed and the results are presented graphically. The results of the temperature, displacement components and stress components have been verified numerically and are represented graphically. Comparisons are made with the results predicted by the presence and absence of reinforcement.

Thermal shock problem of a three-layered functionally graded zirconia/titanium alloy strip based on a unified generalized thermoelasticity theory

ABSTRACTThe one-dimensional thermoelastic analysis of a three-layered ceramic/metal functionally graded strip under thermal loading is considered, utilizing classical and generalized coupled thermoelasticity theories. A finite element code is developed for the numerical study of the transient fields of displacement, temperature change, and stress. The code is verified through the analytical and the numerical results available in the literature. The applications presented imply that the classical coupled thermoelasticity theory offers unsatisfactory description of the thermomechanical response of a solid under fast transient thermal loading. Instead, generalized thermoelasticity theories such as the Lord–Shulman theory or the Green–Lindsay theory should be used.

Generalized coupled non-Fickian/non-Fourierian diffusion-thermoelasticity analysis subjected to shock loading using analytical method

In this article, the generalized coupled non-Fickian diffusion-thermoelasticity analysis is carried out using an analytical method. The transient behaviors of field variables, including mass concentration, temperature and displacement are studied in a strip, which is subjected to shock loading. The governing equations are derived using generalized coupled non-Fickian diffusion-thermoelasticity theory, which is based on Lord-Shulman theory of coupled thermoelasticity. The governing equations are transferred to the frequency domain using Laplace transform technique and then the field variables are obtained in analytical forms using the presented method. The field variables are eventually determined in time domain by employing the Talbot technique. The dynamic behaviors of mass concentration, temperature and displacement are studied in details. It is concluded that the presented analytical method has a high capability for simulating the wave propagation with finite speed in mass concentration field as well as for tracking thermoelastic waves. Furthermore, the obtained results are more realistic than that of others.

On the completeness of a set of potential functions for the solution of generalized thermoelastic problems based on Lord–Shulman theory

This article has been retracted

A GDQ approach to thermally nonlinear generalized thermoelasticity of disks

ABSTRACTGeneralized thermoelasticity response of an annular disk subjected to thermal shock on its inner surface is analyzed in this research. The Lord–Shulman theory, which accounts for one relaxation time in the conventional Fourier law, is used to avoid the infinite speed of thermal wave propagation. Unlike the other available works in which the first law of thermodynamics is linearized, the nonlinearity arising from the temperature change is taken into consideration. The first law of thermodynamics in this case becomes nonlinear and the analysis under such formulation is called thermally nonlinear. Two coupled equations, i.e., the radial displacement wave equation and temperature wave propagation equation are obtained. These equations and the associated boundary conditions are discreted through the generalized differential quadrature method. Solution of the time-dependent system of equations is obtained using the Newmark time marching scheme and the successive Picard method. Numerical results are provided for both thermally linear and thermally nonlinear temperature and radial displacement wave propagations. Parametric studies reveal that at higher temperature levels, thermally nonlinear first law of thermodynamics should be considered instead of thermally linear one. Furthermore, the higher the coupling parameter and/or relaxation time, the higher the divergence of thermally nonlinear-/linear-based results.