A Two-Dimensional Problem for Two Media in the Generalized Theory of Thermoelasticity

The two-dimensional problem for an elastic half-space with a thick layer on its top is considered with the context of the theory of generalized thermoelasticity with one relaxation time. The half-space and the thick layer are composed of different elastic materials. The surface of the upper layer is traction free and subjected to the effect of a thermal shock. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transform.     

A Two-Dimensional Generalized Thermoelastic Diffusion Problem for a Thick Plate Subjected to Thermal Loading Due to Laser Pulse

A two-dimensional problem for a thermoelastic thick plate is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface of the thick plate is taken to be traction free and heated by a laser beam. The chemical potential is also assumed to be a known function of time on the bounding plane. The lower surface is taken to be in contact with a rigid surface and is thermally insulated and impermeable. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms; numerical results are discussed and represented graphically.     

A Two-Dimensional Generalized Electromagneto-Thermoelastic Problem for a Half-Space

A two-dimensional coupled problem in electromagneto-thermoelasticity for a thermally and electrically conducting half-space solid whose surface is subjected to a time-dependent heat is studied in this paper. The problem is in the context of the Green and Lindsay's generalized thermoelasticity theory with two relaxation times. There acts an initial magnetic field parallel to the plane boundary of the half-space. The medium deforms because of heat, and due to the application of the magnetic field, there results an induced magnetic field and an induced electric field in the medium. The Maxwell's equations are formulated and the electromagneto-thermoelastic coupled governing equations are established. The normal mode analysis is used to obtain the exact expressions for the considered variables. The distributions of the considered variables are represented graphically. From the distributions, it can be found the wave type heat propagation in the medium.     

Normal Mode Method for Two-Temperature Generalized Thermoelasticity Under Thermal Shock Problem

The theory of two-temperature generalized thermoelasticity, based on Youssef's theory, was used to solve boundary value problems of one-dimensional generalized thermoelasticity half-space by heating its boundary with different types of heating. The governing equations are solved using new mathematical methods within the purview of the Lord-?hulman (L-S) theory and the classical dynamical coupled theory (CD). The general solution obtained is applied to a specific problem of a half-space subjected to one type of heating—thermal shock type. The separation of variables method is used to get the exact expressions for distributions of displacement, the stresses, and temperature distribution. Variations of the considered functions through the horizontal distance are illustrated graphically. Comparisons are made with results between the two theories. Numerical work is also performed for a suitable material and results are discussed, specifically the conductive temperature, the dynamical temperature, and the stress and strain distributions are shown graphically when discussed.     

Two-Dimensional Thermo-Mechanical Wave Propagations in Generalized Coupled Theory of Thermoelasticity

This article deals with the numerical treatment for two-dimensional thermo-mechanical wave propagations in the thermoelastic solids. The development of the formulation is based on the generalized dynamic coupled theory of thermoelasticity. The numerical procedure is developed by using the method of characteristics, yielding the bicharacteristics and bicharacteristic equations. The difference equations for computing the solutions of two-dimensional thermo-mechanical waves are then derived. The method is applied to the thermal and mechanical wave propagations in finite plates subjected to the impulsive surface heating. The numerical results are shown in the graphical form.     

A Problem on Elastic Half Space Under Fractional Order Theory of Thermoelasticity

The present work is concerned with the solution of a problem on fractional order theory of thermoelasticity for an elastic medium. We investigate the thermoelastic interactions inside the medium by employing the fractional order theory of thermoelasticity, recently advocated by Sherief et al. (Int. J. Solids Struct., 47, 269–275, 2010). State space approach together with the Laplace transform technique is used to obtain the general solution of the problem. The general solution is then applied to three specific problems on an elastic half space, whose boundary is subjected to (i) a thermal shock (i.e., a step input in temperature and zero stress), (ii) a normal load (i.e., a step input in stress and zero temperature change) and (iii) a ramp type increase in temperature and zero stress. To observe the variations of displacement, temperature and stress inside the half-space we compute the numerical values of the field variables for a particular material by utilizing a numerical method of Laplace inversion. The effects of fractional order parameter on the variations of different fields inside the medium are analyzed graphically.     

A Generalized Thermoelastic Problem of Functionally Graded Spherical Cavity

In the context of G–L theory, a generalized thermoelastic problem is considered for an infinite functionally graded and temperature-dependent isotropic spherical cavity. The surface of the sphere is subjected to (i) a ramp-type compression and (ii) maintain at a constant temperature. The Laplace transform for time variable is used on the basic equations and then solved by the potential function approach. The inversion of Laplace transform is carried out numerically by the Zakian method. Finally, numerical computations of the displacement and stress components as well as temperature distribution have been made and are presented graphically.     

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

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 A. E. Green and P. M. Nagdhi , On Undamped Heat Waves in an Elastic Solid , J. Thermal Stresses , vol. 15 , pp. 253 – 264 , 1992 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. 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 H. W. Lord and Y. Shulman , A Generalized Dynamical Theory of Thermoelasticity , J. Mech. Phys. Solids , vol. 15 , pp. 299 – 309 , 1967 .[Crossref], [Web of Science ®] , [Google Scholar]], the thermoelasticity without energy dissipation (Green and Nagdhi [3 A. E. Green and P. M. Nagdhi , Thermoelasticity Without Energy Dissipation , J. Elasticity , vol. 31 , pp. 189 – 209 , 1993 .[Crossref], [Web of Science ®] , [Google Scholar]]) and thermoelasticity with energy dissipation (thermoelasticity type III [1 A. E. Green and P. M. Nagdhi , On Undamped Heat Waves in an Elastic Solid , J. Thermal Stresses , vol. 15 , pp. 253 – 264 , 1992 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]) 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.     

Effects of Three Phase Lags on Generalized Thermoelasticity for an Infinite Medium with a Cylindrical Cavity

The present work is concerned with the effects of three-phase lags on thermoelastic interactions due to step input in temperature on the stress free boundary of a cylindrical cavity in an unbounded medium. The problem is studied in the context of theory of generalized thermoelasticity with three phase lags (Roychoudhuri, [24 S. K. Roychoudhuri , On a Thermoelastic Three-Phase-Lag Model , J. Thermal Stresses , vol. 30 , pp. 231 – 238 , 2007 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]) and the theory of thermoelasticity with energy dissipation (Green and Naghdi, [4 A. E. Green and P. M. Naghdi , On Undamped Heat Waves in an Elastic Solid , J. Thermal Stresses , vol. 15 , pp. 253 – 264 , 1992 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]) in a unified way. Solution of the problem is obtained by using the Laplace transform technique. Significant dissimilarities between two models showing the effects of phase lags are pointed out on the basis of analytical as well as numerical results of the problem. A numerical method for the inversion of Laplace transform is employed.     

Generalized Magnetothermoelasticity for Isotropic Media

The theory of generalized magnetothermoelasticity is applied to a two-dimensional problem of a homogeneous isotropic perfectly conducting thick plate with heat source subjected to a time-dependent compression under constant magnetic and electric intensities. Double integral transforms (Laplace transform for time variable and Fourier transform for space variable) are used, and the resulting equations are written in the form of a vector-matrix differential equation, which was solved by the eigenvalue approach. The inversion of the Laplace transform is carried out numerically by the Zakian method. Finally, numerical computations of the displacement and stress components, temperature distribution, induced magnetic and electric fields have been done and are represented graphically.     

Three-Dimensional Fractional Order Generalized Thermoelastic Problem Under the Effect of Rotation in a Half Space

The current article is concerned with three-dimensional problems for a homogeneous, isotropic thermoelastic half-space subjected to a time-dependent heat source on the surface, which is traction-free under the effect of rotation. The governing equations are taken in the context of fractional order generalized thermoelasticity (Green–Lindsay) theory. Normal mode analysis technique and eigenvalue approaches have been used to solve the non-dimensional equations. Numerical results for temperature, thermal stress and displacement distributions are presented graphically and discussed.     

On the Fundamental Solutions of Generalized Thermoelasticity with Three Phase-Lags

The aim of the present work is to derive the fundamental solutions in the linear theory of generalized thermoelasticity with three phase-lags, recently introduced by Roychoudhuri (2007). We consider the case of homogeneous and isotropic bodies and present a solution of the Galerkin-type of field equations. The solution is then used to determine the effects of concentrated loads and heat sources in an unbounded medium. The fundamental solutions of the field equations in case of steady vibrations are also derived.     

A Domain of Influence Theorem for Thermoelasticity with Three-Phase-Lag Model

We establish the theory of domain of influence for a potential-temperature disturbance under the generalized theory of thermoelasticity with three-phase-lag model. For a finite time t > 0, it has been proved that the potential due to the displacement and the temperature fields does not produce any disturbances outside a bounded domain and the domain of influence is dependent on the support of load, the thermoelastic coupling constant and the phase-lag parameters. It has been shown that the domain of influence of the present case reduces to the domain of influence for classical thermoelasticity theory and the thermoelasticity theory with one relaxation parameter.     

Transient response analysis of a spherical shell embedded in an infinite thermoelastic medium based on a memory-dependent generalized thermoelasticity

In the context of a memory-dependent generalized thermoelasticity, the thermal-induced transient response in an infinite elastic body containing a spherical shell is investigated. A thermal shock is applied on the inner surface of the spherical shell. The infinite body and the spherical shell are assumed to be isotropic but two dissimilar materials. By using an analytical technique based on the Laplace transform along with its numerical inversion, the governing equations of the problem are solved and the non-dimensional physical quantities in the two materials, i.e., temperature, displacement and stress, are obtained and illustrated graphically respectively. In simulation, the accuracy of memory-dependent derivative (MDD) is verified by degrading the present model into L-S model to compare the results obtained from the cases with interfacial thermal resistance and without interfacial thermal resistance. In addition, the effects of the different kernel functions as well as the ratios of the two materials, including the ratios of the density, the thermal-conductivity and the time-delay, on the distributions of the considered variables are obtained and demonstrated graphically respectively.     

Thermoelastic Analysis of an Annulus Using the Green-Naghdi Model

ABSTRACT

The linear theory of thermoelasticity of Green-Naghdi (GN) types II and III for homogeneous and isotropic materials are employed to study the thermal and mechanical waves in an annulus domain. The disturbances are generated by sudden application of temperature to the boundary. The nondimensional form of the governing equations are solved utilizing the Laplace transform method. Locally transversal linearization (LTL) technique, and a numerical inverse Laplace transform method are used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermomechanical wave propagation and reflection from the boundary are investigated and the influence of the damping parameter on temperature, displacement, and stress fields in the Green-Naghdi type III is discussed.     

Generalized Thermoelastic Problem of a Spherically Isotropic Infinite Elastic Medium Containing a Spherical Cavity

This paper is concerned with the determination of thermoelastic stresses and temperature in a spherically isotropic infinite elastic medium having a spherical cavity in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). The surface of the cavity is stress free and is subjected to a thermal shock. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by the eigenvalue approach. The numerical inversion of the transforms is carried out using the Bellman method. The stresses and temperature are computed and presented graphically. A comparison with isotropic body has also been studied.     

Newton's Problem of Minimal Resistance for Bodies Containing a Half-Space

We consider Newton's problem of minimal resistance for unbounded bodies in Euclidean space ^d, d 2. A homogeneous flow of noninteracting particles of velocity v falls onto an immovable body containing a half-space {x : (x, n) < 0} ^d, (v, n) < 0. No restriction is imposed on the number of (elastic) collisions of the particles with the body. For any Borel set A {v} of finite measure, consider the flow of cross-section A: the part of initial flow that consists of particles passing through A.We construct a sequence of bodies that minimize resistance to the flow of cross-section A, for arbitrary A. This sequence approximates the half-space; any particle collides with any body of the sequence at most twice. The infimum of resistance is always one half of corresponding resistance of the half-space.     

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

The Effect of Gravity Field on the Plane Waves in a Fiber-Reinforced Two-Temperature Magneto-Thermoelastic Medium Under Lord-Shulman Theory

The aim of the present work is to investigate the influence of the gravity field on the plane waves in a linearly fiber-reinforced magneto-thermoelastic isotropic medium in the context of the two-temperature theory of generalized thermoelasticity under Lord-Shulman model. The problem has been solved numerically using the normal mode analysis together with the eigenvalue approach. Numerical results for the conductive temperature, thermodynamic temperature, displacement components and the stresses are represented graphically and the results are analyzed. The graphical results indicate that the effect of the two-temperature parameter, gravity field and the magnetic field on the plane waves in the fiber-reinforced thermoelastic medium are very pronounced. Comparisons are made with the results in the presence and absence of the two-temperature parameter, gravity field and the magnetic field. Such problems are very important in many dynamical systems.     

A memory response in the vibration of a microscale beam induced by laser pulse

Abstract

In the application of pulsed laser heating, such as the laser hardening of metallic surfaces, the conduction limited process is the dominant mechanism during the laser-workpiece interaction. As a consequence, time unsteady analysis of this problem becomes necessary. The present study examines the effect of ultra-short-pulsed laser heating in the problem of coupled thermoelastic vibration of a microscale beam resonator. Due to the shortcomings of power law distributions in fractional derivatives, the usage of some other forms of derivatives with some other kernel functions is proposed. With this motivation, the heat transport equation is defined in an integral form of a common derivative on a slipping interval by incorporating the three-phase-lag memory-dependent heat transfer. Finite sinusoidal Fourier and Laplace transform techniques are then employed to determine the lateral vibration of the beam and the temperature increment within the medium. According to the graphical representations corresponding to the numerical results, conclusions about the new theory are drawn. An excellent predictive capability is demonstrated due to the presence of the energy absorption depth, memory dependent derivative, and delay time.