Green Lindsay Theory Research Articles

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.

A two-dimensional problem with rotation and magnetic field in the context of four thermoelastic theories

The aim of the present paper is to study the effect of rotation and magnetic field on the general model of the equations of generalized thermoelasticity for a homogeneous isotropic elastic half-space. The formulation is applied under four theories of generalized thermoelasticity: the coupled theory (CT), Lord-Schulman (LS) theory, Green-Lindsay (GL) theory as well as Green-Naghdi (GN) theory. By employing normal mode analysis, the analytical expressions for the temperature, displacement components and the (Mechanical and Maxwell’s) stresses distribution are obtained in the physical domain. These expressions are also calculated numerically and corresponding graphs are plotted to illustrate and compare theoretical results. The effect of rotation and magnetic field are also studied.

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.

Modeling of fractional magneto-thermoelasticity for a perfect conducting materials

A unified mathematical model of the equations of generalized magneto-thermoelasticty based on fractional derivative heat transfer for isotropic perfect conducting media is given. Some essential theorems on the linear coupled and generalized theories of thermoelasticity e.g. the Lord- Shulman (LS) theory, Green-Lindsay (GL) theory and the coupled theory (CTE) as well as dual-phase-lag (DPL) heat conduction law are established. Laplace transform techniques are used. The method of the matrix exponential which constitutes the basis of the state-space approach of modern theory is applied to the non-dimensional equations. The resulting formulation is applied to a variety of one-dimensional problems. The solutions to a thermal shock problem and to a problem of a layer media are obtained in the present of a transverse uniform magnetic field. According to the numerical results and its graphs, conclusion about the new model has been constructed. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

Influence of the gravitational field on a piezothermoelastic rotating medium with G-L theory

In the present paper, two different theories (coupled theory and Green-Lindsay theory with two relaxation times) are applied to study the deformation of a generalized piezothermoelastic rotating medium under the influence of gravity. The normal mode analysis is used to obtain the expressions for the displacement components, the temperature, the stress, the stress components, the electric potential and the electric displacements. Numerical results for the field quantities are given in the physical domain and illustrated graphically. Comparisons are made with the results predicted by coupled and Green-Lindsay theories in the presence and absence of rotation as well as of gravity.

Interaction Between Mechanical Forces and Waves Propagation in Magneto-Thermo-Elastic for Fiber-Reinforced Wafer

The objective of this paper is to investigate the surface waves in fibre-reinforced anisotropic elastic medium subjected to magnetic and thermal fields. We introduce the coupled theory (CD), Lord-Shulman (LS) theory and Green-Lindsay (GL) theory to study the influence of magnetic field on 2D problem of a fibre-reinforced thermoelastic. The analytical expressions for displacement components and force stress are obtained in the physical domain by using the harmonic vibrations. The wave velocity equations have been obtained in different cases. Numerical results for the temperature, displacement, and thermal stress components are given and illustrated graphically in the presence and absence of the magnetic field of the material medium. A comparison also is made between the three theories in the presence and absence of fiber-reinforced.

Reflection and Refraction of Plane Waves at the Interface of Two Visco-thermoelastic Liquid Layers in Presence of Magnetic Field and Compressional Stress, with the Outer Core Inside the Earth as a Model

This paper is devoted to investigate the effects of viscosity, magnetic field and compressional stress on reflection and refraction of plane waves while propagating through the interface of two visco-thermoelastic liquid layers. The basic dynamical equations of motion for visco-thermoelastic liquid medium in presence of magnetic field and the heat conduction equation in the context of Green–Lindsay theory have been solved to derive the wave equations for the reflected and refracted P, SV and thermal waves. Using appropriate boundary conditions, equations containing the ratios of amplitude of various reflected and transmitted components to that of incident wave have been obtained in matrix form. These amplitude ratios have been computed numerically for P-wave incidence only and results obtained are presented graphically.

Mode-I Crack Problem in Generalized Thermo-microstretch with Harmonic Wave under Three Theories

A general model of the equations of generalized thermo-microstretch for an infinite space weakened by a finite linear opening Mode-I crack are solving. The material is homogeneous isotropic elastic half space. The crack is subjected to prescribed temperature and stress distribution. The formulation is applied to generalized thermoelasticity theories, using mathematical analysis with the purview of the Lord-Åžhulman (LS involving one relaxation time) and Green-Lindsay (GL includes two relaxation times) theories with respect to the classical dynamical coupled theory (CD). The harmonic wave method has been used to getting the exact expression of Normal displacement, Normal stress force, couple stresses, microstress and temperature distribution. The variations of the considered fields with the horizontal distance are explained graphically. A comparison also is made between the three theories and for different depths.

SV-waves incidence at interface between solid-liquid media under electromagnetic field and initial stress in the context of three thermoelastic theories

ABSTRACTIn this article, we studied the propagation of SV-waves under the effect of electromagnetic field and initial stress for three models in thermoelasticity: The couple and Green–Lindsay theories as well as the dual-phase-lag theory. The problem of reflection and transmission of thermoelastic waves at a solid–liquid interface in the presence of electromagnetic fields and initial stress we investigated subjected to certain boundary conditions. The appropriate expressions to find the amplitude ratios of the incident waves (SV-waves) were obtained. The reflection and transmission coefficients for the incident SV-waves were computed numerically. The effect of electric field, magnetic field, and initial stress were illustrated graphically. Comparisons were made with the obtained results in the presence and absence of considered variables and displayed graphically. The results indicate that the effect of electric field, magnetic field, and initial stress on SV-waves incidence at the interface between solid–liquid media are very pronounced.

Internal Heat Source in Thermoelastic Microelongated Solid Under Green Lindsay Theory

Abstract The present study deals with two dimensional deformation, due to internal heat source in a thermoelastic microelongated solid. A mechanical force is applied along the interface of elastic half space and thermoelastic microelongated half space. The problem is in the context of Green Lindsay (GL) theory. The analytic expressions for displacement component, normal force stress, temperature distribution and microelongation have been derived. The effect of internal heat source and microelongation on the derived components have been depicted graphically.

Application of fractional order theory of magneto-thermoelasticity to an infinite perfect conducting body with a cylindrical cavity

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. 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. The solution is obtained by a direct approach by using the thermoelastic potential function. 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.

Reflection and Refraction of P Wave at the Interface Between Thermoelastic and Porous Thermoelastic Medium

Reflection and refraction phenomenon due to an obliquely incident longitudinal wave at a plane interface between an isotropic, homogeneous, thermoelastic medium and a porous thermoelastic medium is studied. Firstly, the generalized thermoelastic theory (G-TE) of wave propagation in a saturated porous thermoelastic medium is developed in the context of the coupled thermoelastic theories of Biot, the theory of generalized thermoelasticity of Lord–Shulman (with one relaxation time) and that of Green–Lindsay (with two relaxation times). Then, the expressions of amplitude ratios of various reflected and refracted waves to that of incident P wave are derived. Numerical results are obtained and used to analyze the difference of reflection amplitude of four kinds of reflection waves, and refraction amplitude of three kinds of refraction waves among the G-TE theory, the Lord–Shulman theory and the Green–Lindsay theory in the case of porous thermoelasticity. The effect of relaxation time $$\tau _{_1} $$ and thermal expansion coefficient $$a_\mathrm{s}$$ on the reflection and refraction of the incident P-wave is also discussed.

Reflection and transmission at the boundary surface of modified couple stress thermoelastic media

In this paper the reflection and transmission at a plane interface in modified couple stress generalized thermoelastic solid half spaces in the context of Loard-Shulman (LS) and Green-Lindsay (GL) theories in welded contact are investigated. Amplitude ratios of various reflected and transmitted waves are obtained due to incidence of a set of coupled longitudinal waves and coupled transverse waves. It is found that the amplitude ratios of various reflected and transmitted waves are functions of the angle of incidence, frequency and are affected by the couple stress properties of the media. Some special cases are deduced from the present formulation.

Plane Strain Deformation In A Thermoelastic Microelongated Solid With Internal Heat Source

The purpose of this paper is to study the two dimensional deformation due to an internal heat source in a thermoelastic microelongated solid. A mechanical force is applied along an overlaying elastic layer of thickness <i>h</i>. The normal mode analysis has been applied to obtain the exact expressions for the displacement component, force stress, temperature distribution and microelongation. The effect of the internal heat source on the displacement component, force stress, temperature distribution and microelongation has been depicted graphically for Green-Lindsay (GL) theory of thermoelasticity.

Reflection and refraction of magneto-thermoelastic plane wave at the pre-stressed liquid-solid interface in generalized thermoelasticity under three theories

The thermomagnetic effect on plane wave propagation at the liquid-solid interface with nonclassical thermoelasticity is investigated. It is assumed that liquid-solid half-space is under initial stress. Numerical computations are performed for the developed amplitude ratios of P, SV and thermal waves under Cattaneo-Lord-Shulman theory, Green-Lindsay theory and classical thermoelasticity. The system of developed equations is solved by the application of the MATLAB software at different angles of incidence for Green and Lindsay model. The effect of initial stress and magnetic field in the lower half-space are discussed and comparison is made in LS, GL and CT models of thermoelasticity. In the absence of magnetic field, the obtained results are in agreement with the same results obtained by the relevant authors. This study would be useful for magneto-thermoelastic acoustic device field.

An analytical solution for the effect of initial stress, rotation, magnetic field and a periodic loading in a thermo-viscoelastic medium with a spherical cavity

ABSTRACTIn this article, an analytical solution for the effect of initial stress, the rotation, and the relaxation times in a magneto-thermo-viscoelastic homogeneous medium with a spherical cavity subjected to periodic loading is presented. The distribution of displacements, temperature, and stresses in a homogeneous medium, in the context of generalized thermo-elasticity using the Green-Lindsay (GL) theory is discussed and obtained in analytical form. The results are displayed graphically to illustrate the effect of initial stress, the rotation, relaxation times, magnetic field, and viscoelasticity. Comparisons are made with the previous work in the absence of rotation and initial stress.

Wave propagation in a generalized piezothermoelastic rotating bar of circular cross section

Purpose – The purpose of this paper is to study the wave propagation in a generalized piezothermoelastic rotating bar of circular cross-section using three-dimensional linear theory of elasticity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in a generalized piezothermelastic rotating bar of circular cross-section by using Lord-Shulman (LS) and Green-Lindsay (GL) theory of thermoelasticity. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been derived by using the thermally insulated/isothermal and electrically shorted/charge free boundary conditions prevailing at the surface of the circular cross-sectional bar. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots. Findings – In order to include the time requirement for the acceleration of the heat flow and the coupling between the temperature and strain fields, the analytical terms have been derived for the non-classical thermo-elastic theories, LS and GL theory. The computed physical quantities such as thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency have been presented in the form of dispersion curves. From the graphical patterns of the structure, the effect of thermal relaxation times and the rotational speed as well as the anisotropy of the of the material on the various considered wave characteristics is more significant and dominant in the flexural modes of vibration. The effect of such physical quantities provides the foundation for the construction of temperature sensors, acoustic sensor and rotating gyroscope. Originality/value – In this paper, the influence of thermal relaxation times and rotational speed on the wave number with thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency has been observed and are presented as dispersion curves. The effect of thermal relaxation time and rotational speed on wave number for the case of generalized piezothermoelastic material of circular cross-section was never reported in the literature. These results are new and original.

Propagation of p-, T-, and SV-waves at the interface between two solid–liquid media with magnetic field and initial stress in the context of two thermoelastic theories

In this paper, we study the effects of magnetic field and initial stress on p-, T-, and SV-waves propagation. We investigated the problem of reflection and refraction of thermoelastic waves at a magnetized solid–liquid interface in the presence of initial stress. In the context of classical theory and Green–Lindsay theory of thermoelasticity, the problem has been solved. The boundary conditions applied at the interface are: (i) displacement continuity; (ii) vanishing tangential displacement; (iii) continuity of normal force per unit initial area; (iv) tangential force per unit initial area must vanish; and (v) continuity of temperature. The amplitudes ratios for the incident p-, T-, and SV-waves have been obtained. The reflection and transmittion coefficients from the incident waves are computed numerically, considering the initial stress and magnetic field effect and the results obtained have been presented graphically.

Propagation of Plane Waves in GeneralizedPiezo-thermoelastic Medium: ComparisonOf Different Theories

In this paper, a general solution for the propagation of plane waves in generalized piezo-thermoelastic medium for two-dimensional problem under the different thermoelastic theories is investigated. We have included Coupled theory (CT), Lord-Schulman (L-S) and Green-Lindsay (G-L) theories. The normal mode analysis is used to obtain the exact expressions for the considered variables. The results of the physical quantities have been illustrated graphically by comparison between (CT), (L-S) and (G-L) theories.

On Reflection and Transmission of p- and Sv-Waves Phenomena at the Interface Between Solid-Liquid Media with Magnetic Field and Two Thermal Relaxation Times

In this research, we studied the reflection and transmission of thermoelastic wave at a solid-liquid interface under influence of two thermal relaxation times and magnetic field. The governing equations are introduced taking into consideration Green-Lindsay theory and Maxwell's stresses. After solving the governing equations, we found the two reflections and transmission coefficients of incident p-(primary) and SV- (Shear Vertical) waves in the presence of thermal relaxation times and a magnetic field. The boundary conditions at the interface have been applied. The appropriate expressions to find the amplitude ratios for the two incidence waves (p- and SV-waves) have been obtained. A numerical calculation is made for the reflection and transmitted coefficients of the incident waves, in which we study the effect of thermal relaxation times and magnetic field. The results obtained are presented graphically for the effect of magnetic field and relaxation times to display the phenomena physical meaning.