Green And Naghdi Research Articles

A Study on Thermoelastic Interaction in a Poroelastic Medium with and without Energy Dissipation

In the current work, a new generalized model of heat conduction has been constructed taking into account the influence of porosity on a poro-thermoelastic medium using the finite element method (FEM). The governing equations are presented in the context of the Green and Naghdi (G-N) type III theory with and without energy dissipations. The finite element scheme has been adopted to present the solutions due to the complex formulations of this problem. The effects of porosity on poro-thermoelastic material are investigated. The numerical results for stresses, temperatures, and displacements for the solid and the fluid are graphically presented. This work provides future investigators with insight regarding details of non-simple poro-thermoelasticity with different phases.

Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.

Thermoelastic interaction in a thermally conducting cubic crystal subjected to ramp-type heating

In this paper, the thermoelastic interactions in a homogeneous, thermally conducting cubic crystal, elastic half-plane has been studied. A linear temperature ramping function is used to more realistically model. The general solution obtained is applied to a specific problem of a half space subjected to ramp-type heating. The components of displacement, stresses, and temperature distribution are obtained by applying a numerical finite element method. Some particular cases are also discussed in the context of the problem. The comparison in Lord and Shulman (LS), Green and Lindsay (GL) and Green and Naghdi (GN) theories have been shown graphically to estimate the effect of ramping parameter of heating for isothermal boundary.

Wave propagation in an initially stressed transversely isotropic thermoelastic half-space

AbstractThe present paper deals with the study of reflection waves in an initially stressed transversely isotropic medium, in the context of Green and Naghdi (GN) thermoelasticity theory type II and III. The components of displacement, stresses and temperature distributions are determined through the solution of the wave equation by imposing the appropriate boundary conditions. Numerically simulated results are plotted graphically with respect to frequency in order to show the effect of anisotropy.

On thermodynamic functions in thermoelasticity without energy dissipation

Using a systematic procedure based on convex/concave functions and Legendre transforms , the consistent set of the thermodynamic functions in the framework of the Green and Naghdi (GN) thermoelasticity without energy dissipation is addressed. Starting from the free energy of GN, the thermodynamic potentials , i.e. internal energy, enthalpy and Gibbs free energy , are derived. Moreover it is shown that four more thermodynamic potentials, named alternative, can be provided. The relations between these functions are provided and the constitutive relations are obtained extending the general methodology used for the classical thermoelastic behaviour. • Thermodynamic functions in coupled nonlinear GN thermoelasticity are derived. • The alternative thermodynamic functions for GN model are derived. • General procedure based on Legendre transforms is used to derive constitutive relations. • Relations between the thermodynamic potentials and the thermal work are provided. • An example showing the relations with the classic thermoelastic model is presented.

Rotational and voids effect on the reflection of P waves from stress-free surface of an elastic half-space under magnetic field and initial stress without energy dissipation

The influence of rotation, magnetic field, voids and initial stress on the reflection of P waves in thermoelasticity without energy dissipation are studied. The basic governing equations for isotropic and homogeneous thermoelastic half-space with voids, rotation, are based on Green and Naghdi (GN) theory under the effect of initial stress, where there is an initial magnetic field parallel to the plane boundary of the half-space. The solution of the problem in xz-plane indicates the existence of four plane waves; P1;P2;P3 and P4 and an SV waves. The relevant boundary conditions at stress-free insulated surface are satisfied by appropriate solutions in the half-space to obtain the reflection coefficients for the incidence of P waves. The numerical values of the complex modulus of the reflection coefficients are visualized graphically to display the effect of the rotation, magnetic field, initial stress and voids parameters.

A GN model for thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a circular hole

In this work, we have constructed the equations for generalized thermoelasticity of an unbounded fiber-reinforced anisotropic medium with a circular hole. The formulation is applied in the context of Green and Naghdi (GN) theory. The thermoelastic interactions are caused by (I) a uniform step in stress applied to the boundary of the hole with zero temperature change and (II) a uniform step in temperature applied to the boundary of the hole which is stress-free. The solutions for displacement, temperature and stresses are obtained with the help of the finite element procedure. The effects of the reinforcement on temperature, stress and displacement are studied. Results obtained in this work can be used for designing various fiber-reinforced anisotropic elements under mechanical or thermal load to meet special engineering requirements.

Generalized Thermoelasticity of Thermal-Shock Problem in a Non-homogeneous Isotropic Hollow Cylinder with Energy Dissipation

A solution of a thermal-shock problem of generalized thermoelasticity of a non-homogeneous isotropic hollow cylinder using a finite-element method is developed. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory of type II and type III by an appropriate choice of parameters. The problem has been solved numerically using a finite-element method. Numerical results for the distributions of displacement, temperature, radial stress, and hoop stress are represented graphically. The results indicate that the effects of non-homogeneity are very pronounced. The effects of non-homogeneity are presented with the two types of the Green and Naghdi theory.

Thermoelastic plane waves for an elastic solid half-space under hydrostatic initial stress of type III

In this paper, we constructed the equations of generalized thermoelastic isotropic and homogeneous half-space under hydrostatic initial stress in the context of the Green and Naghdi (GN) theory of types II and III. Normal mode analysis is used to obtain the exact expressions of temperature, displacement and stress. Comparisons are made with the results predicted by GN theory of types II and III in the presence and absence of the hydrostatic initial stress. The temperature, displacement and stress distributions are represented graphically.

The effect of diffusion on two-dimensional problem of generalized thermoelasticity with Green–Naghdi theory

The present paper is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory under the effect of diffusion. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using the normal mode analysis. These expressions are calculated numerically for a copper-like material and depicted graphically. Effect of presence of diffusion is analyzed theoretically and numerically. Comparisons are made with the results predicted by the type II and type III in the presence and absence of diffusion.

Magneto-thermo-elastic waves in an infinite perfectly conducting elastic solid with energy dissipation

The generalized thermo-elasticity theory, i.e., Green and Naghdi (G-N) III theory, with energy dissipation (TEWED) is employed in the study of time-harmonic plane wave propagation in an unbounded, perfectly electrically conducting elastic medium subject to primary uniform magnetic field. A more general dispersion equation with complex coefficients is obtained for coupled magneto-thermo-elastic wave solved in complex domain by using the Leguerre’s method. It reveals that the coupled magneto-thermoelastic wave corresponds to modified dilatational and thermal wave propagation with finite speeds modified by finite thermal wave speeds, thermo-elastic coupling, thermal diffusivity, and the external magnetic field. Numerical results for a copper-like material are presented.

Generalized magneto-thermoviscoelastic plane waves under the effect of rotation without energy dissipation

A new model of the equations of generalized thermoviscoelasticity for a thermally, isotropic and electrically conducting half-space solid whose surface is subjected to a thermal shock is given. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory under the effect of rotation, where there is an initial magnetic field parallel to the plane boundary of the half-space. The medium is deformed because of thermal shock and due to the application of the magnetic field, it results in induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the expressions for the variables considered. The distributions of temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by the types II and III in the presence and absence of rotation.