GENERALIZED MAGNETO- THERMOELASTICITY AND HEAT CONDUCTION ON AN INFINITE MEDIUM WITH SPHERICAL CAVITY

In this paper we will discuss the problem of distribution of thermal stresses and temperature in a generalized magneto–thermo-viscoelastic solid spherical cavity of radius R according to GreenNaghdi (G-N II) and (G-N III) theory. The surface of the cavity is assumed to be free traction and subjected to a constant thermal shock. The Laplace transform technique is used to solve the problem. The state space approach is adopted for the solution of one dimensional problem. Solution of the problem in the physical domain are obtained by using a numerical method of MATLAP Programmer and the expression for the temperature, strain and stress are obtained. Numerical computations are carried out for a particular material for illustrating the results. Finally the results obtained are presented graphically to show the effect of time on the field.


INTRODUCTION
The theory of Thermo elasticity deals with the effect of mechanical and thermal disturbances on an elastic body. In the nineteenth century, Duhamel (1837) was the first to consider elastic problems with heat changes. In 1855 Neumann (1841) re derived the equations obtained by Duhamel using a different approach. Their theory, the theory of uncoupled thermo elasticity consists of the heat equation which is independent of mechanical effects and the equation of motion which contains the temperature as a known function. There are two defects of the theory. First, the fact that the mechanical state of the elastic body has no effect on the temperature. This is not in accord with true physical experiments. Second, the heat equation being parabolic, predicts an infinite speed of propagation for the temperature which again contradicts physical observations. Biot (1956) introduced the coupled theory of thermo elasticity. In this theory the equations of elasticity and of heat conduction are coupled, which agree with physical experiments since any change of the temperature gives a certain amount of strain in an elastic body and vice versa. In most cases, the solutions which, are obtained by the classical theory, differ little from those obtained by using the theory of coupled thermoelectricity. This theory is useful in many problems. The equations of this theory consist of the equation of motion, which is hyperbolic partial differential equation, and of the equation of energy conservation, which is parabolic. There is a defect in this theory. The second equation being parabolic implies that if and elastic medium extending to infinity is subjected to a thermal or a mechanical disturbance, the effect will be felt instantaneously at infinitely distant points, which contradict experiments. Among the contributions in the subject of coupled thermo elasticity are the works of Nowacki who solved a problem for a half-* Corresponding author. Email: aaelbary@aast.edu space with heat sources (1957) and Ignaczak who solved a onedimensional problem for a spherical cavity (1959). Hetnarski solved a one-dimensional thermal shock problem (1961) and obtained the fundamental solution of the coupled theory (1964). Bahar and Hetnarski (1978) obtained the state space approach to the theory while Takeuti and Tsuji have solved a problem for a place due to rolling (1982). Uniqueness of the solution was proved by Weiner (1957) and some variation principles were obtained by Nickell and Sackman (1968).
Two theories of generalized thermo elasticity have been developed to oppose this last contradiction. One such generalization was obtained by Lord and Shulman (1976) and the other by Green and Lindsay (1972). In 1967 Lord and Shulman (1976) introduced the theory of generalized, thermo elasticity with one relaxation time for the special case of an isotropic body. This theory was extended by by Sherief (1980), Dhaliwal and Sherief (1980) in 1980 to include the anisotropic case. In this theory a modified law of heat conduction including both the heat flux and its time derivative replaces the conventional Fourier's law. The heat equation associated with this theory is hyperbolic and hence eliminates the paradox of infinite speeds of propagation inherent in both uncoupled and coupled theories of thermo elasticity. Green and Lindsay (1972) developed the theory of generalized thermo elasticity with two relaxation times based on a generalized inequality of thermodynamics. This theory does not violate Fourier's law of heat conduction when the body under consideration has a center of symmetry. In this theory both the equations of motion and of heat conduction are hyperbolic. The equation of motion is modified and differs from that of the coupled thermo elasticity theory.
The study of viscoelastic behavior is of interest in several contexts. First, materials used in engineering applications may exhibit viscoelastic behavior as an unintentional side effect. Second, the mathematics underlying viscoelasticity theory within the applied mathematics community. Third, viscoelasticity is of interest in some branches of material science, metallurgy and solid-state-physics. Fourth, the causal links between viscoelasticity and microstructure is exploited in the use of viscoelastic tests as an inspection tools. In reality all materials deviate from Hooke's law in various ways, for example, by exhibiting viscouslike as well as characteristics. Viscoelastic materials are those for which the relationship between stress and strain depends on time. All materials exhibit some viscoelastic response. In common metals such as steel, aluminum, copper etc. At room temperature and small strain, the behavior does not deviate much from linear elasticity. Synthetic polymer, wood as well as metals at high temperature display significant viscoelastic effects.
Modeling of viscoelastic response has a long tradition for describing a variety of phenomena (such as creep, relaxation and energy dissipation or damping) in structural analysis. One particular issue, which has been discussed extensively, is how complex. The linear viscoelastic model must be, i.e., what is the minimum number of material parameters that is required for an accurate description of the observed material behavior.
Most engineering materials exhibit a weak frequency dependence of the damping characteristics, which is difficult to describe with classic linear viscoelasticity that is based on integer rate laws for the pertinent internal variables. However, if the integer time derivative is replaced by a fractional order derivative operator, the number of parameters to describe damping in an accurate fashion can be significantly reduced. It has been argued, Bagley and Torvik (1983), the linear viscoelastic model together with fractional derivative have shown to be very flexible also for describing quasi static response, such as creep and stress relaxation. The Kelvin -Voigt model is one of the macroscopic mechanical models often used to describe the viscoelastic behavior of material. This model represents the delayed elastic response subjected to stress where the deformation is time dependent. Mishra et.al (1992) studied magnetothermo-mechanical interaction in an isotropic viscoelastic cylinder subjected to periodic loading considering Kelvin-Voigt model of linear viscoelasticity.
The theory of electro-magneto-thermo-viscoelasticity has aroused much interest in many industrial applications, particularly in nuclear devices, where there exists a primary magnetic field. Various investigations have been carried out by considering the interaction between magnetic, thermal and strain fields. Analyses of such problem also influence various applications in biomedical engineering as well as in different geomagnetic studied.
Due to the complicated nature of the governing equations for generalized electro-magneto-thermo viscoelasticity with thermal relaxation times, few attempts have been made to solve two-dimensional problems in this field. The propagation of electro magneto thermo elastic waves in an electrically and thermally conducting solid has been studied by many authors. Paria (1962Paria ( , 1972 discussed the propagation of plane magneto-thermo elastic wave in an isotropic unbounded medium under the influence of a uniform thermal field and with magnetic field acting transversely to the direction of propagation, Paria used the classical Fourier's law of heat conduction and neglected the electric displacement together with charge density, the coupling between the current density and the heat flow density and the coupling between the temperature gradient and the electric current. Result of important experiments determining the mechanical properties of viscoelastic material were involved by Koltunov's (1976).  studied electro-thermoelasticity theory with memorydependent derivative heat transfer.  linked the solution of two two-dimensional magneto-thermoelasticity with fractional order heat transfer in a medium of perfect conductivity.  obtained the solution of thermoelastic material response due to laser pulse heating in context of four theorems of thermoelasticity. Bary et al.
(2016) derived a new definition for linear Kelvin -Voigt model. Ismail (2014) analyzed state space approach for generalized thermoelastic medium subjected to non-uniform heat.  presented Vibration of Gold Nano Beam in Context of Two-Temperature, Generalized Thermoelasticity Subjected to Laser Pulse. The solution of Two-temperature theory of magneto-thermo-viscoelasticity with fractional derivative and integral orders heat transfer, were obtained by .  made great strides in the last decade in finding solutions for electro-magneto interaction in fractional Green-Naghdi thermoelastic solid with a cylindrical cavity. Many Applications of state space approach developed for different type of problems in thermoelasticity (2009-2018) some of them consider one temperature and other discussed two temperature.

BASIC GOVERNING EQUATIONS
We shall consider a homogeneous isotropic thermo-viscoelastic medium occupying the region ≤ < ∞ of a perfect electrically conductivity permeated by an initial constant magnetic field , where is the radius of the shell.
Due to the effect of this magnetic field there arises in the conducting medium an induced magnetic field and induced electric field . Also, there arises a force (the Lorentz Force). Due to the effect of this force, points of the medium undergo a displacement , which gives rise to a temperature. The linearized equations of electromagnetism for slowly moving media are: (4) The above field equations are supplemented by constitutive equations which consist first of ohm's law: = − × (5) The second constitutive equation is the one for the Lorenz force which is = × . (6) The third constitutive equation is the stress -displacement -temperature relation for viscoelastic medium of Kelvin -Voigt type: The equation of motion is given by: The generalized heat conduction equation is given by Together with the previous equations, constitute a complete system of generalized -magneto-thermo-viscoelasticity equations for a medium with a perfect conductivity. Let ( , , ) denote the radial coordinates, the co-latitude, and the longitude of a spherical coordinates system, respectively. Due to spherical symmetry, all the considered function will be functions of and only. The components of the displacement vector will be taken the form: = ( , ), = = 0 (11) The strain tensor components are thus given by = , = = , = = 0 It follow that the cubical dilatation is of the form ( 2 ) (13) From equation (7) we obtain the components of the stress tensor as Assume now that the initial magnetic field acts in the -direction and has the components(0,0, ). The induced magnetic field h will have one component h in thedirection, while the induced electric field E will have one component E in the -direction. Then, equations (1), (2) and (5) = .
(19) From equations (17) and (6), we get that the Lorentz force has only one component in the -direction: = 2 (20) Also, we arrived at Equation (21) is to be supplemented by the constitutive equation (13) and the heat conduction equation Where ∇ 2 is Laplace's operator in spherical coordinates which is given by = , where, = , 1 2 = + 2 Equation (23) can be written in the form:

LAPLACE TRANSFORM DOMAIN
Taking the Laplace transform of equations (23)-(31) by using homogeneous initial conditions, defined and denoted as

STATE SPACE FORMULATION
Choosing as state variables the temperature of heat conduction ̅ and the strain components ̅ then equations (35) and (38)  , ≥ (41) We will use the will-known Cayley-Hamiltonian theorem to find the form of the matrix The characteristic equation of the matrix ( ) can be written as 2 − ( 1 + 2 ) + ( 1 2 − 2 1 ) (42) The roots of this equation namely 1 and 2 satisfy the relations 1 + 2 = 1 + 2 (43) 1 2 = 1 2 − 2 1 (44) The Tailor's series expansion for the matrix exponential of Using Cayley -Hamiltonian theorem, we can express 2 and higher orders of the matrix in terms of and where is the unit matrix of second order. Thus, the infinite series in equation (42) can be reduced to Where and 1 are some coefficients depending on and . By Cayley-Hamiltonian theorem, the characteristic roots 1 and 2 of the matrix must satisfy equation (42), thus we have �−� 2 ( − )� = + 1 2 (48) Solving the above linear system of equations, we get where,

APPLICATIONS
In order to evaluate the unknown parameters ̅ ( , )and � ( , ), we shall use the boundary conditions on the internal surface of the shell, = which are given by: (I) Thermal boundary condition at = ( , ) = Taking the Laplace transform, this is defined as following:

(II) Mechanical boundary condition
The internal surface = has a rigid foundation, which is rigid enough to prevent any strain ( , ) = 0 Taking the Laplace transform, this is defined as following: ̅ ( , ) = ̅ = 0 (57) Using the conditions (49) and (50) into equation (34) and using equations (44)-(48), we get To find the displacement, taking Laplace transform for equation (23) and using equations (51) and (52), we get To find the stresses, from equations (36), (37) and (58)-(60) we get where is a sufficiently large integer chosen such that Where is a reselected small positive number that corresponds to the degree of accuracy to be achieved Formula (49) is the numerical inversion formula valid for0 ≤ ≤ 1 . In particular, we choose = 1 , getting

NUMERICAL RESULTS AND DISCUSSIONS
The copper material was chosen for purposes of numerical evaluations and constants of the problem were taken as following In order to study the effect of time on temperature, radial stress, shear stress, displacement and strain, we now present our results in the form of graphs (Figs. 1-5). Fig. 1 shows the variation of temperature θ against rfor wide range ofr,1 ≤ r ≤ 3 for different values of time ( = 0.5, = 0.05 = 0.005) and we have noticed that, the time has significant affects on temperature. The increasing of the value of causes decreasing of the value of temperature, and temperature vanishes more rapidly.  0.05, = 0.09) and we have noticed that, the time t has significant effects on stress, for wide range of (1 ≤ ≤ 2.7) the increasing of the value of causes decreasing of the value of the radial stresses and for wide range of (2.7 ≤ ≤ 8) the increasing of the value of causes increasing of the value of the radial stresses and radial stresses vanishes more rapidly. Fig. 3 shows the variation of the shear stress against for wide range of (1 ≤ r ≤ 3), for different values of time ( = 0.5, = 0.05, = 0.005) and we have noticed that, the time has significant effects on stress. The increasing of the value of causes decreasing of the value of the shear stress, and the shear stress vanishes more rapidly. Fig. 4 shows the variation of strain against for wide range of ,1 ≤ ≤ 3, for different values of time ( = 0.5, = 0.05, = 0.005) and we have noticed that, the time has significant effects on strain, for the wide range (1 ≤ ≤ 1.4)the increasing of the value of causes decreasing of the value of strain , for the wide range(1.4 ≤ ≤ 3) the increasing of the value of t causes increasing of the value of strain and strain e vanishes more rapidly. Fig. 5 shows the variation of displacement against for wide range of (1 ≤ r ≤ 3), for different values of time ( = 0.5, = 0.05, = 0.005) and we have noticed that, the time has significant effects on displacement . The increasing of the value of causes increasing of the value of displacement , and displacement vanishes more rapidly.

CONCLUSIONS
According to this work, many researchers in the field of generalized thermo elasticity have applied Green-Naghdi theory (III-II) for thermo elastic problem and very few of them can successfully applied for magneto thermo Viscoelastic problem. In this paper we deduce that the magnitude of the all physical quantities depend on time .It can be noted that the speed of propagation of all physical quantities is finite and coincide with the physical behavior of Viscoelastic material.