Effect of modified Ohm's and Fourier's laws on magneto thermoviscoelastic waves with Green-Naghdi theory in a homogeneous isotropic hollow cylinder

This paper deals with the modified Ohm's law, including the temperature gradient and charge density effects, and the generalized Fourier's law, including the present current density impact, the problem of conveyance of thermal stresses and temperature in a generalized Magneto–Thermo-Viscoelastic Solid Cylinder of radius L. The formulation is applied to the generalized thermoelasticity dependent on the Green-Naghdi (G-N II) hypothesis. The Laplace change system is utilized to solve the problem. At last, the outcomes got are introduced graphically to show the impact of Magnetic Field and time and on the field variables.


Introduction
*A viscoelastic material is, as the name suggests, one that shows a combination of Newtonian viscous fluid and Hookean elastic solid effects. The viscous term leads to energy dissipation and the elastic term to energy storage. . Amorphous polymers such as plastics and synthetic rubber, fibrous materials (e.g., silk and cellulose), glasses, ceramics, biomaterials (e.g., skin and cellulose), glasses, ceramics, biomaterials (e.g. skin and mussel), and nonmetals, in general, may frequently be considered to behave in a linear viscoelasticity manner. . For a long time mechanics of deformable bodies has been based upon Hooke's law-that is, upon the assumption of linear elasticity. It was well known that most engineering materials like metals, concrete, soil are not linearly elastic or, are so within limits too narrow to cover the range of practical interest. Nevertheless, almost all routine stress analysis is still based on Hooke's law because of its simplicity.
Interactions between strain and electromagnetic fields are being endeavored on account of its various applications in various pieces of science and advancement. The advancement of magneto elasticity likewise instigates us to consider different issues of geophysics, seismology, and related themes. Without delving into the subtleties of the exploration work distributed so far in the fields of magneto elasticity, magneto-thermo-elasticity, magnetothermo viscoelasticity we notice some ongoing papers. Acharya and Roy (1978) examined the magneto-thermoelastic surface waves in initially stressed conducting media. Chaudhary et al. (2004) researched the reflection/Transmission of plane SH wave through a self-reinforced elastic layer between two half-spaces. Plane SH-wave response from elastic slab interposed between two different selfreinforced elastic solids studied by Chaudhary et al. (2006). Examined the surface waves in magnetoelastic initially stressed conducting media been illustrated by De and Sengupta (1971). Othman and Song (2006) examined the effect of rotation on the reflection of magneto-thermoelastic waves under thermoelasticity without energy dissipation. Magneto-thermoelastic wave propagation at the interface between two micropolar viscoelastic media examined by Song et al. (2006). Tianhu et al. (2004) contemplated a two-dimensional generalized thermal shock problem for a half-space in electromagneto-thermoelasticity. Verma et al. (1988) examined the magneto-elastic transverse surface waves in self-reinforced elastic solids. Effect of the rotation on an infinite generalized magnetothermoelastic diffusion body with a spherical cavity concentrated by Abd-Alla et al. (2011). Bayones (2012) contemplated the influence of diffusion on the generalized magneto-thermo-viscoelastic problem of homogenous isotropic material. Viscoelastic materials are those for which the connection among anxiety relies upon a time. All materials display some viscoelastic reaction. In like manner metals, for example, steel, aluminum, copper and so on. Abd-Alla and Abo-Dahab (2009) explored time-harmonic sources in a generalized magnetothermo-viscoelastic continuum with and without energy dissipation. Roychoudhuri and Banerjee (1998) researched the magneto-thermoelastic interactions in an infinite viscoelastic cylinder of temperature rate-dependent material subjected to a periodic loading. Spherically symmetric thermoviscoelastic waves in a viscoelastic medium with a spherical cavity are talked about by Banerjee and Roychoudhuri (1995). The problem of magnetothermo-viscoelastic interactions in an unbounded body with a spherical cavity subjected to a periodic loading is examined by Abd-Alla et al. (2004).
Linear viscoelasticity has been a significant zone of research since the period of Maxwell, Boltzmann, Voigt, and Kelvin. Important data with respect to the linear viscoelasticity hypothesis might be gotten in the books of Gross (1968) and Staverman et al. (1956). Many research like Biot (1955), Gurtin and Sternberg (1962), Liioushin and Pobedria (1970), and Tanner (1988) have contributed prominently to Thermo viscoelasticity.
The Kelvin-Voigt model is one of the naturally visible mechanical models regularly used to portray the viscoelastic conduct of material. This model speaks to the deferred elastic reaction exposed to stress when the deformation is time subordinate however recoverable. The dynamic interaction of thermal and mechanical fields in solids has incredible handy applications in present-day aviation atomic reactors and high vitality molecule quickening agents, for instance. Biot (1956) defined the coupled thermoelasticity theory to wipe out the conundrum inalienable in the old-style uncoupled hypothesis that elastic deformation has no impact on the temperature. The field equations for both the hypotheses are of a mixed parabolic-hyperbolic type, which foresees limitless paces for thermoelastic signs, in spite of physical perceptions. Hetnarski and Ignaczak (1999) analyzed five generalizations to the coupled theory of thermoelasticity.
The first generalization is because of Lord and Shulman who planned the generalized thermoelasticity theory including one thermal relaxation time. This hypothesis is alluded to as the L-S hypothesis or broadened thermoelasticity in which the Maxwell Cattaneo law replaces the Fourier law of heat conduction by presenting a solitary parameter that goes about as a relaxation time.
The second generalization to the coupled thermoelasticity hypothesis is because of Green and Lindsay (1972), called the G-L hypothesis or the temperature-rate dependent theory, which includes two relaxation times.
The third generalization to the coupled thermoelasticity is known as low-temperature thermoelasticity presented by Hetnarski and Ignaczak (1996), called H-I theory. This model is portrayed by a system of non-linear field equations. Low-temperature non-linear models of heat conduction foresee wave-like thermal signals and which are assumed and concentrated in certain works by Kosiński (1989).
The fourth generalization to the coupled theory is worried about the thermoelasticity theory without energy dissipation presented by Green and Naghdi (1991), alluded to as the G-N hypothesis of type II in which the old-style Fourier law is supplanted by heat flux rate-temperature gradient relation. The heat transport equation doesn't include a temperaturerate term and in that capacity, this model concedes undamped thermoelastic wave in thermoelastic material. With regards to linearized adaptation of this hypothesis, a hypothesis on the uniqueness of arrangement has been built up by Chandrasekharaiah (1986). The fourth generalization of the thermoelasticity hypothesis created by Green and Naghdi additionally includes a heat conduction law, which the conventional law and one that involves the thermal displacement gradient among the constitutive variables. This model is alluded to as the GN model III, which includes the dispersal of vitality as a rule and concedes damped thermoelastic waves.
The fifth generalization to the thermoelasticity theory is known as the dual-phase lag thermoelasticity created by Tzou (1995) and Chandrasekharaiah and Krivoshapko (1998). Tzou (1995) considered micro-structural effects into the delayed response in time in the macroscopic formulation by taking into account that the increase of the lattice temperature is delayed to phononelectron interactions on the macroscopic level. A macroscopic lagging (or delayed) response between the temperature gradient and the heat flux vector seems to be a possible outcome due to such progressive interactions. Tzou (1995) introduced two-phase lags to both the heat flux vector and the temperature gradient and considered a constitutive equation to describe the lagging behavior in the heat conduction in solids. Here the classical Fourier law is replaced by an approximation to a modification of the law with two different translations for the heat flux vector and the temperature gradient.

Basic governing equations
The governing equations in the context of the theory of generalized thermo viscoelasticity with magnetic field for the isotropic and homogeneous elastic medium are considered as:  The equation of motion: where, = ×  Heat conduction equation:  Constitutive relations: We take the linearized Maxwell's equations governing the electromagnetic field for a perfectly conducting medium as, We derive the analytical formulation of the problem in cylindrical coordinates( , , ) with the z-axis coinciding with the axis of the cylinder. We consider the strain axis symmetrical about the z-axis. We have only the radial displacement = ( , ), the circumferential displacement = 0 and the longitudinal displacement = 0. we consider a perfectly conducting isotropic homogeneous generalized thermo viscoelastic cylinder subjected to a constant magnetic field (0, , 0) which produce an induced magnetic field ℎ(0, ℎ , 0) and induced electric field (0,0, ).We assume one-dimensional motion for which all the field quantities are functions of and .
The displacement components take the form, The strain components become, The components of magnetic field vectors are, The electric intensity vector is parallel to the current density vector . Hence components of and are given by, Now, the Maxwell's Eqs. 6-9 provide the following results: Using Eq. 11 and Eqs. 14-16 into the relation,

= ×
we obtain, From Eq. 3 we obtain the components of the stress tensor as, Also, we arrived at: Now, we shall use the following non-dimensional variables: Eqs. 14-16 and Eqs. 18-22 take the following form (dropping the primes for convenience).

Solution of the problem
Now to solve Eqs. 35 and 36 put = ̅ and ̅ = we get, ∇ 4 − ( 11 + 22 )∇ 2 + ( 11 22 − 21 12 ) = 0 Eq. 40 can be written as, where 1 2 and 2 2 are the roots of the characteristic equation, 4 − ( 11 + 22 ) 2 + ( 11 22 − 21 12 ) = 0 The solution of Eq. 41 takes the form, where, 1 satisfy the equation, The solution of Eq. 43 takes the form: where, ( 1 )Is the Modified Bessel function of the first kind of order zero; ( 1 )Is the Modified Bessel function of the second kind of order zero; 1, 1 Are parameters depending on s only to be determined from the boundary conditions. For boundedness, we take 1 = 0 then, by the same way we get: then, finally, we get, in a similar manner, we get,

Numerical inversion of Laplace transforms
In order to invert the Laplace transforms in the above equations we shall use a numerical technique based on Fourier expansions of functions. Let ̅ ( ) be the Laplace transform of a given function ( ). The inversion formula of Laplace transforms states that, where d is an arbitrary positive constant greater than all the real parts of the singularities of ̅ ( ). Taking = + , we get, This integral can be approximated by, Taking ∆ = 1 we obtain: For numerical purposes, this is approximated by the function, 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 for 0 ≤ ≤ 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 follows: = 386 ⁄ , = 17.8(10) −5 −1 , = 383.1 2 ⁄ , = 293 , = 8954 2 , = 3.86(10) 10 2 ⁄ ⁄ = 7.76(10) 10 2 ⁄ , = 0.001, = 6.8831(10) −13 , 1 = 6.8831(10) −13 , = 1, 1 = 1 In order to study the effect of time t and study the comparison between two theories on temperature, radial stress, displacement, and strain, we now present our results in the form of graphs (Figs. 1-8).  1 shows the variation of temperature for the coefficient of Ohm and Fourier laws. It is noticed that the modified Fourier and Ohm laws influence is significant, the temperature in the modified model records values higher than those in the old model. Fig. 2 shows the variation of temperature against for wide range of , 1 ≤ ≤ 3 for different values of time ( = 0.01, = 0.05) and we have noticed that the time has significant effects on temperature. The increasing of the value of causes increasing of the value of temperature, and temperature vanishes more rapidly. Fig. 3 shows the variation of radial stress for the coefficient of Ohm and Fourier laws. It is noticed that the modified Fourier and Ohm model affects the radial stress by increasing their values. Fig. 4 shows the variation of the radial stress against for wide range of ,(1 ≤ ≤ 8), for different values of time ( = 0.01, = 0.05) and we have noticed that the time has significant effects on stress, for a 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. 5 shows the variation of strain for the coefficient of Ohm and Fourier laws. It is noticed that the modified Fourier and Ohm laws influence is significant, the strain in the modified model records values higher than those in the old model. Fig. 6 shows the variation of strain against for wide range of , (1 ≤ ≤ 3), at fractional parameter = 0.5 and for different values of time ( = 0.1 = 0.5) and we have noticed that the time has significant effects on the 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 causes increasing of the value of strain and strain vanishes more rapidly. Fig. 7 is plotted to shows variation of displacement for the coefficient of Ohm and Fourier laws. It is noticed that the modified Fourier and Ohm laws influence is significant, the displacement in the modified model records values higher than those in the old model. Fig. 8 shows the variation of displacement against for wide range of , 1 ≤ ≤ 3 , for different values of time ( = 0.5, = 1) 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.

Conclusion
As per this work, numerous analyses in the field of generalized thermoelasticity have applied Green-Naghdi for the elastic problem and not many of them can effectively be applied for magneto thermo Viscoelastic problem. From the above discussion, one can conclude that the new model of generalized magneto-thermo-viscoelasticity predicts new qualities for the temperature, displacement, stresses, and strain. The impact of the temperature gradient at adequately low temperatures may cause new sorts of the magneto-thermo-viscoelastic wave with explicit strength properties. The expansion in the estimations of temperature might be clarified as the lost heat-producing from the development of electric flow; this heat might be the primary motivation behind why the deformation of the medium will, in general, be typical and the magnetic field records esteem more prominent than the qualities in the old model.
It very well may be noticed that the speed of spread of every single physical amount is limited and correspond to the physical conduct of Viscoelastic material.