Wave Propagation in a Magneto-Micropolar Thermoelastic Medium with Two Temperatures for Three-Phase-Lag Model

Samia M.Said

Wave Propagation in a Magneto-Micropolar Thermoelastic Medium with Two Temperatures for Three-Phase-Lag Model

Samia M.Said1

The present paper is concerned with the wave propagation in a micropolar thermoelastic solid with distinct two temperatures under the effect of the magnetic field in the presence of the gravity field and an internal heat source.The formulation of the problem is applied in the context of the three-phase-lag model and Green-Naghdi theory without dissipation.The medium is a homogeneous isotropic thermoelastic in the half-space.The exact expressions of the considered variables are obtained by using normal mode analysis.Comparisons are made with the results in the two theories in the absence and presence of the magnetic field as well as the two-temperature parameter.A comparison is also made in the two theories for different values of an internal heat source.

Green-Naghdi theory,internal heat source,magnetic field,micropolar,three-phase-lag model,two-temperature.

1 Introduction

The comprehensive review on the micropolar elasticity was given by Eringen(1966,1970);Nowacki(1986).Chandrasekharaiah(1986)developed a heat flux dependent micropolar thermoelsticity.Kumar and Singh(1998)studied the reflection of plane waves from the flat boundary of a micropolar generalized thermoelastic with stretch.Kumar(2000)investigated the reflection coefficient in a micropolar viscoelastic generalized half-space.Singh(2007)discussed the wave propagation in an orthotropic micropolar elastic solid.A new theory of the generalized thermoelasticity reinforcement has been constructed by taking into account the deformation of a micropolar generalized thermoelastic medium with voids are discussed by Othman,Lotfy,Said,and Anwar Bég(2012).The deformation due to thermomechanical sources in a homogeneous isotropic micropolar thermoelastic medium with void are discussed by Abbas,Kumar,Sharma,and Garg(2015).

The investigation of the interaction between the magnetic field,stress,and strain in a thermoelastic solid is very important due to its many applications in diverse field such as geophysics(for understanding the effect of the Earth’s magnetic field on seismic waves),damping of acoustic waves in a magnetic field,designing machine elements like heat exchangers,boiler tubes(where the temperature induced elastic deformation occurs),biomedical engineering(problems involving thermal stress),emissions of the electromagnetic radiations from nuclear devices,development of a highly sensitive super conducting magnetometer,electrical power engineering,plasma physics,etc.Many studies in a generalized magneto-thermoelasticity can be found in the literatures by Youssef(2006);Othman and Kumar(2009);Othman and Atwa(2011);Othman and Abass(2015);Abbas and Zenkour(2015).

A theory of heat conduction in deformable bodies which depends upon two distinct temperatures,the conductive temperature and the thermodynamic temperature,has been established by Chen and Gurtin(1968a);Chen and Williams(1968b);Chen,Gurtin,and Williams(1969).In time-independent problems,the difference between these two distinct temperatures is proportional to the heat supply and in the absence of any heat supply;these two temperatures are identical as Chen and Williams(1968b).In time-dependent situations and of the wave propagation problems,in particular,the two-temperatures are in general different,regardless of the presence of a heat supply.Warren and Chen(1973)have studied the wave propagation in the two-temperature theory of thermoelasticity.Youssef(2005)has proposed a theory in the context of the generalized theory of thermoelasticity with two-temperature.Several problems have been solved by Das and Kanoria(2012);Abbas and Zenkour(2014);Zenkour and Abouelregal(2015)applying the twotemperature theory of thermoelasticity.

It is well known that the usual theory of heat conduction based on Fourier’s law predicts an infinite heat propagation speed.It is also known that heat transmission at low temperature propagates by means of waves.These aspects have caused intense activity in the field of heat propagation.Extensive reviews on the second sound theories(hyperbolic heat conduction)are given in Hetnarski and Ignaczak(1999,2000).A two-phase-lag to both the heat flux vector and the temperature gradient was introduced by Tzou(1995).According to this model,classical Fourier’s lawhas been replaced bywhere the temperature gradient∇Tat a pointPof the material at timet+τTcorresponds to the heat fl ux vectorat the same point at timet+τq.HereKis the thermal conductivity of the material.The delay time τTis interpreted as that caused by the micro-structural interactions and is called the phase-lag of the temperature gradient.The other delay time τqis interpreted as the relaxation time due to the fast transient effects of thermal inertia and is called the phase-lag of the heat flux.For τq= τT=0,Fourier’s law in a two-phase-lag model is identical with classical Fourier’s law.If τq= τ and τT=0,Tzou(1995)refers to the model as a single-phase-lag model.Recently Choudhuri(2007)has proposed a three-phase-lag(3PHL)thermoelasticity which is able to contain all the previous theories at the same time.In this case Fourier’s lawhas been replaced bywhereis the thermal displacement gradient andK∗is the additional material constant and τνis the phase-lag for the thermal displacement gradient.The purpose of the work of Choudhuri(2007)is to establish a mathematical model that includes the 3PHL in the heat flux vector,the temperature gradient and in the thermal displacement gradient.For this model,we can consider several kinds of Taylor approximations to recover the previously cited theories.In particular the models of Green and Naghdi(1991,1992,1993)are recovered.The introduction of the 3PHL model provides a general theoretical heat conduction model with different micro-structural considerations in order to enable scientists in the field of heat conduction with a multi-scale model to predict accurately the thermal behavior of structures.The three-phase-lag model is very useful in the problems of nuclear boiling,exothermic catalytic reactions,phonon-electron interactions,phonon-scattering,etc.Quintanilla and Racke(2008);Quintanilla(2009);Kar and Kanoria(2009);Kumar,Chawla,and Abbas(2012);Abbas(2014);Kumar and Kumar(2015)have solved different problems applying the 3PHL model.

In the present work,we shall formulate a two-temperature magneto-micropolar thermoelastic problem in the presence of the gravity field for a medium with an internal heat source that is moving with a constant speed.Normal mode analysis is used to obtain the exact expressions for displacement components,force stresses and temperatures.The distributions of the considered variables are given and represented graphically.Comparisons are conducted between the considered variables as calculated from the 3PHL model and Green-Naghdi theory without dissipation(GN II)in the presence and absence of a magnetic field as well as a two-temperature parameter.A comparison is also made in the two theories for different values of an internal heat source.

2 Formulation of the problem and basic equations

We consider the problem of an isotropic homogeneous micropolar thermoelastic half-space(x≥0).The generalized thermoelastic medium is permeated into a uniform magnetic field with a constant intensitywhich is acting parallel to they-axis.We are interested in a plane strain in thexz-plane[displacement vector

When thez-axis is positive downward the body force components are given by Othman,Elmaklizi,and Said(2013).

The field equations and constitutive relations for a micropolar generalized thermoelastic medium in the absence of body forces and body couples can be written as Eringen(1970),Choudhuri(2007)and Youssef(2005)in the context of generalized thermoelasticity as follows:

The constitutive law of the theory of generalized thermoelasticity is

where σijare the components of stress,eijare the components of strain,ekkis the dilatation,λ,µ are the elastic constants,γ=(3λ +2µ)αt,αtis the thermal expansion coefficient,=T-T0,whereTis the temperature above the reference temperatureT0,εijris the alternate tensor and δijis the Kronecker delta.The strains can be expressed in terms of the displacementuias

We restrict our analysis parallel to thexz-plane with the micro-rotation vector φ =(0,Φ2,0).In the above equations a comma followed by a suffix denotes partial derivative with respect to the corresponding coordinates.

Eq.(2),then yields

whereA1=λ+2µ+k.

2.1 Equation of motion:

whereF1,F3are the Lorentz force and are given in the form,

The variations of the magnetic and electric fields are perfectly conducting slowly moving medium and are given by Maxwell’s equation as Othman and Atwa(2011).

where µ0is the magnetic permeability,ε0is the electric permeability,is the current density vector,is the particle velocity of the medium,and the small effect of the temperature gradient onis also ignored.The dynamic displacement vector is actually measured from a steady-state deformed position and the deformation is assumed to be small.Due to the application of the initial magnetic fi eldthere are an induced magnetic field=(0,h,0)and an induced electric field,as well as the simplified equations of electrodynamics of a slowly moving medium for a homogeneous,thermal and electrically conducting,elastic solid.Expressing the vectorin terms of the displacement by eliminating the quantitiesandfrom Eq.(10),thus yields,

Substituting Eq.(14)into Eq.(9),we get

2.2Heat conduction equation

The relation between the conductive temperature and the thermodynamics temperature is

whereK∗is the additional material constant,K1is the coefficient of thermal conductivity,ρ is the mass density,CEis the specific heat at constant strain,Qis a moving internal heat source,τTand τqare the phase-lag of temperature gradient and the phase-lag of heat flux respectively.Alsowhere τνis the phase-lag of thermal displacement gradient.

2.3 The equations of micropolar materials

Where α,β,γ1,kare the material constants.J0is micro-inertia andmijis the couple stress tensor.

Introducing Eqs.(4)-(6)and(15)in Eqs.(7),(8)we get

whereA2=λ+µ+k.

Usingh=-H0e,then introducing the following non-dimension quantities in the above equation(dropping the primes for convenience):

Thus we get,

where,

Introducing potential functions defined by

whereq(x,z,t),and ψ(x,z,t),are scalar potential functions.

Introducing Eq.(28)in Eqs.(23)-(26),we get

WhereR1=1+RH.

3 Normal mode analysis

The solution of the considered physical variable can be decomposed in terms of normal modes as the following form:

wheremis a complex constant,,ais the wave number in thex-direction,v0is the velocity of a moving internal heat source,Q0is the magnitude of an internal heat source andarethe amplitudes of the field quantities.

Introducing Eqs.(33)in Eqs.(29)-(32)and Eq.(27),we obtain

where

Equation(39)can be factored as

The solution of Eq.(39),which is bound asx→∞,is given by

In a similar manner,we get that

whereMnare parameters,

Introducing Eq.(44)in Eq.(38),this yields

Introducing Eq.(42)and(43)in Eq.(28),this yields

where,H5n=ia-H1nkn,H6n=-kn-iaH1n.

Introducing Eqs.(22)and(33)in Eqs.(5)and(6),we get

Introducing Eqs.(45)-(48)in Eqs.(49)and(50),this yields

where,

From Eqs.(22)and(33)in Eqs.(19),we get

Introducing Eqs.(45)in Eqs.(53),this yields

4 Application

We consider a generalized two-temperature magneto-micropolar thermoelastic problem for a medium with an internal heat source that is moving with a constant speed in the presence of the gravity field which fills the region Ω defined as follows:

In the physical problem,we should suppress the positive exponentials that are unbounded at infinity.The constantsMn(n=1,2,3,4)have to be chosen such that the boundary conditions on the surface atz=0 are as follows:

Where,f(x,t)is an arbitrary function ofx,t,andf∗is the magnitude of the mechanical force.Using the expressions of the variables considered into the above boundary conditions(Eqs.(55)),we can obtain the following equations satisfied with the parameters:

Invoking Eqs.(56)-(59),we obtain a system of four equations.After applying the inverse of matrix method,we have the values of the four constantsMn(n=1,2,3,4).Hence,we obtain the expressions of the considered variables.

5 Particular cases and special cases of thermoelastic theory

i.The corresponding equations for atwo-temperature micropolar thermoelastic medium with an internal heat source(Q0=5.5)in the presence of the gravity field from the above mentioned cases by takingH0to vanish.

ii.The corresponding equations for a magneto-micropolar thermoelastic medium with an internal heat source(Q0=5.5)in the presence of the gravity field from the above mentioned cases by taking δ to vanish.

iii.The corresponding equations for a two-temperature magneto-micropolar thermoelastic medium in the presence of the gravity fi eld for different values of an internal heat source from the above mentioned cases by takingQ0=5.5,1.

iv.Equations of the 3PHL model whenK,τT,τq,τν＞ 0 and the solutions are always(exponentially)stable ifas in Quintanilla and Racke(2008).

v.Equations of the G-N II theory whenK= τT= τq= τν=0.

6 Numerical calculation and discussion

In order to illustrate the theoretical results obtained in the preceding section,and to compare these in the context of the 3PHL model and the GN-II theory,we now present some numerical results for the physical constants as

The computations were carried out for a value of timet=1.2.The vertical displacement componentw,the thermodynamic temperature θ,the conductive temperature Φ,the stress components σzz,σxz,the tangential couple stressmzyand micro-rotation component Φ2with distancezfor the value ofx,namelyx=1.5,were substituted in performing the computations.The results are shown in figures 1-21.The graphs show the four curves predicted by two different theories of thermoelasticity.In these figures,the solid lines represent the solution in the 3PHL model,and the dashed lines represent the solution derived using the G-N II theory.

Figure1:Vertical displacement distribution w in the absence and presence of a magnetic field.

Figure2:Conductive temperature distribution Φ in the absence and presence of a magnetic field.

Figure 3:Thermodynamic temperature distribution θ in the absence and presence of a magnetic field.

Figure 4:Distribution of the stress component σzzin the absence and presence of a magnetic field.

Figures 1-7 show comparisons between the vertical displacementw,the thermodynamic temperature θ ,the conductive temperature Φ,the stress components σzz,σxz,the tangential couple stressmzyand micro-rotation component Φ2in the absence(H0=0)and presence(H0=140)of a magnetic field with a two-temperature parameter(δ =2× 10-14)and an internal heat source(Q0=5.5).

Figure 5:Distribution of the stress component σxzin the absence and presence of a magnetic field.

Figure 6:Distribution of the tangential couple stress mzyin the absence and presence of a magnetic field.

Figure 7:Distribution of the micro-rotation component Φ2in the absence and presence of a magnetic field.

Figure 1 depicts that the distribution of the vertical displacementwbegins from positive values.In the context of the two theories and in the presence of a magnetic field,wstarts with decreasing to a minimum value,then increases,and also moves in the wave propagation.However,in the context of the two theories and in the absence of a magnetic field,wstarts with decreasing,then increases,after then becomes nearly constant.The magnetic field decreases the magnitude ofwthen increases it.Figure 2 exhibits the distribution of the conductive temperature Φ and demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,in the absence and presence of a magnetic field,Φ decreases in the range 0≤x≤10.The magnetic field decreases the magnitude of Φ .Figure 3 explains the distribution of the thermodynamic temperature θ .In the context of the two theories,in the absence and presence of a magnetic field,θ starts with increasing to a maximum value,then decreases to a minimum value and also moves in the wave propagation.The magnetic field increases the magnitude of θ,then decreases,again increases,and in the last decreases it.Figure 4 explains that the distribution of the stress component σzzbegins from a negative value and satisfies the boundary condition atx=0.In the context of the two theories,in the absence and presence of a magnetic field,σzzstarts with decreasing to a minimum value,then increases to a maximum value,and also moves in the wave propagation.The magnetic field decreases the magnitude of σzz,then increases,again decreases,and in the last increases it.Figure 5 depicts the distribution of the stress component σxzand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,in the absence and presence of a magnetic field,σxzstarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation.The magnetic field increases the magnitude of σxz,then decreases,again increases,and in the last decreases it.Figure 6 depicts the distribution of the tangential couple stressmzyand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories and in the presence of a magnetic field,mzystarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation.However,in the context of the two theories and in the absence of a magnetic field,mzystarts with decreasing to a minimum value,then increases to a maximum value,and also moves in the wave propagation.The magnetic field increases the magnitude ofmzy,then decreases,again increases,and so on.Figure 7 describes the distribution of the micro-rotation component Φ2.In the context of the two theories and in the presence of a magnetic field,Φ2starts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation.However,in the context of the two theories and in the absence of a magnetic field,Φ2starts with decreasing to a minimum value,then increases to a maximum value,and also moves in the wave propagation.The magnetic field increases the magnitude of Φ2,then decreases,again increases,and in the last decreases it.Figures 1-7 demonstrate that the values of all the physical quantities converge to zero by increasing the distancez,the behavior of two theories are similar.These trends obey elastic and thermoelastic properties of the solid.

Figures 8-14 show comparisons between the vertical displacementw,the thermodynamic temperature θ ,the conductive temperature Φ ,the stress components σzz,σxz,the tangential couple stressmzyand micro-rotation component Φ2for one temperature(δ =0)and two temperature(δ =2 × 10-14)in the presence of a magnetic field(H0=140)and an internal heat source(Q0=5.5).

Figure8:Vertical displacement distribution w for one and two-temperature.

Figure9:Conductive temperature distribution Φ for one and two-temperature.

Figure 10:Thermodynamic temperature distribution θ for one and twotemperature.

Figure 12:Distribution of the stress componentσxzfor one and twotemperature.

Figure 13:Distribution of the tangential couple stress mzyfor one and twotemperature.

Figure 14:Distribution of the microrotation component Φ2for one and twotemperature.

Figure 15:Vertical displacement distribution w for different values of an internal heat source.

Figure 8 explains that the distribution of the vertical displacementwbegins from positive values.In the context of the two theories,wstarts with decreasing to a minimum value,then increases,and then becomes nearly constant for δ=0.Figure 9 exhibits the distribution of the conductive temperature Φ and demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,Φ starts with increasing to a maximum value,then decreases to a minimum value,and again increases for δ=0.Figure 10 exhibits that the distribution of the thermodynamic temperature θ begins from positive values.In the context of the two theories,θ starts with increasing to a maximum value,then decreases to a minimum value,and again increases for δ=0.Figure 11 explains that the distribution of the stress component σzzbegins from a negative value and satisfies the boundary condition atx=0.In the context of the two theories,σzzstarts with decreasing to a minimum value,then increases,and again decreases for δ=0.Figure 12 shows the distribution of the stress component σxzand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,σxzstarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation for δ=0.Figure 13 depicts the distribution of the tangential couple stressmzyand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,mzystarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation for δ=0.Figure 14 describes the distribution of the micro-rotation component Φ2.In the context of the two theories,Φ2starts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation for δ=0.

Figures 15-21 show comparisons between the vertical displacementw,the thermodynamic temperature θ ,the conductive temperature Φ,the stress components σzz,σxz,the tangential couple stressmzyand micro-rotation component Φ2for a two-temperature magneto-microploar medium(H0=140,δ=2×10-14)and for different values of an internal heat source(Q0=5.5,Q0=1).

Figure 16:Conductive temperature distribution Φ for different values of an internal heat source.

Figure 17:Thermodynamic temperature distribution θ for different values of an internal heat source.

Figure 18:Distribution of the stress component σzzfor different values of an internal heat source.

Figure 19:Distribution of the stress component σxzfor different values of an internal heat source.

Figure 20:Distribution of the tangential couple stress mzyfor different values of an internal heat source.

Figure 21:Distribution of the microrotation component Φ2for different values of an internal heat source.

Figure 15 explains that the distribution of the vertical displacementwbegins from positive values.In the context of the two theories,wstarts with decreasing to a minimum value,then increases,and again decreases forQ0=1.Figure 16 exhibits the distribution of the conductive temperature Φ and demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,Φ decreases in the range 0≤x≤10 forQ0=1.Figure 17 exhibits that the distribution of the thermodynamic temperature θ begins from positive values.In the context of the two theories,θ starts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation forQ0=1.Figure 18 explains that the distribution of the stress component σzzbegins from a negative value and satisfies the boundary condition atx=0.In the context of the two theories,σzzstarts with decreasing to a minimum value,then increases,and also moves in the wave propagation forQ0=1.Figure 19 shows the distribution of the stress component σxzand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,σxzstarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation forQ0=1.Figure 20 depicts the distribution of the tangential couple stressmzyand demonstrates that it reaches a zero value and satisfies the boundary condition atx=0.In the context of the two theories,mzystarts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation forQ0=1.Figure 21 describes the distribution of the micro-rotation component Φ2.In the context of the two theories,Φ2starts with increasing to a maximum value,then decreases to a minimum value,and also moves in the wave propagation forQ0=1.

Figure 22:Vertical displacement distribution w based on the 3PHL model.

Figure 23:Distribution of the stress component σzzbased on the 3PHL model.

Figures 22-26 are giving 3D surface curves for the physical quantities,i.e.,the vertical displacementw,the stress components σzz,σxz,the tangential couple stressmzyand micro-rotation component Φ2to study the effect of a magnetic field on the wave propagation within a two-temperature micropolar thermoelastic isotropic medium with an internal heat source in the context of the 3PHL model.These figures are very important to study the dependence of these physical quantities on the vertical component of distance.The curves obtained are highly depending on the vertical distance from origin,all the physical quantities satisfy boundary condition and are moving in the wave propagation.

Figure 24:Distribution of the stress component σxzbased on the 3PHL model.

Figure 25:Distribution of the tangential couple stress mzybased on the 3PHL model.

Figure 26:Distribution of the micro-rotation component Φ2based on the 3PHL model.

7 Concluding remarks

A rigorous mathematical study of thermoelasticity in solid materials has been conducted utilizing two different,robust,well-formulated theories,namely the 3PHL model and the Green-Naghdi theory without dissipation.The cases of a magnetic field presence and absence have been addressed as well as a two-temperature paramter.Analytical solutions based upon normal mode analysis for thermoelasticity in solids have been developed and utilized.The computations have revealed that:

1)There are significant differences in the field quantities under the GN-II theory and the 3PHL model due to the phase-lag of temperature gradient and the phase-lag of heat flux.

2)The magnetic field,two-temperature parameter and magnitude of an internal heat source have important roles in the distributions of the field quantities.

3)Deformation of a body depends on the nature of the applied force as well as the type of boundary conditions.

4)The curves in the context of the 3PHL model and the GN-II theory decrease exponentially with increasingz;this indicates that the thermoelastic waves are un-attenuated and non-dispersive,while purely thermoelastic waves undergo both attenuation and dispersion.

5)The vertical distance plays a significant role on all the physical quantities.

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1Department of Mathematics,Faculty of Science,Zagazig University,P.O.Box 44519,Zagazig,Egypt.E-mail:samia_said59@yahoo.com