EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 786-808 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Seebeck Effect on a Magneto-Thermoelastic Long Solid Cylinder with Temperature-Dependent Thermal Conductivity A. M. Zenkour1,∗, A. E. Abouelregal2, K. A. Alnefaie3 , N. H. Abu-Hamdeh3 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia 1 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Egypt 2 Department of Mathematics, Faculty of Science, Mansoura University, Egypt 3 Department of Mechanical Engineering, Faculty of Engineering, King Abdulaziz University, Saudi Arabia Abstract. This article considers problem of generalized magneto-thermo-elasticity with dual- phase-lags in an infinitely long solid cylinder with variable thermal conductivity. Modified Ohm’s law that includes effects of temperature gradient (Seebeck’s phenomenon) and charge density as well as generalized Fourier’s law with current density is introduced. Curved surface of cylinder is under thermal shock and placed in uniform axial magnetic field. Laplaces transform and its inversion techniques are applied to solve present problem. Different results for field quantities like temperature, displacement, flexural moment, and stress distributions are presented. In addition, the induced magnetic and electric fields are displayed in some plots. Effects of Seebeck parameter, variability of thermal conductivity parameter and applied magnetic field are also investigated. 2010 Mathematics Subject Classifications: 73C Key Words and Phrases: Magneto-thermo-elasticity, dual-phase-lag, Seebeck effect, solid cylin- der, variable thermal conductivity 1. Introduction Classical thermoelasticity theories predict infinite speed of propagation for thermal field. Elastic change has no effects on temperature in classical uncoupled thermoelasticity theories. However, coupled theory of thermoelasticity eliminates paradox of uncoupled one. In addition, generalized thermoelasticity theories like Lord and Shulman [13] pre- sented, instead of classical Fourier’s law, wave-type heat conduction contains heat flux vector and its time-derivative as well as new thermal relaxation. Green and Lindsay [8] developed another generalized theory with two thermal relaxations. In addition to modifi- cation of Lord and Shulman they also modified all equations of coupled theory. Green and ∗Corresponding author. Email addresses: zenkour@kau.edu.sa (A. M. Zenkour), ahabogal@gmail.com (A. E. Abouelregal), kalnefaie@kau.edu.sa (K. A. Alnefaie), nabuhamdeh@kau.edu.sa (N. H. Abu-Hamdeh) http://www.ejpam.com 786 c© 2017 EJPAM All rights reserved. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 787 Naghdi [9] provide another theory that not accommodates dissipation of thermal energy. Recently, Zenkour [27] discussed transient thermal shock by presenting unified generalized thermoelasticity theory in context of all other theories. Tzou [24, 25] proposed a model that known as dual-phase-lag (DPL). It would be convenient to use this model to investi- gate heat transfer in micro-structures. Chandrasekharaiah [6] proposed a DPL model to modify classical thermoelasticity one. Allam et al. [14, 15] used Green and Naghdi model to deduce 1-D and 2-D problems of time-dependent heat source for homogeneous perfectly conducting electro-magneto- thermo-elastic plate and infinitely long hollow cylinder. Sherief and Ezzat [23] used the Lord-Shulman theory to discuss the response of infinitely long magneto-thermo-elastic con- ducting annular cylinders. Santwana and Roychoudhuri [21] presented magneto-thermoelastic interactions in infinitely cylinders under periodic loading. Tianhu et al. [10] used Lord and Shulman theory to investigate electro-magneto-thermo-elastic interactions in perfectly conducting solid cylinder. Sadeq et al. [16] investigated the steady state thermoelasticity of hollow nanospheres. Abouelregal and Abo-Dahab [4] presented DPL model to investi- gate magneto-thermo-elastic solid with spherical cavity. Abouelregal and Abo-Dahab [5] used DPLs model to study the diffusion on electro-magneto-thermo-elastic solid cylinder. Abbas and Zenkour [2] and Zenkour and Abbas [30] used different theories to discuss electro-magneto-thermoelastic effects on bending of functionally graded cylinders. If, instead of heating a single junction, one heats both junctions of circuit equally, the current stops flowing. Again, however, the current density will increase with an increasing temperature difference between the junctions [18]. The flow of electricity caused by heating does not necessarily depend on presence of two different metals. As shown by Thomson, a similar phenomenon is observed in a homogenous material if the latter is subjected to non-uniform temperature field. As a first approximation, it is permissible to assume a linear dependence of the current induced by heating on the temperature gradient, due to the fact that in most particle situations the gradient is small [12]. Temperature-dependent measurements of material properties should be taken into ac- count due to progress in different fields in techno-structures [1, 3, 7, 19, 26, 28, 29, 31, 32]. All mentioned investigations are restricted to only electromagneto-thermoelasticity anal- ysis. No efforts have been made to consider effects of induced magnetic and electric fields taking into consideration the effects of modified Ohm’s and Fouriers laws and Seebeck coefficient. The present article investigates the DPL model to discuss electro-magneto- thermo-elastic interactions in an isotropic infinite conducting cylinder with temperature- dependent thermal conductivity. The cylinder is placed in primary magnetic field and its curved surface is considered traction free. The cylinder is deforming due to thermal shock and magnetic field to produce induced magnetic and electric fields. Displacement, temperature and thermal moment are numerically illustrated at different positions of the medium. Induced magnetic, electric and perturbed magnetic fields in surrounding free space are also represented. All considered variables are graphically displayed and results are comprehensively discussed. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 788 2. Basic equations in magneto-thermoelasticity Equations of electrodynamics are linearly simplified for a perfect homogeneous con- ducting elastic solid due to Maxwell’s equations as ∇× h = J + ∂D ∂t , ∇ ·B = 0, B = µ0H, ∣∣∣∣∣∣ ∇×E = −∂B ∂t , ∇ ·D = ρe, D = ε0E, (1) where µ0 denotes magnetic permeability and ε0 denotes electric permeability, J denotes current density vector, E denotes induced electric field, B and D represent magnetic and electric induction vectors, ρe is charge (electric current) density, H = H0 + h in which H0 denotes applied magnetic field and h denotes perturbation occurred in total magnetic field by induction. In addition to the above field equations, the first hypothesis of Ohm’s law for moving media represents one of the constitutive equations J = σ0 ( E + µ0 ∂u ∂t ×H0 ) , (2) which turns out to be true only isothermal conditions; if a conductor (say, a metal) is heated non-uniformly, the relation becomes more complicated. Suppose, for example, that one heat a single junction of closed circuit formed of two pieces of wire, each of a different material, welded together in series. It found that a current starts flowing through the circuit, as manifested by, say, the deviation of a compass needle (Seebeck’s effect). Generally, the creation of an electromotive field be locally described Seebeck effect as [20] E = −k0∇θ, (3) where k0 is the Seebeck coefficient (also known as thermopower or thermoelectric sen- sitivity) and θ = T − T0 denotes thermodynamic temperature in which T represents temperature and T0 is the environment one. Modified generalized Ohm’s law for any solid with finite conductivity can be written as [5] J = σ0 ( E + µ0 ∂u ∂t ×H0 ) + ρe ∂u ∂t − k0∇θ, (4) where u represents displacement vector and σ0 denotes electric conductivity. Neglecting some unused items like inner heat sources and volume forces, and considering Lorentz force, then equations of motion become σij + Fi = ρ ∂2ui ∂t2 , (5) where σij are the stress components, ρ denotes mass density and Fi represent Lorentz force components as Fi = µ0 (J×H)i . (6) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 789 The corresponding Maxwell equations in adjoining free space of cylinder are ∇× h0 = ε0 ∂E0 ∂t , ∇ · h0 = 0, ∇× E0 = µ0 ∂h0 ∂t , ∇ · E0 = 0, (7) in which µ0 and ε0 represent magnetic and electric permeabilities in free space, E0 denotes induced electric field, h0 denotes magnetic field in free space and ∇ represents gradient operator. It is assumed that relative permeability in cylinder and permeability in free space are equivalent. The remaining constitutive equations of Duhamel-Neumann are given by σij = µ (ui,j + uj,i) + [λ (∇ · u)− γθ] δij , (8) in which λ and µ represent Lamé’s constants, δij denotes Kroneckers delta, γ = (3λ+ 2µ)αt, and αt represents linear thermal expansion coefficient. The energy equation in context of DPL model including current density effect is expressed as [5]( 1 + τθ ∂ ∂t ) (Kθ,i),i = ( ∂ ∂t + τq ∂2 ∂t2 ) (ρCeθ + γT0∇ · u) + π0∇ · J, (9) in which K represents thermal conductivity, τθ is PL of heat flux, τq represents PL of gradient of temperature, (τq > τθ > 0) and Ce denotes specific heat at uniform strain. In above equations, π0 represents coefficient connecting current density with heat flow density. 3. Problem formulation A long, homogeneous, solid cylinder of radius a with perfect conductivity is initially placed in an axial magnetic field H0 ≡ (0, 0, H0) acting directed parallel to z-axis as shown in Figure 1. Assuming that surface of cylinder be subjected to a thermal shock and traction free. Figure 1: Geometry of a long conducting solid cylinder in an axial magnetic field vector The initial magnetic field H0 is applied to the medium and generates induced magnetic h and electric E fields. Cylindrical coordinates system (r, ϕ, z) is considered for the present A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 790 axisymmetric problem and so all variables depend on r and t only. Then, ur = u (r, t) and uϕ = uz = 0. Components of strain tensor eij are given as err = ∂u ∂r , eϕϕ = u r , ezz = erz = ezϕ = erϕ = 0. (10) Cubic strain dilatation e = ekk is thus represented as e = err + eφφ + ezz = ∂u ∂r + u r = 1 r ∂ ∂r (ru). (11) Thermoelastic stresses σij become σrr = 2µ ∂u ∂r + λe− γθ, σϕϕ = 2µ u r + λe− γθ, σzz = λe− γθ, σrϕ = σrz = σϕz = 0. (12) It can be easily seen that vectors E and J have their components only in ϕ-direction. That is E ≡ (0, E, 0) , J ≡ (0, J, 0) . (13) From Eqs. (1) and (3), we have ∂h ∂r = −J − ε0 ∂E ∂t , (14) 1 r ∂ ∂r (Er) = −µ0 ∂h ∂t , (15) J = σ0 ( E − µ0H0 ∂u ∂t ) − k0 ∂θ ∂r . (16) Eliminating J between Eqs. (14) and (16) and E between Eqs. (15) and (16), leads to ∂h ∂r = σ0µ0H0 ∂u ∂t − ( σ0E + ε0 ∂E ∂t ) + k0 ∂θ ∂r , (17) ( ∇2 − σ0µ0 ∂ ∂t − µ0ε0 ∂2 ∂t2 ) h = σ0µ0H0 ∂e ∂t + k0∇2θ, (18) where ∇2 = ∂2 ∂r2 + 1 r ∂ ∂r represents Laplace operator. Equation of motion, Eq. (4), reduces to ∂σrr ∂r + σrr − σϕϕ r + Frr = ρ ∂2u ∂t2 , (19) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 791 where Frr represents Lorentz force which will, after applying the initial magnetic field vector, be Frr = µ0 (J×H)r = −µ0H0 ( ∂h ∂r + ε0 ∂E ∂t ) . (20) Thus, from Eqs. (11), (12), (19) and (20), it is found that (λ+ 2µ) ∂e ∂r − γ ∂θ ∂r − µ0H0 ( ∂h ∂r + ε0 ∂E ∂t ) = ρ ∂2u ∂t2 . (21) It is better to apply the operator ( 1 r ) ( ∂ ∂r ) (r) to both sides of Eq. (21) to get (λ+ 2µ)∇2e− γ∇2θ − µ0H0∇2h+ µ20H0ε0 ∂2h ∂t2 = ρ ∂2e ∂t2 . (22) Maxwell electromagnetic stress tensor in cylinder is given due to induced fields in the form [7] Mij = µ0 (Hihj +Hjhi − δijHkhk) , (23) which are reduced to two components Mrr and M0 rr. They read Mrr = −µ0H0h, M0 rr = −µ0H0h 0. (24) From Eqs. (6), the induced field components in adjoining free space will be E0 ≡ ( 0, E0, 0 ) , h0 ≡ ( 0, 0, h0 ) . (25) Using the relation ∇× (∇×A) = ∇ (∇ ·A)−∇2A, then Eq. (7) and (25) give ∇2h0 − µ0ε0 ∂2h0 ∂t2 = 0. (26) The thermal conductivity K is temperature-dependent in the linear form [19] K = K(θ) = K0 + β0θ, (27) where K0 denotes thermal conductivity at T0 and β0 = K0β1 represents slope of thermal conductivity-temperature curve in which β1 is constant. Now, we introduce the mapping [17] ψ = 1 K0 ∫ θ 0 K(θ)dθ, (28) in which ψ is new function of heat conduction. Substituting Eq. (27) into Eq. (28) gives [31] ψ = θ ( 1 + 1 2 β1θ ) . (29) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 792 Differentiating Eq. (28) with respect to r, one obtains K0ψ,r = K(θ)θ,r. (30) Also, differentiating Eq. (30) once again with respect to r, one gets K0ψ,rr = [K(θ)θ,r],r . (31) In addition, the first derivative of mapping with respect to t gives K0ψ̇,rr = K(θ)θ̇. (32) Using Eqs. (31) and (32), Eq. (9) becomes( 1 + τθ ∂ ∂t ) ∇2ψ = ( 1 + τq ∂ ∂t )( 1 k ∂ψ ∂t + γT0 K0 ∂e ∂t ) , (33) where ρCe = K/k and k is the diffusivity. For linearity, and remember that θ = T − T0 with |θ/T0| << 1, then Eqs. (22), (17) and (18) take the forms (λ+ 2µ)∇2e− γ∇2ψ − µ0H0∇2h+ µ20H0ε0 ∂2h ∂t2 = ρ ∂2e ∂t2 . (34) ∂h ∂r = σ0µ0H0 ∂u ∂t − σ0E − ε0 ∂E ∂t + k0 ∂ψ ∂r , (35) ( ∇2 − σ0µ0 ∂ ∂t − µ0ε0 ∂2 ∂t2 ) h = σ0µ0H0 ∂e ∂t + k0∇2ψ. (36) The constitutive relations given in Eqs. (12), for linearity will be in the forms σrr = 2µ ∂u ∂r + λe− γψ, σϕϕ = 2µ u r + λe− γψ, σzz = λe− γψ. (37) 4. Solution of the problem For convenience, one can dimensionless all quantities as follows [24] u ′ = gc1ηu, r ′ = c1ηr, a ′ = c1ηa, t ′ = c21ηt, τ ′ q = c21ητq, M ′ rr = gMrr µ , E ′ = ηgE σ0µ20H0c1 , ψ ′ = ψ T0 , J ′ = ηgJ σ20µ 2 0H0c1 , τ ′ θ = c21ητθ, h ′ = ηgh σ0µ0H0 , e ′ = ge, K ′ 1 = T0K1, σ ′ ij = gσij µ , c21 = λ+ 2µ ρ , η = ρCe K , g = γ ρCe . (38) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 793 In terms of dimensionless quantities, Eqs. (33)-(37) and Eq. (30) (suppressing primes for simplicity in the notation), are reduced to ∂h ∂r = ∂u ∂t − v1E − V 2∂E ∂t + S ∂ψ ∂r , (39) ( ∇2 − v1 ∂ ∂t − V 2 ∂ 2 ∂t2 ) h = ∂e ∂t + S∇2ψ, (40) ∇2e− ε∇2ψ − v1g2∇2h+ v1g2V 2∂h ∂t = ∂2e ∂t2 , (41) ( 1 + τθ ∂ ∂t ) ∇2ψ = ∂ ∂t ( 1 + τq ∂ ∂t ) (ψ + e) , (42) ∇2h0 − V 2∂ 2h0 ∂t2 = 0, (43) 1 r ∂ ( E0r ) ∂r = −∂h 0 ∂t , (44) σrr = β2e− 2 u r − b1ψ, σϕϕ = β2e− 2 ∂u ∂r − b1ψ, σzz = ( β2 − 2 ) e− b1ψ, (45) Mrr = −b2h. (46) where v1 = σ0µ0 η , V 2 = c21 c2 , c2 = 1 µ0ε0 , S = k0ηgT0 σ0µ0H0 , g2 = µ0H 2 0 ρc21 , ε0 = γgT0 ρc21 , β2 = λ+ 2µ µ , b1 = γT0 µ , b2 = β2v1g2. (47) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 794 5. Laplace transforms domain solutions The problem may be solved after applying its initial and boundary conditions. Here, the initial conditions are assumed to be homogeneous as u(r, 0) = ∂u(r, t) ∂t ∣∣∣∣ t=0 = 0, ψ(r, 0) = ∂ψ(r, t) ∂t ∣∣∣∣ t=0 = 0, E(r, 0) = ∂E(r, t) ∂t ∣∣∣∣ t=0 = 0, h(r, 0) = ∂h(r, t) ∂t ∣∣∣∣ t=0 = 0. (48) The above conditions are supplemented by adding boundary conditions. Firstly, it is assumed that the transverse components of E are continuous across the cylindrical surface, i. e., E(r, t) = E0(r, t), r = a, t > 0, (49) in which E0 denotes electric field intensity component in free surrounding cylinder in the direction of ϕ. Also, the same components of h are also continuous across the cylindrical surface, i.e., h(r, t) = h0(r, t), r = a, t > 0, (50) in which h0 denotes induced magnetic field intensity component in free surrounding cylin- der in z-direction. Finally, the cylindrical surface is considered traction free, σrr(a, t) = 0, (51) and lies under a thermal shock, θ(a, t) = θ0H(t), (52) in which H(t) denotes unit-step function. Now, it is known that Laplace transform is defined by f̄(r, s) = ∫ ∞ 0 e−stf(r, t)dt, (53) and it will be applied into Eqs. (39)-(44) under initial conditions given in Eqs. (48), leads to ∂h̄ ∂r = sū− ( v1 + V 2s ) Ē + S ∂ψ̄ ∂r , (54) ( ∇2 − v1s− V 2s2 ) h̄ = sē+ S∇2ψ̄, (55) ∇2ē− ε∇2ψ̄ − v1g2∇2h̄+ sv1g2V 2h̄ = s2ē, (56) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 795 (1 + τθs)∇2ψ̄ = s (1 + τqs) (ψ̄ + ē), (57) ∇2h̄0 − V 2s2h̄0 = 0, (58) 1 r ∂(Ē0r) ∂r = −sh̄0. (59) Eliminating ē and h̄ from Eqs. (55) and (56), the following sixth order differential equation for ψ̄ is formed as follows ( ∇6 −A∇4 +B∇2 − C ) ψ̄ = 0, (60) in which A, B and C are some coefficients given as A = q2 + s2 + g2qv1 ( K1 + s q ) + s ( sV 2 + v1 ) + qε, B = qs2 + s2 ( S + s q ) V 2 + g2qsv1 + s ( sV 2 + v1 ) ( q2 + s2 + qε ) , C = g2qs 3V 2v1 + qs3 ( sV 2 + v1 ) , q = s+ τqs 2 1 + τθs . (61) In a similar manner, we can show that ē and h̄ satisfy the equation( ∇6 −A∇4 +B∇2 − C ) {ē, h̄} = 0, (62) Introducing mi (i = 1, 2, 3) into Eq. (60), we find( ∇2 −m2 1 ) ( ∇2 −m2 2 ) ( ∇2 −m2 3 ) ψ̄ = 0, (63) where m2 i are the roots of the characteristic equation m6 −Am4 +Bm2 − C = 0. (64) These roots are given by [10] m2 1 = 1 3 [2p sin(q) +A] , m2 2 = −p 3 [√ 3 cos(q) + sin(q) ] + A 3 , m2 3 = −p 3 [√ 3 cos(q)− sin(q) ] + A 3 , (65) in which q = 1 3 sin−1(R), p = √ A2 − 3B, R = −2a3 − 9AB + 27C 2p3 . (66) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 796 The solution of ψ̄ will be ψ̄ = 3∑ i=1 (AiI0(mir) +BiK0(mir)) , (67) where Ai and Bi (i = 1, 2, 3) represent parameters depending on s and I0(·), K0(·) repre- sent Modified Bessels function of first and second kinds of order zero, respectively. Since the cylinder is solid, then solution should be continuous everywhere. So, Bi should be vanished and the solution for dimensionless ψ̄ in Laplace domain is ψ̄ = 3∑ i=1 Ai(s)I0(mir). (68) In a similar manner {ē, h̄} = 3∑ i=1 {A′i(s), A ′′ i (s)}I0(mir). (69) The compatibility between Eqs. (55) and (57) and Eqs. (69), gives A ′ i = m2 i − q1 q1 Ai = ΓiAi, A ′′ i = s(m2 i − q1) + q1Sm 2 i q1(m2 i − v1s− V 2s2) Ai = ΩiAi. (70) The displacement in Laplace domain comes as follows ū = 3∑ i=1 1 mi A ′ i(s)I1(mir), (71) which is derived by using the well-known relation of Bessel function,∫ zI0(z)dz = zI1(z). (72) If we differentiate Eq. (71) and using the formula d dz [In(z)] = In−1(z)− n z In(z), (73) we obtain ∂ū ∂r = 3∑ i=1 A ′ i(s) ( I0(mir)− 1 mir I1(mir) ) . (74) Substituting from Eqs. (68), (69) and (71) into Eq. (54), leads to Ē = 3∑ i=1 Ci(s)I1(mir), (75) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 797 where Ci = sΓi + SAi − Ωi mi(v1 + V 2s) Ai = φiAi. (76) Now, one can substitute solutions for ū and ψ̄ in Eqs. (45)-(48) to obtain stress and moment components as σ̄rr = 3∑ i=1 [ (β2Γi − b1)I0(mir)− 2Γi mir I1(mir) ] Ai(s), (77) σ̄ϕϕ = 3∑ i=1 [ (β2Γi − b1 − 2Γi)I0(mir) + 2Γi mir I1(mir) ] Ai(s), (78) σ̄zz = 3∑ i=1 [ Γi(β 2 − 2)− b1 ] I0(mir)Ai(s), (79) M̄rr = −b2 3∑ i=1 ΩiI0(mir)Ai(s). (80) Also, the induced fields in free space h̄0 and Ē0, which are bounded at infinity, maybe deduced from Eqs. (60) and (61) in the form h̄0 = K0(V sr)A4(s), (81) Ē0 = 1 V K1(V sr)A4(s), (82) where A4(s) is a parameter depending on s only. The boundary conditions given in Eqs. (48) may be written, after using Laplace transform, in the form Ē(a, s) = Ē0(a, s), (83) h̄(a, s) = h̄0(a, s), (84) σ̄rr(a, s) = 0, (85) ψ̄(R, s) = θ0 ( 1 s + K1 2s ) = Ḡ(s). (86) A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 798 After applying the above boundary conditions we obtain four equations in unknown pa- rameters Aj as 3∑ i=1 Ci(s)I1(mia)−A4(s)K0(V sa) = 0, (87) 3∑ i=1 φiAi(s)I0(mia)−A4(s)K0(V sa) = 0, (88) 3∑ i=1 [ (β2Γi − b1)I0(mia)− 2Γi mia I1(mia) ] Ai(s) = 0, (89) 3∑ i=1 Ai(s)I0(mia) = Ḡ(s). (90) It is easy to find the constants Aj due to the above equations and then the solution in Laplace transform domain will be completed. Moreover, temperature increment θ̄ can be deduced in terms of ψ̄ by using Eq. (29) as θ̄(r, s) = −1 + √ 1 + 2K1ψ̄ K1 . (91) 6. Numerical results and discussions Laplace inversions for all obtained variables in Laplace transform domain should be used to get the final solutions in physical domain. Numerical inversion method based on Fouriers series expansion [11] obvious that inversion g(r, t) of Laplace transform ḡ(s) maybe approximated as g(t) = ect t1 [ g(c) 2 + Re { N∑ k=1 eikπt/t1g ( c+ ikπt t1 )}] , 0 ≤ t ≤ 2t1, (92) in which N presents number of terms in truncated infinite Fourier series. It must be chosen such that ectRe { eiNπt/t1g ( c+ iNπt t1 )} ≤ ε1, (93) where ε1 represents persecuted small positive number, c represents positive free parameter that must be greater than real parts of all singularities of ḡ(s) [28]. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 799 Now, numerical results are carried out for Cooper, which has the following material constants [22]. K = 368 Wm−1 K−1, αt = 1.78× 10−5 K−1, Ce = 383.1 JKg−1 K−1, ρ = 8954 Kgm−3, λ = 7.76× 1010 Nm−2, µ = 3.86× 1010 Nm−2, η = 8886.73 sm−2, T0 = 293 K, ε = 0.0168, g = 1.61, σ0 = 5.7× 107, ε0 = 10−9/36π Fm−1, µ0 = 4π × 10−7 Hm−1, H0 = 107/4π Am−1. (94) Numerical results for physical quantities for dimensionless time t = 0.1 can be con- ducted using this data. Distributions of dimensionless temperature θ, radial displacement u, stress component σrr, Maxwell’s stress Mrr, induced magnetic field h and induced elec- tric field component E, induced field components in adjoining h0 and E0 distributions were evaluated and are presented graphically. These distributions are shown in Figs. 2- 17. Comparisons among different cases may also be made with the help of these figures. From the figures, it is deduced that the field quantities depend not only on state and space variables t and r, but also depend on variability thermal conductivity parameter, the modified Fourier’s and Ohm’s laws. Numerical computations were carried out for two cases as illustrated in the following: Case I investigates dimensionless temperature, displacement, stresses Maxwell’s stress, induced magnetic field induced electric field component, induced field components in the adjoining with different values of variability thermal conductivity parameter K1 when See- beck parameter S, PL of heat flux τq and PL of temperature gradient τθ remain constants. Case II illustrates dimensionless field quantities with different values of Seebeck param- eter S when variability thermal conductivity parameter K1, τq and τθ remain constants. In case I, three different values of variability thermal conductivity parameter K1 are considered to discuss effect of temperature on thermal conductivity. We take K1 = −0.25,−0.5 for variable thermal conductivity and K1 = 0 when thermal conductivity is temperature-independent. Seebeck parameter and other parameters are fixed to be S = 0.5, τq = 0.2 and τθ = 0.1. It is obvious from Figs. 2-9 that K1 has significant effects on all field quantities. Figure 2 describes variation of temperature θ versus radial distance r. It is seen that θ is not vanished only in a bounded region of space at a given instant. However, outside this region, θ is vanished and this means that region has not felt thermal disturbance yet. Figure 3 plots displacement u against r. The displacement gets its maximum magnitudes at the boundary where thermal shock is applied. It is seen that medium adjacent to cylindrical surface undergoes expansion deformation due to thermal shock while others compressive deformation. In Fig. 4, radial stress σrr at cylindrical surface is always zero since the surface is traction free. The medium close to cylindrical surface suffers from tensile stress. Also, it is clear that tensile stress region becomes larger while compressed becomes smaller with passage of time. Moreover, Fig. 4 shows that at some instant non-zero region of stress is finite which indirectly proves wave effect of heat. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 800 Figure 2: Distribution of θ along the radial direction of the solid cylinder for different values of K1 Figure 3: Distribution of u along the radial direction of the solid cylinder for different values of K1 Figure 4: Distribution of σrr along the radial direction of the solid cylinder for different values of K1 Figures 5 and 6 illustrate the induced magnetic field and electric field distribution effects. It is well known that electromagnetic medium is placed in initial magnetic field, and it deforms due to thermal shock. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 801 Figure 5: Distribution of h along the radial direction of the solid cylinder for different values of K1 Figure 6: Distribution of E along the radial direction of the solid cylinder for different values of K1 Figure 7: Distribution of h0 along the radial direction of the solid cylinder for different values of K1 Figures 7 and 8 give the variations of induced field components in the adjoining h0 and E0 distributions against distance for various values of variability thermal conductivity coefficient. It is observed from the gures that these distributions are increase with distance A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 802 Figure 8: Distribution of E0 along the radial direction of the solid cylinder for different values of K1 and takes negative values. Figure 9: Distribution of Mrr along the radial direction of the solid cylinder for different values of K1 Figure 9 depicts the variation of Maxwell’s stress Mrr along the radial distance r of the cylinder. This figure shows the difference between generalized theory of thermoelasticity with constant thermal conductivity and those of variable thermal conductivity. It is noted that as parameter K1 increases effective region is decreased. It is obvious that change of thermal conductivity has very small effect on induced electric field component E. Its effect on the other functions considered is more noticeable. The increases in conductivity tends to increase absolute value of temperature θ, radial stress σrr, induced electric field component E, induced field components in the adjoining h0 and E0 and Maxwell’s stress Mrr. In case II, the distributions of temperature θ, radial displacement u, stress component σrr, Maxwell’s stress Mrr, induced magnetic h and electric E fields, and induced field components in adjoining h0 and E0 distributions are evaluated and presented graphically in Figures 10-17. Two different values of Seebeck parameter S were considered (S = 0.5, 0.25). Additional results for field quantities are given without Seebeck effect (S = 0). A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 803 Figure 10: Distribution of θ along the radial direction of the solid cylinder for different values of K1 Figure 11: Distribution of u along the radial direction of the solid cylinder for different values of S Figure 12: Distribution of σrr along the radial direction of the solid cylinder for different values of S In this case variability thermal conductivity parameter is fixed to be K1 = −0.5. It is noticed that Seebeck parameter S is more pronounced and has significant effects on all considered functions. A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 804 Figure 13: Distribution of h along the radial direction of the solid cylinder for different values of S Figure 14: Distribution of E along the radial direction of the solid cylinder for different values of S Figure 15: Distribution of h0 along the radial direction of the solid cylinder for different values of S 7. Conclusions In this article we present new model of equations of generalized electro-magneto- thermoelasticity for thermally, isotropic and electrically conducting infinitely solid cylinder A. M. Zenkour et al. / Eur. J. Pure Appl. Math, 10 (4) (2017), 786-808 805 Figure 16: Distribution of E0 along the radial direction of the solid cylinder for different values of S Figure 17: Distribution of Mrr along the radial direction of the solid cylinder for different values of S whose surface is subjected to thermal shock. The influences of phase-lags, magnetic field, thermal shock, Seebeck’s coefficient and variable thermal conductivity are considered un- der variable thermal conductivity and magnetic field. The problem has been solved by means of Laplace transform and numerical Laplace inversion. According to above, it is concluded that variability thermal conductivity pa- rameter has significant effects on speed of wave propagation of the studied fields. The temperature-dependent thermal conductivity has significant effect on thermal and me- chanical interactions. Seebeck parameter has significant influence on all distributions. 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