Recently, more attentions have been considered by researchers and engineers to the thermoelasticity theory to release the confliction of infinite speed due to the thermal signals, because of the importance in diverse fields as geophysics, acoustics, engineers, plasma physics, and industries. Biot [1] is the prior who presented the classical coupled thermoelasticity theory due to the coupled interaction between the thermal field and strain. The generalized thermoelasticity models have been introduced by Lord et al. [2] considering one relaxation time. Chen et al. [3] and Chen et al. [4,5] investigated the theory of heat conduction depending on two temperatures. Youssef [6] investigated the theory of two-temperature generalized thermoelasticity. Green et al. [7] who inserted two relaxation times and advocating finite wave speed thermal in solids by correcting the energy equation and Neuman-Duhamel relation or modifying Fourier’s conduction heat equation. Chandrasekharaiah et al. [8] studied the thermoelastic interactions without energy dissipation due to a point heat source. Chandrasekharaiah et al. [9] studied the temperature-rate-dependent thermo-elastic interactions due to a line heat source. The magnetoelastic earth’s material nature may affect on the wave propagation, especially surface waves. A literature review of the earliest contributions to the subject has been discussed in details by Puri [10]. Nayfeh et al. [11] who studied the plane wave propagation under the electromagnetic field in a solid medium. Choudhuri et al. [12] discussed the rotation effect on magneto-thermoelastic media in an elastic medium. Ezzat et al. [13] investigated the electromagneto-thermoelastic plane waves with two relaxation times in a medium of perfect conductivity Ezzat et al. [14] studied the electromagneto-thermoelastic plane waves with thermal relaxation in a medium of perfect conductivity. Zhuang et al. [15] studied the explicit phase field method for brittle dynamic fracture. Rabczuk et al. [16] investigated the nonlocal operator method for partial differential equations with application to the electromagnetic waveguide problem. Bahar et al. [17] introduced the formulation of state space in thermoelastic problems which also developed in Sherief [18] including the heat sources effectively. Sherief et al. [19] studied the two dimensional generalized thermoelasticity problem for an infinitely long cylinder. Youssef et al. [20] analysis a generalized thermoelastic infinite layer problem with the state space approach considering three models. Formulation of state space for the vibration of gold nano-beam in femtoseconds scale pointed out by Elsibai et al. [21]. Biot [22] clears that under stress free state would be fundamentally different from initial stresses states acoustic propagation and obtained the longitudinal and transverse wave velocities along the coordinate axes only. Chattopadhyay et al. [23] explained the plane wave reflection and refraction in an unbounded medium under initial stresses. Montanaro [24] studied the linear thermoelasticity problem with hydrostatic initial stress. Othman et al. [25] studied the reflection waves in a generalized thermoelastic medium from a free surface under hydrostatic initial stress under different thermoelastic theories. Youssef [26] studied the problem of generalized thermoelastic infinite medium with a cylindrical cavity subjected to a ramp-type heating and loads.

The main purpose of the present investigation is intended to demonstrate the magnetic field, relaxation time, hydrostatic initial stress, and two temperature on the thermal shock problem. The governing equations are formulated in the context of Lord-Shulman theory with the presence of body force, two temperatures, thermal shock, and hydrostatic initial stress. We obtained the exact solution using the normal mode technique with appropriate boundary conditions. The field quantities are calculated analytically and displayed graphically under thermal shock problem with effect of external parameters respect to space coordinates.

Considering that the medium is a perfect electric conductor and the absence of the displacement current (SI) [10], the linearized Maxwell equations governing the electromagnetic field as the form as shown in Fig. 1 (1)curlh→=J→curlE→=−μe∂h→∂tdivh→=0,divE→=0where h→=curl(u→×H→0), H→=H→0+h→(x,y,t) in which h→ is the perturbed magnetic field over the primary magnetic field, J→ represents the electric current density, μe represents the magnetic permeability, H0 represents the constant primary magnetic field, and u→ represents the displacement vector, H→=(0,0,H0),u→=(u,v,0).

The equation of heat conduction given [15] as (2)K(∂2φ∂x2+∂2φ∂y2)=(∂∂t+τ0∂2∂t2)ρCET+γT0(∂∂t+τ0∂2∂t2)(∂u∂x+∂v∂y)

The stress–displacement relations for the isotropic material are (3)σxx=(2μ+λ)∂u∂x+λ∂v∂y−γT−P (4)σyy=(2μ+λ)∂v∂y+λ∂u∂x−γT−P (5)σxy=(μ+12P)∂u∂y+(μ−12P)∂v∂x,σyx=(μ+12P)∂v∂x+(μ−12P)∂u∂y

The Maxwell’s equation formulated as (6)τij=μe[Hihj+Hjhi−Hk.hkδij]

The motion equation splits to (7)ρ∂2u∂t2=(λ+2μ+μeH02)∂2u∂x2+(λ+μ+12P+μeH02)∂2v∂x∂y+(μ−12P)∂2u∂y2−γT0∂T∂x (8)ρ∂2v∂t2=(λ+2μ+μeH02)∂2v∂y2+(λ+μ+12P+μeH02)∂2u∂x∂y+(μ−12P)∂2v∂x2−γT0∂T∂y

The heat conduction and dynamical heat related by the form (9)φ−T=a(∂2φ∂x2+∂2φ∂y2)

The non-dimensional variables for simplifying gives as (x′,y′,u′,v′)=c0η(x,y,u,v),(t′,τ0′)=c02η(t,τ0),(θ′,φ′)=(T,φ)−T0T0, (10)σij′=σij2μ+λ,h′=h2μ+λ,P′=P2μ+λ,τ′=τ2μ+λ, where η=ρCEK, C22=μρ and c02=2μ+λρ

By dropping the dashed for convenience, and substitute Eq. (10), then Eqs. (2) and (9) take the following form (11)∇2φ−(1+τ0∂∂t)∂θ∂t−ε(1+τ0∂∂t)∂e∂t=0 (12)φ−θ=β(∂2φ∂x2+∂2φ∂y2) where ε=γρCE and β=aη2c02

Assuming the scalar and vector potential functions Π(x,y,t) and ψ(x,y,t) in the non-dimensional form defined as: (13)u=∂Π∂x+∂ψ∂y,v=∂Π∂y−∂ψ∂x

By using (15) and (10) in Eqs. (7) and (8), we obtain. (14)[∇2−1a2∂2∂t2]Π−a∗θ=0 (15)(∇2−1a3∂2∂t2)ψ=0 where a0=γT0ρC02, a1=ρC02μ, a2=1+Rh2, a∗=a0a2, a3=μ−12PρC02

Rh2 is the Alfven speed.

Also Eq. (11) tends to (16)∇2φ−(1+τ0∂∂t)∂θ∂t−ε(1+τ0∂∂t)∂∂t(∇2Π)=0

The solution of the previous physical variables can be decomposed in terms of normal mode technique in the exponential harmonic form (17)[Π,ψ,φ,θ,σij](x,y,t)=[u∗(x),φ∗(x),θ∗(x),σij∗(x)]exp⁡(ωt+iby) where i=−1, b be a wave number, ω is the time constant, and u∗(x),φ∗(x),θ∗(x) and σij∗(x) are the amplitudes of the physical field quantities.

Using Eq. (17), into Eqs. (12) and (14)–(16), we obtain (18)[D2−A1]Π∗−A2θ∗=0 (19)[D2−b2]φ∗−Aθ∗+B[D2−b2]Π∗=0 (20)[D2−A3]φ∗=−β∗θ∗ (21)[D2−m2]ψ∗=0 where A=ω(1+ωτ0),A1=b2+ω2a2,A2=a∗,A3=(βb2+1)/β,B=−εA, β∗=1β,m2=b2+ω2a3 and D=ddx

Solving Eqs. (18)–(20) by eliminating θ∗(x),Π∗(x), and φ∗(x), we obtain the partial differential equation satisfied by θ∗(x) (22)[D4−ED2+F]θ∗(x)=0 where (23)E=A1+b2+βA(A1+A3)−βBA2(A3+b2)1+βA−βBA2, (24)F=b2A1+βA3(AA1−BA2b2)1+βA−βBA2,

Similarly, we get (25)[D4−ED2+F](Π∗,φ∗)(x)=0

which can be factorized to (26)(D2−k12)(D2−k22)θ∗(x)=0 where kn2(n=1,2) are the roots of the characteristic equation (27)k4−Ek2+F=0

as x→∞, the solution of Eq. (26) is given by (28)θ∗(x)=∑n=12Mn(b,β,ω)exp⁡(−knx)

Similarly (29)φ∗(x)=∑n=12Mn′(b,β,ω)exp⁡(−knx) (30)Π∗(x)=∑n=12Mn′′(b,β,ω)exp⁡(−knx)

The solution of Eq. (21) has the form (31)ψ∗(x)=M3e−mx. (32)u∗(x)=DΠ∗+ibψ∗ (33)v∗(x)=ibΠ∗−Dψ∗ (34)e∗(x)=Du∗+ibv∗

To get the amplitudes of the displacements u and v, which bounded as x→∞, then Eqs. (32) and (33) tend to (35)u∗(x)=−∑n=12Mn′′(b,β,ω)kne−knx+ibM3e−mx (36)v∗(x)=∑n=12ibMn′′(b,β,ω)e−knx+mM3e−mx where Mn,Mn′ and Mn′′ are parameters depend on β,b and ω.

From Eqs. (28)–(30) into Eqs. (18)–(20), we get (37)Mn′(b,β,ω)=H1nMn(b,β,ω),n=1,2 (38)Mn′′(b,β,ω)=H2nMn(b,β,ω),n=1,2 where (39)H1n=(β∗A3−kn2),n=1,2 (40)H2n=A2(kn2−A1),n=1,2

Thus (41)φ∗(x)=∑n=12H1nMn(b,β,ω)exp⁡(−knx) (42)Π∗(x)=∑n=12H2nMn(b,β,ω)exp⁡(−knx)

Substitution of Eqs. (35) and (36) into Eqs. (3)–(5), we obtain (43)σxx∗=∑n=12hnMn(b,β,ω)exp⁡(−knx)−q1M3exp⁡(−mx)−P (44)σyy∗=∑n=12hn′Mn(b,β,ω)exp⁡(−knx)+q1M3exp⁡(−mx)−P (45)σxy∗=∑n=12hn′′Mn(b,β,ω)exp⁡(−knx)−q2M3exp⁡(−mx) (46)τxx∗=∑n=12gnMn(b,β,ω)exp⁡(−knx),τyy∗=∑n=12gnMn(b,β,ω)exp⁡(−knx) where (47)hn=−(−H2nkn2+b2λH2n2μ+λ+γT02μ+λ) (48)hn′=−(b2H2n−λH2nkn22μ+λ+γT02μ+λ) (49)hn′′=−2ibμH2n2μ+λkn (50)q1=ibm(1−λ2μ+λ) (51)q2=μ(m2+b2)2μ+λ−12P(m2−b2) (52)gn=−(−μeH022μ+λH2nkn2+μeH022μ+λb2H2n)

The normal mode analysis is, in fact, to look for the solution in Fourier transformed domain. Assuming that all the field quantities are sufficiently smooth on the real line such that normal mode analysis of these functions exists.

Now we will obtain the parameters Mn(n=1,2,3), we will suppress the positive exponentials that are unbounded at infinity. The constants M1,M2,M3 must choose such that the boundary conditions on the surface atx=0 take the form: 1)

Boundary conditions for the thermal at surface under thermal shock

2)

Boundary condition for the mechanic at surface under initial stress

3)

Boundary condition for the mechanic at the surface is traction free

Substitute into the above boundary conditions in the physical quantities, we obtain (56)∑n=12Mn(b,β,ω)=f∗(y,t) (57)∑n=12(hn+gn)Mn(b,β,ω)−q1M3=0 (58)∑n=12hn′′Mn(b,β,ω)−q2M3=0

In the context of the boundary conditions in Eqs. (56)–(58) at the surface x=0, we get a system of three Algebraic equations, we will apply the inverse matrix method, we will get the three constants Mj, j=1,2,3, after that by substituting into the main expressions to obtain the displacements, temperature and other physical quantity.

To illustrate the analytical variable obtained earlier, we will consider a numerical example consider copper material. The results display the variation of displacements, temperature and stress in the context of LS theory. λ=7.59×109 N/m2,μ=3.86×1010 kg/ms2,CE=383.1 J/(kgk),ε=0.0168 α=−1.28×109 N/m2,ρ=7800 kg/m2,K=386 N/Ks,τ0=0.02,f∗=1αt=1.78×10−5 N/m2,a=1,T0=293 K,ω=ω0+iξ,ω0=2,ξ=1,η=8886.73 m/s2

We took the constants: y=−1, time t=0.1, b=0.25,H=105,P=1010,β=0.1,τ=0.1 for all computations, and we used for the real part of the displacement u,v, strain e and the stresses (σxx,σyy,σxy), thermal temperature θ and conductive temperature ϕ. All the field quantities don’t depend only on space x and time t, also on the relaxation time τ and in dimensionless form:

The output is plotted in Figs. 2–7.

Fig. 2 displays the values of the conductive temperature ϕ, which decreases with increasing of axial x. They indicate that the medium along the axial x tends to zero as x tends to infinity. The effect of time t, wave number b, longitudinal wave velocity, β, relaxation time τ, magnetic field H and initial stress P on the conductive temperature. The conductive temperature increases with increasing of time and longitudinal wave velocity, while it decreases with increasing of wave number and relaxation time, also, there isn’t any effect due to the magnetic field and initial stress on the conductive temperature.

Fig. 3 plots the values of the thermodynamic temperature θ, which decreases with increasing of axial x. These figures indicate that the medium along axial x. The effect of time t, wave number b, longitudinal wave velocity β, relaxation time τ, magnetic field H and initial stress P on the conductive temperature. The thermodynamic temperature increases with increasing of time and longitudinal wave velocity, while it decreases with increasing of wave number and relaxation time, as well, there is no effect of magnetic field and initial stress on the thermodynamic temperature.

Fig. 4 shows the values of displacement u, which has an oscillatory behavior in the whole range of axial x under the effects of time t, wave number b, longitudinal wave velocity β, relaxation time τ, magnetic field H and initial stress P. In these figures, it is clear that the displacement has a nonzero value only in the bounded region of space, while it increases with increasing of time, wave number and longitudinal wave velocity, as well it decreases with increasing relaxation time and magnetic field, while there is no effect of initial stress on the displacement.

Fig. 5 displays the value of displacement v which has an oscillatory behavior in the whole range of axial x under the effects of time t, wave number b, longitudinal wave velocity, β, relaxation time τ, magnetic field H and initial stress P. In these figures, it is clear that the displacement has a non-zero value only in the bounded region of space, while it increases with increasing of time, wave number and longitudinal wave velocity, as well it decreases with increasing relaxation time and magnetic field, while there is no effect of initial stress on the displacement.

Fig. 6 clears the values of stress, σxy which has an oscillatory behavior in the whole range of axial x under the effects of time t, wave number b, longitudinal wave velocity, β, relaxation time τ, magnetic field H and initial stress P. In these figures, it is clear that the stress has a nonzero value only in the bounded region of space, while it increases with increasing of time, wave number and longitudinal wave velocity, as well it decreases with increasing temperature, magnetic field and initial stress.

Fig. 7 shows the values of the stress σyy, which increases with increasing of axial x. These figures indicate that the medium along axial x the effect of time t, wave number b, longitudinal wave velocity, β, relaxation time τ, magnetic field H and initial stress P on the stress. The stress decreases with increasing of time, wave number, longitudinal wave velocity and initial stress, while it increases with increasing relaxation time, as well, there is no effect of magnetic field on the stress.

The results and conclusions can be summarized as follows 1)

Normal mode analysis of the problem of magneto-thermoelastic solid has been applied and developed.