Recently, more attention has been given to the uncoupled classical thermoelastic theory, which predicts that two phenomena have a confliction and do not agree with the physical laboratory results. While the heat conduction equation is the first phenomenon without any elastic terms, the second is the prediction of infinite propagation speed for heatwaves due to the thermal signals in the equation of heat in a parabolic type. Biot [1] is the prior who presented the coupled thermoelasticity theory between the motion equation and equation of heat to overcome and release the confliction of the first shortcoming. However, the equation of heat for the theory of coupled thermoelasticity in a parabolic form. Thus, both theories have shared and addressed the second shortcoming.

During the last five decades, more different theories developed the thermoelasticity theory to the generalized theory of thermoelasticity. Lord et al. [2] who is the first introduced the theory by the heat conduction equation based on the modified law of Fourier’s considering one relaxation time that transforms the equation of heat conduction in a parabolic form to hyperbolic form the equation of heat conduction. Green et al. (GL) [3] who modified the equation of energy and the relation of Duhamel-Neumann, taking into account two thermal relaxation times. The third was by Green et al. (GN) [4] who introduced a generalized theory of thermoelasticity. They included the ‘displacement gradient-thermal’ within the constitutive variables independence but without dissipation accommodate of the thermal energy.

The material characteristics at high temperatures, such as Poisson’s ratio, the elasticity modulus, the coefficient of thermal expansion, and the thermal conductivity are not constants any more [5]. Recently and because of scientific and technological development, the need to understand the actual actions of the material features has become actual [6]. Budaev et al. [7] discussed the dependence temperature of shear elasticity for some liquids which indicated that the increase of temperature depends on the decrease of shear modulus and is explainable by increasing the fluctuation free volume. Rishin et al. [8] employed the method of dynamic resonance to define the dependence temperature of the elasticity modulus of some materials that have plasma-sprayed. The increasing results of test temperature in a monotonic decrease in elasticity modulus. Honig et al. [9] discussed a method for the numerical inversion of the Laplace transform.

The equations of fractional derivatives and fractional differential were applied to obtain solutions to some problems in viscoelasticity, fluid mechanics, physics, engineering, biology, signal processing, mechanical engineering, systems identification, control theory, electrical engineering, finance, and fractional dynamics [10]. It describes anomalous diffusion (subdiffusion, superdiffusion, diffusion of non-Gaussian) that does not conform to the classical law of Fick in [11,12]. Numerically, the experiments show that in various one-dimensional systems that have total momentum conservation, the thermal conduction does not form in the Fourier law. Moreover, the thermal conductivity matches the size of the system in [13].

Podlubny [14] introduced an important overview of different fractional calculus applications in science, engineering, and technology. Ross [15] and Miller et al. [16] introduced a historical brief of the progress of fractional order calculus. Youssef et al. [17] discussed and made a new model for generalized thermoelasticity theory in fractional order. Sherief et al. [18] presented a fractional calculus-based thermoelasticity theory. Povstenko [19] studied axisymmetric stresses from pulses instantaneous and diffusion sources in an infinite space in a diffusion of time-fractional equation in a two-dimensional. Povstenko [20] applied Caputo time-fractional derivative technique on the thermal stresses theory based on the equation of thermal conduction to explore thermal stresses in a circular cylindrical hole an infinite body. Allam et al. [21] examined the interactions between the electromagneto-thermoelastic in a body with a spherical cavity an infinite perfectly conducting considering the GN model. The elasticity modulus is considered a linear function of temperature. Abouelregal [22] investigated the generalized thermo-piezoelectric semi-infinite under the assumption of the fractional-order with temperature-dependent and ramp-type heating. Structural continuous dependence in micropolar porous bodies is discussed in [23]. Study of heat and mass transfer in the Eyring–Powell model of peristaltically fluid propagating through a rectangular compliant channel has been discussed by Riaz et al. [24]. Some new related works have been discussed in [25–28].

In this work, we studied the thermoelasticity theory of a thin slim strip under the variable thermal conductivity in the fractional-order form is solved. Thermal stress theory considering the equation of heat conduction based on the time-fractional derivative of Caputo of order α is applied to obtain a solution. We assumed that the strip surface is to be free from traction and impacted by a thermal shock. The transform of Laplace (LT) and numerical inversion techniques of Laplace were considered for solving the governing basic equations. The inverse of the LT was applied in a numerical manner considering the Fourier expansion technique. The numerical results for the physical variables were calculated numerically and displayed via graphs. The parameter of fractional order effect and variation of thermal conductivity on the displacement, stress, and temperature were investigated and compared with the results of previous studies. The results indicated the strong effect of the external parameters, especially the time-fractional derivative parameter on a thermoelastic thin slim strip phenomenon. The obtained results are deduced to special case if thermal conductivity and thermal shock neglect.

Considering an isotropic homogeneous thermoelastic thin slim strip, the generalized thermoelastic governing differential equations in the fractional-order form [29] consist of:

(i) The motion equations, if the body forces were neglected

(1) ρui..=(λ+μ)uj,ji+μui,jj−γTi

(ii) The constitutive (stress-strain) equations

(2) σij=λekkδij+2μeij−γTδij

where λ and μ are the Lamé parameters, σij is the tensor of stress, γ =αt(3λ+2μ) , αt is the thermal expansion coefficient, and ρ is the medium density, eij are the elements of the strain tensor, T0 is the temperature reference, T is the temperature, and ui are the elements of the displacement vector.

(iii) Assuming series of Taylor of time-fractional order α is the new fractional form in the present paper [21] presented the time-nonlocal dependence between the the heat flux vector and temperature gradient discussed concerning fractional integrals and derivatives as follows:

(3) qi+τ0αα!∂α∂tαqi=−KT,i

where K is the heat conductivity, qi is the vector of thermal flux, ∂α∂tα is the fractional derivative of Caputo, and α(0<α≤1) is the fractional-order parameter.

(iv) The equation of heat conduction as time-fractional form takes the form [29]

(4) (KT,i),i=(δ+τ0αα!∂α∂tα)(ρCE∂T∂t+γT0∂ekk∂t)

where CE is the unit mass specific heat.

Eq. (4) is the fractional derivatives of generalized energy equation considering the relaxation time τ0 . Some of the theories of the law of thermal conduction follow different values of α and τ0 as limit cases. The theories of coupled or generalized thermoelasticity with one relaxation time and the generalized theory of thermoelasticity without energy dissipation (TWOED) adopt restricted cases according to the value of δ , τ0 and α .

The temperature Eq. (4), α→0 and δ=1 tends to:

(5) (KT,i),i=(∂∂t+τ0∂2∂t2)(ρCET+γT0e)

this is the generalized theory that has a thermal relaxation time.

In the restricted case, when α→0,τ0=1 and δ=0, the heat conduction Eq. (4), tends to

(6) (KT,i),i=ρCE∂2T∂t2+γT0∂2e∂t2

This is the GN generalized theory without energy dissipation.

Eq. (4) in the thermoelasticity coupled theory in the limiting case α→0,δ=1 and τ0→0 as

(7) (KT,i),i=ρcE∂T∂t+γT0∂e∂t.

Variations in mechanical properties due to an imposed temperature field are not the only ones that accompany heating. Similar variations are observed in the thermal properties characterized by such coefficients as the thermal linear expansion coefficients of αt , conductivity of thermal K and others. An acceptable approximation in limited temperature interval is obtained by considering the thermal conductivity to depend linearly on the change of temperature.

The function of thermal conductivity formed as a linear function of temperature is given as [30]

(8) K=K(T)=K0(1+K1T),Kk=ρCE

where K(T) is thermal conductivity as temperature-dependent, K1 is the thermal conductivity variation (usually negative experimental coefficient), K0 denotes the thermal conductivity at T = T₀, k is the diffusivity, and t is the time.

Using Eq. (4) with Eq. (8), we get

(9) (KT,i),i=(δ+τ0αα!∂α∂tα)(Kk∂T∂t+γT0∂2e∂t)

Using the mapping

(10) Θ=1K0∫0T⁡K(τ)dτ

where Θ is the mapping function.

Differentiating with respect to the coordinates, Eq. (10) tends to

(11) Θ,i=(1+K1T)T,i

Redifferentiating concerning the coordinates axis, we obtain

(12) Θ,ii=[(1+K1T)T,i],i

Similarly, by differentiating with respect to time, the mapping is

(13) Θ˙=(1+K1T)T˙

From Eqs. (12) and (13), Eq. (9) is given as

(14) Θ,ii=∂∂t(δ+τ0αα!∂α∂tα)(Θk+γT0K0e)

From Eqs. (10) and (11), we have

(15) Θ=T+K12T2

Substituting from Eq. (11) into Eq. (1), we obtain

(16) ρ∂2ui∂t2=(λ+μ)uj,ij+μui,jj−γ(1+K1T)Θ,i

For the linearity governing partial differential equations, considering the condition |T−T0|T0<<1 , give the approximating function of the thermal conductivity K(T) .

Then, Eq. (16) takes the form

(17) ρ∂2ui∂t2=(λ+μ)uj,ij+μui,jj−γΘ,i

Using Eq. (15) and neglecting the small values of temperature, the constitutive relation reduces to

(18) σij=λekkδij+2μeij−γΘδij

Taking into account a thin rod semi-infinite with the half-space region x≥0 , the x− axis perpendicular to the layer, parallel to oyz plane, the one-dimension displacement vector is given as

ux=u(x,t),uy=uz=0

The strain components are

e=exx=∂u∂x

The heat equation is

(19) ∂2Θ∂x2=∂∂t(δ+τ0αα!∂α∂tα)(1kΘ+γT0K0e)

The equation of motion is

(20) ∂2u∂t2=(λ+2μ)∂2u∂x2−γ∂Θ∂x

The constitutive relation takes the form

(21) σxx=σ=(λ+2μ)∂u∂x−γΘ

Considering a half-space x≥0 that primarily rests and has an initial temperature T0 with zero velocity of temperature, the original conditions are

u(x,0)=∂u(x,0)∂t=0

(22) T(x,0)=∂T(x,0)∂t=0

We assume that when x=0 , the surface becomes free from stresses and is put in sudden heating. The boundary conditions take the form

The thermal boundary conditions:

(23) T=T0H(t),forx=0

(24) Θ=υ0H(t),forx=0

where H(t) is function of Heaviside unit step

υ0=T0(1+K12T0)

The boundary conditions concerning mechanical stress, displacement, and temperature are

(25) σxx=0,forx=0

and

0.$$]]>{u(x,t),T(x,t),Θ(x,t)}→0,asx→∞,t>0.

To simplify the physical quantities, we put them in the following non-dimensional forms

x′=c1kx,u′=c1ku,t′=c12kt

(26) σxx′=σxxρc12,τ0′=c12kτ0,

Θ′=γρc12Θ,c12=(λ+2μ)ρ

From Eq. (26), the governing equations (eliminating the primes for convenience) take the forms

(27) σ=∂u∂x−Θ=e−Θ

(28) ∂2u∂t2=∂2u∂x2−∂Θ∂x=De−DΘ=Dσ

(29) ∂2e∂t2=D2e−D2Θ=D2σ

(30) D2Θ=∂∂t(δ+τ0αα!∂∂t)(Θ+εe)

where D=∂/∂x and ε=γ2T0k/(ρc12K0)

If we apply the following LT

f¯(s)=∫0∞⁡e−stf(t)dt

Applying it in Eqs. (27)–(30), and using the initial conditions in Eq. (22), we get

(31) σ¯=e¯−Θ¯

(32) D2σ¯=s2e¯

(33) (D2−g)Θ¯=gεe¯,g=s(δ+τ0αα!sα)

The boundary conditions in Eqs. (24) and (25) and the regularity condition in the LT domain take the forms

(34) Θ¯=υ0sforx=0

(35) σ¯=0forx=0

{u(x,s),T(x,s)}→0,asx→∞

By eliminating e¯ , we get

(36) (D2−s2)σ¯=s2Θ¯

(37) (D2−g(1+ε))Θ¯=gεσ¯

By eliminating σ¯ between Eqs. (36) and (37), we get

(38) D4−(s2+g(1+ε))D2+gεs2]Θ¯=0

Also, we can show that σ¯ satisfies

(39) [D4−LD2+M]σ¯=0

where

L=(s2+g(1+ε)),M=εs2g

The solution of Eqs. (38) and (39) takes the form

(40) σ¯=A1s2exp(−m1x)+A2s2exp(−m2x)

(41) Θ¯=A1(m12−s2)exp(m1x)+A2(m22−s2)exp(m2x)

where the parameters m1 and m2 satisfy the equation

m4−Lm2+M=0

We can get the displacement using Eq. (28), such that

u¯=1s2Dσ¯

Thus, we obtain

(42) u¯=A1m1exp(m1x)+A2m2exp(m2x)

The temperature increment T¯ is given by providing a solution to (15) to give

(43) T¯=−1+1+2K1Θ¯K1.

We utilize the problem’s boundary conditions to evaluate the A1 and A2 parameters. Eqs. (34) and (35) with Eqs. (40) and (41), they immediately give

(44) A1+A2=0

(45) A1(m12−s2)+A2(m22−s2)exp(m2x)=υ¯0s

The solution of the former system of the linear equations provides the parameters A1 and A2 in the form

(46) A1=−υ¯0s(m22−m12),A1=υ¯0s(m22−m12)

Hence,

(47) σ¯=−υ¯0s(m22−m12)exp(−m1x)+υ¯0s(m22−m12)exp(−m2x)

(48) Θ¯=−υ¯0(m12−s2)s(m22−m12)exp(m1x)+υ¯0(m22−s2)s(m22−m12)exp(m2x)

(49) u¯=−υ¯0m1s(m22−m12)exp(m1x)+υ¯0m2s(m22−m12)exp(m2x)

Accordingly, the problem is solved in the transformed domain completely.

It is too difficult to obtain the analytical inverse the LT of the intricate solutions to the temperature, displacement, stress, and strain in the LT domain. The method of numerical inversion is outlined to solve the problem in the physical domain. Durbin [31] obtained the approximation formula

(50) f(t)=2estt1(−12Re[F(s)]+Re∑n=0N⁡[Re(F(s+2inπt1))cos(2nπt1)−Im(F(s+2inπt1))sin(2nπt1)])

It is worth noting that choosing the free parameters N and st1 is significant for the accurate results and applying the method’s Korrecktur and the methods of convergence acceleration. The values of the parameters in Eq. (50) are defined as t1=20 , s=0.25, and N=1000 .

To calculate the analytical procedure, we take into account a numerical physical example. The findings depict the variations of the non-dimensional values of temperature, displacement, and thermal stresses. Thus, we consider the following values material constants (Copper material and the type 316) as shown in Tab. 1.

The computations for the results obtained are carried out for the time t=0.15 to obtain the displacement u , temperature T, and stress σxx . They were conducted for several x(0≤x≤5) , for different values of the parameter of fractional-order α with a wide range of (0<α≤1) that contains both cases of conductivity; (0<α<1) for low conductivity and α=1 for normal conductivity. Here, the numerical results are displayed graphically in Figs. 1–3.

It should be pointed out that, the increasing value of α decreases the speed of wave propagation of the stress and the temperature, whereas the displacement increases. We have noticed that the α value has a strong effect on all distributions. From these figures, the surface stress equals zero and matches the prescribed boundary condition.

In Figs. 4–6, we presented the stress, temperature, and displacement, respectively with different values of K1 . The thermal conductivity parameter has a significant impact on all fields. Physically, for the variable K , the temperature is a linear function with negative values of K1 , the heat conductivity decreases with the arising of the temperature, and the distance between the particles increases. The wave speed progress of all fields is slower. Thus, the values of all fields of quantities decrease.

The physical field variables numerical values were calculated and presented by graphs in Figs. 7–9 respect to the axis x to clear the variations nature of the field in the context of different thermoelastic models. The figures illustrate that all variables nearly have the same nature for the (LS), (GN), and (CD) models. Their behaviors are significantly different. The wave propagation with finite speeds is manifested in all figures for theories of (LS) and (GN). It differs for the case for considering the coupled equation of heat conduction (CD) model, in which the mechanical and thermal effects fill the entire space.

The main observations from these figures are organized as:

The LT technique is applied to derive the temperature, displacement, and stress due to the mechanical and thermal shock temperatures.