The present paper paper, we estimate the theory of thermoelasticity 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

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 [

During the last five decades, more different theories developed the thermoelasticity theory to the generalized theory of thermoelasticity. Lord et al. [

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 [

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 [

Podlubny [

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

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

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

(ii) The constitutive (stress-strain) equations

where

(iii) Assuming series of Taylor of time-fractional order

where

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

where

The temperature

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

In the restricted case, when

This is the GN generalized theory without energy dissipation.

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

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

where _{0},

Using

Using the mapping

where

Differentiating with respect to the coordinates,

Redifferentiating concerning the coordinates axis, we obtain

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

From

From

Substituting from

For the linearity governing partial differential equations, considering the condition

Then,

Using

Taking into account a thin rod semi-infinite with the half-space region

The strain components are

The heat equation is

The equation of motion is

The constitutive relation takes the form

Considering a half-space

We assume that when

The thermal boundary conditions:

where

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

and

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

From

where

If we apply the following LT

Applying it in

The boundary conditions in

By eliminating

By eliminating

Also, we can show that

where

The solution of

where the parameters

We can get the displacement using

Thus, we obtain

The temperature increment

We utilize the problem’s boundary conditions to evaluate the

The solution of the former system of the linear equations provides the parameters

Hence,

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 [

It is worth noting that choosing the free parameters

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

Parameter | Value | Parameter | Value |
---|---|---|---|

The computations for the results obtained are carried out for the time

It should be pointed out that, the increasing value of

In

The physical field variables numerical values were calculated and presented by graphs in

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

The parameter

Considering the new models applied, we introduced a novel classification for the materials based on their fractional order parameter

The results motivate the investigation of the conducting thermoelastic.

The graphs illustrate the significant effect of the thermal conductivity on all the quantities fields and in different materials that we take into account in any analysis of heat conduction.

The field quantities, displacement, temperature, stress, and do not depend only on the state and the space variables

The different thermoelasticity theories, i.e., Lord and Shulman, GN, and classical dynamical coupled theories were compared.

In the generalized thermoelasticity, the wave propagation with a finite speed is evident in all these figures. This is not the case of the theory of coupled thermoelasticity, where the considered function has non-vanishing values for all of

Taif University Researchers Supporting Project number (TURSP-2020/164), Taif University, Taif, Saudi Arabia.