Higher Order Generalized Thermoelastic Model with Memory Responses

in Nonhomogeneous Elastic Medium due to Laser Pulse

SOUMEN SHAW1,*, AKTAR SEIKH2

1Department of Mathematics,

Dinabandhu Andrews College, Kolkata-700084,

INDIA

2Indian Institute of Engineering Science and Technology,

Shibpur, Howrah-711103,

INDIA

*Corresponding Author

Abstract: - The present article deals with the thermoelastic behavior of a nonhomogeneous isotropic material. This

study is carried out in the context of an advanced thermoelastic model involving a higher order memory dependent

derivative (MDD) with dual time delay terms. The thermoelastic interactions and evolved stresses into the medium

are analyzed subject to external mechanical load as well as laser-type heat source. It is observed that the material

moduli of the medium have a significant impact on its thermodynamic behavior. The analytical expression of the

field functions is obtained in the integral transform domain. To know the nature of the field functions in the space-

time domain, a discretized form of the inverse integral transformations is applied and depicted graphically for

various kernel functions and empirical constants.

Key-Words: - Nonhomogeneous medium, memory-dependent derivative, fractional derivative, generalized

thermoelasticity, material moduli, continuous load, instantaneous load.

1 Introduction

A conventional method that has been widely used to

investigate the thermoelastic behavior of a material is

Fourier’s law of heat conduction. Even though

Fourier’s law is well established, it predicts the

infinite speed of propagation of thermal signals, and

one of its potential drawbacks is its accuracy, which

leads to failure in situations involving incredibly low

temperatures, extremely high heat flow, and very

short periods. To surmount the drawbacks, various

modified generalized thermoelasticity models have

been introduced. The hyperbolic type heat transport

equation present in generalized thermoelasticity

theory predicts that thermal signals will propagate

with a finite speed in a wavelike fashion. Numerous

generalized thermoelasticity models have been

developed by different researchers, the widely

acceptable models are mentioned in the articles, [1],

[2], [3], [4], [5], [6], [7], etc. Under different kinds of

circumstances, the unique property of the solutions

was established by many researchers, [8], [9], [10].

The monograph, [11], describes a brief history of

generalized thermoelasticity. The Dual-phase lag

model, which was proposed by Tzou, is one of these

newly presented models. [12], introduced a three

phase lag generalized thermoelasticity model.

Recently, a modified G-L theory with strain rate has

been introduced in the article [13] and a modified L-

S theory with decomposed heat flux has been

introduced in [14].

Non-homogeneous properties of the material

moduli played an important role in the field of

mechanics. Variation in temperature distribution is

another significant domain and received great

attention in the industry connection with the rapid

growth of society. Material response at low

temperatures and at high flux rates is extremely

important in fields like the aviation industry, nuclear

explosion, etc. In the thermoelastic domain, most of

Received: April 14, 2023. Revised: February 17, 2024. Accepted: March 11, 2024. Published: April 17, 2024.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

21

Volume 19, 2024

the researchers studied deformation and temperature

distribution based on the medium having constant

material moduli. Discarding the influence of elastic

deformation of the medium as well as the temperature

variations on the material moduli leads the analysis

towards a smaller region in terms of applicability.

Practically, the material moduli viz., modulus of

elasticity, thermal conductivity, and coefficient of

linear thermal expansion is no longer constant in the

high-temperature range, [15]. By considering

temperature-dependent properties, Noda, in his article

[16], examined the characteristics of thermal stresses

in a thermoelastic material. [17], dealt with a

generalized thermoelastic problem by taking the

dependency of material moduli on the reference

temperature. The [18], reports that mechanical

rigidity and chemical inertness varies with the

temperature dependence of Young’s modulus of a

single-crystal diamond. Furthermore, Young’s

modulus of a Si-crystal plays an important role in

determining thermal stress inside the ingot during the

cooling process for crystal growth, [19].

In 2014, implementing the idea of derivatives

with sleeping intervals, Wang and Li developed the

idea of memory-dependent derivative in the

generalized thermoelasticity in terms of the fractional

derivative due to [20], [21], [22]. In this modern

generalized thermoelasticity model, the time delay

factor is conveniently used as the length of the

slipping time interval. The first order memory

dependent derivative can be expressed as an integral

form of a common derivative using a freely chosen

kernel function k(t − ξ) on the sliding interval [t − τ,t]

as follows:

where τ > 0 is the time delay.

Memory-dependent derivatives outperform

fractional order generalized thermoelasticity at

establishing the memory effect (the preceding state

affects the immediate rate of change). It is simple to

define in terms of the physical environment, and the

memory-dependent differential equation signifies a

higher level of expressiveness.

The heat transport equation has been modified in

the perspective of MDD. The differential equations

that emerge from using memory-dependent methods

are more effective in practical applications because

the definition of MDD is more intuitive in how

physical importance is observed. Other efforts have

recently been attempted to modify the conventional

Fourier law to improve on earlier models using

governing equations that incorporate higher-order

derivatives. In 2021, based on MDD, a new model of

generalized thermoelasticity has been proposed by

considering the time delay factor, [23]. An advanced

model in the field of thermoelasticity from the

perspective of higher order MDD with dual time

delay factors has been introduced in [24]. The

Memory-dependent derivatives in magneto-

thermoelastic transversely isotropic media with two

temperatures has been discussed in [25]. The impacts

of stiffness and memory on energy ratios at the

interface of different media have been investigated in

[26]. Thermal wave propagation in an unbounded

medium with a cavity has been discussed in [27].

Recently published few articles have shown

significant developments in-terms of capturing the

nonlocal response of the materials, [28], [29], [30],

[31], [32], [33], [34].

Moreover, several aspects of the thermoelastic

medium have been studied in [35], [36], [37], [38],

[39], [40]. Modified Moore–Gibson–Thompson on

rotating semiconductors and on orthotropic hollow-

cylinder have been studied in the articles [41], [42].

Recently quick thermal processes using an ultra-

short laser pulse are receiving attention from a

thermoelasticity viewpoint. The primary reason for

this attraction is that it requires a study of the coupled

temperature and deformation fields. This suggests

that the energy absorption of the laser pulse causes a

localized temperature increase that causes thermal

expansion and prompts fast reactions inside the

structure of the elements, increasing the vibration of

the structure. There have been several significant

studies on the impact of laser pulses on generalized

thermoelasticity, some of which are given in [43],

[44], [45], [46].

The main objective of the present article is to

analyze the impact of the inhomogeneity of material

characteristics interacting with generalized

thermoelastic models containing higher order

memory dependent derivatives. The homogeneity of

the medium varies in the form of variable material

moduli. The nature of the deformations and

distribution of the temperature are predicted through

this present analysis. The variation of stresses,

displacement, and temperature are depicted

graphically for different kernel functions and

empirical constants.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

22

Volume 19, 2024

2 Statement of the Problem

In the present problem, we are taking into

consideration a thermoelastic problem for an

isotropic medium with a laser pulse heat source in the

x1 −x3 plane.

The constitutive equation: The stress-strain-

temperature relation is given by [11],

 (1)

The equation of motion: The equation of motion in

absence of body force is given by [11],

󰇘 (2)

The heat conduction equation: The generalized

heat conduction equation from the perspective of

higher order memory-dependent derivative with dual

time delay factors as given in [24] is:

,

(3)

we shall take the heat Q input as given in [47].

Q(x1,x3,t) = I0J(t)g1(x1)g2(x3), (4)

where

Here r1, and ν0 are the radius of the beam and

absorption depth of heating energy respectively.

The operator 

󰇛󰇜 is defined as:

(6)

with the kernel function k(t − ξ) as:

󰇛󰇜

󰇛󰇜

󰇛󰇜











 





󰇣

󰇤

(7)

where ϵij is the strain component defined as:

is

Kronecker delta, γ = (3λ + 2µ)αt, λ and µ are

material moduli, αt is the coefficient of thermal

expansion, τij is the stress component, T is the

absolute temperature, T0 is reference temperature, Tˆ =

T − T0, ρ is the material density, k is the coefficient of

thermal conductivity, τθ, and τq are two-time delay

factors. For the detailed discussion about the heat

conduction models on kernel functions, [48].

The equation (3) is simpler to comprehend in

terms of its physical significance, and the equivalent

differential equations based on memory dependence

have stronger expressive capabilities. When N1 = N2,

it represents a diffusive behavior, and when N2 = N1 +

1 it explains a wave behavior. For N1 = 1 and N2 = 2,

it gives a dual phase-lags model. In this case, the

system is exponentially stable when τq < 2τθ and

unstable when τq > 2τθ. For N1 = 2 and N2 = 2, it also

represents a dual phase-lags model. In this case, the

system is exponentially stable when τq > (2 − )τθ.

Thus, the values of the parameters N1 and N2 are not

allowed to be selected arbitrarily but rather

accurately depending upon their stability.

3 Governing Equations

We shall consider the material moduli in the way

described below as given in [49].

γ = γtg(T), µ = µtg(T), λ = λtg(T) (8)

where g(T) is a dimensionless function of

temperature given by:

g(T) = 1 − α∗T0 (9)

where the α∗ is the empirical material constant when

α∗ = 0 it implies temperature independent material

moduli.

Now by introducing (1) and (8) in equation (2)

we have the following equations of motion:

(10)

where

(11)

We shall take the dimensionless quantities as follows:

(12)

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

23

Volume 19, 2024

where

By introducing (12) in equations (10) and (11) the

following equations are obtained:

(13)

(14)

where and

The nondimensional form of the heat conduction

equation (3) becomes:

󰇭







 

󰇮󰇭







 

󰇮󰇗󰇗











(15)

where

Now we shall introduce the potential functions ϕ and

ψ as follows:

(16)

By introducing (16) in the equations (13),(14),

and (15) we obtained the following system of

equations:



, (17)



, (18)

󰇭







 

󰇮󰇭







 

󰇮󰇗󰇗











(19)

where 







.

Now we shall introduce the Laplace transform

defined as:

󰇛󰇜󰇛󰇜

∞

 (20)

By applying the Laplace transform to the

memory-dependent derivative 

 we have:



󰇛󰇜󰇣󰇛󰇜󰇛󰇜



󰇤󰇛󰇜󰇛󰇜 (21)

where

)

(22)

Now by taking the Laplace transform on both

sides of equations(17),(18), and(19) and considering

homogeneous initial conditions the following system

of equations is obtained :

 (23)

 (24)

Now we shall use the Fourier transform as

described below:

(26)

By applying the Fourier transform on both sides

of equations (23), (24) and (25) we obtained the

following system of equations:

󰇛󰇜, (27)

󰇛󰇜, (28)

󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜, (29)

where

 

, 󰇛󰇜, 󰇡

󰇢, 󰇡

󰇢, 



, 󰇛󰇜

, 󰇛󰇜󰇡

󰇢.

Thus, ϕ˜ satisfies the following ODE:

,

(30)

where are the roots of the equation:

m4 + (E1 − d3 + d4)m2 − (E1d3 − d4ζ2) = 0.

Here we assume that the solutions are bounded as

x3 → ∞ and is not a root of the above quadratic

equation of m2. The solutions for ϕ˜, θ˜ and ψ˜ are

obtained as:

(31)

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

24

Volume 19, 2024

(32)

(ζ,x3,s) = H3exp(−m3x3), (33)

where H1, H2, H3 are arbitrary,

,

Now introducing (20), (26) in (16) and using

(31), (33) we have the following solutions for

displacement components:

The nondimensional forms of shearing stress τ13

and normal stress τ33 are:





 (36)



󰇣



󰇤 (37)

Introducing (20), (26) in (36), (37) and using

(32), (34), (35) we have:

w

where

4 Mechanical Conditions

(i) τ13 acting on the plane x1 = constant, here x1 = 0,

along the direction of x3 axis. (ii) τ33 is the

longitudinal stress along x3 axis. On the line x3 =

0,

τ13 = 0,

θ = 0, (40) (40)

τ33 = −χ0δ(x1)χ(t)

where χ0 is a constant and δ(x1) is the Dirac-delta

function. It represents a point load with intensity χ0 at

the origin.

The applied mechanical loads can be

characterized based on the involved functions. For

example; (i) a continuous load:

(ii) a continuous point load:

For computational purposes, we shall consider

two different forms of loads in the equation (40) on

the boundary plane, as follows:

󰇛󰇜󰇛󰇜

󰇛󰇜 (41)

where H(t) is the Heaviside step function.

4.1 Continuous Load

In this case, after introducing (20), (26) in (40) we

obtained the following system of equations:

(42)

Using the matrix inverse method we have:

(43)

4.2 Impact Load

Here, in this case, we obtain the following system of

equations after introducing (20), (26) in (40):

N11H1 + N12H2 + N14 = 0

N21H1 + N22H2 + N23H3 + N24 =0

N31H1 + N32H2 + N33H3 + N34 = 

Using the matrix inverse method we have:

(44)

(45)

5 Numerical Analysis

The solutions have been determined in the Laplace-

Fourier transform domain and are functions of x3, as

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

25

Volume 19, 2024

well as the parameters s and ζ. With the help of the

numerical inversion methods as described in [50],

[51], the solutions of temperature, displacement

components, and stress components in the physical

domain are obtained numerically. Numerical

simulations have been carried out based on

analytically determined solutions that illustrate the

nature of temperature, displacement components, and

stress components.

We consider a copper-like material for the

numerical purpose. The values of the used parameters

are as follows, [14]:

ρ = 8954 kg/m3, Cv = 383.1 m2/s2K, µt = 3.86 ×

1010N/m2, λt = 7.76 × 1010 N/m2, k = 386 W/mK, αt =

1.78 × 10−5 K−1, ν0 = 0.5 m−1, T0 = 293K, I0 = 105j, r1 =

100 µm, τq = 0.07s, τθ = 0.01s.

The nature of physical quantities for various

empirical constants and kernel functions are

presented graphically. For the case of various

empirical constants, the kernel function is taken as b

= 1, a = 1 in (7) whereas in the case of different

kernel functions the empirical constants have the

value α∗ = 0.1, N1 = 5, N2 = 6.

To demonstrate the characteristics of temperature,

displacement components, and stress components

under continuous and impact loads in the presence of

a laser pulse heat source, numerical simulations are

performed.

5.1 The shearing Stress (τ13)

Figure 1, Figure 2, Figure 3 and Figure 4 in

Appendix show how the shearing stress (τ13) varies

under continuous and impact load in various

situations. Figure 1 and Figure 2 (Appendix) depict

characteristics of shearing stress (τ13) under

continuous load using various kernel functions and

various empirical constants respectively.

Figure 1 and Figure 2 (Appendix) clarify that the

shearing stress (τ13) under continuous load has a small

variation in 0 ≤ x3 ≤ 0.5 and then varies highly in the

region 0.5 ≤ x3 ≤ 1.5. Thereafter in the region x3 > 1.5

it eventually vanishes.

Figure 3 and Figure 4 (Appendix) depict

characteristics of the shearing stress (τ13) under

impact load using different types of kernel functions

and various empirical constants respectively. Figure 3

and Figure 4 (Appendix) clarify that the shearing

stress (τ13) under impact load varies highly at the very

bottom of the x3 axis and then up to x3 = 1.5 it has a

significant variation. Eventually, the shearing stress

(τ13) converges to zero in the higher region x3 > 1.5.

5.2 The Normal Stress (τ33)

Figure 5, Figure 6, Figure 7 and Figure 8 (Appendix)

show how the normal stress (τ33) varies under

continuous and impact load in various situations.

Figure 5 and Figure 6 (Appendix) show the

characteristics of the normal stress (τ33) under

continuous load, respectively, using different kernel

functions and empirical constants.

It is noticed from Figure 5 and Figure 6

(Appendix) that the normal stress (τ33) under

continuous load varies slowly in the region 0 ≤ x3 ≤

0.5 and then it has a large variation in the region 0.5

≤ x3 ≤ 1.5. Thereafter in the region x3 > 1.5 the

normal stress (τ33) eventually goes to zero.

Figure 7 and Figure 8 (Appendix) depict, using

different kernel functions and different empirical

constants, the characteristics of the normal stress

(τ33) under impact load. It is noticed from Figure 7

and Figure 8 (Appendix) that the normal stress (τ33)

under impact load varies significantly in the region 0

≤ x3 ≤ 0.5 and then it varies rapidly in the region 0.5 ≤

x3 ≤ 1.5. Thereafter in the region x3 > 1.5 the normal

stress (τ33) eventually goes to zero.

5.3 The Displacement Component u1

The nature of the displacement component u1 varies

under continuous load and impact load in various

situations, as shown in Figure 9, Figure 10, Figure 11

and Figure 12 (Appendix).

Figure 9 and Figure 10 (Appendix) illustrate the

nature of the displacement component u1 under

continuous load, respectively, using different kernel

functions and different empirical constants. It is

clarified from Figure 9 and Figure 10 (Appendix) that

the displacement component u1 under continuous load

gradually varies in the region 0 ≤ x3 ≤ 0.5 and then

variations occur quickly in 0.5 ≤ x3 ≤ 1.5. After that,

the displacement component u1 ultimately converges

to zero in the higher region x3 > 1.5.

From Figure 11 and Figure 12 (Appendix) it is

clarified that under the impact load, the displacement

component u1 gradually decreases in the very lower

region of x3 thereafter up to x3 = 1.5 the displacement

component u1 varies highly. After that, the

displacement component u1 ultimately converges to

zero in the higher region x3 > 1.5.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

26

Volume 19, 2024

5.4 The Displacement Component u3

Figure 13, Figure 14, Figure 15 and Figure 16

(Appendix) illustrate the nature of the displacement

component u3 under continuous load and impact load

in various cases. Figure 13 and Figure 14 (Appendix)

describe the nature of the displacement component u3

under continuous load using different kernel

functions and empirical constants, respectively.

In Figure 13 and Figure 14 (Appendix), it has

been noticed that the displacement component u3

under continuous load has a slowly varying nature in

the region 0 ≤ x3 ≤ 0.5, and then a rapid variation

occurs in the region 0.5 ≤ x3 ≤ 1.5. Eventually, in the

higher region x3 > 1.5, the displacement component

u3 converges to zero.

Figure 15 and Figure 16 (Appendix) describe the

nature of the displacement component u3 under

impact load using different kernel functions and

empirical constants, respectively. In Figure 15 and

Figure 16 (Appendix), it has been noticed that in 0 ≤

x3 < 1, the displacement component u3 under impact

load firstly increases to a higher magnitude thereafter

it varies slowly. In 1 ≤ x3 ≤ 2, the displacement

component u3 varies rapidly thereafter in the higher

region x3 > 2 it finally converges to zero.

5.5 The Temperature (θ)

Figure 17, Figure 18, Figure 19 and Figure 20

(Appendix) describe the nature of the temperature (θ)

varies under continuous load and impact load for

various cases. Figure 17 and Figure 18 (Appendix)

describe the nature of the temperature (θ) under

continuous load using various kernel functions and

various empirical constants, respectively.

In Figure 17 and Figure 18 (Appendix) it is

observed that under continuous load, in 0 ≤ x3 ≤ 0.8

the temperature (θ) has a small variation, and then in

0.8 < x3 ≤ 1.5 it varies rapidly. Finally, in the higher

region x3 > 1.5, the temperature (θ) eventually goes to

zero.

Figure 19 and Figure 20 (Appendix) describe the

nature of the temperature (θ) under impact load using

different kernel functions and empirical constants,

respectively. In Figure 17 and Figure 18 (Appendix)

it is noticed that in 0 ≤ x3 ≤ 0.8 the temperature (θ)

varies slowly thereafter in the region 0.8 < x3 ≤ 1.5 it

has high variation. Finally, in the higher region, x3 >

1.5 the temperature (θ) goes to zero.

5.6 Effects of Kernel Functions and Empirical

Constants

The considered empirical constant and different

kernel functions have their advantages in the analysis

of thermal thermal deformation of the solids, [52],

[53]. Three different kernel functions are used in the

current study. The findings are demonstrated clearly.

The identical behavior of all the thermodynamic field

functions is an important and notable fact. Moreover,

all the field variables depend significantly on the

kernel functions as well as an empirical constant in

the analysis of thermoelasticity. These findings show

that different kernel function structures reflect

various memory influences, allowing one to select a

kernel function that improves the effects of the MDD.

6 Conclusions

In this study, a non-homogeneous thermoelastic

medium under mechanical loads with a laser pulse

type heat source has been considered. Non-

homogeneity with temperature-dependent material

moduli responses has been investigated in the context

of an advanced thermoelasticity model involving

dual time delay factors and higher order memory-

dependent derivatives.

The evaluation of theoretical and numerical

results can lead to the following conclusions:

i) The kernel functions and empirical constants

have a significant impact on the physical

quantities, therefore their selection can be

made on the situation as well as the

requirement.

ii) In the case of instantaneous load (impact load),

when values of the elastic moduli are decreased

deformations and temperature fluctuations

increase. Thus, the values of the materials’

elastic moduli are inversely proportional to the

entropy changes of the system. In the case of

continuous load, this phenomenon is observed

steadily.

iii) All of the computed results indicate that the

physical quantities are non-zero in a finite zone

and converge to zero at the outside of that

region. This property of the physical quantities

leads to the characteristics of the behavior of

the hyperbolic type thermoelasticity models.

iv) Thermoelastic model with Non-linear kernel

function can predict more accurate values of

the field functions.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

27

Volume 19, 2024

v) Higher-order differential parameters have a

considerable influence on all of the field

variables under investigation.

The present study may be applicable in various

fields in the aviation industry and places where the

deformation of the medium varies widely with

temperature. This analysis can be extended in the

nonlinear deformation of continuum solid as well as

non-Fourier thermal conduction environment.

References:

[1] H. Lord and Y. Shulman (1967), “A

generalized dynamical theory of ther-

moelasticity”, J. Mech. Phys. Solid, 15(5),

299–309.

[2] Green AE and Lindsay KA (1972),

“Thermoelasticity”, J. Elast., 2(1), 1–7.

[3] Green, A. E. and Naghdi, P. M. (1991), “A Re-

examination of the Basic Postulates of

Thermomechanics”. Proceedings of the Royal

society, London A, 432(1885), 171–194.

[4] A. E. Green and P. M. Naghdi (1992), “On

Undamped heat waves in an elastic solid”, J.

Therm. Stress., 15(2),253-264.

[5] Green, A.E. and Naghdi, P.M. (1993),

“Thermoelasticity without energy dissipation”,

J. Elast., 31(3), 189-209.

[6] Tzou, D.Y. (1995a), “A unified approach for

heat conduction from macro-to micro-scales”,

J. Heat Transfer., 117(1), 8-16.

[7] Tzou, D.Y. (1995b), “Experimental support for

the Lagging behavior in heat propagation”, J.

Thermophys. Heat Transfer., 9(4), 686-693.

[8] Ignaczak, J. (1982), “A note on uniqueness in

thermoelasticity with one relaxation time”, J.

Therm. Stress., 5(3-4), 257–263.

[9] Sherief, H.H. and Dhaliwal, R.S. (1980), “A

uniqueness theorem and a variational principle

for generalized thermoelasticity”, J. Therm.

Stress., 3(2), 223–230.

[10] Sherief, H.H. (1987), “On uniqueness and

stability in generalized thermoelasticity”,

Quart. Appl. Math., 45(4), 773–778.

[11] Chandrasekharaiah, D.S. (1998), “Hyperbolic

thermoelasticity: A review of recent literature”,

Appl. Mech. Rev., 51(12), 705-729.

[12] Roy Choudhuri, S.K. (2007), “On A

Thermoelastic Three-Phase-Lag Model”, J.

Therm. Stress., 30(3), 231-238.

[13] Yu, Y.J., Zhang-Na Xue and Xiao-Geng Tian

(2018), “A modified Green–Lindsay

thermoelasticity with strain rate to eliminate

the discontinuity”, Meccanica, 53, 2543–2554.

[14] Shaw, S. (2020),”A thermodynamic analysis of

an enhanced theory of heat conduction model:

Extended influence of finite strain and heat

flux”, Int. J. Eng. Sci., 152, 103277.

[15] Lomakin, A. (1976), “The Theory of Elasticity

of Non Homogeneous Bodies”, Moscow State

University Press, Moscow.

[16] N. Noda (1991), “Thermal stresses in materials

with temperature-dependent properties”, Appl.

Mech. Rev., 44(9), 383-397.

[17] H. M. Youssef (2005), “Dependence of

modulus of elasticity and thermal conductivity

on reference temperature in generalized

thermoelasticity for an infinite material with a

spherical cavity”, Appl. Math. Mech., 26(4),

470-475.

[18] Xiulin Shen, Kongping Wu, Huanying Sun,

Liwen Sang, Zhaohui Huang, Masataka Imura,

Yasuo Koide, Satoshi Koizumi, Meiyong Liao

(2021), ”Temperature dependence of Young’s

modulus of single-crystal diamond determined

by dynamic resonance”, Diamond and Related

Materials, 116, 108403.

[19] Kamiyama, E. and Sueoka, K. (2023), ”Method

for estimating elastic modulus of doped

semiconductors by using ab initio

calculations—Doping effect on Young’s

modulus of silicon crystal”, AIP Advances,

13(8), 085224.

[20] Yu YJ, Hu W, Tian XG (2014), “A novel

generalized thermoelasticity model based on

memory-dependent derivative”, Int J Eng Sci.,

81, 123–134.

[21] Wang, J.L. and Li, H.F. (2011), “Surpassing

the fractional derivative: concept of the

memory-dependent derivative”, Comput. Math.

Appl., 62, 3, 1562–1567.

[22] Caputo, M. and Mainardi, F. (1971), “A new

dissipation model based on memory

mechanism”, Pure and appl. Geophys., 91,

134-147.

[23] Abouelregal, A. E., Moustapha, M. V., Nofal,

T. A., Rashid, S., and Ahmad, H. (2021).

“Generalized thermoelasticity based on higher-

order memory-dependent derivative with time

delay”, Results in Phys., 20, Article: 103705.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

28

Volume 19, 2024

[24] Abouelregal, A. E. (2022), “An advanced

model of thermoelasticity with higher-order

memory-dependent derivatives and dual time-

delay factors”, Waves in Random and Complex

Media, 32(6), 2918-2939.

[25] Kaur, I., Lata, P. and Singh K. (2020),

“Memory-dependent derivative approach on

magneto-thermoelastic transversely isotropic

medium with two temperatures”, Int J Mech

Mater Eng, 15, 10.

[26] Barak, M.S., Kumar, R., Kumar, R. and Gupta,

V. (2023), “The effect of memory and stiffness

on energy ratios at the interface of distinct

media”, Multidiscipline Model. Mat. and

Struct., 19(3), 464-492.

[27] Banerjee, S., Shaw, S., and Mukhopadhyay, B.

(2019), “Memory response on thermal wave

propagation emanating from a cavity in an

unbounded elastic solid”, J. Therm. Stress.,

42(2), 294-311.

[28] Jin-Liang Wang, Hui-Feng Li (2021),

”Memory-dependent derivative versus

fractional derivative (I): Difference in temporal

modeling”, Journal of Computational and

Applied Mathematics, 384, 112923.

[29] Kaur, I., Singh, K.(2022), “Functionally graded

nonlocal thermoelastic nanobeam with

memory-dependent derivatives”, SN Appl. Sci.

4, 329.

[30] Kaur, I., Lata, P., and Singh, K. (2020), “Effect

of memory dependent derivative on forced

transverse vibrations in transversely isotropic

thermoelastic cantilever nano-Beam with two

temperature”, Appl. Math. Model., 88, 83-105.

[31] Kaur, I. and Singh, K. (2021), “Effect of

memory dependent derivative and variable

thermal conductivity in cantilever nano-Beam

with forced transverse vibrations”, Forces in

Mechanics, 5, 100043.

[32] Shaw, S. (2019),”Theory of generalized

thermoelasticity with memory-dependent

derivatives”, Journal of Engineering

Mechanics, 145(3), 04019003.

[33] Ezzat MA, El-Karamany AS, El-Bary AA

(2016), “Modeling of memory-dependent

derivative in generalized thermoelasticity”, Eur

Phys J Plus., 131, Article No.372.

[34] Kumar, R., Tiwari, R., Kumar, R. (2022),

”Significance of memory-dependent derivative

approach for the analysis of thermoelastic

damping in micromechanical resonators”,

Mechanics of Time-Dependent Materials,

26(1), 101-118

[35] Gupta, V. and Barak, M.S. (2023), ”Quasi-P

wave through orthotropic piezo-thermoelastic

materials subject to higher order fractional and

memory-dependent derivatives”, Mechanics of

Advanced Materials and Structures,

doi.org/10.1080/15376494.2023.2217420.

[36] Barak, M.S. and Gupta, V. (2023),”Memory-

dependent and fractional order analysis of the

initially stressed piezo-thermoelastic medium”,

Mechanics of Advanced Materials and

Structures,

doi.org/10.1080/15376494.2023.2211065.

[37] Gupta, V., Kumar, R., Kumar, R. and Barak,

M.S. (2023),”Energy analysis at the interface

of piezo/thermoelastic half spaces”, Int. J. of

Numerical Methods for Heat Fluid Flow,

33(6), 2250-2277.

[38] Barak, M.S., Kumar, R., Kumar, R. and Gupta,

V. (2023), ”Energy analysis at the boundary

interface of elastic and piezothermoelastic half-

spaces” Indian J. Phys., 97, 2369–2383.

[39] Gupta, V., Kumar, R., Kumar, M., Pathania, V.

and Barak, M.S.

(2023),”Reflection/transmission of plane

waves at the interface of an ideal fluid and

nonlocal piezothermoelastic medium”, Int. J. of

Numerical Methods for Heat Fluid Flow, Vol.

33(2), 912-937.

[40] Yadav, A.K., Barak, M.S. and Gupta, V.

(2023),”Reflection at the free surface of the

orthotropic piezo-hygro-thermo-elastic

medium”, Int. J. of Numerical Methods for

Heat Fluid Flow, Vol. 33(10), 3535-3560.

[41] Abouelregal, A.E., Ahmad, A., Elagan, S.K.

and Alshehri, N.A. (2021),”Modified Moore–

Gibson–Thompson photo-thermoelastic model

for a rotating semiconductor half-space

subjected to a magnetic field”, Int. J. of

Modern Physics C, 32(12), 2150163.

[42] Abouelregal, A.E., Ahmad, H., Badr, S.K.,

Elmasry, Y., Yao, S-W. (2022),”Thermo-

viscoelastic behavior in an infinitely thin

orthotropic hollow cylinder with variable

properties under the non-Fourier MGT

thermoelastic model”, Z Angew Math Mech.

102:e202000344.

[43] Wang, X., and Xu, X. (2001), “Thermoelastic

wave induced by pulsed laser heating”, Appl.

Phys. A, 73(1), 107–114.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

29

Volume 19, 2024

[44] Wang, X., and Xu, X. (2002), “Thermoelastic

wave in metal induced by ultrafast laser

pulses”, J. Therm. Stress., 25(5), 457–473.

[45] Li, Y., Zhang, P., Li, C., and He, T. (2019),

“Fractional order and memory-dependent

analysis to the dynamic response of a bi-

layered structure due to laser pulse heating”,

Int. J. Heat Mass Trans., 144, 118664.

[46] Kumar, R., Tiwari, R., Singhal, A. and

Mondal, S. (2021), “Characterization of

thermal damage of skin tissue subjected to

moving heat source in the purview of dual

phase lag theory with memory-dependent

derivative.”, Waves in Random and Complex

Media, DOI:10.1080/17455030.2021.1979273.

[47] Al-Qahtani, H. M., and S.K. Datta. (2008),

“Laser-generated thermoelastic waves in an

anisotropic infinite plate: Exact analysis”, J.

Therm. Stress., 31 (6), 569–83.

[48] Shaw, S. and Mukhopadhyay, B. (2017), “A

discontinuity analysis of generalized

thermoelasticity theory with memory

dependent derivatives”, Acta mech., 228, 2675-

2689.

[49] M. A. Ezzat, A. El-Karamany, A. A. Samaan

(2004), “The dependence of the modulus of

elasticity on reference temperature in

generalized thermoelasticity with thermal

relaxation”, Appl. Math. Comput., 147(1), 169-

189.

[50] Shaw, S. and Mukhopadhyay, B. (2013),

“Moving heat source response in micropolar

half-space with two-temperature theory”,

Continuum Mechanics and Thermodynamics,

25, 523-535.

[51] G. Honig, U. Hirdes (1984),”A method for the

numerical inversion of Laplace transforms”, J.

Comput. Appl. Math.”, 10(1), 113-132.

[52] Marin, M (1998),”A temporally evolutionary

equation in elasticity of micropolar bodies with

voids”, Bull. Ser. Appl. Math. Phys, 60(3), 3-

12.

[53] Marin, M (1998), “Contributions on

uniqueness in thermoelastodynamics on bodies

with voids”, Cienc. Mat. (Havana), 16(2), 101-

109.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Soumen Shaw and Aktar Seikh carried out the

theoretical model and simulation.

- Aktar Seikh has implemented the Algorithm in

MatLab.

- Soumen Shaw has organized from the formulation

of the problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Author(s) thankfully acknowledges Department of

Science and Technology-INSPIRE, Government of

India (No. DST/INSPIRE

Fellowship/2017/IF170307) for the financial support

to carry out this research work.

Conflict of Interest

The authors declare that there is no conflict of

interest.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_

US

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

30

Volume 19, 2024

APPENDIX

Fig. 1: Shear stress distribution on continuous load

for different kernel functions at t = 0.25

Fig. 2: Shear stress distribution on continuous load

for different values of α∗ at t = 0.25

Fig. 3: Shear stress distribution on impact load for

different kernel functions at t = 0.25

Fig. 4: Shear stress distribution on impact load for

different values of α∗ at t = 0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

−4

−3

−2

−1

0

1

2

3

x

axis

St

re

ss

(

τ

13

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−4

−3

−2

−1

0

1

2

3

4

x

axis

St

re

ss

(

τ

13

)

b=1, a=1

b=0, a=1/2

b=0, a=0

0

0.5

1

1.5

2

2.5

3

3.5

4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

x

axis

St

re

ss

(

τ

1

3

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−2

−1.5

−1

−0.5

0

0.5

1

x

axis

St

re

ss

(

τ

13

)

b=1, a=1

b=0, a=1/2

b=0, a=0

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

31

Volume 19, 2024

Fig. 5: Normal stress distribution on continuous load

for different kernel functions at t = 0.25

Fig. 6: Normal stress distribution on continuous load

for different values of α∗ at t = 0.25

Fig. 7: Normal stress distribution on impact load for

different kernel functions at t = 0.25

Fig. 8: Normal stress distribution on impact load for

different values of α∗ at t = 0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

−2

−1

0

1

2

3

4

x

axis

St

re

ss

(

τ

33

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−3

−2

−1

0

1

2

3

x

axis

St

re

ss

(

τ

33

)

b=1, a=1

b=0, a=1/2

b=0,a=0

0

0.5

1

1.5

2

2.5

3

3.5

4

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x

axis

St

re

ss

(

τ

3

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

axis

St

re

ss

(

τ

3

)

b=1, a=1

b=0, a=1/2

b=0, a=0

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

32

Volume 19, 2024

Fig. 9: Displacement component (u1) on continuous

load for different kernel functions at t = 0.25

Fig. 10: Displacement component (u1) on continuous

load for different values of α∗ at t = 0.25

Fig. 11: Displacement component (u1) on impact load

for different kernel functions at t = 0.25

Fig. 12: Displacement component (u1) on impact load

for different values of α∗ at t = 0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

axis

Di

sp

la

ce

m

e

nt

(u

1

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

1

2

3

4

5

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

axis

Di

sp

la

ce

m

e

nt

(u

1

)

b=1, a=1

b=0, a=1/2

b=0, a=0

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

x

axis

Di

sp

la

ce

m

e

nt

(u

1

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

x

axis

Di

sp

la

ce

m

e

nt

(u

1

)

b=1, a=1

b=0, a=1/2

b=0, a=0

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

33

Volume 19, 2024

Fig. 13: Displacement component (u3) on continuous

load for different kernel functions at t = 0.25

Fig. 14: Displacement component (u3) on continuous

load for different values of α∗ at t = 0.25

Fig. 15: Displacement component (u3) on impact load

for different kernel functions at t = 0.25

Fig. 16: Displacement component (u3) on impact load

for different values of α∗ at t = 0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.15

−0.1

−0.05

0

0.05

x

axis

Di

sp

la

ce

m

en

t

(u

3

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x

axis

Di

sp

la

ce

m

en

t

(u

3

)

b=1, a=1

b=0, a=1/2

b=0, a=0

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

x

axis

Di

sp

la

ce

m

en

t

(u

3

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

x

axis

Di

sp

la

ce

m

en

t

(u

3

)

b=1, a=1

b=0, a=1/2

b=0, a=0

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

34

Volume 19, 2024

Fig. 17: Temperature distribution on continuous load

for different kernel functions at t = 0.25

Fig. 18: Temperature distribution on continuous load

for different values of α∗ at t = 0.25

Fig. 19: Temperature distribution on impact load for

different kernel functions at t = 0.25

Fig. 20: Temperature distribution on impact load for

different values of α at t = 0.25

0

0.5

1

1.5

2

2.5

3

3.5

4

−1.5

−1

−0.5

0

0.5

1

1.5

x

axis

Te

m

p

er

at

ur

e

(

θ

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−4

−3

−2

−1

0

1

2

3

x

axis

Te

m

pe

rat

ur

e (

θ

)

b=1, a=1

b=0, a=1/2

b=0, a=0

0

0.5

1

1.5

2

2.5

3

3.5

4

−1.5

−1

−0.5

0

0.5

1

1.5

x

axis

Te

m

pe

ra

tu

re

(

θ

)

α

*

=0.1

α

*

=0.05

α

*

=0.005

0

0.5

1

1.5

2

2.5

3

3.5

4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

axis

Te

m

pe

ra

tu

re

(

θ

)

b=1, a=1

b=0, a=1/2

b=0, a=0

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.3

Soumen Shaw, Aktar Seikh

E-ISSN: 2224-3429

35

Volume 19, 2024