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Perm Polytech Style: Lamba N.K., Deshmukh K.C. Memory dependent response in an infinitely long thermoelastic solid circular cylinder. PNRPU Mechanics Bulletin, 2024, no. 1, pp. 5-12. DOI: 10.15593/perm.mech/2024.1.01
ВЕСТНИК ПНИПУ. МЕХАНИКА
№ 1,2024 PNRPU MECHANICS BULLETIN
http s://ered.p stu .ru/index.php/mechanics/index
Научная статья
DOI: 10.15593/perm.mech/2024.1.01 УДК
MEMORY DEPENDENT RESPONSE IN AN INFINITELY LONG THERMOELASTIC SOLID CIRCULAR CYLINDER
N.K. Lamba1, K.C. Deshmukh2
1Department of Mathematics, Shri Lemdeo Patil Mahavidyalaya, Mandhal, India 2Ex-Profesoor, Department of Mathematics, R.T.M. Nagpur University, Nagpur, India
О СТАТЬЕ
АННОТАЦИЯ
Получена: 24 октября 2023 г. Одобрена: 05 февраля 2024 г. Принята к публикации: 12 марта 2024 г.
Ключевые слова:
memory-dependent derivatives (MDD), solid circular cylinder, generalized thermoelasticity, laplace transform, thermal stresses.
Memory-dependent derivatives (MDD) have physical meaning, and compared to fractional derivatives, they are more suitable and convenient for temporal remodeling. In this study, the temperature and stress distributions in an infinitely extended generalized thermally elastic solid circular cylinder have been investigated by utilizing the concept of a memory-dependent heat conduction model. The homogeneous, isotropic, infinitely long solid circular cylinder is considered to have a lateral surface that is free of traction and is subjected to a known surrounding temperature. In the domain of the integral Laplace transform, the problem is worked out, and its complex inversion is performed numerically using the Fourier series expansion method. The material properties of copper metal are chosen for the purpose of numerical computation, and the thermoelastic impact of time delay on temperature distribution, displacement distribution, and thermal stresses are represented graphically. Also, time delay's effect on temperature history, displacement history, and thermal heat transfer stress history are shown, respectively. This study could also benefit mathematicians and researchers involved in the development of thermoelasticity, as it accounts for the memory-related derivatives that are useful in explaining the behaviour of a variety of physical processes. The thermal fluctuations captured by various factors with memory-dependent responses are used in engineering applications to realistically design machines or structures.
© ПНИПУ
© Dr. Kishor C. Deshmukh - was working as Professor in the P.G. Department of Mathematics, R.T.M. Nagpur University, Nagpur, e-mail: kcdeshmukh2000@rediffmail.com, : 0000-0001-8291-9106
Dr. Navneet Kumar Lamba - is working as an assistant professor in the Department of Mathematics at Shri Lemdeo Patil Mahavidyalaya, Mandhal. He has received his M.Sc. and Ph.D. from R.T.M. Nagpur University, e-mail: navneetkumarlamba@gmail.com, : 0000-0002-6283-1681.
Эта статья доступна в соответствии с условиями лицензии Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0)
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0)
Introduction
In this current decade, many researchers have started a new contribution on memory-based derivatives. In 2011, Wang and Li [1] gave the concept of MDD (memory-dependent derivatives) by surpassing the fractional derivative. Currently, MDD has developed as an emerging field of fractional calculus that is continually expanding due to its ability to reflect memory-dependent responses in various physical processes. Despite being specified on an interval, the fractional derivative (FD) mainly captures local change. The physical meaning of MDD is significantly more noticeable than that of FD. The MDD-dependent weight is reflected by the kernel function, while the time delay demonstrates the duration of the memory effect. Available research shows that the memory-related derivative is more appropriate for temporal modelling which is useful in describing the thermal effect of solid bodies. Memory-dependent derivatives are capable of replacing fractional derivatives and having useful applications in the field of thermoelasticity, thermoelectricity, particle physics and vibration mechanics etc. For more details on the fractional-related theory of thermal response, refer to the contents developed by many renowned researchers, as shown in references [2-13].
El-Karamany and Ezzat [14] constructed generalized thermoelastic theory with time delay and investigated MDD problem for a half space Cartesian body. Sun and Wang [15] reconstructed the memory dependent heat model based on the assumption that temperature increases slowly after heat transfer. Xue et al. [16] investigated thermal stresses in hollow cylinder containing internal and surface crack by using integral transform technique by considering heat conduction model with memory dependent derivatives. Ma and Gao [17] studied dynamic response of infinite hollow cylinder under thermal shock based on generalized nonlocal thermoelastic theory and MDD theory. Recently, Verma et al. [18] determined the impact of memory in hygrothermoelastic hollow cylinder by theory of uncoupled-coupled heat and moisture. Marin [19, 20] contributed the thermoelastodynamics study on bodies with void. Abouelregal et al. [21 to 26] elaborated the study of thermoelasticity based on higher-order memory-dependent derivative with time delay and discussed vibrational behaviour in nanobeam. Atta [27] analyzed fractional operator response on thermal diffusion problem of infinite medium. Yu et al. [28] developed a novel thermoelastic model using memory-related derivatives and described its applications. Hussein [29] determined the solution by considering the circular cylinder problem under fractional theory.
Chepurnenko and Turina [30] determined the stressstrain relation for multilayer beams using the finite element method. Ogorodnikov et al. [31] constructed the mathematical model for nonlinear viscoelasticity and discussed the impact of fractional integro-differentiation. Also, various other authors contributed to the field and evaluated the impact of fractional responses on different bodies, as highlighted in [32-38].
According to the aforementioned literature review, the Memory dependent derivative surpasses the fractional derivative on the basis of significant improvements in memory-related response. Thus, the authors have the opportunity to investigate the impact of MDD in an infinitely long, solid circular cylinder whose lateral face is free from traction and exposed to a specified ambient temperature. To obtain the solution of temperature and stress distribution, Laplace transform is utilized and its inversion is carried out numerically by Fourier series expansion method as given by Honig and Hirdes [39]. Obtained thermoelastic responses are represented graphically and the effects of MDD are discussed.
1. Basic Equations
A memory-dependent derivative of function ¥(x, t) is
described by Wang and Li [1] as a weighted integration of the common integer-order derivation on a slipping interval defined as
DaW(x,t)=! | K(t(1)
t-m
The m-th-order memory-dependent derivative of a function ¥ (x, t) that is m times differentiable with respect to t is written as
dm—1 i ' dm^ ( V f)
Dm *(v, t )=dmr d^(V, t )=IJK (t—s)-gfm f
(2)
where, Kernel function K(t-j) and time delay mare chosen arbitrarily in order to capture materials real behaviors. In general, 0 < K (t -j)< 1 for je[t - m,t] and
3y(x, t)
D t )|<
dt
so that the magnitude of memory
dependent derivative Dmf (x, t) is usually smaller than that
dy(x, t)
of the common partial derivative
dt
The kernels function may be chosen as
2b
K (t — f) = 1 — ^(t — f) + ^ (t — f) = CO
1 — (t — f)
1 _(—) CO
1—t—f
CO
a = b = 0, a = 0, b =
CO
a = 0, b =-, 2
a = b = 1,
CO
where, a and b are constants.
2. Problem Formulation
Followed by Hussein [29], we assume a homogeneous, isotropic infinitely long solid circular cylinder having (r, y, z) as cylindrical polar coordinates and radius R . The
cylinder's surface is considered to be traction-free. All the considered functions are dependent on r and time t as a result of the cylindrical symmetry.
Because of the radial symmetry of the problem, the u displacement elements are treated as u = (u (r, t), 0, 0).
The strain components produced due to displacement field are as
du
dr
e = -yy
e = e = e = e = 0,
ry yz rz zz '
where cubic dilatation e is given by
du + u dr r
(4)
(5)
(6)
(7)
Sherief et al. [13] presented the theory of fractional order of thermoelasticity, in which the heat conduction equation was presumed to be:
de q
q+тэ7
= -kVT ; 0 < e <1,
(8)
where q denotes the element of the heat flux vector, т represents the relaxation time, k denotes the thermal material conductivity and I is constant.
Yu et al. [28] recently provided an application of a memory-related derivative. In their study, they substituted the memory-dependent derivative (MDD) for fractional calculus to represent memory-dependence in the rate of heat flux in the Lord-Shulman generalised thermoelasticity theory, as follows:
q + lDaq = -k VT.
(9)
In absence of an inner thermal source, the heat transfer equation for isotropic materials has the following form:
dT
de.
(1 + xDm )( pC£ — +YTo—) = k V2T
dt
dt'
(10)
where CE denotes the specific heat at static strain, p designates the density, T represents the temperature in absolute terms, and T0 is the temperature of the reference that is supposed to be such that |T -T01 << 1.
The motion equation for this theory is formatted as
(Л + 2|) V2u-y^ = p|U dr dt
(11)
where X, | denotes the Lame's constants, and y represents material constant specified by (3X + 2|) a, where a, stands for the linear thermal expansion coefficient;
where the definition of V2, the Laplace's operator in cylindrical polar coordinates, is
V2 = 1 — f r — r dr I dr
We obtain below expression by using the divergence operator on both sides of equation (11) as
(Ь + 2|) V2 e -yV2T = p|^.
dt
(12)
The constitutive relations giving the stresses ay have the form
a„ = 2|e„ + Xe-y(T -Ta), (13)
°„= 2| e„+Xe-y (T - T0), (14)
ezz =Xe-Y (T - To), (15)
ez y = erz = er ¥= 0. (16) Let's introduce the dimensionless variables:
r = ацr, |' = ацu, t' = а2цt,
/ r Grr , Gyy
T = cnt, Grr =-, Oyy =-
yy
| I
9'= Y(T-To), = , F' = - YF
À + 2|
кц c à + 2|ll
R ' = ац R, q' =
y qr
À + 2| кц c '
where
c = JX + 2|, ц=рCe
The governing equations (8), (9), (11) and (12) have the following forms when using the aforementioned non-dimensional variables and dropping the primes out of convenience
(1 + xDm )(— + e— ) = V2 9, dt dt
(17)
V2 e -V2 9 =
de
dt2!
(18)
Grr = 2err +(ß2 -2)e-ß2 9, (19)
Gyy= 2e¥¥+(ß2 -2)e-ß2 9, (20)
ezz =(ß2 -2)e -ß2 9, (21)
ez y= erz = er y= 0, (22)
err =
u
r
where
e =
T„ Y2
( + 2|)k n'
ß2 =
( + 2|)
|
It is assumed that the initial conditions of the problem are of a homogeneous nature, complemented by the following limiting boundary conditions as:
(i) The surface of the cylinder is traction free
(r,t)|r=* = 0; t > (23)
(ii) The boundary condition for the heat convection = H (t ). (24)
99 d r
3. Problem-solving in the Integral Laplace Transform Domain
Using the Laplace transform with relation-defined parameter s
T (s) =
J e" stT (t ) dt.
(25)
The following equations are obtained by applying the transform in equation (25) to both sides of equations (17) through (24).
[V2 - s (1 + G)] 9 = es (1 + G) e, (V2 - s2)ë = V2 9,
ôrr = 2Frr +(ß2 -2) -ß2 9,
ô¥¥= 2 ey¥ + (ß2 - 2 )e-ß2 9, ëzz =(ß2 - 2 )e-ß2 9,
e = e = e = „,
z y rz r y 5
ôrr ( R, s) = „,
d9 1 R — = _; r = R, d r s
(26)
(27)
(28) (29) (3„)
(31)
(32)
(33)
Where
G = 1 |(1 - e"»)|1 - ^ + ^ 1 - f a2 - 2b + ^ | e"» !>■ (34)
ras ra2s2
ra I
ra s
Eliminating e between (26) and (27), we get
|V4 -V2[s(1 + e)(1 + G) + s2] + s3 (1 + G)}9 = 0. (35) The equation (35) can be factorized as
where k2, k22 are the roots of the characteristic equation
k4 — k2[s(1 + e)(1 + G) + s2] + s3 (1 + G) = 0 and satisfying the two relations
k2 + k22 = s (1 + e)(1 + G) + s2, k2k22 = s3 (1 + G).
The solution of Eq. (36) can be written in the form 9 = 9j + 92 where 9i stands for complete integral of the equation
(V2 — k2) 9 = 0 for i = 1,2. (37)
The result of the solution for equation (37) can be expressed as
9=4(s)(k2-s2)I„ (r) + + Bt(s)(k2 - s2)K„ (r); i = 1,2,
(38)
where I0 (z), K0 (z) are the modified Bessel function forms of the first and second kinds, respectively, and Ai (s), Bi (s) , i = 1,2 are parameters that rely only on s.
Thus the general solution of Eq. (36) is given by
2
9 = ^ [Ai (s)(k2 - s2)/0 ( r) + Bi(s) (k2 - s2)K0 ( r)]
i=1
since K0 (z) not bounded at r = 0, we must putting Bi (s) = 0 . Thus,
2
9 = ^A,.(s)(k2-s2)/0 (r), (39)
i=1
p2s2I0 (k2R)-(2kjR)I1 (kjR) (40)
4(s) =
4M =-
i(k12 -s1)k111 (k1 R)[p2s210 (k2 R)-(2kjR)I1 (k1 R)] 1 ' j-(k22 - s2 )k211 (k2 R)[p2s210 (k R) - (2kjR)Il ( R)] j
P2 s210 ( R) - (2kjR)I1 ( R) (41)
i(k22 - s2)k2 Il (k2 R)j[p2s2I0 ( R)-(2k1/R)I1 k R)] 1 ' " [-(# - s2)k111 ( R)jjp2/10 (k2 R)-(2kjR)I1 (k2 R)] j
Similarly, we get
e = k2 A,(s) I0 ( r)+ k22 A2(s) I0 (k2 r). (42)
Using equation (42) as a substitute in (7) and integrating with regard to r gives us
U = kx [A.(s)I1(k1r)] + k2 [A2(s)11 (k2r)], (43)
du
= k A,( s)
(V2 - k2 ) (V2 - k22) 9= „,
(36)
dr
+ k2 A2 (s)
I1( k:r)
r
I,(k2 r )
- k I„(V )
k2I„ (k2 r )
Substituting from equations (4), (5), (39)-(44) into (28) and (29), we obtain the stress components as
2k
o„ = Д(s) p2 s2/„(V)--Il(klr)
L r
' 2k
+ 4(S) p2 S2I„(k2Г)--г-Ix(k,Г)
r
= {A¡ (s) (2 s2 - 2k2) I„ (V) + I, (V)} +
+ {4 (s) (2 s2 - 2k2 )i„ (k2r) + 2-11, (k2r) j.
4. Numerical Inversion of the Laplace transformation
(45)
(46)
and thermal heat transfer stress history is shown respectively in Fig. 4 to 8. In numerical computation non-dimensional relaxation time T = 0.1 and radius varies from r = 0 to r = R = 1.
Fig. 1. Temperature distribution
Solution of the problem is obtained in Laplace transform domain and its complex inversion is carried out numerically by assuming process assumed by Honig and Hirdes [39]. The Laplace transform of ) is defined as
y(s) = je st v(t) dt.
(47)
Its numerical inversion is as given in [39] is utilized for calculation.
¥(t ) = -
2 v(c)+Re ¿ иг v (c+ijf-
ct = 4.7.
(48)
5. Numerical Results and Discussions
Materials properties of copper metal mentioned in [29] are chosen for the purpose of numerical computation with the physical data:
Symbol Value Unit
P 8954 (kg )(m )-3
X 7.76 (10)10 (m )-1 (s )-2
3.86 (10 )10 (m )-1 (s )-2
n 8886.73 (kg )(m )-1 (K )-1 (s )-
k 386 (kgm )(K-1 )()
T 293 (K)
CE 383.1 (m2)(-1 )(s -2)
In this part, our focus is to investigate the impact of time delay rn = 0.01, 0.02, 0.03 on temperature distribution, displacement distribution and thermal stresses, whose graphical plotting is as shown in Fig. 1 to 4 respectively. Also, time delay's effect on temperature history, displacement history
1.2 1.0 0.8 13 0.6
0.4
0.2 0.0
\
/ / \ to = 0.02 Ч ^Чч
; // \ Lü = 0.01
// ^ h Cü = 0.03
0.0 0.2
0.4 0.6 r
Fig. 2. Displacement distribution
1.0 0.5 0.0 lb -0.5 -1.0 -1.5 -2.0
ZOJ=t
_/
Fig. 4. Hoop stress distribution
0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
r
Fig. 3. Radial stress distribution
t
Fig. 5. Effect on temperature history
0.4; 0.3; 13 0.2; 0.1 -
0.0 .....................
0.0 0.2 0.4 0.6 0.8 1.0 t
Fig. 6. Effect on the displacement history
Fig. 7. Effect on radial stress history
1
!".,.-■■ \ u-0.02 / :' \ №0.01
0.0 0.2 0.4 0.6 0.8 1.0
t
Fig. 8. Effect on past hope stress
From Fig. 1 the great impact of time delay on the temperature distribution function is seen along radial direction at t = 0.03. The smoother distribution of temperature is
observed for larger time delay or we can say that temperature decreases with increase in value of parameter m. The obtained result totally matches with [16] for Kernel function K (t - j) = 1. Also temperature goes on decreasing towards
outward radial direction for different value of time delay m = 0.01, 0.02, 0.03 . Further we can say that thermal waves
are varying continuously, smoothly and reach to steady state depending of time delay.
Fig. 2 presents the variation of displacement distribution for different values of time delay m along radial direction r . Displacement distribution is zero at r = 0 and reaches to its peak at r = 0.3 then it slight decreases towards outer radii. Here large distribution in displacement is noted for small values of parameter m. Hence we can say that displacement is also significantly affected by values time delay parameter.
Fig. 3 shows the radial stress distribution along radial direction with the impact of time delay parameters m = 0.01, 0.02, 0.03 at t = 0.03 . The influence of time delay strongly affects the stress distribution and it is found that stresses decreases in the range 0 < r < 0.1 and then goes on increasing gradually towards outer radii. For higher value of parameter m, large distribution in stress function is observed. Hence we can say a time delay dependent stress wave varies continuously and smoothly along radial direction. Also the value of radial stresses becomes zero at r = 1 which matches with the prescribed free-end mathematical condition as defined in equation (23).
Fig. 4 shows the continuous and smooth variation of hope stress (ovv) distribution with impact of time delay in
radial direction. The stress functions are compressive throughout the radial direction and large distributions in stresses are noted for small value of parameter m. Further maximum stress distribution is obtained at r = 0.2 then after it gradually increases towards outer radii.
The effect of time delay on temperature history at r = 0.5 is depicted in Fig. 5. Initially temperature is zero and goes on increasing with time for different value of m. Also it is noticed that temperature varying smoother for large time dealy, hence we can say that effect of time delay on temperature history plays an important role which can be used in classification of materials on the basis of ability to conduct heat.
Fig. 6 indicates that the displacement distribution is also significantly affected by different values of time parameter delay m on temperature history. For large value of time delay small distribution in displacement history is observed for infinitely long circular cylinder.
The effects of the history of radial stress (orr) at
r = 0.5 for time delay are depicted in Fig. 7. It is evident that with increase in time radial stresses gradually increasing towards outer radii and maximum distribution in stress is observed for small value of m. It is noticeable that radial stresses are tensile in initial time period 0 < t < 0.2 and becomes compressive in range 0 < t < 0.2 . This indicates that
radial stress history significant effected by the impact of time delay.
Fig. 8 shows the impact of hoop stress (o¥¥) history with impact of time delay r = 0.5 . Initially the stress (ovv)
decreases rapidly in the range 0 < t < 0.1 and gradually increases with increase in time. Also large variation in hoop stress distribution exists for higher value of parameter ra. The finite speed of wave propogation is seen for different time delay parameter and obtained stress curve is compressive throughout the variation.
Conclusion
MDD is the extension part of FD theory. Fractional order theory reflects the memory effect which is given by Caputo fractional derivative and applicable to wave equation and diffusion equation. While MDD is the timeframe reflects the duration of the memory effect and the kernel feature reflects the dependent weight. The heat flow solution is obtained employing the fractional order theory as a long tail solution that corresponds to the kernel. In this paper we modify the work
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• The variation of temperature, displacement and stress functions are significantly affected by different value of time delay parameter ra.
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Memory dependent derivatives may be used for classification of materials on the basis of time delay parameters which has ability to describe memory impact on temperature, displacement and stress history.
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Финансирование. Исследование не имело спонсорской поддержки. Конфликт интересов. Авторы заявляют об отсутствии конфликта интересов. Вклад авторов равноценен.
Funded. The study was not sponsored.
Conflict of interest. The authors declare no conflict of interest. The contribution of the authors is equivalent.
