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Generalized thermo-viscoelasticity with memory-dependent derivatives
Author's Accepted Manuscript
Generalized thermo-viscoelasticity memory–dependent derivatives
with
M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary
www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(14)00342-7 http://dx.doi.org/10.1016/j.ijmecsci.2014.10.006 MS2836
To appear in:
International Journal of Mechanical Sciences
Cite this article as: M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary, Generalized thermo-viscoelasticity with memory–dependent derivatives, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2014.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Generalized thermo-viscoelasticity with memory–dependent derivatives 3
M. A. Ezzat 1,*, A. S. El-Karamany2, A. A. El-Bary 1
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
2
Department of Mathematical and Physical Sciences, Nizwa University, Nizwa -611, P. O. Box 1357, OMAN, [email protected] 3
Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt
*
Corresponding Author: [email protected]
ABSTRACT A new generalized thermo-viscoelasticity theory with memory–dependent derivatives is constructed. The governing coupled equations with time-delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to one- dimensional problem of a half-space. The bounding surface is taken traction free and subjected to a time dependent thermal shock. Laplace transforms technique is used to obtain the general solution in a closed form. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory are given. The predictions of the theory are discussed and compared with dynamic classical coupled theory.
Keywords:
Fourier's
Law;
thermo-viscoelasticity
theory;
Memory-dependent
derivative; Fractional calculus; Time-delay; Kernel function; Laplace transforms; Numerical results.
Nomenclature
,
Lame' constants
mass density
t
time
CE
specific heat at constant strains
Ko
= (2 / 3) , bulk modulus
Cο2
=
ε ij
components of strain tensor
K
, longitudinal wave speed
2 e ij
components of strain deviator tensor
ij
components of stress tensor
S ij
components of stress deviator tensor
e
= ii , Dilatation
thermal conductivity
T
absolute temperature
ui
components of displacement vector
T
coefficient of linear thermal expansion
3K T
ij
Kronecker’s delta.
To
reference temperature
Co
= ( 2 )/ 1/2 , speed of propagation of isothermal elastic waves
o
= CE /
2T 0 = , Thermal coupling parameter k o C 02
T T 0 , such that / T0 1,
Q
strength of applied heat source per unit mass
Γ(.)
Gamma function
1. Introduction The linear theory of elasticity is of paramount importance in the stress analysis of steel, which is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e.g., concrete, wood, and coal. However, the theory does not apply to the behavior of many of the new synthetic materials of the elastomer and polymer type, e.g., polymethyl-methacrylate (Perspex), polyethylene, and polyvinyl chloride. Linear viscoelastic materials are rheological materials that exhibit time temperature rate-of-loading dependence. When their response is not only a function of the current input, but also of the current and past input history, the characterization of the viscoelastic response can be expressed using the convolution (hereditary) integral.
3 Tschoegl [1] has presented a general overview of time-dependent material properties. Gross investigated the mechanical-model representation of linear viscoelastic behavior results [2]. One can refer to Atkinson and Craster [3] for a review of fracture mechanics and generalizations to the viscoelastic materials, and Rajagopal and Saccomandi [4] for non-linear theory. Physical observations and results of the conventional coupled dynamic thermoelasticity theories involving infinite speed of thermal signals, which were based on the mixed parabolic-hyperbolic governing equations of Biot [5] are mismatched. To remove this paradox, the conventional theories of thermoelasticity have been generalized, where the generalization is in the sense that these theories involve a hyperbolic-type heat transport equation supported by experiments which exhibit the actual occurrence of wave type heat transport in solids, called second sound effect. The first is due to Cattaneo [6] who obtained a wave-type heat equation by postulating a new law of heat conduction to replace the classical Fourier law. Several generalizations to the coupled theory are introduced. One can refer to Ignaczak [7] and Chandrasekharaiah [8] for a review. Hetnarski and Ignaczak [9] described the modern approaches to the analytical treatment of dynamical thermoelasticity. Within the theoretical contributions to thrmo-viscoelasticity theory are the proofs of uniqueness theorems under different conditions by Ezzat and El Karamany [10, 11] and the boundary element formulation was done by El-Karamany and Ezzat [12, 13]. The fundamental solutions for the cylindrical region were obtained by Ezzat [14]. Ezzat at el. [15] solved some problems in thrmo-viscoelasticity with thermal relaxation by using state space approach [16]. Ezzat [17] investigated the relaxation effects on the volume properties of an electrically conducting viscoelastic material. In the last decade, considerable interest in fractional calculus has been stimulated by the applications in different areas of physics and engineering. Recently, some efforts have been done to modify the classical Fourier law of heat conduction by using the fractional calculus [18-23]. Diethelm [24] has developed Caputo [25] derivative to be: t
Da f (t ) K (t ) f ( m) ( ) d a
with
(1)
4
K (t )
(t ) m 1 . (m )
(2)
Where K (t ) is the kernel function and f ( m ) denotes the common m-order derivative, which has specific physical meaning. The memory-dependent derivative is defined in an integral form of a common derivative with a kernel function on a slipping interval. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory dependent differential equation has more expressive force. Wang and Li [26] introduced a memory-dependent derivative (MDD), the first order memory-dependent derivative of function f is simply defined in an integral form of a common derivative with a kernel function on a slipping interval, in the form D f ( x, t )
1
t
t
K (t ) f ( x, ) d
(3)
where is the time delay and K (t ) is the kernel function in which they can be chosen freely. Wang and Li indicated that the memory effect requires weight 0 K (t ) 1 for [t , t ) so that the magnitude of memory-dependent
derivative, derivative
D f ( x, t ) is usually smaller than that of the common partial
f ( x, t ) . The kernel form K (t ) can also be chosen freely, such as 1, t
( t ) t 1, 1 , where p 0.25,1, 2 , etc. which may be more practical. They p
are a monotone function with K 0 for the past time t and K =1 for the present time t. In case, K (t ) 1 , we have
D f ( x, t )
1
t
f( x, ) d
t
f ( x , t ) f ( x, t )
This means that the common partial derivative So that, D f ( x, t )
f ( x, t ) . t
is the limit of D as 0 . t
f ( x, t ) f ( x, t ) f ( x, t ) lim 0 t
(4)
5 Sherief et al. [27] introduced the fractional order theory of thermoelasticity, in which the heat conduction equation was assumed to be the form qi
qi k,i t
Recently, an interesting application of memory-dependent derivative is given by Ya-Jun Yu et al [28]. They introduced the memory-dependent derivative (MDD) instead of fractional calculus, into the rate of heat flux in Lord-Shulman generalized thermoelasticity theory [29], to denote memory-dependence, as, q q k , i i ,i
(5)
Equation (5) has more clear physical meaning. In the current work, a modified law of heat conduction including both the heat flux and its memory-dependent derivative replaces the conventional Fourier's law in thermo-viscoelasticity. The resulting non-dimensional coupled equations
of
generalized thermo-viscoelasticity with memory-dependent derivative together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to thermal shock and traction free surface. A direct approach is introduced to obtain the solutions in the Laplace transform domain for different forms of kernel functions. The inversion of Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier expansion techniques proposed by Honig and Hirdes [30]. In this theory, the coupled thermo-viscoelastic model is used. This implies infinite speeds of propagation of thermo-viscoelastic waves. The solutions are represented graphically for different values of time-delay and different forms of kernel function.
2. Differentiation of memory dependent derivative Let n N , I :[a, b] and f : I R be such that f and all its derivatives up to f ( n ) are continuous on I , and f ( n 1) exists on (a, b) . Then,
Dn f ( x, t )
1
t
K (t ) f
t
( n)
( x, ) d
(6)
3. Derivation of heat conduction equation with time-delay in thermoelasticity. The classical Fourier’s law, in which relates the heat flux vector q to the temperature gradient q( x, t ) . T ( x, t )
(7)
6 The energy equation in terms of the heat conduction vector q in the context of thermoelasticity theory is given by [5] CE T ( x, t ) To e( x, t ) . q( x, t ) + Q( x, t ) . t
(8)
Using relation (5), we get the generalized heat conduction law for the considered new generalized theory with time-delay q( x, t ) q( x, t ) + D q( x, t )
(9)
From a mathematical viewpoint, Fourier law (7) in the theory of generalized heat conduction with time-delay, is given by q( x, t ) D q( x, t ) T ( x, t )
(10)
Taking the memory-time derivative of Eq. (8) (suppressing x for convenience), we get D CET To e . D q Dw Q t
(11)
Multiplying Eq. (11) by and adding to Eq. (8), we obtain
1 D CE
e T To . q D q 1 D Q t t
(12)
Substituting from Eq. (10), we get
1 D CE
e T To 2T 1 D Q t t
(13)
Eq. (13) is the generalized energy equation with memory-dependent derivative, taking into account the time- delay . The dynamic coupled theory of heat conduction law follows as the limit case when 0 . This model is more intuitionistic for understanding the physical meaning and the corresponding memory dependent differential equation is more expressive.
4. The formulation of the physical problem The governing equations for generalized thermo-viscoelasticity consist of: 1- The equation of motion in the absence of body forces
2ui ji , j . t 2
2-The constitutive equation [17]
(14)
7
eij ( x, τ)
t
Sij R(t τ)
τ
0
dτ R eij ,
(15)
where Sij ij
kk 3
ij ,
(16)
and R(t) is relaxation function given by t R(t ) 2 1 A* e * t t *1 dt , 0
(17)
where *, * and A* are non-dimensional empirical constants and ( *) is the Gamma function, 0 * 1, * 0, 0 A *
* d , R (t ) 0, R (t ) 0 . * ( ) dt
3- The kinematic relations 1 2
e 3
ij (ui , j u j ,i ), eij ij ij , e kk .
(18)
4- The stress-strain temperature relation:
Ko e 3T T To
(19)
where
kk 3
, ij ji
Substituting from (25) into (21) we obtain
e
ij R ij ij K o e ij ij , 3 where T To and
(20)
1 . To 5- The energy equation with memory-dependent derivative in the absence of heat
sources
1 D CE
e To 2 t t
(21)
6- The strain –displacement relations e ij
1 u i . j u j ,i . 2
(22)
In the above equations a comma denotes material derivatives and the summation convention are used.
8 We shall consider a viscoelastic solid occupying the region x 0 , where x-axis is taken perpendicular to the bounding plane of half-space pointing inwards. For the one-dimensional problems, all the considered functions will depend only on the space variables x and t. The displacement components
u x u ( x, t ), u y u z 0.
(23)
The strain component e xx
u . x
(24)
The equation of motion
2u 2 2u ( R K ) . o 2 2 t 3 x x
(25)
The energy equation
2 ( x, t ) ( x, t ) 2u ( x, t ) C T E o x 2 t t x 2 ( x, ) 3u ( x, ) K (t ) CE T d o 2 2 x t t
(26)
The constitutive equation 2 3
xx ( R Ko )
u . x
(27)
We assume that the boundary conditions have the form
(0, t ) f (t ), (, t ) 0, (0, t ) (, t ) 0, t 0
(28)
where f (t ) is a known function of t . The initial conditions are taken as u ( x, 0) u ( x, 0) ( x, 0) ( x, 0) ( x, 0) ( x, 0) 0, t 0.
Let us introduce the following non-dimensional variables, 1 2 , , qi qi , R R 2 3 Ko Co Co0 3K o The equations (25) –(27) in non-dimensional form yield: x Coo x, u Coou , t Co2ot ,
1 R xu tu x ,
(29)
( x, t ) 2 2u ( x, t ) 1 D , x 2 x t t
(30)
2
2
2
2
9
1 R
u , x
(31)
t 4 * * t *1 R(t ) dt . 1 A e t 3K o 0
(32)
From now on, the kernel function form K (t ) can be chosen freely as: 1 if m n 0 (t ) 1 1 if , m 0, n 2 2 2n m (t ) 2 K (t ) 1 (t ) 2 1 (t ) if m 0 n / 2 t 2 ) if m n 1, (1
(33)
where m and n are constants.
5. Solution in the Laplace transform domain Appling the Laplace transform with parameter s defined by the formulas
L{g( x, t )} g( x, s) e st g( x, t ) dt , 0
on both sides of Eqs. (30)-(32), we get
D
s 2 u D ,
(34)
D 2 s (1 G ) ( Du ),
(35)
2
1 u , x
(36)
and the boundary conditions (28) become
(0, s) f ( s), (0, s) 0 , where D
2u 2u 4 , L R 2 sR ( s ) 2 , 1/ 1 sR , R ( s) x x x 3 sK o
G ( s) (1 e s )(1
2b 2a 2 2a 2 s 2 2 ) (a 2 2b )e , s s s
(37)
A * ( *) , 1 * ( s *)
10 [(1 e s )], m n 0 [1 1 (1 e s )], m 0, n 1 s 2 L{ D f (t )} F ( s ) 1 , s s s [(1 e ) (1 e ) e ], m 0, n s 2 2 2 s [(1 s ) 2 s 2 (1 e )], m n 1 F ( s) L{(
2u )} s( Du ) . t xt
Eliminating u between Eqs. (34) and (35), we obtain
D
4
s 2 s (1 G)(1 ) D 2 s 3 (1 G) 0 .
(38)
In a similar manner, we can show that u satisfies the equation
D
4
s 2 s (1 G)(1 ) D 2 s 3 (1 G ) u 0 .
(39)
The solutions of Eqs. (38) and (39) which are bounded for x≥0 have the form
( x, s ) C1 e k x C2 e k x ,
(40)
u ( x, s ) C3 e k1x C4 e k2 x ,
(41)
1
2
where k1 and k 2 are the roots with positive real parts of the characteristic equation k 4 s 2 s (1 G)(1 ) k 2 s3 (1 G) 0,
(42)
satisfying the relations
k12 k22 s 2 s (1 G)(1 )
(43)
k12 k22 s 3 (1 G )
and Ci , i 1, 2,3, 4 are parameters depending on s to be determined from the boundary conditions of the problem. Substitution from Eqs. (40) and (41) into Eq. (34), we obtain the following relations C3
k1
s k 2
2 1
C1 , C4
k2
s 2 k22
C2 .
(44)
Substitution from Eq. (44) into Eq. (41), we have u ( x, s )
k1
s k 2
2 1
C1e k1 x
k2
s k 2
2 2
C2 e k 2 x .
Substitution from Eqs. (40) and (45) into Eq. (36), we have
(45)
11
C1 C e k1x 2 2 2 e k2 x . 2 k2 s k s
( x, s) s 2
(46)
2 1
In order to determine the C1 and C2 , we shall use the boundary conditions (37), we obtain
C1
k12 s 2 k22 s 2 f ( s ), C f ( s) . 2 k12 k22 k12 k22
(47)
Plugging Eq. (47) into Eqs. (40), (45) and (46), the final mathematical expressions for the dimensionless form of strain, conductive temperature and stress fields can be deduced as
( x, s )
1 (k12 s 2 ) e k1x (k22 s 2 )e k2 x f ( s ) , k k22
u ( x, s )
( x, s)
(48)
2 1
k e
k k
2 2
s2
e
2 1
k12 k22
k1 x
1
k1 x
k 2 e k2 x ,
(49)
e k2 x .
(50)
From Eq. (24) and (49), one can obtain the strain distribution as e ( x, s )
k k22 2 1
k
2 1
e k1 x k 22 e k2 x .
(51)
In order to obtain the non-dimensional heat flux component, we shall use the Laplace transform of Eq. (10), we have q ( x, s )
1 ( x, s) 1 G x
1 k k12 s 2 e k1x k2 k22 s 2 e k2 x f ( s). 2 2 1 1 G k1 k2
(52)
This completes the solution in the Laplace transform domain.
6. Inversion of the Laplace transforms We shall now outline the method used to invert the Laplace transforms in the above equations. Let f ( s ) be the Laplace transform of a function f(t). The inversion formula for Laplace transforms can be written as Honig and Hirdes [30]: f (t )
edt 2
e
ity
f (d iy ) dy ,
12 where d is an arbitrary real number greater than all the real parts of the singularities of f ( s ) . Expanding the function h(t ) exp(dt ) f (t ) in a Fourier series in the interval [0, 2L], we obtain the approximate formula: f (t ) f N (t )
N 1 c0 ck , for 0 t 2L 2 k 1
(53)
where edt ck Re eik t / L f d ik / L . L
(54)
Two methods are used to reduce the total error. First, the ‘Korrektur’ method is used to reduce the discretization error. Next, the ε-algorithm is used to reduce the truncation error and therefore to accelerate convergence. The Korrektur-method uses the following formula to evaluate the function f (t ) f (t ) f NK (t ) f N (t ) e 2 dL f N (2 L t ) .
(55)
We shall now describe the ε-algorithm that is used to accelerate the convergence of m
the series in (53). Let N be an odd natural number and let sm ck , be the sequence k 1
of partial sums of (53).We define the ε-sequence by
0,m 0, 1,m sm , m 1, 2,3,... and n1,m n1,m1 1/ n,m1 n,m ,
n, m 1, 2,3,....
It can be shown from Honig and Hirdes [30], that the sequence 1,1 , 3,1 , ..., N ,1 ,... converges to f (t ) c0 / 2 faster than the sequence of partial sums.
7. Numerical results and discussion In this section, we aim to illustrate numerical results of the analytical expressions obtained in the previous section and elucidate the influence of time-delay on the behavior of the field quantities. In order to interpret the numerical computations, we consider
material
properties
of
a
Polymethyl
Methacrylate
(Plexiglas) material. Following the values of physical constants are shown in Table 1 [17]. The calculations were carried out for the function f (t), which represents a time dependent thermal shock [31]
13
t sin f (t ) a 0
0t a
or
otherwise
f ( s)
a 1 e as a2s2 2
.
Considering the above physical data, we have evaluated the numerical values of the field quantities with the help of a computer program developed by using MATLAB software. The accuracy maintained was 7 digits for the numerical program. The computations were performed for a value of time, namely t = 0.1 and for time-delay 0.0009, 0.009, 0.09 . The numerical technique outlined above was used to obtain the conductive temperature, thermo-dynamical temperature, stress, displacement and strain distributions as well as the heat flux distribution for different forms of kernel function. The results are displayed graphically at different positions of x as shown in Figs. 1–5. In these figures, it is noticed that the time-delay parameter has a significant effect on all the fields. Fig. 1, exhibits the space variation of the heat flux distribution. In this figure, solid line represents the solution obtained in the frame of dynamic coupled theory (Biot theory, 0 ) and the other lines represent the solutions obtained in the new case. We notice from this figure that, at t 0.1 and 0.09, the solution to heat flux considered vanishes identically outside a bounded region of space surrounding the heat source at a distance from it equal to x*(t), say x*(t) is a particular value of x depending only on the choice of t and is the location of the wave front. This demonstrates clearly the difference between the solution corresponding to the classical use of the Fourier heat equation (Biot model, 0.0 ) and to the use of generalized case ( 0 ). In the first and older theory, the waves propagate with infinite speeds, so the value of any of the functions is not identically zero (though it may be very small) for any large value of y. In the non- Fourier theory, the response to the thermal and mechanical effects does not reach infinity instantaneously but remains in a bounded region. Fig. 2 indicates the variation in temperature distribution for a kernel function K (t , ) 1, at different values of time-delay 0.0009, 0.009, 0.09 . We noticed that the temperature fields has been affected by the time-delay , where the increasing of the value of the parameter causes decreasing in temperature fields. The thermal waves are continuous functions, smooth and reach to steady state depending on the value of time-delay , which means that the particles transport the
14 heat to the other particles easily and this makes the decreasing rate of the temperature greater than the other ones. Also, we learn from these figures that the thermal waves cut x-axis more rapidly when increases. Fig.
3
display
the
stress
function K (t , ) 1 (t ) /
distribution at
with
different
distance values
for of
a
kernel
time-delay
0.0009, 0.009, 0.09 . We observe that the stress field has the same behavior as the temperature at and the absolute value of the maximum stress decreases. Figs. 4 and 5 show the variation of displacement and strain distributions for two different kernel functions namely, K(t, ξ) = [1- (t- ξ)/ω]² and K(t, ξ) =1- (t- ξ), respectively, at different values of . It was found that the change of time-delay has a small effect on the displacement and strain.
8. Conclusion ●
The main goal of this work is to introduced a generalized model for Fourier law of heat conduction with time-delay and kernel function by using the definition for reflecting the memory effect (instantaneous change rate depends on the past state).
●
According to this new theory, we have to construct a new classification for materials according to their , time-delay where this parameter becomes a new indicator of its ability to conduct heat in conducting medium.
References [1] Tschoegl N. Time dependence in material properties: An overview. Mech Time-Depend Mater 1997; 1: 3–31. [2] Gross B. Mathematical Structure of the Theories of Viscoelasticity. Hemann, Paris; 1953. [3] Atkinson C, Craster R. Theoretical aspects of fracture mechanics. Prog Aerospace Sci 1995; 31: 1–83. [4] Rajagopal K, Saccomandi G. On the dynamics of non-linear viscoelastic solids with material moduli that depend upon pressure. Int J Eng Sci 2007; 45: 41– 54. [5] Biot M. Thermoelasticity and irreversible thermodynamics. J Appl Phys 1956; 27: 240–253.
15 [6] Cattaneo C. Sur une forme de l’équation de la Chaleur éliminant le paradoxe d’une propagation instantaneée. CR Acad Sci 1958, 2; 47: 431-433. [7] Ignaczak J. Generalized thermoelasticity and its applications. In: Hetnarski R. Thermal Stresses III Elsevier, New York; 1989: 279-354. [8] Chandraskharaiah D. Hyperbolic thermoelasticity, a review of recent Literature. Appl Mech Rev 1998; 51: 705–729. [9] Hetnarski R, Ignaczak J. Noncalssical dynamical thermoelasticity. Int J Solid Struct 2000; 37: 215-224. [10] Ezzat MA, El-Karamany AS. On uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation. Can J Phys 2003; 81: 823–833. [11] Ezzat MA, El-Karamany AS. The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 2002; 40: 1275–1284. [12] El-Karamany AS, Ezzat MA. On the boundary integral formulation of thermo – viscoelasticity theory. Int J Eng Sci 2002; 40: 1943–1956. [13] El-Karamany AS, Ezzat MA. Boundary integral equation formulation for the generalized thermoviscoelasticity with two relaxation times. J Appl Math Comput 2004; 151: 347–362 [14] Ezzat M. Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region. Int J Eng Sci 2004; 42: 1503–1519. [15] Ezzat MA, El-Karamany AS, Samaan A. State space approach to generalized thermo-viscoelasticity with thermal relaxation. J Therm Stress 2001; 24: 823– 846. [16] Ezzat MA. State space approach to solids and fluids. Can J Phys Rev 2008; 86: 1241-1250. [17] Ezzat MA. The relaxation effects of the volume properties of electrically conducting viscoelastic material. Mater Sci Eng B 2006; 130: 11–23. [18] Povstenko Y. Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech Res Commun 2010; 37: 436–440. [19] Sherief HH, El-Said A, Abd El-Latief A. Fractional order theory of thermoelasticity. Int J Solid Struct 2010; 47: 269-275.
16 [20] Ezzat MA. Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B 2010; 405: 4188-4194. [21] Ezzat MA, El-Karamany AS. Fractional order heat conduction law in magnetothermoelasticity involving two temperatures. ZAMP 2011; 62: 937-952. [22] Ezzat MA, El-Karamany AS. Theory of fractional order in electrothermoelasticity. Euro J Mech A/Sol 2011; 30: 491-500. [23] Ezzat MA, El-Karamany AS. Fractional thermoelectric viscoelastic materials. Appl Polym Sci 2012; 124: 2187–2199. [24] Diethelm K. Analysis of Fractional Differential Equation: An ApplicationOriented Exposition Using Differential Operators of Caputo Type. SpringerVerlag. Berlin, Heideberg, 2010. [25] Caputo M. Linear models of dissipation whose Q is almost frequency independent II. Geophys. J R Astron Soc 1967; 13: 529-539. [26] Wang J, Li H., Surpassing the fractional derivative: Concept of the memory – dependent derivative. Comp Math Appli 2011; 62: 1562-1567. [27] Sherief, H. H., El-Sayed, A. M. A., & Abd El-Latief, A. M. (2010). Fractional order theory of thermoelasticity. Intl J Solids and Structures, 47(2) 269-275 [28] Yu, Y.-J., Hu, W., Tian, X.-G. (2014): A novel generalized thermoelasticity model based on memory-dependent derivative, Int J Eng Sci 2014; 811:23-134. [29] Lord, H., & Shulman, Y., A generalized dynamic theory of thermoelasticity. J Mech Phys Solids 1967; 15: 299-309. [30] Honig G, Hirdes U. A method for the numerical inversion of the Laplace transform. J Comp Appl Math 1984; 10:113-132. [31] Sherief HH, Abd El-Latief A. Effect of variable thermal conductivity on a halfspace under the fractional order theory of thermoelasticity. Int J Mech Sci 2013; 74: 185-189.
17 Table 1
1.2 103 kg / m3
k 0.55 J / m.sec .K
E 525x 107 N / m 2
CE 1.4x103 J / kg.K
453.7 107 N / m2
194 107 N / m2
210 104 N / m2 K
0 3.36 106 sec/ m2 0.35
Co 2200 m / sec
0.12
T 13 105
18
Highlights •
We derive the heat conduction equation with time-delay in thermoelasticity theory.
•
The new model of thermo-viscoelasticity is applied to one- dimensional problem of a half-space.
•
The memory-dependent derivatives are better than the fractional one for reflecting the memory effect.
Generalized thermo-viscoelasticity memory–dependent derivatives
with
M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary
www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(14)00342-7 http://dx.doi.org/10.1016/j.ijmecsci.2014.10.006 MS2836
To appear in:
International Journal of Mechanical Sciences
Cite this article as: M.A. Ezzat, A.S. El-Karamany, A.A. El-Bary, Generalized thermo-viscoelasticity with memory–dependent derivatives, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2014.10.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Generalized thermo-viscoelasticity with memory–dependent derivatives 3
M. A. Ezzat 1,*, A. S. El-Karamany2, A. A. El-Bary 1
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
2
Department of Mathematical and Physical Sciences, Nizwa University, Nizwa -611, P. O. Box 1357, OMAN, [email protected] 3
Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt
*
Corresponding Author: [email protected]
ABSTRACT A new generalized thermo-viscoelasticity theory with memory–dependent derivatives is constructed. The governing coupled equations with time-delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to one- dimensional problem of a half-space. The bounding surface is taken traction free and subjected to a time dependent thermal shock. Laplace transforms technique is used to obtain the general solution in a closed form. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory are given. The predictions of the theory are discussed and compared with dynamic classical coupled theory.
Keywords:
Fourier's
Law;
thermo-viscoelasticity
theory;
Memory-dependent
derivative; Fractional calculus; Time-delay; Kernel function; Laplace transforms; Numerical results.
Nomenclature
,
Lame' constants
mass density
t
time
CE
specific heat at constant strains
Ko
= (2 / 3) , bulk modulus
Cο2
=
ε ij
components of strain tensor
K
, longitudinal wave speed
2 e ij
components of strain deviator tensor
ij
components of stress tensor
S ij
components of stress deviator tensor
e
= ii , Dilatation
thermal conductivity
T
absolute temperature
ui
components of displacement vector
T
coefficient of linear thermal expansion
3K T
ij
Kronecker’s delta.
To
reference temperature
Co
= ( 2 )/ 1/2 , speed of propagation of isothermal elastic waves
o
= CE /
2T 0 = , Thermal coupling parameter k o C 02
T T 0 , such that / T0 1,
Q
strength of applied heat source per unit mass
Γ(.)
Gamma function
1. Introduction The linear theory of elasticity is of paramount importance in the stress analysis of steel, which is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e.g., concrete, wood, and coal. However, the theory does not apply to the behavior of many of the new synthetic materials of the elastomer and polymer type, e.g., polymethyl-methacrylate (Perspex), polyethylene, and polyvinyl chloride. Linear viscoelastic materials are rheological materials that exhibit time temperature rate-of-loading dependence. When their response is not only a function of the current input, but also of the current and past input history, the characterization of the viscoelastic response can be expressed using the convolution (hereditary) integral.
3 Tschoegl [1] has presented a general overview of time-dependent material properties. Gross investigated the mechanical-model representation of linear viscoelastic behavior results [2]. One can refer to Atkinson and Craster [3] for a review of fracture mechanics and generalizations to the viscoelastic materials, and Rajagopal and Saccomandi [4] for non-linear theory. Physical observations and results of the conventional coupled dynamic thermoelasticity theories involving infinite speed of thermal signals, which were based on the mixed parabolic-hyperbolic governing equations of Biot [5] are mismatched. To remove this paradox, the conventional theories of thermoelasticity have been generalized, where the generalization is in the sense that these theories involve a hyperbolic-type heat transport equation supported by experiments which exhibit the actual occurrence of wave type heat transport in solids, called second sound effect. The first is due to Cattaneo [6] who obtained a wave-type heat equation by postulating a new law of heat conduction to replace the classical Fourier law. Several generalizations to the coupled theory are introduced. One can refer to Ignaczak [7] and Chandrasekharaiah [8] for a review. Hetnarski and Ignaczak [9] described the modern approaches to the analytical treatment of dynamical thermoelasticity. Within the theoretical contributions to thrmo-viscoelasticity theory are the proofs of uniqueness theorems under different conditions by Ezzat and El Karamany [10, 11] and the boundary element formulation was done by El-Karamany and Ezzat [12, 13]. The fundamental solutions for the cylindrical region were obtained by Ezzat [14]. Ezzat at el. [15] solved some problems in thrmo-viscoelasticity with thermal relaxation by using state space approach [16]. Ezzat [17] investigated the relaxation effects on the volume properties of an electrically conducting viscoelastic material. In the last decade, considerable interest in fractional calculus has been stimulated by the applications in different areas of physics and engineering. Recently, some efforts have been done to modify the classical Fourier law of heat conduction by using the fractional calculus [18-23]. Diethelm [24] has developed Caputo [25] derivative to be: t
Da f (t ) K (t ) f ( m) ( ) d a
with
(1)
4
K (t )
(t ) m 1 . (m )
(2)
Where K (t ) is the kernel function and f ( m ) denotes the common m-order derivative, which has specific physical meaning. The memory-dependent derivative is defined in an integral form of a common derivative with a kernel function on a slipping interval. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory dependent differential equation has more expressive force. Wang and Li [26] introduced a memory-dependent derivative (MDD), the first order memory-dependent derivative of function f is simply defined in an integral form of a common derivative with a kernel function on a slipping interval, in the form D f ( x, t )
1
t
t
K (t ) f ( x, ) d
(3)
where is the time delay and K (t ) is the kernel function in which they can be chosen freely. Wang and Li indicated that the memory effect requires weight 0 K (t ) 1 for [t , t ) so that the magnitude of memory-dependent
derivative, derivative
D f ( x, t ) is usually smaller than that of the common partial
f ( x, t ) . The kernel form K (t ) can also be chosen freely, such as 1, t
( t ) t 1, 1 , where p 0.25,1, 2 , etc. which may be more practical. They p
are a monotone function with K 0 for the past time t and K =1 for the present time t. In case, K (t ) 1 , we have
D f ( x, t )
1
t
f( x, ) d
t
f ( x , t ) f ( x, t )
This means that the common partial derivative So that, D f ( x, t )
f ( x, t ) . t
is the limit of D as 0 . t
f ( x, t ) f ( x, t ) f ( x, t ) lim 0 t
(4)
5 Sherief et al. [27] introduced the fractional order theory of thermoelasticity, in which the heat conduction equation was assumed to be the form qi
qi k,i t
Recently, an interesting application of memory-dependent derivative is given by Ya-Jun Yu et al [28]. They introduced the memory-dependent derivative (MDD) instead of fractional calculus, into the rate of heat flux in Lord-Shulman generalized thermoelasticity theory [29], to denote memory-dependence, as, q q k , i i ,i
(5)
Equation (5) has more clear physical meaning. In the current work, a modified law of heat conduction including both the heat flux and its memory-dependent derivative replaces the conventional Fourier's law in thermo-viscoelasticity. The resulting non-dimensional coupled equations
of
generalized thermo-viscoelasticity with memory-dependent derivative together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to thermal shock and traction free surface. A direct approach is introduced to obtain the solutions in the Laplace transform domain for different forms of kernel functions. The inversion of Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier expansion techniques proposed by Honig and Hirdes [30]. In this theory, the coupled thermo-viscoelastic model is used. This implies infinite speeds of propagation of thermo-viscoelastic waves. The solutions are represented graphically for different values of time-delay and different forms of kernel function.
2. Differentiation of memory dependent derivative Let n N , I :[a, b] and f : I R be such that f and all its derivatives up to f ( n ) are continuous on I , and f ( n 1) exists on (a, b) . Then,
Dn f ( x, t )
1
t
K (t ) f
t
( n)
( x, ) d
(6)
3. Derivation of heat conduction equation with time-delay in thermoelasticity. The classical Fourier’s law, in which relates the heat flux vector q to the temperature gradient q( x, t ) . T ( x, t )
(7)
6 The energy equation in terms of the heat conduction vector q in the context of thermoelasticity theory is given by [5] CE T ( x, t ) To e( x, t ) . q( x, t ) + Q( x, t ) . t
(8)
Using relation (5), we get the generalized heat conduction law for the considered new generalized theory with time-delay q( x, t ) q( x, t ) + D q( x, t )
(9)
From a mathematical viewpoint, Fourier law (7) in the theory of generalized heat conduction with time-delay, is given by q( x, t ) D q( x, t ) T ( x, t )
(10)
Taking the memory-time derivative of Eq. (8) (suppressing x for convenience), we get D CET To e . D q Dw Q t
(11)
Multiplying Eq. (11) by and adding to Eq. (8), we obtain
1 D CE
e T To . q D q 1 D Q t t
(12)
Substituting from Eq. (10), we get
1 D CE
e T To 2T 1 D Q t t
(13)
Eq. (13) is the generalized energy equation with memory-dependent derivative, taking into account the time- delay . The dynamic coupled theory of heat conduction law follows as the limit case when 0 . This model is more intuitionistic for understanding the physical meaning and the corresponding memory dependent differential equation is more expressive.
4. The formulation of the physical problem The governing equations for generalized thermo-viscoelasticity consist of: 1- The equation of motion in the absence of body forces
2ui ji , j . t 2
2-The constitutive equation [17]
(14)
7
eij ( x, τ)
t
Sij R(t τ)
τ
0
dτ R eij ,
(15)
where Sij ij
kk 3
ij ,
(16)
and R(t) is relaxation function given by t R(t ) 2 1 A* e * t t *1 dt , 0
(17)
where *, * and A* are non-dimensional empirical constants and ( *) is the Gamma function, 0 * 1, * 0, 0 A *
* d , R (t ) 0, R (t ) 0 . * ( ) dt
3- The kinematic relations 1 2
e 3
ij (ui , j u j ,i ), eij ij ij , e kk .
(18)
4- The stress-strain temperature relation:
Ko e 3T T To
(19)
where
kk 3
, ij ji
Substituting from (25) into (21) we obtain
e
ij R ij ij K o e ij ij , 3 where T To and
(20)
1 . To 5- The energy equation with memory-dependent derivative in the absence of heat
sources
1 D CE
e To 2 t t
(21)
6- The strain –displacement relations e ij
1 u i . j u j ,i . 2
(22)
In the above equations a comma denotes material derivatives and the summation convention are used.
8 We shall consider a viscoelastic solid occupying the region x 0 , where x-axis is taken perpendicular to the bounding plane of half-space pointing inwards. For the one-dimensional problems, all the considered functions will depend only on the space variables x and t. The displacement components
u x u ( x, t ), u y u z 0.
(23)
The strain component e xx
u . x
(24)
The equation of motion
2u 2 2u ( R K ) . o 2 2 t 3 x x
(25)
The energy equation
2 ( x, t ) ( x, t ) 2u ( x, t ) C T E o x 2 t t x 2 ( x, ) 3u ( x, ) K (t ) CE T d o 2 2 x t t
(26)
The constitutive equation 2 3
xx ( R Ko )
u . x
(27)
We assume that the boundary conditions have the form
(0, t ) f (t ), (, t ) 0, (0, t ) (, t ) 0, t 0
(28)
where f (t ) is a known function of t . The initial conditions are taken as u ( x, 0) u ( x, 0) ( x, 0) ( x, 0) ( x, 0) ( x, 0) 0, t 0.
Let us introduce the following non-dimensional variables, 1 2 , , qi qi , R R 2 3 Ko Co Co0 3K o The equations (25) –(27) in non-dimensional form yield: x Coo x, u Coou , t Co2ot ,
1 R xu tu x ,
(29)
( x, t ) 2 2u ( x, t ) 1 D , x 2 x t t
(30)
2
2
2
2
9
1 R
u , x
(31)
t 4 * * t *1 R(t ) dt . 1 A e t 3K o 0
(32)
From now on, the kernel function form K (t ) can be chosen freely as: 1 if m n 0 (t ) 1 1 if , m 0, n 2 2 2n m (t ) 2 K (t ) 1 (t ) 2 1 (t ) if m 0 n / 2 t 2 ) if m n 1, (1
(33)
where m and n are constants.
5. Solution in the Laplace transform domain Appling the Laplace transform with parameter s defined by the formulas
L{g( x, t )} g( x, s) e st g( x, t ) dt , 0
on both sides of Eqs. (30)-(32), we get
D
s 2 u D ,
(34)
D 2 s (1 G ) ( Du ),
(35)
2
1 u , x
(36)
and the boundary conditions (28) become
(0, s) f ( s), (0, s) 0 , where D
2u 2u 4 , L R 2 sR ( s ) 2 , 1/ 1 sR , R ( s) x x x 3 sK o
G ( s) (1 e s )(1
2b 2a 2 2a 2 s 2 2 ) (a 2 2b )e , s s s
(37)
A * ( *) , 1 * ( s *)
10 [(1 e s )], m n 0 [1 1 (1 e s )], m 0, n 1 s 2 L{ D f (t )} F ( s ) 1 , s s s [(1 e ) (1 e ) e ], m 0, n s 2 2 2 s [(1 s ) 2 s 2 (1 e )], m n 1 F ( s) L{(
2u )} s( Du ) . t xt
Eliminating u between Eqs. (34) and (35), we obtain
D
4
s 2 s (1 G)(1 ) D 2 s 3 (1 G) 0 .
(38)
In a similar manner, we can show that u satisfies the equation
D
4
s 2 s (1 G)(1 ) D 2 s 3 (1 G ) u 0 .
(39)
The solutions of Eqs. (38) and (39) which are bounded for x≥0 have the form
( x, s ) C1 e k x C2 e k x ,
(40)
u ( x, s ) C3 e k1x C4 e k2 x ,
(41)
1
2
where k1 and k 2 are the roots with positive real parts of the characteristic equation k 4 s 2 s (1 G)(1 ) k 2 s3 (1 G) 0,
(42)
satisfying the relations
k12 k22 s 2 s (1 G)(1 )
(43)
k12 k22 s 3 (1 G )
and Ci , i 1, 2,3, 4 are parameters depending on s to be determined from the boundary conditions of the problem. Substitution from Eqs. (40) and (41) into Eq. (34), we obtain the following relations C3
k1
s k 2
2 1
C1 , C4
k2
s 2 k22
C2 .
(44)
Substitution from Eq. (44) into Eq. (41), we have u ( x, s )
k1
s k 2
2 1
C1e k1 x
k2
s k 2
2 2
C2 e k 2 x .
Substitution from Eqs. (40) and (45) into Eq. (36), we have
(45)
11
C1 C e k1x 2 2 2 e k2 x . 2 k2 s k s
( x, s) s 2
(46)
2 1
In order to determine the C1 and C2 , we shall use the boundary conditions (37), we obtain
C1
k12 s 2 k22 s 2 f ( s ), C f ( s) . 2 k12 k22 k12 k22
(47)
Plugging Eq. (47) into Eqs. (40), (45) and (46), the final mathematical expressions for the dimensionless form of strain, conductive temperature and stress fields can be deduced as
( x, s )
1 (k12 s 2 ) e k1x (k22 s 2 )e k2 x f ( s ) , k k22
u ( x, s )
( x, s)
(48)
2 1
k e
k k
2 2
s2
e
2 1
k12 k22
k1 x
1
k1 x
k 2 e k2 x ,
(49)
e k2 x .
(50)
From Eq. (24) and (49), one can obtain the strain distribution as e ( x, s )
k k22 2 1
k
2 1
e k1 x k 22 e k2 x .
(51)
In order to obtain the non-dimensional heat flux component, we shall use the Laplace transform of Eq. (10), we have q ( x, s )
1 ( x, s) 1 G x
1 k k12 s 2 e k1x k2 k22 s 2 e k2 x f ( s). 2 2 1 1 G k1 k2
(52)
This completes the solution in the Laplace transform domain.
6. Inversion of the Laplace transforms We shall now outline the method used to invert the Laplace transforms in the above equations. Let f ( s ) be the Laplace transform of a function f(t). The inversion formula for Laplace transforms can be written as Honig and Hirdes [30]: f (t )
edt 2
e
ity
f (d iy ) dy ,
12 where d is an arbitrary real number greater than all the real parts of the singularities of f ( s ) . Expanding the function h(t ) exp(dt ) f (t ) in a Fourier series in the interval [0, 2L], we obtain the approximate formula: f (t ) f N (t )
N 1 c0 ck , for 0 t 2L 2 k 1
(53)
where edt ck Re eik t / L f d ik / L . L
(54)
Two methods are used to reduce the total error. First, the ‘Korrektur’ method is used to reduce the discretization error. Next, the ε-algorithm is used to reduce the truncation error and therefore to accelerate convergence. The Korrektur-method uses the following formula to evaluate the function f (t ) f (t ) f NK (t ) f N (t ) e 2 dL f N (2 L t ) .
(55)
We shall now describe the ε-algorithm that is used to accelerate the convergence of m
the series in (53). Let N be an odd natural number and let sm ck , be the sequence k 1
of partial sums of (53).We define the ε-sequence by
0,m 0, 1,m sm , m 1, 2,3,... and n1,m n1,m1 1/ n,m1 n,m ,
n, m 1, 2,3,....
It can be shown from Honig and Hirdes [30], that the sequence 1,1 , 3,1 , ..., N ,1 ,... converges to f (t ) c0 / 2 faster than the sequence of partial sums.
7. Numerical results and discussion In this section, we aim to illustrate numerical results of the analytical expressions obtained in the previous section and elucidate the influence of time-delay on the behavior of the field quantities. In order to interpret the numerical computations, we consider
material
properties
of
a
Polymethyl
Methacrylate
(Plexiglas) material. Following the values of physical constants are shown in Table 1 [17]. The calculations were carried out for the function f (t), which represents a time dependent thermal shock [31]
13
t sin f (t ) a 0
0t a
or
otherwise
f ( s)
a 1 e as a2s2 2
.
Considering the above physical data, we have evaluated the numerical values of the field quantities with the help of a computer program developed by using MATLAB software. The accuracy maintained was 7 digits for the numerical program. The computations were performed for a value of time, namely t = 0.1 and for time-delay 0.0009, 0.009, 0.09 . The numerical technique outlined above was used to obtain the conductive temperature, thermo-dynamical temperature, stress, displacement and strain distributions as well as the heat flux distribution for different forms of kernel function. The results are displayed graphically at different positions of x as shown in Figs. 1–5. In these figures, it is noticed that the time-delay parameter has a significant effect on all the fields. Fig. 1, exhibits the space variation of the heat flux distribution. In this figure, solid line represents the solution obtained in the frame of dynamic coupled theory (Biot theory, 0 ) and the other lines represent the solutions obtained in the new case. We notice from this figure that, at t 0.1 and 0.09, the solution to heat flux considered vanishes identically outside a bounded region of space surrounding the heat source at a distance from it equal to x*(t), say x*(t) is a particular value of x depending only on the choice of t and is the location of the wave front. This demonstrates clearly the difference between the solution corresponding to the classical use of the Fourier heat equation (Biot model, 0.0 ) and to the use of generalized case ( 0 ). In the first and older theory, the waves propagate with infinite speeds, so the value of any of the functions is not identically zero (though it may be very small) for any large value of y. In the non- Fourier theory, the response to the thermal and mechanical effects does not reach infinity instantaneously but remains in a bounded region. Fig. 2 indicates the variation in temperature distribution for a kernel function K (t , ) 1, at different values of time-delay 0.0009, 0.009, 0.09 . We noticed that the temperature fields has been affected by the time-delay , where the increasing of the value of the parameter causes decreasing in temperature fields. The thermal waves are continuous functions, smooth and reach to steady state depending on the value of time-delay , which means that the particles transport the
14 heat to the other particles easily and this makes the decreasing rate of the temperature greater than the other ones. Also, we learn from these figures that the thermal waves cut x-axis more rapidly when increases. Fig.
3
display
the
stress
function K (t , ) 1 (t ) /
distribution at
with
different
distance values
for of
a
kernel
time-delay
0.0009, 0.009, 0.09 . We observe that the stress field has the same behavior as the temperature at and the absolute value of the maximum stress decreases. Figs. 4 and 5 show the variation of displacement and strain distributions for two different kernel functions namely, K(t, ξ) = [1- (t- ξ)/ω]² and K(t, ξ) =1- (t- ξ), respectively, at different values of . It was found that the change of time-delay has a small effect on the displacement and strain.
8. Conclusion ●
The main goal of this work is to introduced a generalized model for Fourier law of heat conduction with time-delay and kernel function by using the definition for reflecting the memory effect (instantaneous change rate depends on the past state).
●
According to this new theory, we have to construct a new classification for materials according to their , time-delay where this parameter becomes a new indicator of its ability to conduct heat in conducting medium.
References [1] Tschoegl N. Time dependence in material properties: An overview. Mech Time-Depend Mater 1997; 1: 3–31. [2] Gross B. Mathematical Structure of the Theories of Viscoelasticity. Hemann, Paris; 1953. [3] Atkinson C, Craster R. Theoretical aspects of fracture mechanics. Prog Aerospace Sci 1995; 31: 1–83. [4] Rajagopal K, Saccomandi G. On the dynamics of non-linear viscoelastic solids with material moduli that depend upon pressure. Int J Eng Sci 2007; 45: 41– 54. [5] Biot M. Thermoelasticity and irreversible thermodynamics. J Appl Phys 1956; 27: 240–253.
15 [6] Cattaneo C. Sur une forme de l’équation de la Chaleur éliminant le paradoxe d’une propagation instantaneée. CR Acad Sci 1958, 2; 47: 431-433. [7] Ignaczak J. Generalized thermoelasticity and its applications. In: Hetnarski R. Thermal Stresses III Elsevier, New York; 1989: 279-354. [8] Chandraskharaiah D. Hyperbolic thermoelasticity, a review of recent Literature. Appl Mech Rev 1998; 51: 705–729. [9] Hetnarski R, Ignaczak J. Noncalssical dynamical thermoelasticity. Int J Solid Struct 2000; 37: 215-224. [10] Ezzat MA, El-Karamany AS. On uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation. Can J Phys 2003; 81: 823–833. [11] Ezzat MA, El-Karamany AS. The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 2002; 40: 1275–1284. [12] El-Karamany AS, Ezzat MA. On the boundary integral formulation of thermo – viscoelasticity theory. Int J Eng Sci 2002; 40: 1943–1956. [13] El-Karamany AS, Ezzat MA. Boundary integral equation formulation for the generalized thermoviscoelasticity with two relaxation times. J Appl Math Comput 2004; 151: 347–362 [14] Ezzat M. Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region. Int J Eng Sci 2004; 42: 1503–1519. [15] Ezzat MA, El-Karamany AS, Samaan A. State space approach to generalized thermo-viscoelasticity with thermal relaxation. J Therm Stress 2001; 24: 823– 846. [16] Ezzat MA. State space approach to solids and fluids. Can J Phys Rev 2008; 86: 1241-1250. [17] Ezzat MA. The relaxation effects of the volume properties of electrically conducting viscoelastic material. Mater Sci Eng B 2006; 130: 11–23. [18] Povstenko Y. Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech Res Commun 2010; 37: 436–440. [19] Sherief HH, El-Said A, Abd El-Latief A. Fractional order theory of thermoelasticity. Int J Solid Struct 2010; 47: 269-275.
16 [20] Ezzat MA. Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B 2010; 405: 4188-4194. [21] Ezzat MA, El-Karamany AS. Fractional order heat conduction law in magnetothermoelasticity involving two temperatures. ZAMP 2011; 62: 937-952. [22] Ezzat MA, El-Karamany AS. Theory of fractional order in electrothermoelasticity. Euro J Mech A/Sol 2011; 30: 491-500. [23] Ezzat MA, El-Karamany AS. Fractional thermoelectric viscoelastic materials. Appl Polym Sci 2012; 124: 2187–2199. [24] Diethelm K. Analysis of Fractional Differential Equation: An ApplicationOriented Exposition Using Differential Operators of Caputo Type. SpringerVerlag. Berlin, Heideberg, 2010. [25] Caputo M. Linear models of dissipation whose Q is almost frequency independent II. Geophys. J R Astron Soc 1967; 13: 529-539. [26] Wang J, Li H., Surpassing the fractional derivative: Concept of the memory – dependent derivative. Comp Math Appli 2011; 62: 1562-1567. [27] Sherief, H. H., El-Sayed, A. M. A., & Abd El-Latief, A. M. (2010). Fractional order theory of thermoelasticity. Intl J Solids and Structures, 47(2) 269-275 [28] Yu, Y.-J., Hu, W., Tian, X.-G. (2014): A novel generalized thermoelasticity model based on memory-dependent derivative, Int J Eng Sci 2014; 811:23-134. [29] Lord, H., & Shulman, Y., A generalized dynamic theory of thermoelasticity. J Mech Phys Solids 1967; 15: 299-309. [30] Honig G, Hirdes U. A method for the numerical inversion of the Laplace transform. J Comp Appl Math 1984; 10:113-132. [31] Sherief HH, Abd El-Latief A. Effect of variable thermal conductivity on a halfspace under the fractional order theory of thermoelasticity. Int J Mech Sci 2013; 74: 185-189.
17 Table 1
1.2 103 kg / m3
k 0.55 J / m.sec .K
E 525x 107 N / m 2
CE 1.4x103 J / kg.K
453.7 107 N / m2
194 107 N / m2
210 104 N / m2 K
0 3.36 106 sec/ m2 0.35
Co 2200 m / sec
0.12
T 13 105
18
Highlights •
We derive the heat conduction equation with time-delay in thermoelasticity theory.
•
The new model of thermo-viscoelasticity is applied to one- dimensional problem of a half-space.
•
The memory-dependent derivatives are better than the fractional one for reflecting the memory effect.