Thermo-viscoelastic materials with fractional relaxation operators

Applied Mathematical Modelling xxx (2015) xxx–xxx

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Thermo-viscoelastic materials with fractional relaxation operators M.A. Ezzat a,⇑, A.S. El-Karamany b, A.A. El-Bary c a

Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt Department of Mathematical and Physical Sciences, Nizwa University, Nizwa -611, P.O. Box 1357, Oman c Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt b

a r t i c l e

i n f o

Article history: Received 3 December 2013 Received in revised form 21 February 2015 Accepted 10 March 2015 Available online xxxx Keywords: Viscoelastic material Fractional relaxation operators Laplace transforms Coupled theory Ramp-type heating Numerical results

a b s t r a c t The new model of linear thermo-viscoelasticity for isotropic media taking into consideration the rheological properties of the volume with fractional relaxation operators is given. The governing equations are taken in a unified system from which some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained. The resulting formulation is applied to several concrete problems, a thermal shock problem and a problem for a half-space subjected to ramp-type heating as well as a problem of a layer media. Laplace transform techniques are used. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effect of fractional relaxation operators and ramping parameter of heating with different theories of thermoelasticity. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction It is well known that the usual theory of heat conduction based on Fourier’s law predicts infinite heat propagation speed. It is also known that heat transmission at low temperature propagates by means of waves. These aspects have caused intense activity in the field of heat propagation. Extensive reviews on the second sound theories (hyperbolic heat conduction) are given in Chandrasekharaiah [1] or Hetnarski and Ignaczak [2]. Due to the recent large-scale development and utilization of polymers and composite materials, the linear-viscoelasticity remains an important area of research. 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. A general overview of timedependent material properties has been presented by Tschoegl [3]. The mechanical-model representation of linear viscoelastic behavior results was investigated by Gross [4]. One can refer to Atkinson and Craster [5] for a review of fracture mechanics and generalizations to the viscoelastic materials, and Rajagopal and Saccomandi [6], for non-linear theory.

⇑ Corresponding author. Present address: Department of Mathematics, Faculty of Science and Letter in Al Bukayriyyah, Al-Qassim University, Al-Qassim, Saudi Arabia. E-mail addresses: [email protected] (M.A. Ezzat), [email protected] (A.S. El-Karamany), [email protected] (A.A. El-Bary). http://dx.doi.org/10.1016/j.apm.2015.03.018 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.

Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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Nomenclature k; l CE K C 2o eij eij

rij Sij e E k R (t, b) T ui

aT c dij To

m

Lame’ constants specific heat at constant strains k þ ð2=3Þl; bulk modulus K q ; longitudinal wave speed components of strain tensor components of strain deviator tensor components of stress tensor components of stress deviator tensor eii, Dilatation modulus of elasticity thermal conductivity relaxation functions absolute temperature components of displacement vector coefficient of linear thermal expansion 3K aT Kronecker’s delta. reference temperature Poisson’s ratio

e

c2 T 0 kgo qC 20

H

T  T0, such that |H/T0|  1 strength of applied heat source per unit mass mass density the ratio of the shear viscosity to Young’s modulus two relaxation times fractional orders time Gamma function

Q

q s

so ; t a; b t

C(.)

; Thermal coupling parameter

Within the theoretical contributions to the generalized thermoviscoelasticity theory are the proofs of uniqueness theorems under different conditions by Ezzat and El Karamany [7,8] and the boundary element formulation was done by ElKaramany and Ezzat [9,10]. The model of the equation of generalized thermo-viscoelasticity, ignoring the relaxation effects of the volume, with one relaxation time and with two relaxation times are established by Ezzat et al. [11–13]. Ezzat [14] investigated the relaxation effects on the volume properties of an electrically conducting viscoelastic material. Recently, Ezzat et al. [15] established the equations of the linear theory of generalized thermo-viscoelasticity for an electrically conducting isotropic media permeated by a primary uniform magnetic field, taking into account the rheological properties of the volume. Fractional calculus has been used successfully to modify many existing models of physical processes. Caputo and Mainardi [16] and Caputo [17] found good agreement with experimental results when using fractional derivatives for description of viscoelastic materials and established the connection between fractional derivatives and the theory of linear viscoelasticity. Adolfsson et al. [18] constructed a newer fractional order model of viscoelasticity. Recently, Ezzat [19–21] established a new model of fractional heat conduction equation using the new Taylor series expansion of time-fractional order which developed by Jumarie [22]. Ezzat and El-Karamany [23] studied a problem of thermo-viscoelastic for a perfect conducting half-space in the context of fractional magneto-generalized thermoelasticity. Povistinko [24] obtained the fundamental solutions to time-fractional heat conduction equations in two joint half-lines. Ezzat et al. [25] derived a new fractional relaxation operator using the methodology of fractional calculus. Abbasat el. [26] solved thermoelastic interactions problems in anisotropic media in the context of the theory of fractional order generalized thermoelasticity. Sherief and Abd El-Latief applied the fractional order theory of thermoelasticity to a 1D thermal shock problem for a half-space [27]. A state-space method for the calculation of dynamic response of systems made of viscoelastic materials with exponential type relaxation kernels was introduced by Menon and Tang [28]. Extension of thermo-viscoelastic and magneto-thermo-viscoelastic problems in generalized theory are found to be present in the works of many researchers out of which Mukhopadhyay and Bera [29], Ezzat [30]and El-Karamany and Ezzat [31]. This article introduces a new model for the linear theory of generalized thermo-viscoelasticity, taking into consideration the fractional relaxation operators effects of the volume. The resulting formulation is applied to several concrete one-dimensional problems. The formulation is applied to the generalized thermoelasticity theories: Lord–Shulman [32], Green–Lindsay [33] and to the dynamic coupled theory, Biot [34]. Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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3

The Laplace transform and direct approach techniques are used. 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 [35]. Numerical results for the temperature; the stress and displacement distributions are given and illustrated graphically for given problems. 2. The mathematical model The system of governing equations of the generalized fractional linear thermo-viscoelasticity theory with two relaxation times consists of the following: (1) The constitutive equation Rabotnov [36]

Z

Sij ðx; tÞ ¼

t

Rb ðt  nÞ

0

rðx; tÞ ¼

rkk ðx; tÞ 3

¼

@ eij ðx; nÞ ^ b ðeij Þ; dn ¼ R @n

Z

t

Ra ðt  nÞ

0

ð1Þ

@ ^ ^ a ðeij  3aT TÞ; ^ nÞdn ¼ R ðeij  3aT TÞðx; @n

ð2Þ

where Rb(t) and Ra(t) are the relaxation modules functions such that R(1) > 0,

Sij ¼ rij  Rb ðt; bÞ ¼ Ra ðt; aÞ ¼

rkk 3

1 2

e 3

dij ;

eij ¼ ðui;j þ uj;i Þ; eij ¼ eij  dij ; e ¼ ekk ; x ¼ ðx1 ; x2 ; x3 Þ;

R1

ðt=sÞb ;

0 < b 6 1;

ð3Þ

ðt=sÞa ;

0 < a 6 1;

ð4Þ

Hðx; tÞ ¼ Tðx; tÞ  T 0 ;

ð5Þ

Cð1  bÞ R2

Cð1  aÞ

_ ðx; tÞ; ^ tÞ ¼ Hðx; tÞ þ t H Tðx; Hence,

^ b ðeij Þ; Sij ðx; tÞ ¼ R

0 < b 6 1;

ð6Þ

^ 0 < a 6 1; rðx; tÞ ¼ R^ a ðe  3aT TÞ;

ð7Þ

where

^ b ðf Þ ¼ R

^ a ðf Þ ¼ R

R1 sb Cð1  bÞ R2 sa Cð1  aÞ

Z

t

ðt  nÞb

@ f ðx; nÞ dn; @n

ðt  nÞa

@ f ðx; nÞ dn; @n

0

Z

t

0

0 < b 6 1;

0 < a 6 1;

(2) The stress–strain temperature relation:  e 

^ dij : rij ¼ R^ b eij  dij þ R^ a ðe  3aT TÞ 3

ð8Þ

ð9Þ

ð10Þ

(3) The equation of motion

q

  @ 2 ui ^ 1 2 1 ^ a ðe;i  cT^ ;i Þ: þR R ¼ r u þ e b i ;i 2 6 @t 2

ð11Þ

(4) The heat equation

kH;ii ¼ q C E

! ! @H @2H @e @2e ^ þ so 2 þ 3aT T o Ra þ n0 s0 2 : @t @t @t @t

ð12Þ

Now, we shall consider a homogeneous isotropic thermo-viscoelastic solid occupying half-space x P 0, which is initially quiescent and where all the state functions depend only on the dimension x and the time t. The displacement vector has components

ux ¼ uðx; tÞ;

uy ¼ uz ¼ 0:

ð13Þ

The strain component takes the form:

e ¼ exx ¼

@u : @x

ð14Þ

Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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The equation of motion takes the form

q

! ! @2u 2 ^ @2u @2u @ T^ ^ R R ¼  c þ ; b a @x @x2 @x2 @t 2 3

0 < b 6 1; 0 < a 6 1:

ð15Þ

The constitutive equation yield:

2 3

rxx ¼ R^ b

    @u ^ a @ u  3aT T^ þR @x @x

0 < b 6 1;

0 < a 6 1:

ð16Þ

The heat conduction equation is given by:

! ! @2H @H @2H @2u @3u ^ k 2 ¼ qC E þ so 2 þ 3aT T o Ra þ n 0 s0 ; @x@t @x @t @x@t 2 @t

0 < a 6 1:

ð17Þ

The previous equations constitute a complete system of generalized thermo-viscoelasticity with two relaxation times and fractional relaxation operators in the absence of heat sources. In the above equations a comma denotes material derivatives. The summation notation is used. Let us introduce the following non-dimensional variables:

x ¼ C o go x;

H ¼

t  ¼ C 2o go t;

u ¼ C o go u;

so ¼ C 2o go so ; t ¼ C 2o go t; e ¼

c2 T 0 ; kqC 20 g0

cH K qC E rxx 2 R2 ; C 2o ¼ ; go ¼ ; rxx ¼ ; R1 ¼ R1 ; R2 ¼ : 2 3K q k K K q Co

In terms of these non-dimensional variables, we have (dropping asterisks for convenience):

@2H ¼ @x2

! ! Z t @H @2H eR2 sa @ 2 uðx; nÞ @ 3 uðx; nÞ a @ dn; þ so 2 þ ðt  nÞ þ n0 s0 @n @x@n @t Cð1  aÞ 0 @t @x@n2

@2u R2 sa ¼ 2 C ð1  aÞ @t

rxx ¼

R2 sa Cð1  aÞ

Z

t

ðt  nÞ

a

0

Z

t 0

ðt  nÞa

ð18Þ

! ! Z t ^ nÞ @ @ 2 uðx; nÞ @ Tðx; R1 sb @ 2 uðx; nÞ b @ dn þ dn;  ðt  nÞ @x @n @x2 @n @ x2 Cð1  bÞ 0

ð19Þ

    Z t @ @uðx; nÞ ^ R1 sb @ @uðx; nÞ dn:  Tðx; nÞ dn þ ðt  nÞb @n @x @n @x Cð1  bÞ 0

ð20Þ

Limiting cases (i) In the theory of coupled thermoelasticity 1- Eqs. (18)–(20) in the limiting case n0 = 0, b ¼ 0;

so ¼ t ¼ 0; R1 ¼ 2l; R2 ¼ K transforms to the works of Biot [34].

(ii) In the theory of coupled thermo-viscoelasticity 1- Eqs. (18)–(20) in the limiting case b ¼ 0; a ¼ 0; and Pobedria [38].

so ¼ t ¼ 0; n0 = 0 we obtain the works of Ilioushin and Pobedria [37]

(iii) In the theory of generalized thermoelasticity with two relaxation times Eqs. (18)–(20) in the limiting case b ¼ 0; a ¼ 0;

so > 0; t > 0; R1 ¼ 2l; n0 = 0, R2 = K the work of Ezzat [39] results.

(iv) In the theory of generalized thermo-viscoelasticity with two relaxation times Eqs. (18)–(20) in the limiting case b ¼ 0; a ¼ 0;

so > 0; t > 0; R1 ¼ 2l; n0 = 0 yields the work of El-Karamany and Ezzat [40].

(v) In the theory of generalized thermoelasticity with one relaxation time Eqs. (18)–(20) in the limiting case b ¼ 0; a ¼ 0;

so > 0; t ¼ 0; n0 ¼ 1; lead to the works of Ezzat et al. [41,42].

(vi) In the theory of generalized thermo-viscoelasticity with fractional relaxation operator Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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Eqs. (18)–(20) in the limiting case b > 0; a ¼ 0;

so > 0; t ¼ 0; n0 ¼ 1; transforms to the works of Ezzat et al. [25].

3. The analytical solutions in the Laplace-transform domain Performing the Laplace transform defined by the relation

gðsÞ ¼

Z

1

est gðtÞ dt;

0

of both sides Eqs. (18)–(20), with the homogeneous initial conditions

_ ðx; 0Þ ¼ 0: _ 0Þ ¼ rðx; 0Þ ¼ r_ ðx; 0Þ ¼ Hðx; 0Þ ¼ H uðx; 0Þ ¼ uðx;  ¼ LfRðtÞg and omitting the bars, Eqs. (18)–(20) can be written in the form denoting RðsÞ

 ¼ -2 e sð1 þ n0 s0 sÞDu ; ðD2  s  so s2 ÞH

ð21Þ

;  ¼ -1 -2 ð1 þ tsÞD H ðD2  -1 s2 Þu

ð22Þ

1

r xx ¼ where

-1

;   -2 ð1 þ tsÞH Du

ð23Þ

a -b 1b 1 -1 ¼ R1 ðssÞb þR , the Laplace transform of the relaxation modulus can be a ; -2 ¼ R2 ðssÞ , since L{t } = C(1  b)/s 2 ðssÞ

written as

LfR1 ðt; bÞg ¼ R1 ðssÞb

  1 ; s

LfR2 ðt; aÞg ¼ R2 ðssÞa

  1 : s

ð24Þ

 between Eqs. (21) and (22), we obtain Eliminating H

n o    ¼ 0: D4  -1 s2 þ sð1 þ so sÞ þ e-1 -22 sð1 þ tsÞð1 þ no so sÞ D2 þ -1 s3 ð1 þ so sÞ u

ð25Þ

The general solution of Eq. (25) which is bounded at infinity can be written as

 ðx; sÞ ¼ k1 C 1 ek1 x  k2 C 2 ek2 x ; u

ð26Þ

where C1 and C2 are parameters depending on s only and k1 and k2 are the roots with positive real parts of the characteristic equation 4

k 





-1 s2 þ sð1 þ so sÞ þ e-1 -22 sð1 þ t sÞð1 þ no so sÞ k2 þ -1 s3 ð1 þ so sÞ ¼ 0:

ð27Þ

From Eqs. (21) and (26), we get

 ðx; sÞ ¼ H

h     i 1 2 2 C 1 k1  -1 s2 ek1 x þ C 2 k2  -1 s2 ek2 x : -1 -2 ð1 þ tsÞ

ð28Þ

Substituting from Eqs. (26) and (28) into Eq. (23), we obtain

r ðx; sÞ ¼ s2 ðC 1 ek1 x þ C 2 ek2 x Þ;

ð29Þ

4. Applications (i) Prescribed boundary conditions We shall consider a viscoelastic medium occupying a semi-infinite region x P 0: We assume that a thermal shock of the form:

 ð0; sÞ ¼ f ðsÞ; Hð0; tÞ ¼ f ðtÞ or H

ð30Þ

where f(t) is a known function of t, is applied to the boundary plane x = 0 at time t = 0, and the boundary plane x = 0 is taken to be traction free, i.e.

rð0; tÞ ¼ 0 or r ð0; sÞ ¼ 0:

ð31Þ

In order to determine C1, C2 we shall use the boundary conditions (30) and (31) to obtain

C 1 ¼ C 2 ¼

-1 -2 ð1 þ tsÞ  2

2

k1  k2

f ðsÞ;

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Eqs. (26), (28) and (29) become

 ðx; sÞ ¼  u h  ðx; sÞ ¼ H

r ðx; sÞ ¼

-1 -2 ð1 þ tsÞ  2 k1



2 k2

k1 ek1 x  k2 ek2 x f ðsÞ;

  i 2 2 ðk1  -1 s2 Þek1 x  k2  -1 s2 ek2 x 2

2

k1  k2 s2 -1 -2 ð1 þ tsÞ  2

2

k1  k2

ð32Þ

f ðsÞ;

ek1 x  ek2 x f ðsÞ:

ð33Þ

ð34Þ

(ii) A problem for a half-space subjected to ramp-type heating Consider a half-space homogeneous elastic medium occupying the region x P 0: We assume that a thermal shock of the form:

 ð0; sÞ ¼ gðsÞ: Hð0; tÞ ¼ gðtÞ or H

ð35Þ

The bounding plane x = 0 has a constant displacement, that is,

 0 ð0; sÞ ¼ 0: u0 ð0; tÞ ¼ 0 or u

ð36Þ

The parameters C1, C2 can obtain by using the boundary conditions (35) and (36), hence

C1 ¼

-2 k22 ð1 þ tsÞ  2 s2 ðk1



2 k2 Þ

g ðsÞ;

C2 ¼ 

-2 k21 ð1 þ tsÞ  2

2

s2 ðk1  k2 Þ

g ðsÞ:

Eqs. (26), (28) and (29) become

 ðx; sÞ ¼  u

k1 k2 -2 ð1 þ tsÞ  k1 x   k2 e  k1 ek2 x gðsÞ; 2 2 s2 k1  k2

h     i 2 2 2 2 k2 k1  -1 s2 ek1 x  k1 k2  -1 s2 ek2 x  ðx; sÞ ¼   H gðsÞ; -1 s2 k21  k22 h

r ðx; sÞ ¼

-2 ð1 þ tsÞ k22 ek1 x  k21 ek2 x 2

2

k1  k2

ð37Þ

ð38Þ

i gðsÞ:

ð39Þ

(iii) A problem for a layered medium Consider a layer of thickness X whose lower surface rests on a rigid base, while its upper surface is traction free. We choose the coordinate axes such that the upper plane lies at x = 0 and the x-axis pointing downwards. The mechanical boundary conditions can be written as

rð0; tÞ ¼ 0 or r ð0; sÞ ¼ 0;

ð40Þ

 ðX; sÞ ¼ 0: uðX; tÞ ¼ 0 or u

ð41Þ

The thermal boundary conditions are assumed to be

  ð0; sÞ ¼ hðsÞ; Hð0; tÞ ¼ hðtÞ or H

ð42Þ

ðX; sÞ ¼ 0; qðX; tÞ ¼ 0 or q

ð43Þ

where q denotes the component of the heat flux vector perpendicular to the surface of the layer. Condition (42) means that the upper surface is acted on by a constant thermal shock at time t = 0, while condition (43) signifies that the lower rigid surface is thermally insulated. This problem is somewhat similar to a one treated by Ezzat in [44] for generalized thermoelasticity theory. Using the Fourier’s law of heat conduction, which is valid for generalized thermoelasticity theory [45], Eq. (43) reduces to

 0 ðX; sÞ ¼ 0: H

ð44Þ

Eqs. (23), (40) and (42) can be combined to give

  0 ð0; sÞ ¼ -1 -2 ð1 þ tsÞhðsÞ u

ð45Þ

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The general solution of Eq. (25) for a bounded region is assumed to be

 ðx; sÞ ¼ A cosh k1 x þ B sinh k1 x þ C cosh k2 x þ D sinh k2 x; u

ð46Þ

where A, B, C and D are some parameters depending on s and X. From Eqs. (21) and (46), we get

 ðx; sÞ ¼ H

" # 2 2 2 2 1 Aðk1  -1 s2 Þ Bðk1  -1 s2 Þ Cðk2  -1 s2 Þ Dðk2  -1 s2 Þ sinh k1 x þ cosh k1 x þ sinh k2 x þ cosh k2 x : k1 k1 k2 k2 -1 -2 ð1 þ tsÞ ð47Þ

Using the boundary conditions (40)–(45) in Eqs. (46) and (47), the parameters A, B, C and D can be obtained as

A¼ C¼

-1 -2 k1 ð1 þ tsÞh 2 k1



2 k2

-1 -2 k2 ð1 þ tsÞ h 2

2

k1  k2

tanh k1 X;

tanh k2 X;

B¼

-1 -2 k1 ð1 þ tsÞh

D¼

2

2

k1  k2

;

-1 -2 k2 ð1 þ tsÞ h 2

2

k1  k2

ð48Þ :

Substituting from Eq. (48) into (46) and (47), we have

 ðx; sÞ ¼  u

 ðx; sÞ ¼ H

-1 -2 k1 ð1 þ tsÞ 2

1 2 k1

2

k1  k2



2 k2

 sinh k1 ðX  xÞ sinh k2 ðX  xÞ  hðsÞ; k1  k2 cosh k1 X cosh k2 X

ð49Þ

 cosh k1 ðX  xÞ cosh k2 ðX  xÞ  2 2 hðsÞ: ðk1  -1 s2 Þ  ðk2  -1 s2 Þ cosh k1 X cosh k2 X

ð50Þ

Substituting equations (49) and (50) into Eq. (23), one obtains

r ðx; sÞ ¼





-1 -2 s2 ð1 þ tsÞ cosh k1 ðX  xÞ cosh k2 ðX  xÞ  hðsÞ:  2

2

cosh k1 X

k1  k2

ð51Þ

cosh k2 X

5. Numerical results and discussion The method based on a Fourier series expansion proposed by Honig and Hirdes [35] and is developed in detail in many texts such as Ezzat [43] and Sherief and Abd El-Latief [27] is adopted to invert the Laplace transform in Eqs. (32)–(34), (37)–(39) and (49)–(51). The numerical code has been prepared using Fortran 77 programming language. The analysis is conducted for a Polymethyl Methacrylate (Plexiglas) material. Following the values of physical constants are shown in Table 1. The calculations were carried out for three cases of functions f (t), g(t) and h(t). Problem I: Thermal shock at the boundary [45]

( f ðtÞ ¼

 sin pat 0 6 t 6 a 0

paðs þ e Þ or f ðsÞ ¼ 2 2 : a s þ p2 as

otherwise

Problem II: A problem for a half-space subjected to ramp-type heating [46]

8 > <0 gðtÞ ¼ h1 tto > : h1

06 t 0 6 t 6 to t>0

or gðsÞ ¼

h1 ð1  esto Þ : t o s2

Problem III: A problem for a layered medium [47]

1 hðtÞ ¼ HðtÞ or f ðsÞ ¼ : s For each problem, we apply the following procedure:

Table 1 Values of the constants.

q ¼ 1:2  103 kg=m3

k ¼ 0:55 J=m s K

E = 525  107 N/m2

C E ¼ 1:4  103 J=kg K c = 210  104 N/m2 K e = 0.12

k ¼ 453:7  107 N=m2 g0 = 3.36  106 s/m2 m = 0.35

l = 194  107 N/m2 C o ¼ 2200 m=s aT = 13  105

Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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6 Coupled theory G-L theory L-S theory 5

Temperature, Θ

4

α = β = 0.0

3

α = β = 0.5

2

1

0 0

0.2

0.4

0.6

0.8

1

1.2

-1 Distance, x Fig. 1. Temperature distribution for t = 0.1, case I.

0.05 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.05 Coupled theory G-L theory L-S theory

-0.1

Stress, σ

-0.15 -0.2 -0.25

α = β = 0.0

-0.3 -0.35

α = β = 0.5

-0.4

-0.45 -0.5 Distance, x Fig. 2. Stress distribution for t = 0.1, case I.

Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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0.2 Coupled theory G-L theory L-S theory 0.1

Displacement, u

0 0

0.2

0.4

0.6

0.8

1

1.2

-0.1

-0.2

-0.3

-0.4 Distance, x Fig. 3. Displacement distribution for t = 0.1, case I.

0.4 Coupled theory G-L theory L-S theory

0.35

Temperature, Θ

0.3

0.25

0.2

0.15

0.1

0.05

0 0

0.1

0.2

0.3

0.4

0.5

0.6

-0.05 Distance, x Fig. 4. Temperature distribution for t = 0.1, case II.

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M.A. Ezzat et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx 0 0

0.2

0.4

0.6

0.8

-0.02

1

1.2

Coupled theory G-L theory L-S theory

Stress, σ

-0.04

-0.06

α = β = 0.0, t = 0.1 α = β = 0.5, t = 1.0

-0.08

-0.1

-0.12

-0.14

Distance, x Fig. 5. Stress distribution for t = 0.1, case II.

0.007 Coupled theory G-L theory L-S theory

0.006

Displacement, u

0.005

0.004

0.003

0.002

0.001

0 0.65

0.8

0.95

1.1

1.25

1.4

1.55

Distance, x Fig. 6. Displacement distribution for t = 0.1, case II.

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1.2 Coupled theory G-L theory L-S theory 1

0.8

α = β = 0.0

Temperature, Θ

α = β = 0.5 0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

-0.2 Distance, x Fig. 7. Temperature distribution for t = 0.1, case III.

0.15 Coupled theory G-L theory L-S theory

0.05

Stress, σ

-0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α = β = 0.0

-0.15

-0.25

α = β = 0.5

-0.35

-0.45

-0.55 Distance, x Fig. 8. Stress distribution for t = 0.1, case III.

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0.03 Coupled theory G-L theory L-S theory

0.02 0.01 0 Displacement, u

0

0.2

0.4

0.6

0.8

1

1.2

-0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 Distance, x Fig. 9. Displacement distribution for t = 0.1, case III.

The computations were carried out for one value of time, namely t = 0.1 and two values of a and b, namely, a = b = 0.0 and a = b = 0.5 (new theory). The temperature, stress and displacement distributions are obtained and plotted. Problem I is shown in Figs. 1–3 and problem II is shown in Figs. 4–6, while problem III is shown in Fig. 7–9. In these figures, solid lines represent the solution obtained in the frame of Biot theory and dashed lines represent Green–Lindsay theory, while dotted lines represent Lord–Shulman theory. For the different three cases, we observe the following: 1- In all Figs. it is noticed that the fractional order a and b have significant effect stress and displacement except temperature distribution. 2- For a = b = 0.0, the results coincide with all the previous results of applications that are taken in the context of the coupled, the generalized thermo-viscoelasticity with one relaxation time (L–S theory) [32] and the generalized thermo-viscoelasticity with two relaxation times (G–L theory) in on the various fields [33]. 3- For a = b = 0.5 the solution seems to behave like the generalized theories of thermo-viscoelasticity, this result is very important since the new theory may preserves the advantage of the generalized theory, i.e. the response to the thermal and mechanical effects does not reach infinity instantaneously but remains in the bounded region of space that expands with the passing of time. 4- In problem II, Figs. 4–6 exhibit the space variation of temperature, stress and displacement at different values of to for three models. We can see from Fig. 4 that, in larger value of to the speed of the wave propagation of the temperature vanishes in smaller value of x. It is notice that from Fig. 5 that the absolute value of the stress increases when the value of the parameter to decreases. One can see from Fig. 6 that the displacements increase when to decreases.

6. Conclusion Owing to the complicated nature of the governing equations for the generalized thermo-viscoelasticity with one or two relaxation times, few attempts have made to solve different problems in this field. These attempts utilized approximate method valid for only a specific range of some parameters [3]. In this work, a simply method is introduced in the field of generalized thermo-viscoelasticity with one and two relaxation times and applied to three different problems. This method gives exact solutions in the Laplace transform domain without any assumed restrictions on either the temperature or the displacement distributions. A numerical method based on a Fourier-series expansion has used for the inversion process [48]. Please cite this article in press as: M.A. Ezzat et al., Thermo-viscoelastic materials with fractional relaxation operators, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.03.018

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The method used in the present work is applicable to a wide range of thermoelasticity problems. It can be applied to problems, which are described by the linearized Navier–Stokes equations for thermoelectric fluid, were the governing equations are coupled [49]. Representative results for the all functions for generalized theory are distinctly different from those obtained for the coupled theory. This due to the fact that thermal waves in the coupled theory travel with an infinite speed of propagation as opposed to finite speed in the generalized case. It is clear that for small values of time the solution is localized in a finite region. This region grows with increasing time and its edge is the location of the wave front. This region is determined by the values of time t and relaxation times so and t. The predictions of the new theory are discussed and compared with dynamic classical coupled theory [50]. According to this new theory, we have to construct a new classification for viscoelastic materials according to their fractional parameters a and b where these parameters become new indicator of their ability to restrict stress and displacement in the same medium. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

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