conditions, is given. The constitutive equations are derived for the linear two -temperature

icity tle bo

equations of the linearized thermoelasticity. The temperature discrepancy aP 0 represents a singular perturbation onperature coupledature theork of Chak

[8] investigated the propagation of harmonic plane waves in a media described by the two-temperature theorytemperature Magneto-Viscoelasticity theory with one relaxation time in a Medium of Perfect Conductivity was estaby Ezzat and El-Karamany [9]. Theoretical aspects of two-temperature thermoelasticity theory such as existence, stconvergence and spatial behavior, were given by Quintanilla [10]. Banik and Kanoria [11] establish the solution of the

http://dx.doi.org/10.1016/j.apm.2014.10.0310307-904X/ 2014 Elsevier Inc. All rights reserved.

Corresponding author.E-mail addresses: aelkaramani@yahoo.com (A.S. El-Karamany), maezzat2000@yahoo.com (M.A. Ezzat).

Applied Mathematical Modelling 39 (2015) 21552171

Contents lists available at ScienceDirect

Applied Mathematical Modellingthe eld equations [4]. The uniqueness, reciprocity theorems and variational principle of the two-temthermoelasticity theory, for homogeneous isotropic solid were given by Iesan [5]. The two-tempergeneralized thermoelasticity theory has been obtained by Youssef [6]. Puri and Jordan [7], extending the wory forrabarti. Two-blishedability,and the thermodynamic temperature. It is assumed that the conductive temperature deviations u and its rst two gradientsare small. In addition, for linearized theory the displacement u, its rst three gradients and the velocity gradient are small.The linearized constitutive relations aremechanically simple in the sense that there is no dependence onrE andrrE, whereE the strain tensor [3].

The temperature deviations is given for an isotropic solid by the relationu h aDu, so the two temperatures coincide ifa 0 or D/ 0, in both cases the constitutive equations of the two-temperature thermoelasticity are just the classicalKeywords:Constitutive equations ofthermoviscoelasticityVariational principleUniqueness and reciprocal theoremsTwo-temperature theoryGreenNaghdi theory of type III

1. Introduction

The two-temperature themoelastheat conduction theory for deformabGreenNaghdi thermoviscoelasticity theory of type III, and the time-independence ofthe conductivity tensors kij and k

ij is proved. An application is given for isotropic

thermoviscoelastic solid and the results are presented graphically. The curves of the stressand temperature distributions are more uniform and the thermodynamic temperature issmaller in magnitude relative to the one-temperature case.

2014 Elsevier Inc. All rights reserved.

heory was formulated by Chen and Gurtin [1], Chen et al. [2,3]. They introduced adies, which depends upon two distinct temperatures, the conductive temperatureTwo-temperature GreenNaghdi theory of type III in linearthermoviscoelastic anisotropic solid

Ahmed S. El-Karamany a,, Magdy A. Ezzat baDepartment of Mathematical and Physical Sciences, Nizwa University, Nizwa -611, P.O. Box 1357, OmanbDepartment of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

a r t i c l e i n f o

Article history:Received 3 February 2013Received in revised form 11 July 2014Accepted 16 October 2014Available online 28 October 2014

a b s t r a c t

The reciprocal theorem is proved and the variational principle is established for the lineartwo-temperature GreenNaghdi theory of type III in an anisotropic and inhomogeneousthermoviscoelastic solid. A proof of a uniqueness theorem for thermoviscoelasticity, withoutrestrictions imposed on the relaxation or thermal conductivity tensors, except symmetry

journal homepage: www.elsevier .com/locate /apm

2156 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171Nomenclature

a; b non-negative constantsAij temperature discrepancy tensorbi mass forceGijkl relaxation tensorf i rjinj surface tractionf^ i prescribed surface tractionFi pseudo-mass forcehi entropy ux vectorh hinikij thermal conductivity tensorkij conductivity rate tensorni the outer unit vector normal to the surfaceQ intensity of applied heat source per unit volumeqi heat ux vectorproblem for the two-temperature thermoelasticity theory in context of the three-phase lag generalized heat conduction law.Two-temperature theory in linear micropolar thermoviscoelasticity was investigated by El-Karamany [12]. Kar and Kanoria[13] investigated the generalized three-phase-lag thermo-visco-elastic problem of a spherical shell.

The heat transport equations of the two-temperature and the classical thermoelasticity theories are of a mixed parabolic-hyperbolic type predicting innite speeds of propagation for thermal waves contrary to physical observations. The general-ized thermoelasticity theories in which the heat transport equation is hyperbolic do not suffer from this paradox. One canrefer to Ignaczak [14], to Chandrasekharaiah [15], Hetnarski and Ignaczak [16] for reviews of different generalized theories.

Many works were devoted to the generalized thermoelasticity theories e.g. [1719]. Green and Naghdi [2022] developeddifferent theories labeled type I, type II, and type III. The GreenNaghdi theory of type I in the linearized theory is equivalentto the classical coupled thermoelasticity theory. The GreenNaghdi thermoelasticity theory of type II does not admit energydissipation. The two-temperature thermoelasticity admits dissipation of energy and the theory of elasticity without energydissipation is valid only when the two-temperatures coincide [23]. The GreenNaghdi theory of type III admits dissipation ofenergy and the heat ux is a combination of type I and type II.

The linear viscoelasticity remains an important area of research not only due the advent and use of polymers, but alsobecause most solids when subjected to dynamic loading exhibit viscous effects. The stressstrain law for many materialssuch as Polycrystalline metals and high polymers can be approximated by the linear viscoelasticity theory. Many works were

q qiniq^ prescribed heat ux on boundaryS entropy per unit volumeT thermodynamic temperaturet timeT0 u0 > 0, reference temperature hj jT0 1U internal energy per unit volumeui components of displacement vectoru^i prescribed displacement on boundaryx positiona thermal displacement _a ua^ prescribed temperature displacementb relaxation functionbi a;i_bi u;icij thermo-elastic relaxation tensor.h T T0, thermodynamic temperature deviation.h^ prescribed temperature deviation from T0 u0n internal rate of production of entropy per unit volumeq mass densityrij components of stress tensorW free energy density/ conductive temperatureu /u0; uj ju0 1eij components of strain tensor2 internal energy per unit mass

and MTh

theore[28].

and Swere pdeni

to moand retional

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2157theorems and convolution variational principle. El-Karamany and Ezzat [38] established the constitutive laws for the three-phase -lag micropolar thermoelasticity theory. The uniqueness and reciprocal theorems are proved and a variational princi-ple is established for a linear micropolar anisotropic and inhomogeneous thermoelastic solid. A continuous dependenceresult is given for isotropic solid.

Ezzat et al. [39] constructed a mathematical model of two-temperature magneto-thermoelasticity where the fractionalorder dual-phase-lag heat conduction law is considered.

In this work, in the frame of the GreenNaghdi theory of type III, a reciprocal theorem is proved for the linear anisotropicand inhomogeneous thermoviscoelastic solid without the use of Laplace Transforms, and the Gurtin variational principle [29]is given for thermoviscoelastic solid. Based on the variational principle a uniqueness theorem is established withoutrestrictions on the relaxation tensors Gijkl; b, and cij or thermal conductivity tensors kij, and k

ij except symmetry conditions.

An isotropic thermoviscoelastic homogeneous semi-space xP 0 is considered, with quiescent initial state and thermal shockapplied to the traction-free boundary plane x 0. The exact solution in the Laplace transform domain is obtained using statespace approach. The thermodynamic temperature h, the conducive temperature u and the stress component r are obtainedby numerical inversion of Laplace transforms and presented graphically. In the Appendix, the constitutive equations arederived for the linear two -temperature GreenNaghdi thermoviscoelasticity theory of type III, and the time-independenceof the conductivity tensors kij and k

ij is proved.

2. The mathematical model

2.1. The assumptions of the model

(1) A linear thermoviscoelastic material that occupies a regular region V with a piecewise smooth boundary surface @Vin the three-dimensional Euclidian space is considered. The rectangular coordinate system x1; x2; x3 is employed.x is the position and t is the time, and all the variables in this paper are considered to be functions of x, and tdened in V V [ @V 0;1. A superposed dot denotes differentiation with respect to time, and the commafollowed by subscript denotes partial differentiation with respect to the space variables xi. The summation notationis used.

(2) We consider the usual linear theory in which the components of displacement vector and temperature deviationsfrom some reference temperature, their space and time derivatives are small.

2.2. The governing equations

The system of governing equations of the linear thermoviscoelasticity theory consists of(i) The continuity equation

dqdt

q _uj;j 0: 1

(ii) The kinematical relations

eij 12 ui;j uj;i inV 0;1: 2

(iii) The initial conditions in V are

uix;0 u0i x _uix; 0 v0i x: 3

ax;0 0 _ax;0 u0x; 4where the functions u0i ; v0i , and u0 are prescribed functions of x in Vdify the Cattaneo heat conduction law and in the context of the two-temperature thermoelasticity theory, uniquenessciprocal theorems are proved, the convolutional variational principle is given for fractional thermoelasticity. For frac-thermoelasticity not involving two temperatures, El-Karamany and Ezzat [37] established the uniqueness, reciprocalEzzat [36] introduced two general models for thermoelasticity theory, where the fractional derivatives and integrals are usedackman [31], Carlson [32], and Lebon [33]. Uniqueness theorems for linear viscoelasticity and thermoviscoelasticityroved by various authors, assuming positive deniteness or strong ellipticity of the relaxation tensors or/and positiveteness of thermal conductivity tensor; e.g., Edelstein and Gurtin [34], Oden and Tadjbakhsh [35]. El-Karamany andVariational principles for the elasticity and viscoelasticity theories have been derived by Gurtin [29], Leitman [30], Nickelloro [26].e importance of reciprocity relations and the variational principles stems from the fact that these principles provide atical basis for the modern numerical techniques such as nite-element methods [27] and boundary element methodsdevoted to the viscoelasticity and thermoviscoelasticity theories, e.g. Gurtin and Sternberg [24], Christensen [25], Fabrizio

f^ i and(v) Laws of thermodynamics

ij ji

namic

Folries, a

where 0

t t

Consid

The co

So, it i

2158 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171whererijx; t rjix; t rijx; t Gijkl ekl _cij h; 17

Sx; t Sex; t _b h _cij eij; 18ij

Aij are linearly dependent and both are positive semi-denite. Therefore, we assume that [23]

Aij bkij; 16where aP 0 and bP 0 are two constants.

(vii) The constitutive laws for the linear thermoviscoelasticity theory [42]

e _Aij akij: 15

s natural to assume in the GreenNaghdi theories involving two temperatures that the conductivity rate tensor k andinite [1]:T / Aiju;j;i: 14

nductivity tensor kij and the temperature discrepancy tensor Aij are linearly dependent and both are positive semi-def-ax;0 0; bix;0 a;ix;0 0: 13(vi) For the theory of heat conduction involving two temperatures we have [41]0 0

s au0t t0 s0; s;i a;i bi; _bi u;i:ering t0 0 we obtaina Z t

ux; sds; _a u; ax; t0 0; bix; t Z t

u;ix; sds; 12Tx; t0 /x; t0 T0 u0; s t0

/x; sds s0 and sx; t0 s0; 11

T0 u and s0 are constants; andZ trji;j qbi qui: 10lowing Green and Naghdi [20], we introduce the thermal displacement in the two-temperature GreenNaghdi theo-ssuming that there exists a reference time t0 such thatq _2 _U rji _eij Q qi;i; 9s (7) yieldshold at every x and every t.Taking into consideration that r r and using the divergence theorem and Eqs. (1) and (2), the rst law of thermody-T / ;iand the local eld equation for balance of entropy [1,20,23]

_S Q n qi

; 8dt Vq 2

2_ui _ui dV

Vqbi _uidV

@Vf i _uidA

VQdV

@VqinidA 7It is required that the rst law of thermodynamics [40]

dZ

1 Z Z Z Zq^ are prescribed functions of x; t, on @Vvc I; where vc r and q.The partitions of the boundary surface @V : @Vu; @Vr, and @Va; @Vq are such that @V @Vr [ @Vu @Vq [ @Va and @Vu \ @Vr @Va \ @Vq.

Where ni nix@V . The functions u^i and u^ are prescribed functions of x; t on @Vv I where (v u and a. The functionsuix; t u^i on @Vu and rjinj f^ ix; t on @Vr; 5

ax; t a^ on @Va and qini q^x; t on @Vq: 6(iv) The boundary conditions on @V I, where I ft : t 2 0;1g are

re x; t re x; t G x;0e x; t c x;0hx; t; 19

and

f t s ; f g f 0gt f g:

The h

The en

whereWe

Using the properties of convolution, we get

Eqs. (1

Eq. (23) can be written in the form

In vie

Introd

Then,

where

Using

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2159;i

R1 u10 Q _Sex; t _bx; t _ax;0 _cijx; teijx; 0:u10 kij _a;j kija;j a _b _eij _cij R1;t S hi;i R 0; 35

R u10 Q S0; S0 Sx;0; h^ u10 q^: 36Eqs. (18), (20), (22) and (23) one obtains the following heat transport equationi 0 i 0 ij j ij jEqs. (22) and (31) take the form

h u1 q u1k b kb ; 34qi u0hi: 33

ucing the entropy ux vector hix; t [45] where

_ h a Aijbj;i: 32t S t u10 Q S0 t u10 qi;i: 31w of Eqs. (12), (13) and (21), we can write Eq. (14) in the formij ijt S b h c e : 30t rij Gijkl ekl cij h; 29Fi t bi tm0i u0i ; 28

wheret _f g t _f g f g f 0t g: 260), (17)(20) and the initial conditions (3), using the preceding equation, lead to [44]

t rij;j qFi qui; 27ij ji ij jik x k x; kx k x in V: 25Gijkl Gklij Gjikl Gijlk; cij cji; in V 0;1; 24Gijklx; t; cijx; t, and bx; t are fourth order, second order and zero order relaxation tensors.assume that the following symmetry relations hold [24,43]u0S Q qi;i; 23

ergy equation is written as follows

_qi kiju;j kija;j kijbj kijbj: 22@t s @s @teat conduction law is [21,23]

_ _ @f t s @f t s @ _f g 0f x; t sgx; sds; f

0f x; sds; 21Z t Z tij ji ijkl kl ij

Sex; t bx;0hx; t cijx;0eijx; t 20

Den

conditions satisfying the symmetry relations; i.e., N is given by the kinematical relations (2), the equations of motion (27),

Den

and (2

whichadditi

Pm m 1;2 [48]

m m m

@Vt f i ui dA

VqFi ui dV

Vt R h dV

@Vu0 t q a dA S21; 39

2160 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171where S1221 indicates the same expression as on the left-hand side except that superscripts 1 and 2 are interchanged [49].

Proof. for the thermoviscoelastic material we haveZV

b h1 h2

dV ZV

b h2 h1

dV : 40

Using the symmetry relations (24) we getp fui ; a g 38are assumed to be the solutions to the mixed problems, given by Denition 1and corresponding to the causes (37). The twosystems of causes Pm and results pm are assumed to satisfy the conditions given in Denitions 2 and 3; then they are con-nected by the following reciprocal relation

Theorem 1. Assume that the symmetry relations (24) and (25) hold, thenZ1 2

Z1 2

Z1 2

Z1 1 2 12Pm : fFmi ; Rm; f^ mi ; u^mi ; q^m; a^m; h^m; u0mi ;v0mi ;u0mg: 37The results3. Reciprocal theorem

Consider two problems where applied mass forces, heat sources, surface tractions, assigned surface displacements, sur-face temperature and surface heat ux are specied differently under different initial conditions. The actions start at t 0and produce in the body displacements ui, temperature increment h and thermal displacement a.

Let the variables involved in these two problems be distinguished by superscripts in parentheses. Thus, the causes are:Denition 5. We denote by A the linear space of all admissible processes endowed with addition and scalar multiplication[29,47]. In particular, if p and ~p are two different admissible processes andx is a scalar, then p1 p ~p ui ~ui; eij ~eij; . . . ;hi ~hi and p2 x~ui;x~eij; . . . ;x~hi are admissible processes.Denition 4 (the thermoviscoelastic process). A thermoviscoelastic process corresponding to the external system of loads1 fQ ; bi; f^ i; q^; u^i; a^g is dened as an admissible process p ui; eij;rji; h;a;u; S;hi; qi that complies with the mixed problemN provided that: (a) The functions fu^i; a^g are prescribed continuous functions on @Vv 0;1 and ff^ i; q^g are prescribedpiecewise regular functions in @Vvc 0;1.

(b) The functions fbi;Qg are prescribed continuous functions in V 0;1 and vanish in V 1;0.satises the fundamental system of the eld equations, given boundary conditions and given initial conditions, inon to the symmetry relations.Denition 3 (solution of the mixed problem). By a solution of the mixed problem, we mean any admissible state fui;a; hg,5) hold. (e) u0 T0 is a strictly positive constant and qx > 0 is a smooth function in V .u; S;hi; qi; Where: (a) The set fui;a; hg is a dynamically admissible state in V 0;1 assuming ui 0; a 0; h 0 onV 1;0. h being once and ui and a twice continuously differentiable functions of x and t (b) The set frij; S; qig is anadmissible system in V 0;1 assuming its elements vanish on V 1;0 and are continuously differentiable functionsof x and t on V 0;1 (c) Gijkl; cij and b are assumed to be continuously differentiable functions of x and t in V 0;1 sothat Gijkl 0; cij 0; b 0 for t 2 1;0 and each component is of bounded variation [24]. (d) The symmetry relations (24)

ition 2 (the admissible process). We mean by the admissible process an ordered array [32,46] p ui; eij;rij; h;a;the constitutive relations (29), (30), (32), (33) and (35), in addition to the symmetry conditions (24) and (25); and the initialconditions (3), (4) and the boundary conditions (5), (6).denote by N the mixed problem constituted by the fundamental system of the eld equations, boundary and initial

ition 1 (the mixed initial-boundary value problem N). On the basis of the governing equations given in (i)(vii) we2.3. Denitions

t S1 h2 t S2 h1 dV cij e1ij h2 cij e2ij h1 dV ; 42

Substi

Apply

Substi

Then,

Using

Using

Z

Eq. (5

Substi

Using

in the

Weticity

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2161establish the convolutional variational Principle [29] for the linear two- temperature GreenNaghdi thermoviscoelas-theory of type III, assuming that the equilibrium thermoviscoelastic modulus b1 is positive [26].4. Variational principleunbounded region, the surface integrals are absent and the reciprocity relation (39) takes the formZV

qF1i u2i

dV ZV

t R1 h2

dV S1221: 53In the particular case of an innite thermoviscoelastic medium, assuming that only the body forces and heat sources act@V V V @V

that hi u10 qi, the Reciprocity Relation (39) results. hZt f 1i u2i

dAZ

qF1i u2i

dV Z

t R1 h2

dV Z

h1 a2

dA S1221: 52V ;i V ;i

tuting from Eq. (48) into Eq. (47), taking into consideration Eqs. (49) and (51), we getZ h1r;r Aijb2j

dV

Z h2r;r Aijb1j

dV : 51V

0) leads to u10 f akijbj;i krsbs;r bkijbj;i k

rsbs;rgdV : 50Eqs. (15), (16), (25) and (34) we obtainZV Aijbj;i hr;rdV u10

ZVf Aijbj;i krsbs;r Aijbj;i k

rsbs;r gdVV hi bi dV

V hi bi dV : 49Eq. (34) and the symmetry relations (25) it follows thatZ1 2

Z2 1

V

hr;r a dV @V

h a dAV

hi bi dV : 48we haveZ1 2

Z1 2

Z1 2

47@V V V V

tuting from Eq. (32) into the last integral of the preceding equation we getZ@V

t f 1i u2i

dAZV

qF1i u2i

dV ZV

t R1 h2

dV ZV

h1r;r a2

dV ZV

h1r;r Aijb2j

;i

dV S1221:Zt f 1i u2i

dAZ

qF1i u2i

dV Z

t R1 h2

dV Z

t h1i;i h2

dV S1221: 46V V V

ing the divergence theorem and substituting from Eq. (27) one obtainsZt r1ji u2i;j dV

Zt R1 h2dV

Zt h1i;i h2dV S1221: 45V V

tuting from Eqs. (2) and (35), one obtainsVt rji eij t rji eij dV

V cij h eij cij h eij dV : 43

Eliminating cij hm emij

from Eqs. (42) and (43) we nd thatZt r1ji e2ij dV

Zt S1 h2

dV S1221: 44V VZ1 2 2 1

Z2 1 1 2

ZV

Gijkl e1kl e2ij

dV ZV

Gijkle2kl e1ij

dV : 41

Using the constitutive Eqs. (29) and (30) we get from Eqs. (40) and (41)Z Z h i

t ijkl kl ij i i ij

1

t rij eijdV a Aija;j;i hr;rdV h a^ adA h adA

Theorin V

If and

Proof

(a)

Z Z Z

(b)

And Efundarequir

Thvariat

2162 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171en, iterating this procedure by making suitable choices of ~p and applying the fundamental lemma of calculus ofion we see that p is a solution to the mixed boundary-value problem N given in Denition 1.And this result implies the boundary condition given by the second equation of (5).dGtfpg 0 t P 0: 58Let dGtfpg 0 t P 0. We must show that p 2 A is a solution to the mixed problem N whenever Eq. (58) holds [50].Choose ~p ~ui;0;0;0;0;0;0;0;0;0 and let ~ui, together with all its space derivatives, vanish on @V 0;1. Then fromEqs. (57) and (58) follows

ZVqui t rij;j qFi ~uidV 0 t 2 0;1: 59

q. (59) must hold for every dynamically admissible ~ui with the foregoing properties. But this fact together with themental lemma of calculus of variation implies the validity of Eq. (27). Next, let ~p ~ui;0;0;0;0; 0;0;0;0;0 but this timees merely that ~ui vanish on @Vu 0;1. Then, Eqs. (57), (58), and (27) yield

t f i f^ i 0 on @Vr 0;1: 60@Vu

t u^i ui ~f idA@Va

a^ a ~hdA@Vr

t f i f^ i ~uidA@Vq

h h^ ~adA: 57

If p is a solution to the mixed boundary- value problem, then using Denition 1 of the mixed problem N we getZ Z Z Z V t ~rij eij 12 ui;j uj;i dV V

bi a;i ~hidV Vf a Aijbj;i h ~hk;kgdVd~pGtf~pg dGtfpg V Gijkl ekl cij h t rji ~eijdV

V b1t S b h cij eij ~S dV

ZVf hi u10 kijbj kij bj ~bigdV

ZVu10 t qi kiju;j kij u;j ~u;idV

ZVt u10 u;i bi ~qidV

ZVt R hi;i S ~hdV

ZVqui t rij;j qFi ~uidVem 2. Assume that the relaxation tensors and the thermal conductivity tensors satisfy the symmetry relations (24) and (25)0;1 and in V respectively. Then,d~pGtf~pg dGtfpg 0 t P 0: 56only if p is a solution to the mixed boundary-value problem N.

. We will present the proof in two parts, (a) and (b) [50]

We assume that p is a solution to the mixed boundary value problem. If p ui; eij;rji; h;a;u; S;hi; qi and~p ~ui; ~eij; ~rji; ~h; ~a; ~u; ~S; ~hi; ~qi are two admissible processes belonging to A, then px~p 2 A for every scalar x. Calcu-lating the rst variation, we obtainZ Z

1 V V 2 @Va @Vq

Z@Vu

t f i u^idAZ@Vr

t f i f^ i uidA: 55V hi bidV

Vt S hi;i R hdV

Vu0 t qi u;idV

Vt rij;j qFi uidVZ Z

1 Z Z

^2 V b1ij

b h h u10 kijbj kij bj bi u10 t kiju;j kij u;j u;i u10 t qi biodVZ Z Z ZG fpg 1Z

G e e qu u 1 t S b h c e S

b1 > 0 where b1x limt!1bx; t: 54

Using Denition 5, we dene the functional Gtfpg in the linear space A for each p ui; eij;rji; h;a;/; S;hi; qi and eacht 2 0;1 by:

Nomodu

Theor

Proof

All eleu; S;hUsingt 2 0;

Then,

For pvariat

suppliTak

dpXtfpg t rij ~eijdV qui ~uidV bi ~hidV t S ~hdV 0: 65

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2163V V V V

where we have used that h a, Eq. (34) and the symmetry relations (25) in obtaining the following identitiesZVt hi ~hdV

ZV ~bi hidV

ZV bi ~hidV : 66

Since dpXtfpg 0 where ~pd f~ui; ~h; ~bi; ~eij; ~hig 2 A, we choose ~pd ~ui;0;0;0; 0 2 A and we obtainZVqui ~uidV 0; t 2 0;1: 67get from the preceding equation for the difference solutionZ Z Z Z

es, then Eq. (63) is satised identically.ing into consideration Eqs. (2), (29), (30), the symmetry relations (24), (25) and applying the divergence theorem, we@Vu @Va

The surface integrals are equal to zero for the difference solutions.Clearly, if pd is the solution to the mixed boundary value problem with null initial and boundary conditions and zeroZ

t u^i ~f idAZ

a^ ~hdA: 64d~pXtf~pg dpXtfpg V Gijkl ekl cij h t rij ~eijdV

Vqui t rij;j ~uidV

Vt S hi;i ~hdV

ZV

t ~rij eij 12 ui;j uj;i

dV ZV

1b1

t S b h cij eij ~SdV@Va

h a a^dA@Vu

t u^i ui f i: 62

d~pXtf~pg dpXtfpg 0 t P 0: 63 ui; eij;rij; h;a; S; hi 2 A and p ~ui; ~eij; ~rij; ~h; ~a; ~S; ~hi 2 A, then px~p 2 A for every scalar x. Calculating the rstion on obtains Z Z Z2 V V V 2b1Z Z

1 hi bidV t rij eijdV

1 t S b h cij eij SdVXtfpg 12 Vft Gijkl ekl eij qui uigdV 12 V

b h hdV Vt rij;j uidV

Vt S hi;i hdV

Z Z Zhd h2 h1;ud u2 u1; edij e2ij e1ij ; . . .n o

:

ments of the external system of loads Pd are zeros, and the set of the difference solutions pd ui; eij;rji; h;a;i; qi satises the fundamental system of the eld equations with null supplies, null initial and boundary conditions.Denitions 5, we dene the following functional Xtfpg in the linear space A for each p ui; eij;rij; h;a; S;hi and each1 by: Z Z Z Zi i

ference solution by

pd udi u2i u1i ; ad a2 a1n o

: 61

Taking into consideration the kinematical relations (2), the constitutive laws (29), (30), (22), and the relation (32) betweenthe conductive and the thermodynamic temperatures, one obtains. We assume that there are two sets of solutions p1 fu1;a1g and p2 fu2;a2g. We denote the set of the dif-mixed problem N given by Denition 1; provided that u0 > 0; qx > 0 , and the symmetry conditions (24) and (25) hold inV 0;1 and in V respectively.em 3. There exists in V V [ @V 0;1 at most one dynamically admissible state p fuix; t; ax; tg solution to the5. Uniqueness theoremw, we use a convolutional variational problem to prove a uniqueness theorem without restrictions on the elasticli tensor or thermal conductivity tensors, except symmetry conditions.

Eq. (6

a 0. Eq. (32) for the difference solution: h a A b leads to h 0. Therefore, all the elements of the dif-ferencimpos

Inmediu

A t

where

(ii)The

(iii)

All th

Th

The st

The h

2164 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 215521710 q qCEC20 kx0 C0gx; u0 C0gu; t0 C0gt; s0 C0gs; r0ij rij=k 2l; h hc=qC0; u0 uc=qC0;

C2 k 2l ; k0 k

; g qCE 1=j:qCEh T0aT3kR^

k 2lR^

le k @3u

@x2@t k @

2u@x2

; 75

where

R^

fg Z t0Rft s @gx; s

@sds; f k;l; 76

where k;l-the Lame constants, q-the density, CE-the specic heat at constant strains, aT-the coefcient of linear thermalexpansion, k-the thermal conductivity, and k-the thermal conductivity rate (a specic constant in GreenNaghdi theories)

Eq. (14) for one-dimensional isotropic solid takes the form

u h a @2u@x2

: 77

We introduce the following non-dimensional variables

2 2 0 2 2xx k l T k l

eat transport equation is written as followse ejj @u@x

: 73

The constitutive equation yields [51]:

r r kR^

2lR^

e a 3kR^

2lR^

h: 74rain component takes the form:u1 ux; t; u2 u3 0: 72e state functions are depending only upon the variable x1 x and the time t.

e displacement vector has components:limx!1

h 0; limx!1

r 0: 71

The regularity conditions:r0; t 0; or r0; s r0 0: 70Mechanical boundary condition:bounding plane x 0 is taken to be traction-free, i.e.s

u0 is a constant and Ht is the Heaviside unit step function.u0; t u0Ht; or u0; s u0 ; 69hermal shock is applied to the boundary plane x 0 in the form(i) Thermal boundary condition:order to show the effect of the two-temperature theory we consider an isotropic semi-space homogeneous viscoelasticm occupying the region xP 0 with quiescent initial state and the following boundary conditions6. Applicationij j ;i

e solutions set are zeros. And the solution is unique. This completes the proof. In this proof, no restrictions areed on the relaxation tensors or thermal conductivity tensor except symmetry.Eq. (68) leads to bi 0, since the initial and boundary conditions are homogeneous for the difference solution we getd d d d d7) leads to udi 0 and consequently edij 0. Choosing ~pd 0;0;0;0; ~hi 2 A , we nd thatZV bi ~hidV 0; t 2 0;1: 68

d

Thconve

Th

where

where e , b aC =j , and j-the diffusivity.

2

where

@x @x

s k k

InvIn

rier se

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2165gt ect

t1

12gc Re

X1k1

eikpt=t1 gc ikp=t1 ; 0 6 t 6 2t1: 89ex; s u0M1 k1e k1x k2 e k2x

s3k1 k2 ; 88

ersion of Laplace transformsorder to invert the Laplace transform in the above equations, we adopt a numerical inversion method based on a Fou-ries expansion [54]. In this method, the inverse gt of the Laplace transform gs is approximated by the relation !" #1 2

p ph irx; s sk1 k2 ; 86

Hx; s uoM1 s2 k2 e

k2

px s2 k1 e

k1

px

h i3 ; 87u0M1 ek1

px e

k2

px

ux; s

uo k1 L1 e k2 L1 esk1 k2 ; 85Hence, by using state space approach [53], one can get under the above boundary conditions, the exact solution in theLaplace transform domain in the following forms:

k2

px

k1

px

h iL1 s 1 esR

s k b0s21 esR; L2 ess k b0s21 esR

;

M1 s21 b0L1; M2 s21sR b0L2

: 842 2e @u@x

; s k @ u@x2

s2h esRs2e; sR 1 A ps b; 81

r rxx sRe h; @2 r@x2

s2e; u h b0@2 u@x2

; 82f s R10 estf tdt. From the preceding system of equations we get@2 u

2 L1 u L2 r;@2 r

2 M1 uM2 r; 83q2C20CE0 0

Performing Laplace transforms we get r@2rxx@x2

@2e@t2

; u h b0@2u@x2

; _qx @2u

@x@t k @u

@x: 79

e calculations will carried out for the case:

Rkt Rlt Rt 1 AZ t0f t dt; 80

f t f 1t ebtta

1;

a; b; A are non-dimensional empirical constants [52]

0 < a < 1; b > 0; 0 6 A < ba

Ca ; Rt > 0;dRtdt

< 0;

c2/0 2r R^

e h; @3u

@x2@t k @

2u@x2

@2h

@t2 e R

^ @2e@t2

; 78

!us, in the preceding non-dimensional variables we get the following system of equations (Suppressing the primes fornience): !

For numerical purposes this is approximated by the function

gNt edt

t1

12gc Re

XNk1

eikpt=t1gc ikp=t1 !" #

; 90

where N is a sufciently large integer representing the number of terms in the truncated innite Fourier series. N must bechosen such that

ectRe eiNpt=t1 gc iNp=t1 6 e1;where e1 is a persecuted small positive number that corresponds to the degree of accuracy to be achieved. The parameter c isa positive free parameter that must be greater than the real parts of all singularities of gs. The optimal choice of c wasobtained according to the criteria described in [54].

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3x

Fig. 1. The conductive temperature distribution.

0.8

1

1.2

2166 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171Fig. 2. The thermodynamic temperature distribution.00 0.5 1 1.5 2 2.5 3

x0.20.4

0.6

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2167Table 1Values of the constants.

k 386W/mK, aT 1:78 105 K1; CE 383:1 m2=s2 K;

g 8886:73 s=m2 l 3:86 1010 N=m2; k 7:76 1010 N=m2;q 8954 kg=m3; T0 u0 293 K; C0 4158 m=s; b0 0:075; k 10

1

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3x

Fig. 3. The stress distribution.7. Numerical results

A copper-like material was chosen for purposes of numerical evaluations and the constants of the problem were takenfrom Ezzat et al. [9] as given in Table 1.

The computations were carried out for t = 0.1. Formula (90) was used to invert the Laplace transforms in Eqs. (85)(87)giving the thermodynamic temperature h, the conducive temperature u and the stress component r. The results arepresented graphically at different x. In Figs. 13, it is noticed that the temperature discrepancy bo has a signicant effecton all the elds, where bo 0 indicates the one temperature case and bo 0:075 indicates the two-temperature case. Inthese gures we noticed that the curves are smaller in magnitude in case bo 0:075. The absolute value of the maximumstress decreases when bo 0:075 relative to the case when bo 0. Previously, the discontinuity of the stress distributionwas a critical situation and no one has explained the reason physically, while in the context of the two-temperature theoryof thermoelasticity the stresses function is continuous. This paper indicates that the two-temperature theory in the general-ized thermo-viscoelasticity describes the behavior of the viscoelastic body more real than the theory of one-temperaturethermo-viscoelasticity. Therefore, the conductive heat wave and the thermodynamical heat wave may be separated.

8. Conclusions

(1) The convolutional variational principle and the reciprocal theorem are established for the linear two-temperaturedynamic thermoviscoelasticity theory in context of the GreenNaghdi theory of type III, in an anisotropic andinhomogeneous solid.

(2) We assume in the GreenNaghdi theory of type III involving two temperatures that the conductivity rate tensor kij andthe temperature discrepancy tensor Aij are linearly dependent and both are positive semi-denite.

(3) The thermodynamic temperature T hu0 and the conductive temperature / uu0 coincide when Aij 0 orD/ 0 , and we get from the given model the equations and the variational principle for the GreenNaghdithermoviscoelasticity theory of type III not involving two temperatures, replacing u and u0 by h and T0 respectively.

(4) The effect of separating the thermodynamic temperature and the conductive temperature is signicant in generalizedthermoelasticity and thermoviscoelasticity. The absolute value of the maximum stress decreases relative to the casewhen the two temperatures coincide. The curves of the stress and temperature distributions are more uniform and thethermodynamic temperature h is smaller in magnitude relative to the one-temperature case.

e 0:0168: a 2 ; b 0:05; A 0:106:

(5) Although the two-temperature theory was proposed since 1968 (see [15]), in the last decade there is an increasinginterest in this theory, especially when it is combined with one of the generalized thermoelasticity theories (see for

The co

A (

Fro

For th

whereTak

kij kijx are time-independent (the proof is given in

2168 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171_W 1

Gijklt s; 0@s

ds@t

1

cijt s;0 @s ds @t 1cijt s;0 @s ds @t

Z t1

bt s; 0 @hs@s

ds

@ht@t

u10 kijbj _bj K; A5

where

K 12

Z t1

@

@tGijklt s; t g @ekls

@s@eijg@g

dsdgZ t1

@

@tcijt s; t g

@eijs@s

@hg@g

dsdg

12

Z t1

@

@tbt s; t g @hs

@s@hg@g

dsdg: A6

We get from (A3)@W@u;i

0 and

rij Z t1

Gijklt s @ekls@s

dsZ t1

cijt s@hs@s

ds; A7Z t @ekls @eijt Z t @hs @eijt Z t @eijs @ht@s @g 2

Carrying out the differentiation with respect to t using Leibnitzs rule, we getW 2 1

Gijklt s; t g@s @g

dsdg1

cijt s; t g @s @g dsdg 2 1bt s; t g

@hs @hgdsdg 1u10 kijbibj: A4A (ii). Then, W can be expressed in the form

1Z t @ekls @eijg Z t @eijs @hg 1 Z tpi u10 qi [20].ing into consideration the energy equation (A2) and representation theorem [25], we consider kij kijx, and/ u0 /2 u00 0 u0 u0 we obtain

@W@h

S

_h @W@eij

rij

_eij @W@bi

pi

_bi @W@u;i

_u;i Tn / Tpi;i 0; A3m Eqs. (8) and (9) one obtains the reduced energy equation [23]

_W _TS Tn rij _eij 1 T/

qi;i Tqiu;i/2

0: A2

e linearized theory we set1 1 ; T 1 and introducing h T u and u /u where h

1, and u

1,where Y feij; T; bi;u;ig is the set of the independent variables.nstitutive equations for two-temperature GreenNaghdi thermoviscoelasticity theory of type iii

i) We introduce the free energy density dened by

WY U TS; A1Appendixuniqueness of solution without restrictions except the symmetry conditions. Using the variational principle a unique-ness theorem is proved with no restrictions imposed on relaxation and thermal conductivity tensors, except symmetryconditions. This result can be used in thermoelasticity. The elasticity tensor, in case of pre-stressed thermoelasticbodies fails to be positive-denite [41]. So, it is worthwhile to propose the method used in this work.(6) Uniqueness theorems for anisotropic viscoelasticity and thermoviscoelasticity were proved by various authors,assuming positive deniteness or strong ellipticity of the relaxation tensors. In this work, one of our points is to proveexample [6,7,911,41]). This is due to the decreasing in magnitude of stresses and thermodynamic temperature,the continuity and uniform distributions of all the state variables obtained in context of this theory.

where

by parWe

For th

where

A (

Carryi

ij ij ij

1

An

Eqs. (1

A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 21692u0

1 @tkijt s; t g @s @g dsdg:

d:@W@bi

Z t1

u10 kijt s

@bj@s ds . Thus, Eq. (A14) takes the form:

@W@bi

pi

_bi _biZ t1

u10 kijt s@u;j@s ds. Using

5) and (16), Eq. (A9) yields n 1u20

n2 u0K, where1 @t @s @g 2 1 @t @s @g1

Z t @ @bis @bjg

Z t @

cijt s; t g@eijs @hgdsdg 1

Z t @bt s; t g @hs @hgdsdgZ t1

bt s; 0 @hs@s

ds

@ht@t

u10Z t1

kijt s;0@bjs@s

ds

_bi K;

where: K 12

Z t1

@

@tGijklt s; t g @ekls

@s@eijg@g

dsdg_W 1

Gijklt s; 0 kl@s

ds@t

1

cijt s;0 @s ds @t 1cijt s;0 @s ds @tZ t @e s @e t Z t @hs @e t Z t @e s @ht2 1 @s @g 2 1 @s @g

ng out the differentiation with respect to t using Leibnitzs rule we get1Z t

bt s; t g @hs @hgdsdg 1u10Z t

kijt s; t g@bis @bjgdsdg:W 12 1

Gijklt s; t g @ekls@s

@eijg@g

dsdg1

cijt s; t g@eijs@s

@hg@g

dsdg;Then, Z t Z t

1 @s 1 @sLet Pi u10Z t

kijt s@bj ds

Z tkijt s

@u;j ds : A14iii) Now, we prove that kij and kij are time-independent n1 kiju;ju;i akiju;j;ikrs/s;r bkij_bj;ik

rsbs;r > 0: A13

Since ft kijbj;ikrsbs;r > 0 and from Eqs. (13) we have f0 0 then ft is increasing function of t, therefore

_ft 2kij _bj;ikrsbs;r > 0.

Eqs. (A11)(A13) lead to KP 0 which agrees with [25].n > 0; A12

The internal rate of production of entropy n for the Green Naghdi theory of type III should be positive [20,21]ij ;j ;i r;r 0 ij j ;i rs s ;r ij j ;i rs s ;r

e linearized theory Eq. (A9) yields

n 1u20

n1 u0K: A11Using Eqs. (15), (16) and (A10), we get for the GreenNaghdi theory of type III

A u p u1ak _b k _b bk _b k b :oviscoelastic moduli are positive and nite [26]: limt!1Gijklt > 0; limt!1cijt > 0, and limt!1bt > 0. Integratingts the integrals in Eqs. (A7) and (A8), we arrive at the constitutive Eqs. (17) and (18).expand pi pieij; h; bi;u;i to get [21]pi u10 kij _bj kijbj: A10By Denition 2 it is assumed that eijt 0; rijt 0; ht 0; b 0; cij 0, and Gijkl 0 for t < 0. The equilibriumtherm@bi pi _bi Aiju;j;i pk;k TnK 0; A9

Gijklt s Gijklt s; 0; cijt s cijt s; 0; and bt s bt s;0:S Z t1

cijt s@eijs@s

dsZ t1

bt s @hs@s

ds; A8

@W

[39] Magdy A. Ezzat, Ahmed S. El-Karamany, Shereen M. Ezzat, Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag

[47] S.27

[48] D.CI

2170 A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171Chirita, M. Ciarletta, Reciprocal and variational principles in linear thermoelasticity without energy dissipation, Mech. Res. Commun. 37 (3) (2010)1275.Iesan, On some reciprocity theorems and variational theorems in linear dynamic theories of continuum mechanics, Memorie dellAcad. Sci. Torino. Sci. Fis. Mat. Nat. Ser. 4 (1974) 17.heat transfer, Nucl. Eng. Des. 252 (2012) 267277.[40] A.E. Green, P.M. Naghdi, On thermodynamics, rate of work and energy, Arch. Rat. Mech. Anal. 40 (1971) 3749.[41] A. Magaa, R. Quintanilla, Uniqueness and growth of solutions in two-temperature generalized thermoelastic theories, Math. Mech. Solids 14 (2009)

622634.[42] Leitman, J.M., Fisher, G.M.C., The linear theory of viscoelasticity. In: Handbuch der Physik, vol. VI a/3. Springer-Verlag, Berlin, 1973, pp. 1031.[43] W.A. Day, Time-reversal and symmetry of the relaxation function of a linear viscoelastic material, Arch. Rat. Mech. Anal. 40 (3) (1971) 155157.[44] J. Ignaczak, A completeness problem for stress equations of motion in linear elasticity theory, Arch. Mech. Stos. 15 (1963) 225234.[45] Naotake Noda, R. Hetnarski, Y. Tanigawa, Thermal Stresses, second ed., Taylor & Francis, New York, 2003.[46] E. Sternberg, On the Analysis of Thermal Stresses in Viscoelastic Solids, in: High Temperature Structures and Materials. Proc. 3d Symp. Naval Struct.

Mech., Pergamon Press, 1964.n2 _biZ t1

kijt s @_bj

@sds akrs _bs;r

Z t1

kijt s @_bj

@sds

!;i

bkrs _bs;rZ t1

kijt s@bj@s

ds

;i

:

And n2 > 0. For xed value of t the instantaneous values of _bi; krs _bs;r and

krs _bs;r and the functionalsR t1 kijt s

@ _bj@s

ds;Z t

1kijt s @

_bj@s

ds;i

andR t1 k

ijt s

@bj@s

ds

;i

respectively, may in

general be of opposite signs since the functionals depend on the entire past history of the temperature and thermal displace-ment gradients [25]. In fact, for a given values of t, _bi; krs _bs;r krs _bs;r and the functionals will have the same signs only if thetensors kij, and k

ij are time-independent, in which case pi u10 kij _bj kijbj.

References

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A.S. El-Karamany, M.A. Ezzat / Applied Mathematical Modelling 39 (2015) 21552171 2171

Two-temperature GreenNaghdi theory of type III in linear thermoviscoelastic anisotropic solid1 Introduction2 The mathematical model2.1 The assumptions of the model2.2 The governing equations2.3 Definitions

3 Reciprocal theorem4 Variational principle5 Uniqueness theorem6 Application7 Numerical results8 ConclusionsAppendixThe constitutive equations for two-temperature GreenNaghdi thermoviscoelasticity theory of type iii

References