THEORY OF GENERALIZED MICROPOLAR ... | ResearchGate

Journal of Thermal Stresses, 32: 923–942, 2009 Copyright © Taylor & Francis Group, LLC ISSN: 0149-5739 print/1521-074X online DOI: 10.1080/01495730903032276

THEORY OF GENERALIZED MICROPOLAR THERMOELASTIC DIFFUSION UNDER LORD–SHULMAN MODEL Moncef Aouadi Rustaq College of Applied Sciences, Department of Mathematics and Computer Science, Rustaq, Sultanate of Oman The general equations of motion and constitutive equations, based on the theory of Lord–Shulman with one relaxation time, are derived for a general homogeneous anisotropic medium with a microstructure, taking into account the effects of heat and diffusion. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases which can be deduced from our generalized model. Keywords: Lord–Shulman model; Micropolar thermoelastic diffusion; Reciprocity; Uniqueness; Variational principle

INTRODUCTION Generalized continuum theories for mechanical behavior developed over the last century admitted degrees of freedom not considered in the classical theory of elasticity. The micropolar elasticity theory takes into consideration the granular character of the medium, and is intended to be applied to materials for which the ordinary classical theory of elasticity fails owing to the microstructure of the material. Within such a theory, solids can undergo macro-deformations and micro-rotations. The motion in this kind of solids is completely characterized by the displacement vector ux t and the microrotation vector x t while in the case of classical elasticity, the motion is characterized by the displacement vector only. The general theory of linear micropolar thermoelasticity was given by Eringen [1–3] and Nowacki [4–6]. The micropolar theory was extended to include thermal effects by Nowacki [4–6], Eringen [7], Tauchert et al. [8], Tauchert [9], and Nowacki and Olszak [10]. One can refer to Dhaliwal and Singh [11] for a review on the micropolar thermoelasticity and a historical survey of the subject, as well as to Eringen and Kafadar [12] in the Continuum Physics series, in which the general theory of micromorphic media has been summed. Received 27 October 2008; accepted 21 January 2009. Address correspondence to Moncef Aouadi, Rustaq Faculty of Education, Department of Mathematics and Computer Science, P.O. Box 10, Sultanate of Oman, Rustaq 329, Oman. E-mail: [email protected] 923

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M. AOUADI

In recent years increasing attention has been directed towards the generalized theory of thermoelasticity, which was found to give more realistic results than the coupled or uncoupled theories of thermoelasticity, especially when short time effects or step temperature gradients are considered. The theory of generalized thermoelasticity with one relaxation time was first introduced by Lord and Shulman [13], who obtained a wave-type heat equation by postulating a new law of heat conduction instead of the classical Fourier’s law. One can refer to Hetnarski and Ignaczak [14] for a review and presentation of the generalized theories of thermoelasticity. In this paper we extend the micropolar theory to include thermal and diffusion effects. In fact, the development of high technologies in the years before, during, and after the second world war pronouncedly affected the investigations in which the fields of temperature and diffusion in solids cannot be neglected. At elevated and low temperatures, the processes of heat and mass transfer play the decisive role in many problems of satellites, returning space vehicles, and landing on water or land. These days, oil companies are interested in the process of thermodiffusion for more efficient extraction of oil from oil deposits. Nowacki [15–18] developed the theory of thermoelastic diffusion. In this theory, the coupled thermoelastic model is used. Recently Sherief and Saleh [19] investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Singh discussed the reflection phenomena of waves from free surface of a thermoelastic diffusion elastic solid with one relaxation time in [20] and with two relaxation times in [21]. Aouadi studied in [22] the generalized thermoelastic diffusion problem with variable electrical and thermal conductivity. Aouadi studied also the interaction between the processes of elasticity, heat and diffusion in an infinitely long solid cylinder [23] and in an infinite elastic body with spherical cavity [24]. Gawinecki and Szymaniec [25] proved a theorem about global existence of the solution for a nonlinear parabolic thermodiffusion problem. Gawinecki et al. [26] proved a theorem about existence, uniqueness and regularity of the solution for the same problem. Uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, was proved by Sherief et al. [27] on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients. Aouadi [28] proved this theorem in the Laplace transform domain, under the assumption that the functions of the problem are continuous and the inverse Laplace transform of each is also unique. Aouadi [29] derived the uniqueness and reciprocity theorems for the generalized thermoelastic diffusion problem in anisotropic media, under the restriction that the elastic, thermal conductivity and diffusion tensors are positive definite. For the coupled problem, the existence of a generalized, regular and unique solution has been proved by Aouadi [30] by means of some results of semigroup of linear operators theory. Recently, Aouadi [31] derived some spatial stability results for the quasi-static problem. DERIVATIONS OF THE GOVERNING EQUATIONS In the first part of this section we present the general balance laws of a continuum with microstructure. Then we derive the field equations of the linear theory of micropolar thermoelastic diffusion bodies.

THEORY OF GENERALIZED MICROPOLAR THERMOELASTIC DIFFUSION

925

We consider a body that at some instant occupies the region V of the Euclidean three-dimensional space and is bounded by the piecewise smooth surface A. The motion of the body is referred to a fixed system of rectangular cartesian axes Oxi i = 1 2 3. We denote by n the outward unit normal of A. Boldface characters stand for tensors of an order p ≥ 1, and if v has the order p, we write vij···s (p subscripts) for the components of v in the cartesian coordinate frame. We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers 1 2 3, summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. In what follows we use a superposed dot to denote partial differentiation with respect to the time t. The system of governing equations of a linear micropolar thermoelasticity consists of [1–6]: (i) Equations of motion (on V × 0 ) jij + Fi = üi

(1)

¨j ijk jk + jij + Gi = Jij

(2)

(ii) The kinematic relations (on V × 0 ) 1 1 uij + uji ri = ipq uqp 2 2 1 1 ij = ij + ji ij = ji − ij 2 2

ij = eij − ijk rk − k eij =

(3) (4)

from which one obtain uij = eij − ijk rk uij = ij − ijk k ji u˙ ij = ji ˙ ij + jki jk ˙ k

(5)

where ui is the displacement vector field, ji is the force stress tensor, eij is the strain tensor, is the reference mass density, Fi is the component of the external forces per unit mass, i is the vector of internal rotations, ji the moment of couple stress tensor, ji is the microcurvature tensor, ijk is the alternating tensor, Jij is the microrotation tensor, Gi is the component of the external applied couple per unit mass, ri is the rotation vector, ji is the micro-strain tensor. Let us consider the diffusion phenomenon combined with the process of heating of a solid body and its deformation. In order to combine the three fields, concentration, temperature and deformation, we depart from the equations of the balance of energy and entropy, assuming that the procedure is thermodynamically irreversible. As the point of departure of the discussion we take the principle of energy, the entropy balance, and the Clausius–Duhem inequality [32, 33]: U˙ = ji ˙ ji + ji ˙ ji + T S˙ + P C˙ qi Pi ˙

= S + − T i T i

(6) (7)

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M. AOUADI

=−

1 qi Ti P + i T >0 T T T i

(8)

In the above equations, i denotes the flow of the diffusing mass vector, qi is the vector of heat flux, U is the internal energy per unit mass, S is the entropy, T is the absolute temperature, P is the chemical potential per unit mass, is a source of entropy and C is the concentration. Here we have made a simplifying assumption, namely that the temperature of both the components is the same. The following equations result from the Clausius–Duhem inequalities [32, 33] qi = −kij Tj − ij Pj i = − ij Tj − dij Pj where kij and dij are the standard coefficients of thermal conductivity and diffusion tensors respectively, ij is the thermal-diffusion coefficient describing the Soret effect and ij is the diffusion-thermal coefficient describing the Dufour effect. The above equations generalize the classic Fourier and Fick laws to the problems of thermodiffusion. From Eqs. (7) and (8), we obtain the entropy equation: T S˙ = Pii − qii

(9)

and from Eqs. (6) and (9), we obtain the conservation of mass equation, namely C˙ = −ii

(10)

With the help of (6), (7), and (9), the inequality (8) becomes: ˙ ji − T S˙ − U˙ + ji ˙ ji + ji

1 Pi >0 qi Ti − T T T i

(11)

We now introduce the Helmholtz free energy function , defined by = U − TS

(12)

Then the inequality (11) can be written in the form 1 Pi ˙ ˙ ˙ >0 − + T S + ji ˙ ji + ji ji − qi Ti − T T T i

(13)

Substituting for U from Eq. (12) into Eq. (6), we get d = ij d ij + ij dij − SdT + PdC

(14)


927

The function (and all other functions under consideration) can be expressed in terms of the independent variables ij ij , T and C. Using the chain rule of differentiation, we obtain d =

dT + dC d ij + dij + ij ij T C

(15)

Comparing Eqs. (15) and (14), we immediately obtain ij =

ij

ij =

ij

S=− T P= C

(16)

We now expand the function in a power series of the independent variables of the form = 0 + a0 + cij ij + b0 C + Dij ij −

cE 2 1 2 1 + C + cijkl ij kl 2T0 2 2

1 + dijkl ij kl + aij ij + bij ij C − C + pij ij + qij ij C + pijkl ij kl (17) 2 where = T − T0 T0 is the temperature of the medium in its natural state assumed to be such that /T0 1, cE is the specific heat at constant strain and Cijkm is the tensor of elastic constants. The constants and are measures of thermodiffusion effects and diffusive effects, respectively. The rest of the parameters are material constants. The following symmetries are evident cijkl = cklij dijkl = dklij Now, in the natural state of the medium, we have 0 = 0 = 0 ij = 0 C = 0 ij = 0 ij = 0 We, therefore, obtain 0 = 0 a0 = 0 b0 = 0 cij = 0 Dij = 0

(18)

Using Eq. (18), the relations (16)–(17) yield ij = cijkl kl + pijkl kl + aij + bij C

(19)

ji = pijkl kl + dijkl kl + pij + qij C

(20)

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M. AOUADI

S = −aij ij − pij ij +

cE + C T0

P = bij ij + qij ij − + C

(21) (22)

which are the constitutive equations of the theory of linear micropolar thermoelastic diffusion. To obtain the equations of motion, we substitute from Eqs. (19) and (20) into Eqs. (1) and (2), and, upon using Eq. (3), we get üi = cjikl klj + pjikl klj + aji j + bji Cj + Fi ¨ j = pjikl klj + djikl klj + pji j + qji Cj Jij + ijk cjkml ml + pjkml ml + ajk + bjk C + Gi ij = eij − ijk rk − k

(23)

1 u + uji 2 ij 1 ri = ipq uqp 2

eij =

The linearized form of Eq. (9) is T0 S˙ = −qii

(24)

−qii = −T0 aij ˙ ij − T0 pij ˙ ij + cE ˙ + T0 C˙

(25)

Using Eq. (21), Eq. (24) reduces to

As was done by Sherief et al. in the derivation of the generalized equations of thermoelastic diffusion theory [27] and thermoelastic micropolar theory [34] under Lord–Shulman’s model [13], we now assume a generalized Fourier’s law of heat conduction of the form qi + 0 q˙ i = −kij j

(26)

where 0 is the thermal relaxation time. Taking the divergence of both sides of (26) and using Eq. (25) and its time derivative, we arrive at the equation of heat conduction in our case, namely ˙ ij + 0 ¨ ij + cE ˙ + 0 ¨ + T0 C˙ + 0 C ¨ = kij ij −T0 aij ˙ ij + 0 ¨ ij − T0 pij (27) Analogous to Eq. (26) for the heat flux vector, we assume a similar equation for the mass flux vector of the form [34]: i + ˙i = −dij Pj

(28)


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where is the diffusion relaxation time. This will ensure that the equation, satisfied by the concentration C, will also predict finite speeds of propagation of matter from one medium to the other. Taking the divergence of both sides of Eq. (28), we get upon using Eq. (10) and its derivative with respect to time C˙ + C¨ = dij Pij

(29)

Substituting from Eq. (22) in Eq. (29), we arrive at C˙ + C¨ = dij bij ij + qij ij − + C+ij

(30)

The governing equations of micropolar thermoelastic diffusion consist of the equations of motion (23), the equation of heat conduction (27) and the equation of mass diffusion (30). These equations are supplemented by the constitutive Eqs. (19)–(22). For the case of isotropic media, we have cijkl = ij kl + + ik jl + − il jk dijkl = ij kl + + ik jl + − il jk aij = −1 ij bij = −2 ij kij = kij

(31)

dij = dij Jij = Jij pij = qij = pijkl = qijkl = 0 where and are Lame’s constants, 1 = 3 + 2 + t and 2 = 3 + 2 + c , where t is the coefficient of linear thermal expansion and c is the coefficient of linear diffusion expansion, and are constants of the theory of micropolar thermoelasticity. Substituting from Eqs. (31) into constitutive Eqs. (19)–(22), (26) and (28) these equations take the form ji = ekk ij + 2 + eij + ijk rk − k − 1 ij − 2 Cij ji = kk ij + ij + ji c S = 1 ekk + E + C T0 P = C − 2 ekk − −ki = qi + 0 q˙ i −dPi = i + ˙i

(32)

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M. AOUADI

ij = eij − ijk rk − k 1 u + uji 2 ij 1 ri = ipq uqp 2

eij =

We shall now formulate a different alternative form that will be useful in proving uniqueness in the next section. In this new formulation, we will use the chemical potential as a state variable instead of the concentration. From Eq. (32)4 , we obtain C = 2 ekk + d + nP

(33)

The alternative form can be written by substituting Eq. (33) into Eqs. (32), ˙ ui = jij + Fi ¨ ¨ j = ijk jk + jij + Gi Jij ji = 2 + eij + ijk rk − k + ij 0 ekk − 1 − 2 P ji = kk ij + ij + ji S = 1 ekk + c + dP T0 S˙ = −qii −ki = qi + 0 q˙ i

(34)

C˙ = −ii −DPi = i + ˙i ij = eij − ijk rk − k 1 u + uji 2 ij 1 ri = ipq uqp 2

eij =

where 1 = 1 +

c 1 2 2 a2 2 = 2 0 = − 2 c = E + d= n= T0

(35)

VARIATIONAL PRINCIPLE The principle of virtual work (with variation of displacements) for the elastic deformable body in the above alternative form can be written in the form ¨ i i dV + fi ui + i i dA Fi − u¨ i ui dV + Gi − J V V A = ji uij + ji ij dV V


931

where for an arbitrary field f = fx t, we have f = f˙ dt. On performing such variations, denoted by the symbol , the operator dtd ≡ s is treated as a constant [35]. On the left-hand side we have the virtual work of mass force Fi , mass couple ¨ i , surface force fi = ji nj and vector Gi , inertial force üi and inertial couple J surface moment i = ji nj , whereas on the right-hand side we have the virtual work of internal forces. Taking advantage of the constitutive equations, we get ¨ i i dV + fi ui + i i dA Fi − u¨ i ui dV + Gi − J V V A = − 1 ekk dV − 2 Pekk dV (36) V

V

where =

1 2 + eij eij + 0 ekk ekk + 2ri − i ri − i 2 V + + ij ij + kk kk + − ij ij dV

In the case when we take into account the coupling of the deformation field with the temperature, there arise the necessity of considering an additional relation characterizing the phenomena of the thermal conductivity. Let us now after Biot [35] define a vector Hi connected with the entropy through the relation S = −Hii

(37)

Combining Eqs. (34)5 , (34)7 and (37), we obtain T0 dHi d 2 Hi + 0 2 + i = 0 k dt dt

(38)

c + 1 ekk + dP + Hii = 0

(39)

Multiplying both sides of Eq. (38) by Hi and integrating over the region of the body, we obtain V

i +

d2 H T0 dHi + 0 2 i Hi dV = 0 k dt dt

Using the divergence theorem, we obtain A

ni Hi dA −

V

d 2 Hi T0 dHi Hii dV + + 0 2 Hi dV = 0 k V dt dt

(40)

Substituting from Eq. (39) into Eq. (40), we get the second variational equation A

ni Hi dA + 1

V

ekk dV + d

V

PdV + + = 0

(41)

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M. AOUADI

where the function of thermal potential is defined by c 2 dV with = c dV 2 V V

=

and the function of thermal dissipation is defined by

T = 0 2k V

d 2 Hi d 2 Hi dHi T0 dHi + 0 2 Hi dV with = + 0 2 Hi dV (42) dt dt k V dt dt

Eq. (42) is the second variational equation. Note that if dtd = s is treated as a positive parameter, is a positive definite functional. In order to obtain the last of the variational equations, we now introduce the vector function Gi defined as follows C = −Gii

(43)

Combining Eqs. (34)89 and (43), we obtain 1 ˙ ¨ i + Pi = 0 G + G D i

(44)

Multiplying Eq. (44) by Gi and integrating over the region of the body, we get 1 dG d 2 Gi i + 2 + Pi Gi dV = 0 dt dt V D Using the divergence theorem, we get A

Pni Gi dA −

V

1 PGii dV + D V

dGi d 2 Gi + 2 Gi dV = 0 dt dt

(45)

Substituting from Eqs. (33) and (43) into Eq. (45), we obtain A

Pni Gi dA +

V

2 Pekk dV + d

V

PdV + + = 0

(46)

where the diffusion potential is defined by =

n 2 P dV with = n PPdV 2 V V

and the function of diffusive dissipation is defined by =

1 2D V

dGi d2 G d2 G 1 dGi + 2 i Gi dV with = + 2 i Gi dV dt dt D V dt dt

Eq. (47) is the third variational equation. is a positive functional if dtd = s > 0.

(47)


933

Eliminating the integrals 1 V ekk dV and 2 V Pekk dV between the Eqs. (36), (41), and (46), we obtain the final variational principle in the following form + + + + + d P dV =

V

¨ i i dV Fi − u¨ i ui dV + Gi − J V + fi ui + i i dA − ni Hi dA − Pni Gi dA V

A

A

(48)

A

On the right-hand side of Eq. (48) we find all the causes, the mass forces and couples, inertial forces and couples, the surface forces and moments, the heating and the chemical potential on the surface A bounding the body. Note that Eq. (48) is a variational principle of the Biot’s type for the diffusive micropolar thermoelasticity with one relaxation time.

UNIQUENESS THEOREM Now we assume that the virtual displacements ui , the virtual microrotations i , the virtual increment of temperature and the virtual increment of chemical potential P correspond to the increments occurring really in the body in question. Then ui =

ui i P ˙ i dt = dt = dt ˙ ˙ dt = u˙ i dt i = dt = P = dt = Pdt t t t t (49)

and Eq. (48) is reduced to the following relation d + + + + + d P dV dt V ¨ i ˙ i dV = Fi − u¨ i u˙ i dV + Gi − V V ˙ i dA − ni H˙ i dA − Pni G ˙ i dA + fi u˙ i + i A

A

(50)

A

is the kinetic energy of the body enclosed by the volume V . Note that =

1 ˙ i ˙ i dV with d = u˙ i u¨ i + J ˙ i ¨ i dV u˙ i u˙ i + J 2 V dt V

(51)

We also have 1 + + d PdV = nP 2 + 2dP + c2 dV 2 V V

(52)

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M. AOUADI

Substituting from Eqs. (51)–(52) into Eq. (50), we arrive at 1 d nP 2 + 2dP + c2 dV ++++ dt 2 V ¨ i ˙ i dV + fi u˙ i + i ˙ i dA = Fi − u¨ i u˙ i dV + Gi − V V A ˙ i dA − ni H˙ i dA − Pni G A

(53)

A

We shall now use Eq. (53) and three results derived in [29] to prove the uniqueness theorem T T d d H˙ H˙ dV = 0 H˙ i H˙ i dV + 0 0 (54) dt k V 2k dt V i i and d ˙ ˙ 1 ˙ ˙ d Gi Gi dV + G G dV = dt D V 2D dt V i i

(55)

If c n and d are constants satisfying the inequality [29] 0 < d2 < cn

(56)

c2 + 2dP + nP 2 > 0

(57)

then

Theorem 1. Assume that (i) T0 2 + k 0 k D 0 + and − are strictly positive, (ii) 0 < d2 < cn, Then the mixed problem of generalized micropolar thermoelastic diffusion has at most one solution, which satisfy the boundary conditions on the surface A fi1 = ij nj i1 = ij nj = 1 P = P1

(58)

and the initial conditions at t = 0 ˙ 0 = 0 ˙ = ˙ 0 P = P 0 P˙ = P˙ 0 ˙i = ui = u0i u˙ i = u˙ 0i i = 0i i

(59)

˙ 0 0 ˙ 0 P 0 and P˙ 0 are known functions. where fi1 i1 1 P1 u0i u˙ 0i 0i i 1

1

2

2

Proof. Let ui i 1 P 1 and ui i 2 P 2 be two solutions sets of Eqs. (34) with zero mass forces and mass couples, the same boundary conditions (58) and the same initial conditions (59). Consider the difference functions 1

2

1

2

ui = ui − ui i = i − i = 1 − 2 P = P 1 − P 2

(60)


935

which satisfy the governing equations with zero mass forces and couples, homogenous initial and boundary conditions. They thus satisfy an equation similar to Eq. (53) with zero right hand side, i.e., d 1 nP 2 + 2dP + c2 dV = 0 (61) ++++ dt 2 V Substituting Eqs. (54) and (55) into (61), we get 1 T 0 0 ˙ ˙ d 2 2 ˙ ˙ H H dV + ++ nP + 2dP + c dV + G G dV dt 2 V 2k V i i 2D V i i T 1 ˙ ˙ + 0 H˙ i H˙ i dV + (62) G G dV = 0 k V D V i i In view of hypothesis (i), we get T ˙ ˙ 1 d nP 2 + 2dP + c2 dV + 0 0 H˙ i H˙ i dV + Gi Gi dV ≤ 0 ++ dt 2 V 2k V 2D V (63) We see that the expression T ˙ ˙ 1 nP 2 + 2dP + c2 dV + 0 0 H˙ i H˙ i dV + G G dV ++ 2 V 2k V 2D V i i

(64)

is a decreasing function of time. We note that the expression V nP 2 + 2dP + c2 dV occurring in the expression (64) is always positive, since 0 < d2 < cn (hypothesis (ii)). Thus the expression (64) vanishes for t = 0, due to the homogenous initial conditions, and it must be always non-positive for t > 0. Using the hypothesis (i), it follows immediately that expression (64) must be identically zero for t > 0. We thus have ui = i = P = = 0

(65)

RECIPROCITY THEOREM We shall consider a homogeneous isotropic generalized micropolar thermoelastic diffusion body occupying the region V and bounded by the surface A. We assume that the force stresses ij , the moment of couple stresses ij , the strains eij and microcurvatures ij are continuous together with their first derivatives whereas the displacements ui , the microrotations i , the temperature and the chemical potential P are continuous and have continuous derivatives up to the second order, for x ∈ V + A, t > 0. The components of surface forces, surface moments, the normal component of the heat flux and the normal component of the chemical flux at regular points of A, are given by fi = ji nj i = ji nj q = ki ni p = DPi ni respectively. We denote by nj the outward unit normal of A

(66)

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To the system of field equation we must adjoin boundary conditions and initial conditions. We consider the following boundary conditions: ui x t = Ui x t i x t = i x t x t = x t Px t = x t x ∈ A t > 0

(67)

and the homogeneous initial conditions ˙ ui x 0 = u˙ i x 0 = 0 i x 0 = ˙ i x 0 = 0 x 0 = x 0 = 0 ˙ Px 0 = Px 0 = 0 x ∈ V t = 0

(68)

We derive the dynamic reciprocity relationship for a generalized micropolar thermoelastic diffusion bounded body V , which satisfies Eqs. (34), the boundary conditions (66) and the homogenous initial conditions (67), and subjected to the action of mass forces Fi x t, mass couples Gi x t, surface traction fi x t, the couple stress vector i x t, the heat flux qx t and the chemical flux px t Performing the Laplace transform defined as f¯ x s = fx t = fx te−st dt 0

over Eqs. (34) and omitting the bars for simplicity, we obtain jij = s2 ui − Fi ijk jk + jij = Js2 j − Gi ji = 2 + eij + ijk rk − k + ij o ekk − 1 − 2 P ji = kk ij + ij + ji S = c + 1 ekk + dP T0 sS = −qii −ki = 1 + 0 sqi

(69)

C = d + 2 ekk + nP sC = −ii −DPi = 1 + si ij = eij − ijk rk − k 1 u + uji 2 ij 1 ri = ipq uqp 2

eij =

Substituting from Eqs. (69)11–13 into (69)3 one obtains ji = + uij + uji + ijk rk + ij 0 ekk − 1 − 2 P

(70)


937

Now consider two problems where applied mass forces and couples, surface forces and moments, surface temperature and chemical potential are specified differently. Let the variables involved in these two problems be distinguished by superscripts 1 1 1 1 in parentheses. Thus, we have ui i ji ij 1 P 1 for the first problem 2 2 2 1 and ui i ji ji 2 P 2 for the second problem. Each set of variables satisfies the system of equations (69). Using the divergence theorem we get

1 2 1 2 ji ui j dV − jij ui dV V V 1 2 1 2 = ji nj ui dV − jij ui dV

1 2

V

ji uij dV =

V

V

Substituting from Eqs. (69)1 and (66)1 into the previous equation, we obtain V

1 2

ji uij dV =

A

1 2

fi ui dA −

1 2

V

s2 ui ui dV +

A similar expression is obtained for the integral together with Eq. (71) it follows that V

1 2

2 1

ji uij − ji uij dV =

A

1 2

2 1

fi ui − fi ui dA +

V

1 2

V

2 1

ji eij dV , from which

1 2

V

(71)

Fi ui dV

2 1

Fi ui − Fi ui dV (72) 2 1

1 2

Using equation (70) and taking into consideration that uij uji − uij uji = 0 we obtain 1 2 2 1 1 2 2 1 2 1 ji uij − ji uij dV = 2 ri i − ri i dV − 1 1 ekk − 2 ekk dV V V V 2 1 − 2 P 1 ekk − P 2 ekk dV (73) V

From Eqs. (73) and (72) we get 1 2 2 1 1 2 2 1 fi ui − fi ui dA + Fi ui − Fi ui dV A V 1 2 2 1 2 1 = 2 ri i − ri i dV − 1 1 ekk − 2 ekk dV V V 2 1 − 2 P 1 ekk − P 2 ekk dV

(74)

V

Using the divergence theorem, equations (69)23 and (66), we get D

1

2

2

1

mji ij − mji ij dV =

1 2 2 1 1 2 2 1 i i − i i dA + Gi i − Gi i dV A V 1 2 2 1 + 2 ri i − ri i dV (75) V

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M. AOUADI 2

1

1

2

Using equation (69)7 and taking into consideration that ij ji − ij ji = 0, we find that the integral in the left-hand side of equation (75) is equal to zero, therefore equation (74) with (75) leads to the first part of the reciprocity theorem in the Laplace transform domain 1 2 1 2 1 2 1 2 fi ui dA + Fi ui dV + i i dA + Gi i dV A V A V 1 2 1 2 12 + 1 ekk dV + 2 P ekk dV = S21 (76) V

V

which contains the mechanical causes of motion Fi and Gi , the prescribed 12 indicates the same displacements and the surface microrotations fi and i . S21 expression as on the left-hand side except that the superscripts (1) and (2) are interchanged. Using Eqs. (69)5−6 reduces −qii = sT0 c + 1 ekk + dP

(77)

Now, taking the divergence of both sides of Eq. (77) and using Eq. (69)7 , we arrive at the equation of heat conduction, namely kii = s + 0 s2 T0 c + 1 ekk + dP

(78)

To derive the second part, we multiply 2 by the corresponding equation (78) for the first problem, 1 by the analogous equation for the second problem, subtracting and integrating over V , we get 1 2 1 2 k ii 2 − ii 1 dV = s + 0 s2 T0 1 ekk 2 − ekk 1 dV V V + s + 0 s2 T0 dP 1 2 − P 2 1 dV (79) V

Since 1

1

1

2

2

2

2

1

ii 2 = i 2 i − i i and ii 1 = i 1 i − i i

Equation (79) can be written, using the divergence theorem and Eqs. (66)3 and (67)3 , in the form 1 2 q 1 2 − q 2 1 dA = s + 0 s2 T0 1 ekk 2 − ekk 1 dV A V + s + 0 s2 T0 d P 1 2 − P 2 1 dV (80) V

Eq. (69) constitutes the second part of reciprocity theorem which contains the thermal causes of motion and q. By the same manner, from Eqs. (69)8−10 , we obtain the equation of chemical potential DPii = s + s2 d + 2 ekk + nP

(81)


and the third part of reciprocity theorem 1 2 p1 2 − p2 1 dA = s + s2 2 eij P 2 − eij P 1 dV A V 2 + s + s d 1 P 2 − 2 P 1 dV

939

(82)

V

which contains the chemical potential causes of motion and 1 p.1 2 2 2 1 2 1 e − e dV , Eliminating the integrals 1 V 2 V P ekk − P ekk dV kk kk 1 2 2 1 and d V P − P dV from Eqs. (76), (80) and (82), we get 1 2 2 1 1 2 2 1 fi ui − fi ui dA + Fi ui − Fi ui dV s1 + 0 s1 + sT0 A V 1 2 2 1 1 2 2 1 + i i − i i dA + Gi i − Gi i dV A V 1 2 2 1 − 1 + s q − q dA − 1 + 0 sT0 p1 2 − p2 1 dA = 0 A

A

(83)

We have thus arrived at a reciprocity theorem in the Laplace transform domain. To invert the Laplace transform in Eqs. (76), (80), (82) and (83) we shall use the convolution theorem t t ft − g d = gt − f d −1 FsGs = 0

0

and the symbolic notations fx fx !f = 1 + 2 fx fx ℵf = 1 + 0 + + 0 2

Lf = 1 + 0

Inverting Eq. (76) we obtain the first part of reciprocity theorem in the final form

t 1 2 1 2 fi x t − ui x d dA + Fi x t − ui x d dV A 0 V 0 t t 1 2 1 2 + i x t − i x d dA + Gi x t − i x d dV A 0 V 0 t t 2 2 12 + 1 1 x t − ekk x d dV + 2 P 1 x t − ekk x d dV = S21 t

V 0

V 0

Inverting Eq. (80) we obtain the second part of reciprocity theorem in the final form 2 t t Lekk x d dV q 1 x t − 2 x d dA + T0 1 1 x t − A 0 V 0 t LP 2 12 + dT0 1 x t − x d dV = S21 V 0

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M. AOUADI

Inverting Eq. (82) we obtain the third part of reciprocity theorem in the final form

t

A 0

p1 x t − 2 x d dA + 2

−d

V

t 0

1 x t −

V

t 0

2

P 1 x t −

!ekk x d dV

2

!P 12 x d dV = S21

Finally, inverting Eq. (83) we obtain the general reciprocity theorem in the form

t

A 0

+

t ℵui ℵui 1 Fi x t − x d dA + x d dV V 0 2

1

fi x t − A 0

t

1

i x t −

2

2 2 t ℵi ℵi 1 x d dA + x d dV Gi x t − V 0

1 t 1 q x t − !2 x d dA T0 A 0 t 12 − p1 x t − L 2 x d dA = S21 −

A 0

CONCLUDING REMARKS The results established in this paper can be summarized as follows: (1) We have derived a linear theory for micropolar thermoelastic diffusion bodies. A microelement of the continuum with microstructure is equipped with the mechanical degrees of freedom for rigid rotations in addition to the classical translation degrees of freedom. (2) The field equations of the theory of homogenous and isotropic solids are presented. Using the variational theorem, the uniqueness theorem of solution of the initial boundary value problem is proved, and the dynamic reciprocity theorem is derived. (3) In deriving variational principles of continuum physics, Biot’s principle is one of the reliable and powerful methods, and it was extensively used in micropolar thermoelasticity [36], porous piezoelectric thermoelasticity [37], generalized thermoelastic diffusion theory [27] and coupled thermo-magnetoelectroelasticity [38]. We show that the application of Biot’s principle leads to a uniqueness theorem. (4) The obtained results are valid for some special cases which can be deduced from our generalized model: a. The coupled micropolar thermoelastic diffusion problem results from Eqs. (34) setting 0 = = 0. b. The generalized micropolar thermoelastic problem results from Eqs. (34)1−7 setting 2 = d = 0.


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c. The generalized thermoelastic diffusion problem results from Eqs. (34)135−911 setting = 0. d. The generalized thermoelastic problem results from Eqs. (34)135−711 setting = 0. (5) The results of this paper will be useful for numerical computation based on variational principles and reciprocity theorem. This is because the established results provide a more complete theoretical basis for modern numerical techniques such as finite element and boundary element methods. REFERENCES 1. A. C. Eringen, A Unified Theory of Thermomechanical Materials, Int. J. Engng. Sci., vol. 4, pp. 179–202, 1966. 2. A. C. Eringen, Linear Theory of Micropolar Elasticity, ONR Technical Report No. 29, (School of Aeronautics, Aeronautics and Engineering Science, Purdue University, 1965). 3. A. C. Eringen, Linear Theory of Micropolar Elasticity, J. Math. Mech., vol. 15, pp. 909–923, 1966. 4. W. Nowacki, Couple Stresses in the Theory of Thermoelasticity I, Bull. Acad. Polon. Sci., Ser. Sci Tech., vol. 14, pp. 129–138, 1966. 5. W. Nowacki, Couple Stresses in the Theory of Thermoelasticity II, Bull. Acad. Polon. Sci., Ser. Sci Tech., vol. 14, pp. 263–272, 1966. 6. W. Nowacki, Couple Stresses in the Theory of Thermoelasticity III, Bull. Acad. Polon. Sci., Ser. Sci Tech., vol. 14, no. 8, pp. 801–809, 1966. 7. A. C. Eringen, Foundation of Micropolar Thermoelasticity, Courses and Lectures, No. 23, CISM, Udine, Springer-Verlag, Vienna and New York, 1970. 8. T. R. Tauchert, W. D. Claus Jr., and T. Ariman, The Linear Theory of Micropolar Thermoelasticity, Int. J. Engng. Sci., vol. 6, pp. 36–47, 1968. 9. T. R. Tauchert, Thermal Stresses in Micropolar Elastic Solids, Acta Mech., vol. 11, pp. 155–169, 1971. 10. W. Nowacki and W. Olszak, Micropolar Thermoelasticity, in Micropolar Thermoelasticity, CISM Courses and Lectures, No 151, Udine, Springer-Verlag, Vienna, 1974. 11. R. S. Dhaliwal and A. Singh, Micropolar Thermoelasticity, in R. Hetnarski (ed)., Thermal Stresses II, Mechanical and Mathematical Methods, ser. 2, North Holland, Amsterdam, 1987. 12. A. C. Eringen and C. B. Kafadar, Polar Field Theories, in A. C. Eringen (ed)., Continuum Physics, vol. 4, Academic Press, New York, 1976 13. H. Lord and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solid, vol. 15, pp. 299–309, 1967. 14. R. B. Hetnarski and J. Ignaczak, Generalized Thermoelasticity, J. Thermal Stresses, vol. 22, pp. 451–476, 1999. 15. W. Nowacki, Dynamical Problems of Thermoelastic Diffusion in Solids I, Bull. Acad. Pol. Sci., Ser. Sci. Tech., vol. 22, pp. 55–64, 1974. 16. W. Nowacki, Dynamical Problems of Thermoelastic Diffusion in Solids II, Bull. Acad. Pol. Sci., Ser. Sci. Tech., vol. 22, pp. 129–135, 1974. 17. W. Nowacki, Dynamical Problems of Thermoelastic Diffusion in Solids I, Bull. Acad. Pol. Sci., Ser. Sci. Tech., vol. 22, pp. 257–266, 1974. 18. W. Nowacki, Dynamical Problems of Thermoelastic Diffusion in Elastic Solids, Proc. Vib. Prob., vol. 15, pp. 105–128, 1974.

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19. H. H. Sherief and H. Saleh, A Half-Space Problem in the Theory of Generalized Thermoelastic Diffusion, Int. J. Solids Struct., vol. 42, pp. 4484–4493, 2005. 20. B. Singh, Reflection of P and SV Waves from Free Surface of an Elastic Solid with Generalized Thermoelastic Diffusion, J. Earth Syst. Sci., vol. 114, pp. 159–168, 2005. 21. B. Singh, Reflection of SV Waves From the Free Surface of an Elastic Solid in Generalized Thermoelastic Diffusion, J. Sound Vib., vol. 29, pp. 764–778, 2006. 22. M. Aouadi, Variable Electrical and Thermal Conductivity in the Theory of Generalized Thermoelastic Diffusion, Z. Angew. Math. Phys., vol. 57, pp. 350–366, 2006. 23. M. Aouadi, A Generalized Thermoelastic Diffusion Problem for an Infinitely Long Solid Cylinder, Int. J. Math. Math. Sci., vol. 2006, pp. 1–15, 2006. 24. M. Aouadi, A problem for an Infinite Elastic Body with a Spherical Cavity in the Theory of Generalized Thermoelastic Diffusion, Int. J. Solids Struct., vol. 44, pp. 5711–5722, 2007. 25. J. A. Gawinecki and A. Szymaniec, Global Solution of the Cauchy Problem in Nonlinear Thermoelastic Diffusion in Solid Body, PAMM, Proc. Appl. Math. Mech., vol. 1, pp. 446–447, 2002. 26. J. Gawinecki, P. Kacprzyk, and P. Bar-Yoseph, Initial Boundary Value Problem for Some Coupled Nonlinear Parabolic System of Partial Differential Equations Appearing in Thermoelastic Diffusion in Solid Body, J. Anal. Appl., vol. 19, pp. 121–130, 2000. 27. H. H. Sherief, F. Hamza, and H. Saleh, The Theory of Generalized Thermoelastic Diffusion, Int. J. Eng. Sci., vol. 42, pp. 591–608, 2004. 28. M. Aouadi, Uniqueness and Reciprocity Theorems in the Theory of Generalized Thermoelastic Diffusion, J. Thermal Stresses, vol. 30, pp. 665–678, 2007. 29. M. Aouadi, Generalized Theory of Thermoelastic Diffusion for Anisotropic Media, J. Thermal Stresses, vol. 31, pp. 270–285, 2008. 30. M. Aouadi, Qualitative Aspects in the Coupled Theory of Thermoelastic Diffusion, J. Thermal Stresses, vol. 31, pp. 706–727, 2008. 31. M. Aouadi, Spatial Stability for the Quasi-Static Problem in Thermoelastic Diffusion Theory, Acta Appl. Math., vol. 106, pp. 307–323, 2009. 32. S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, North-Holland, Amsterdam, 1962. 33. S. R. de Groot, Thermodynamics of Irreversible Process, North-Holland, Amsterdam, 1952. 34. H. H. Sherief and F. A. Hamza, Theory of Generalized Micropolar Thermoelasticity and an Axisymmetric Half-Space Problem, J. Thermal Stresses, vol. 28, pp. 409–437, 2005. 35. M. A. Biot, Thermoelasticity and Irreversible Thermodynamics, J. Appl. Phys. vol. 27, pp. 240–253, 1956. 36. F. Passarella, Some Results in Micropolar Thermoelasticity, Mech. Res. Comm., vol. 23, pp. 349–357, 1996. 37. M. Ciarletta and E. Scarpetta, Some Results on Thermoelasticity for Porous Piezoelectricity Materials, Mech. Res. Comm., vol. 23, pp. 1–10, 1996. 38. M. Aouadi, On the Coupled Theory of Thermo-Magnetoelectroelasticity, Q. J. Mech. Appl. Math., vol. 60, pp. 443–456, 2007.