State space approach to two-dimensional electromagneto–thermoelastic problem with two relaxation times

International Journal of Engineering Science 39 (2001) 1383±1404

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State space approach to two-dimensional electromagneto± thermoelastic problem with two relaxation times Magdy A. Ezzat, Mohamed I. Othman *, Angail A. Smaan Faculty of Education, Department of Mathematics, P.O. Box 1905, Salalah-211, Oman Received 25 January 2000; received in revised form 1 May 2000; accepted 19 June 2000 _ (Communicated by E.S. S ß UHUBI)

Abstract The model of the two-dimensional equations of generalized magneto±thermoelasticity with two relaxation times in a perfectly conducting medium are established. The method of the matrix exponential, which constitutes the basis of the state space approach of modern theory, is applied to the non-dimensional equations. Laplace and Fourier integral transforms are used. The resulting formulation is applied to a problem of a thick plate subject to heating on parts of the upper and lower surfaces of the plate that varies exponentially with time. Numerical results are given and illustrated graphically for the problem considered. A comparison was made with the results obtained in the absence of a magnetic ®eld. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Biot [1] formulated the theory of coupled thermoelasticity to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no e€ect on the temperature. The heat equation for both theories are of the di€usion type predicting in®nite speeds of propagation for heat waves contrary to physical observations. Lord and Shulman [2] introduced the theory of generalized thermo-elasticity with one relaxation time by postulating a new law of heat conduction to replace the classical Fourier's law. This law contains the heat ¯ux vector as well as its time's derivative. It contains also a new constant that acts as a relaxation time. The heat equation of this theory is of the wave-type, ensuring ®nite speeds of propagation for heat and elastic wave. The remaining governing equations for this theory, namely, the equations of motion and the

*

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0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 0 ) 0 0 0 9 5 - 1

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Nomenclature k; l Lame's constants q density speci®c heat at constant strain CE t time T absolute temperature reference temperature chosen so that jT T0 j  1 T0 components of stress tensor rij components of strain tensor eij components of strain vector ui k thermal conductivity magnetic permeability l0 electric permeability e0 …l0 H02 †=q, Alfen velocity a20 2 …k ‡ 2l†=q b0 b20 ‡ a20 c20 c22 …l=q†, square velocity of transverse waves p c 1=…l0 e0 †, the velocity of light a 1 ‡ …a20 =c2 † c20 =c22 b2 a0 ab2 s; m relaxation times e …ou=ox† ‡ …ov=oy†, the dilatation coecient of linear thermal expansion at c …3k ‡ 2l†at e …c2 T0 †=…qCE …k ‡ 3l†† constitutive relation remain the same as those for the coupled and the uncoupled theories. Dhaliwal and Sherief [3] extended this theory to general an isotropic media in the presence of heat sources. Because of the complicated nature of these equations, few attempts have been made to solve them. Sherief [4] solved a spherically symmetric problem with a point source of heat, and Sherief and Anwar [5] solved a cylindrically symmetric problem with a line source of heat. Sherief and Ezzat [6] have studied the fundamental problem of thermoelasticity for an in®nite spherically symmetric space using the method of potentials. All of these problems are one-dimensional. Sherief and Anwar [7] have studied the state space formulation for two-dimensional problem of generalized thermoelasticity with one relaxation time. M uller [8] ®rst introduced the theory of generalized thermoelasticity with two relaxation times. A more explicit version was then introduced by Green and Laws [9], Green and Lindsay [10] and independently by S ß uhubi [11]. In this theory, the temperature rates are considered among the constitutive variables. This theory also predicts ®nite speeds of propagation as in Lord and Shulman's theory. It di€ers from the latter in that Fourier's law of heat conduction is not violated if the body under consideration has a center of symmetry. Erbay and S ß uhubi [12] studied wave

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propagation in a cylinder. Ignaczak [13] studied a strong discontinuity wave and obtained a decomposition theorem [14]. Ezzat [15] has also obtained the fundamental solution for this theory. Many authors have considered the propagation of electromagneto±thermoelastic waves in an electrically and thermally conducting solid. Paria [16] discussed the propagation of plane magneto±thermoelastic waves in an isotropic unbounded medium under the in¯uence of a uniform thermal ®eld and with a magnetic ®eld acting transversely to the direction of the propagation. Paria used the classical Fourier law of heat conduction, and neglected the electric displacement. Wilson [17] extended Paria's results by introducing a component of the magnetic ®eld parallel to the direction of the propagation. A comprehensive review of the earlier contributions to the subject can be found in [18]. Among the authors who considered the generalized magneto±thermoelastic equations are Nayfeh and Nasser [19], who studied the propagation of plane waves in a solid under the in¯uence of an electromagnetic ®eld. They obtained the governing equations in the general case and the solution for some particular cases. Choudhuri [20] extended these results to rotating media. Lately, Sherief [21] solved a problem for a solid cylinder, while Sherief and Ezzat [22] solved a thermal shock half-space problem using asymptotic expansions. Ezzat [23] has studied the problem of generation of generalized magneto±thermoelastic waves by thermal shock in a perfectly conducting half-space. Ezzat [24] studied the state space formulations for one-dimensional problem of generalized magneto±thermoelasticity with two relaxation times in a perfectly conducting medium. Sherief and Ezzat [25] have studied a problem in generalized magneto±thermoelasticity for an in®nitely long annular cylinder. Ezzat and Othman [26] have established the model of two-dimensional equations of generalized magneto±thermoelasticity with two relaxation times in a perfectly conducting medium. In the present paper, we shall formulate the state space approach to two-dimensional electromagneto±thermoelasticity problem with two relaxation times in a perfect conducting medium. Laplace and Fourier transform techniques are used throughout and the inversion of the transforms is carried out using a numerical inversion Honig [27]. The resulting formulation is applied to a problem. The expressions for temperature distribution, thermal stress and displacement components are obtained for this problem considered and represented graphically.

2. Formulation of the problem In our considerations of two-dimensional problems of generalized magneto±thermoelasticity with two relaxation times, we shall make two important restrictions. We assume ®rst that the medium under consideration is a perfect electric conductor and, secondly, that the initial magnetic ®eld vector H is oriented in such a way that propagation of plane cylindrical waves in the xy-plane is possible. Under these assumptions we can obtain very simple expressions for the displacement, temperature and the electromagnetic quantities.We begin our consideration with the linearized equations of electromagnetism, valid for slowly moving media. _ curl h ˆ J ‡ e0 E; _ curl E ˆ l0 h;

…1† …2†

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Eˆ

l0 …u_ ^ H†;

…3†

div h ˆ 0:

…4†

The above equations are supplemented by the displacement equations of the theory of elasticity, taking into account the Lorentz force. q ui ˆ …k ‡ l†uj;ij ‡ lui;jj

c…T ‡ mT_ †;i ‡ l0 …J ^ H †i

…5†

and the heat conduction equation _ …Q ‡ sQ†:

kT;ii ˆ qCE …T_ ‡ sT † ‡ cT0 u_ ;i

…6†

The constitutive equation rij ˆ kekk dij ‡ 2leij

c…T

T0 ‡ mT_ †dij

…7†

and strain±displacement relations 1 eij ˆ …ui;j ‡ uj;i †; 2

…8†

together with the previous equations, constitute a complete system of generalized magneto± thermoelasticity with two relaxation times equations for a medium with a perfect electric conductivity. In the above equations, a dot denotes di€erentiation with respect to time, while a comma denotes material derivatives. The summation notation is used. We shall consider only the simplest case of the two-dimensional problem. We assume that all causes producing the wave propagation are independent of the variable z, and that waves are propagated only in the xy-plane. Thus all quantities appearing in Eqs. (1)±(8) are independent of the variable z. Then the displacement vector has components ‰u…x; y; t†; v…x; y; t†; 0Š. Assume now that the initial conditions are homogeneous and the initial magnetic ®eld has components …0; 0; H0 †. Then relations (1)±(3) yield J ˆ curl h

_ e0 E;

…9†

_ 0†; E ˆ l0 H0 … v_ ; u;

…10†

hˆ

…11†

H0 …0; 0; e†:

We shall consider a thermoelastic medium governed by the equations of generalized electromagneto±thermoelasticity with two relaxation times whose state depends on the space variables x0 , y 0 and the time variable t0 . The initial conditions are taken to be homogeneous.

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We shall use the following non-dimensional variables: x ˆ c 0 g0 x 0 ;

y ˆ c0 g0 y 0 ; u ˆ c0 g0 u0 ; v ˆ c0 g0 v0 ; r0ij c…T 0 T0 † qQ0 ; Qˆ hˆ ; r ˆ ; ij 2 qc0 l kT0 c20 g2

t ˆ c20 g0 t0 ;

s ˆ c20 g0 s0 ;

m ˆ c20 g0 m0 ;

where the dashed quantities denote dimensional variables. In terms of these non-dimensional variables, the equations of motion has the form b2 u;xx ‡ u;yy ‡ …b2 …b2

1†v;xy

1†u;xy ‡ b2 v;yy ‡ v;xx

b2 …h;x ‡ mh;xt † ˆ a0 u;tt ;

…12†

b2 …h;y ‡ mh;yt † ˆ a0 v;tt

…13†

and the components of the stress are:   o 2 2 2 rxx ˆ b0 u;x ‡ …b0 2†v;y b 1 ‡ m h; ot

…14a†

rxy ˆ u;y ‡ v;x ;

…14b†

ryy ˆ …b20

…14c†

  o 2 2 2†u;x ‡ b0 v;y b 1 ‡ m h; ot   o 2 2†e b 1 ‡ m h: ot

rzz ˆ …b20

…14d†

The non-dimensional heat equation is  h;xx ‡h;yy ˆ

 o o2 oe ‡s 2 h‡e ot ot ot



 oQ Q‡s : ot

…15†

These equations will be supplemented with appropriate boundary conditions relevant to the particular application under consideration as will be discussed later on. Taking the Laplace transform de®ned by the relation f…s† ˆ

Z 0

1

e st f …t† dt

to both sides of Eqs. (12)±(15) and using the homogeneous initial conditions, we obtain 1† v;xy

b2 …1 ‡ ms†h;x ˆ a0 s2 u;

…16†

1† u;xy ‡ b2 v;yy ‡ v;xx

b2 …1 ‡ ms†h;y ˆ a0 s2 v

…17†

b2 u;xx ‡ u;yy ‡ …b2 …b2

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and the components of the stress are: rxx ˆ b20 u;x ‡ …b20

 b2 …1 ‡ ms†h;

2† v;y

rxy ˆ u;y ‡ v;x ;

…18a† …18b†

ryy ˆ …b20

2† u;x ‡ b20 v;y

rzz ˆ …b20

2† e

 b2 …1 ‡ ms†h;

b2 …1 ‡ ms†h

…18c† …18d†

and  xx ‡ h;  yy ˆ s…1 ‡ ss†h ‡ es h; e

 …1 ‡ ss†Q:

…19†

We now use the Fourier transform with respect to the space variable x, de®ned by Z 1 1  e iqx f …x† dx: f …q† ˆ p 2p 1 The inversion formula for this transform is Z 1 1 f …x† ˆ p eiqx f  …q† dq: 2p 1 Taking the Fourier transform of both sides of Eqs. (16)±(19), we get b2 q2 u ‡ u;yy ‡ iq…b2 iq…b2

1† u;y ‡ b2 v;yy

rxx ˆ iqb20 u ‡ …b20 rxy

ˆ

u;y

ˆ

…b20

q2 v 2† v;y

iqb2 …1 ‡ ms†h ˆ a0 s2 u ; b2 …1 ‡ ms†h;y ˆ a0 s2 v ;

b2 …1 ‡ ms†h ;



‡ iq v;

ryy ˆ b20 v;y ‡ iq…b20 rzz

1† v;y



2† e

…20† …21† …22a† …22b†

b2 …1 ‡ ms†h ; b2 …1 ‡ ms†h ; 2† u

e q2 h ‡ h;yy ˆ s…1 ‡ ss†h ‡ es

: …1 ‡ ss†Q

…22c† …22d† …23†

3. State space formulation We take as state variables in the physical domain the quantities e; h; De; Dh. In the transformed domain, the state space variables are taken as e ; h ; D e ; Dh where, u ‡ D v : e ˆ iq

…24†

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Eliminating u and v between Eqs. (20), (21) and (23) with the help of Eq. (24), we obtain the following equation:    ; …25† D2 e ˆ as2 ‡ q2 ‡ pe…1 ‡ ms† e ‡ p…1 ‡ ms†h …1 ‡ ms†…1 ‡ ss†Q D2 h ˆ se e ‡ …q2 ‡ p†h

 : …1 ‡ ss†Q

…26†

Eqs. (25) and (26) can be written in matrix form as follows: dV~ …q; y; s† ~ s†V~ …q; y; s† ‡ B…q; ~ y; s†; ˆ A…q; dy

…27†

where 2

0 6 0 ~ s† ˆ 6 A…q; 6 2 2 4 as ‡ q ‡ es…1 ‡ ms† se 3 e …q; y; s† 7 6  6 h …q; y; s† 7 7; 6 ~ V …q; y; s† ˆ 6  e …q; y; s† 7 5 4 D   Dh …q; y; s†

0 0 p…1 ‡ ms† …q2 ‡ p†

2

~ y; s† ˆ B…q;

3 1 0 0 17 7 7; 0 05 0 0

P ˆ s…1 ‡ ss†;

2 3 0 607 7   …q; y; s†6 …1 ‡ ss†Q 6 7: 415 1

The formal solution of system (27) can be written in the form   Z y ~ ~ ~ ~ ~ V …q; y; s† ˆ exp…A…q; s†y† V …q; y0 ; s† ‡ exp… A…q; s†z†B…q; z; s† dz ; y0

…28†

where y0 denotes any arbitrarily chosen initial value for y. In the special case, when there are no heat sources acting inside the medium, Eq. (28) simpli®es to ~ s†y†V~ …q; y0 ; s†: V~ …q; y; s† ˆ exp…A…q; ~ s† is The characteristic equation of the matrix A…q;  2 k4 as ‡ 2q2 ‡ se…1 ‡ ms† ‡ P k 2 ‡ q4 ‡ q2 ‰as2 ‡ se…1 ‡ ms†Š ‡ se…1 ‡ ms† ‡ pas2 ˆ 0:

…29†

…30†

The roots of Eq. (30) satisfy the relations k12 ‡ k22 ˆ as2 ‡ 2q2 ‡ se…1 ‡ ms† ‡ P ;

…31a†

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k12 k22 ˆ q4 ‡ q2 ‰as2 ‡ se…1 ‡ ms†Š ‡ ‰se…1 ‡ ms† ‡ pas2 Š:

…31b†

Using Cayley±Hamilton theorem, the ®nite series representing can be truncated to the following form: ~ s†y† ˆ L…q; ~ y; s† ˆ b0 I~ ‡ b1 A~ ‡ b2 A~2 ‡ b3 A~3 ; exp…A…q;

…32†

where I~ is the unit matrix of order 4 and b0 ; . . . ; b3 are some parameters depending on y, q and s. By Cayley±Hamilton theorem, the characteristic roots k12 and k22 of the matrix A~ must satisfy the equations exp…k1 y† ˆ b0 ‡ b1 k1 ‡ b2 k12 ‡ b3 k13 ; exp… k1 y† ˆ b0

b1 k1 ‡ b2 k12

b3 k13 ;

exp…k2 y† ˆ b0 ‡ b1 k2 ‡ b2 k22 ‡ b3 k23 ; exp… k2 y† ˆ b0

b1 k2 ‡ b2 k22

b3 k23 :

The solution of the above system is given by k12 cosh…k2 y† k22 cosh…k1 y† ; k12 k22  2  1 k1 k22 b1 ˆ 2 sinh…k2 y† sinh…k1 y† ; k1 k22 k2 k1

b0 ˆ

b2 ˆ b3 ˆ

1 k12

k22 1

k12

‰ cosh…k1 y† 

k22

…33†

cosh…k2 y†Š;

1 sinh…k1 y† k1

 1 sinh…k2 y† : k2

Substituting the expressions (33) into (32) and computing A~2 and A~3 we obtain, after repeated ~ y; s† as use of Eqs. (31a) and (31b), the elements …`ij ; i; j ˆ 1; 2; 3; 4† of the matrix L…q; `11 ˆ



1 k12

k22

k12

q2


p cosh…k1 y†

k22

q2

p…1 ‡ ms† ‰ cosh…k1 y† cosh…k2 y†Š; k12 k22  2 1 …k1 q2 p† …k22 q2 `13 ˆ 2 sinh…k y† 1 k1 k22 k1 k2   p…1 ‡ ms† 1 1 `14 ˆ 2 sinh…k1 y† sinh…k2 y† ; 2 k1 k2 k1 k2


 p cosh…k2 y† ;

`12 ˆ

p†

 sinh…k2 y† ;

M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

`21 ˆ `22 ˆ `23 ˆ `24 ˆ `31 ˆ `32 ˆ `33 ˆ `34 ˆ `41 ˆ `42 ˆ `43 ˆ `44 ˆ

es k12

k22



1 k12

k22 es

k12

k22 1

k12

k22 1

k12

‰ cosh…k1 y† …k12

 

q2

cosh…k2 y†Š; p†cosh…k2 y†

1 sinh…k1 y† k1 …k12



q2 k2

k1 k12

k22



1 k22

…k12

q2

es k22 1 k12

p†cosh…k1 y†

es k12

k22

‰ cosh…k1 y† 

1 k12

k1 q ‡ p

k22

…k12

q2

q2

 
pse p ‡ …1 ‡ ms† sinh…k2 y† ; k2

k2 sinh…k2 y†Š; …k22

q2

 p† cosh…k2 y† ;

cosh…k2 y†Š; k2 sinh…k2 y†Š;



k22

 p† cosh…k1 y† ;

 q2 p† sinh…k2 y† sinh…k1 y† ; k1  
pse q2 p ‡ …1 ‡ ms† sinh … k1 y † k2 k22 k1

‰k1 sinh…k1 y† 2

q2

…k22

p†

p…1 ‡ ms† ‰ cosh…k1 y† k12 k22 k12

…k22

 1 sinh…k2 y† ; k2

p…1 ‡ ms† ‰k1 sinh…k1 y† k12 k22 k12

1391

k22




 pse ‡ …1 ‡ ms† sinh…k1 y† k1

 2

k2 q ‡ p

k12




  pse ‡ …1 ‡ ms† sinh…k2 y† ; k2

cosh…k2 y†Š; p†cosh…k2 y†

…k22

q2

 p† cosh…k1 y† :

…34†

It should be noted here that we have repeatedly used Eqs. (31a) and (31b) in order to write (34) in the simplest possible form. Furthermore, it should be noted that the corresponding expressions for generalized thermoelasticity with two relaxation times in the absence of magnetic ®eld can be deduced by setting a ˆ 1 in Eqs. (31a) and (31b).

4. Application We consider the problem of a thick plate of ®nite high 2L and of in®nite extent with heating on a part of the surface. The initial state of the plate is assumed to be quiescent. Choosing the y-axis perpendicular to the surface of the plate with the origin coinciding with the middle plate, the region X under consideration becomes

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X ˆ f…x; y; z† :

1 < x < 1;

L < y < L;

1 < z < 1g:

Two parallel strips width 2L on the upper and lower surface of the plate are heated by time varying heat sources. The surfaces of the plate are taken to be traction free. The boundary conditions of the problem in the transformed domain are; thus …1†

rxy ˆ 0

on y ˆ L;

…35†

…2†

ryy ˆ 0 qn ‡ hh

on y ˆ L;

…36†

…3†



ˆ r …q; s†

on y ˆ L;

…37†

where qn denotes the normal component of the heat ¯ux vector, h is Biot's number and r…x; t† represents the intensity of the applied heat sources. We shall now use the generalized Fourier's law of heat conduction in the non-dimensional form, namely qn ‡ s

qn ˆ ot

oh : on

Taking the Laplace and Fourier transforms, this reduces to qn

ˆ

1 oh : 1 ‡ ss on

Using the last equation, condition (37) reduces to Dh ˆ …1 ‡ ss†…hh

 r † on y ˆ L:

…38†

The temperature and dilatation are given from Eq. (29) as h …q; y; s† ˆ `21 e0 ‡ `22 h0 ‡ `23 e00 ‡ `24 h00 ; e …q; y; s† ˆ `11 e0 ‡ `12 h0 ‡ `13 e00 ‡ `14 h00 : In case of symmetry as in the previous application these equations reduce to h …q; y; s† ˆ

e …q; y; s† ˆ

1 k12

2 X  … 1†i 1 ese0

k22 iˆ1 1

k12

2 X  … 1†i 1 ki2

k22 iˆ1

 2 q ‡ as2 ‡ se…1 ‡ ms†

q2

 ki2 h0 cosh…ki y†;

 p e0 ‡ p…1 ‡ ms†h0 cosh…ki y†:

…39†

…40†

Substituting from Eq. (24) into Eq. (20) we obtain …D2

k32 † u ˆ iq‰b2 …1 ‡ ms†h

where k32 ˆ a0 s2 ‡ q2 .

…b2

1† e Š;

…41†

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Substituting from Eqs. (39) and (40) into the right-hand side of Eq. (41) and solving the resulting di€erential equation, we get u ˆ C cosh…k3 y† ‡

iq k12

 …1 ‡ me† p…b2

2 X … 1†i 1  2 b se ‡ …b2 k22 iˆ1 ki2 k32

1† ‡ b2 q2 ‡ as2 ‡ se…1 ‡ me†

1† q2 ‡ p


 ki2 e0


 ki2 h0 cosh…ki y†:

…42†

Substituting (42) into (24) and integrating the resulting equation, we get v ˆ

2 X iqC 1 … 1†i 1  2 sinh…k3 y† ‡ 2 ki k3 k1 k22 iˆ1 ki2 k32

‡

q2 k12

2 X … 1†i k22 iˆ1 ki2

1

 2 b se ‡ …b2

‡ b2 …q2 ‡ as2 ‡ se…1 ‡ ms†

q2

1†…q2 ‡ p


p e0 ‡ p…1 ‡ ms†h0 sinh…ki y†   ki2 † e0 ‡ …1 ‡ ms† p…b2

1†

 ki2 † h0 sinh…ki y†:

…43†

The stress components can be obtained by substituting from the above equations into Eq. (22a)±(22d). The above approach gives the solution of the problem in the transformed domain in terms of three constants C, e0 and h0 which can be obtained from the boundary conditions of the articulate problem under consideration. rxx ˆ iqb20 C cosh…k3 y† …1 ‡ ms†‰p…b2 ‡



2 q2 b20 X … 1†i 1  2 b se ‡ …b2 k12 k22 iˆ1 …ki2 k32 †

1† ‡ b2 …q2 ‡ as2 ‡ se…1 ‡ ms†

2  …b20 2† X … 1†i 1 …ki2 2 2 k1 k2 iˆ1 2 X i 1  … 1† seb2 ‡ …b2

q2

‡ se…1 ‡ ms† 

ki2 †Šh0 cosh…ki y†

2 X  … 1†i 1 see0 iˆ1

ki2 †Šh0 cosh…ki y†

iq…b20

2†C cosh…k3 y†

q2 …b2 2† p†e0 ‡ p…1 ‡ ms†h0 cosh…ki y† ‡ 2 0 2 k1 k2

1†…q2 ‡ p

iˆ1

 ki2 † e0

1†…q2 ‡ p

 ki2 † e0

…1 ‡ ms†‰p…b2

1† ‡ b2 …q2 ‡ as2

b2 …1 ‡ ms† k12 k22

 2 q ‡ as2 ‡ se…1 ‡ ms†

 ki2 h0 cosh…ki y†;

…44†

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rxy

 2 …q2 ‡ k32 † iq X ki i 1 ˆ C sinh…k3 y† ‡ 2 … 1† ‰b2 se ‡ …b2 2 2 k3 k1 k2 iˆ1 …ki k32 † 1 ‡ ‰ki2 ki

q

2

2

2



2

2

p ‡ q …b se ‡ …b

1†…q ‡ p

 2 X i 1 ki …1 ‡ ms†  … 1† ‰p…b2 2 2 k k i 3 iˆ1 …1 ‡ ms†  p ki

ryy

ˆ

2iqb20 C 2

b 

…ki2

2

2

cosh…k3 y†

q

k12

2

2

1 …k12

k22 †

2

1† ‡ …q

iˆ1

‡ b2 …q2 ‡ as2 ‡ se…1 ‡ ms†

ki2 †

 

2 2

2

b q ‰b se ‡ …b

2†…ki2

2

e0 ˆ

   A1  ‡ A2

2iqA4 cosh…k3 L†

ki2 †Š

e0

  h0 cosh…ki y†;

seb2 …1 ‡ ms†Še0 ‡ …1 ‡ ms†‰…b20

…46†

2†p …47†

A5 2 …k ‡ q2 † sinh…k3 L† k3 3

 A6 2 2 …k ‡ q † sinh…k3 L† k3 3  #

A5 2 …k ‡ q2 † sinh…k3 L† ‡ 2iqA3 cosh…k3 L† k3 3

ki2 †Š

ki2 †Š ‡ b2 p…1 ‡ ms†

ki2 †Šh0 cosh…ki y†;

2iqA3 cosh…k3 L†

…45†

1†…q2 ‡ p ki2 †Š

where "  …1 ‡ ss† r 1 ‡ A2 A1

k22

h0 sinh…ki y†;

1†…q ‡ p

2

p†

iq k12

ki2 †Š

2

1†…q ‡ as ‡ se…1 ‡ ms†

q2

ki2 †Š



2

1† ‡ b2 …q2 ‡ as2 ‡ se…1 ‡ ms† 2

2 X i 1 … 1† ‰…b20

e0 sinh…ki y†

 2 2 2 X q …b 2† 2 i 1 … 1† ‰b se ‡ …b2 2 2 2 …ki k3 † k2 iˆ1

p† ‡ b se…1 ‡ ms†

‡ b …1 ‡ ms†‰q p…b

rzz ˆ

1

q2 …b20 2†…1 ‡ ms† ‰p…b2 …ki2 k32 † 2

2

1† ‡ b …q ‡ as ‡ se ‡ ms†

2

2

ki2 ††Š

1† ‡ b2 …q2 ‡ as2 ‡ se…1 ‡ ms† 2

q …p…b

1†…q2 ‡ p

1

;



M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

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 A5 2  2 2iqA3 cosh…k3 L† h0 ˆ …1 ‡ ss†r …k ‡ q † sinh…k3 L† k3 3    A6 2 2 …k ‡ q † sinh…k3 L† 2iqA4 cosh…k3 L†  A1 k3 3  1  A5 2 ‡ A2 …k3 ‡ q2 † sinh…k3 L† ‡ 2iqA3 cosh…k3 L† : k3 "   …1 ‡ ss† r  k3 A5 2 2 ‡ 2iqA Cˆ cosh…k L† …k ‡ q † sinh…k L† ‰A3 A2 A 3 3 3 3 A1 …k32 ‡ q2 † sinh…k3 L† k3 3    A6 2 2 …k ‡ q † sinh…k3 L†  A1 2iqA4 cosh…k3 L† k3 3  1 #  A5 2 ; ‡ A2 …k ‡ q2 † sinh…k3 L† 2iqA3 cosh…k3 L† k3 3 A1 ˆ A2 ˆ

A3 ˆ

es k12

2 X … 1†i 1 ‰ki sinh…ki L† 2

h…1 ‡ s† cosh…ki L†Š;

2 X … 1†i 1 …es ‡ q2 ‡ s2 2

ki2 †‰h…1 ‡ es† cosh…ki L†

k2 1

k12

k2 iq

k12

A4 A1 Š

iˆ1

iˆ1

 2 2 X …ki q2 † 2 i 1 ‰b es…1 ‡ ms† ‡ …b2 … 1† 2 ki …k12 k32 † k2 iˆ1

ki sinh…ki L†Š;

1†…q2 ‡ p

ki2 †Š

1 2 …q ‡ p ki

 ki2 †

 sinh…ki L†; A4 ˆ

iq k12

 2 2 X …ki q2 † i 1 … 1† ‰ 2 ki2 …k12 k32 † k2 iˆ1

 sinh…ki L†; 2 2q2 X A5 ˆ 2 … 1†i 2 k1 k2 iˆ1

1

 sinh…ki L†; 2 2q2 X A6 ˆ 2 … 1†i 2 k1 k2 iˆ1



1 …ki2

k32 †

2

2

p…b

1†…1 ‡ ms†

‰b es…1 ‡ ms† ‡ …b

2

2

b …q ‡ p

2

1†…q ‡ p

…ki2

ki2 †Š ‡

p …1 ‡ ms† ki

ki2 †Š ‡ b2 …ki2



1 ‰p…b2 1†…1 ‡ s† ‡ b2 …q2 ‡ p k32 †  ‡ b2 ‰p…1 ‡ ms† ‡ se ‡ q2 ‡ s2 ki2 Š sinh…ki L†: 1

2

ki2 †Š

This completes the solution of the problem in the transformed domain.

q

2



p

 se†

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5. Inversion of the transforms In order to obtain the solution of the problem in the physical domain, we have to invert the iterated transforms in (39),(42)±(47). These expressions can be formally expressed as function of y and the parameters of the Fourier and Laplace transforms q and s, of the form f …q; y; s†. First, we invert the Fourier transform using the inversion formula given previously. This gives the Laplace transform expression f…q; y; s† of the function f …q; y; t† as r Z 1 Z 1 1 2 iqx …48† e f …q; y; s† dq ˆ f…q; y; s† ˆ p f cos qxfe ‡ isinqxf0 g dq; p 2p 1 0 where fe and f0 denote the even and odd parts of the function f …q; y; s†; respectively. We shall now outline the numerical inversion method used to ®nd the solution in the physical domain. For ®xed values of q, x and y the function inside braces in the last integral can be considered as a Laplace transform g…s† of some function g…t†. The inversion formula for the Laplace transforms can be written as Z c‡i1 1 est g…s† ds; g…t† ˆ 2pi c i1

Fig. 1. Representation of the horizontal displacement distribution as a function of x on the surface plane.

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where c is an arbitrary real number greater than all the real parts of the singularities of g…s†. Taking s ˆ c ‡ iy, the above integral takes the form ect g…t† ˆ 2

Z

1 1

eity g…c ‡ iy† dy:

Expanding the function h…t† ˆ exp… ct†g…t† in a Fourier series in the interval ‰0; 2LŠ, we obtain the approximate formula (see e.g. [27]) g…t† ˆ g1 …t† ‡ ED ; where 1 X 1 g1 …t† ˆ C0 ‡ Ck 2 kˆ1

for 0 6 t 6 2L;

Fig. 2. Representation of the vertical displacement distribution as a function of x on the surface plane.

…49†

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M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

and Ck ˆ

i ect h ikpt=L g…c ‡ ikp=L† : Re e L

…50†

ED , the discretization error, can be made arbitrary small by choosing enough. As value of g…t† becomes the in®nite series in (49) can be summed up to a ®nite number N of terms, the approximate N X 1 Ck gN …t† ˆ C0 ‡ 2 kˆ1

for 0 6 t 6 2L:

…51†

Using the above formula to evaluate g…t†, we introduce a truncation error ET which must be added to the discretization error to produce the total approximation error. Two methods are used to reduce the total error. First, the Korrecktur method is used to reduce the discretization error. Next, the e-algorithm is used to reduce the truncation error and hence to accelerate convergence.

Fig. 3. Representation of the temperature distribution as a function of x on the surface plane.

M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

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Fig. 4. Representation of the stress distribution rxx as a function of x on the surface plane.

The Korrecktur method uses the following formula to evaluate the function g…t† g…t† ˆ g1 …t†

e

2cL

0 g1 …2L ‡ t† ‡ ED ;

0 j  jED j (see e.g. [27]). Thus, the approximate value of g…t† where the discretization error jED becomes

gNK …t† ˆ gN …t†

e

2cL

gN 0 …2L ‡ t†:

…52†

N 0 is an integer such that N 0 < N . We shall now describe the e-algorithm, which is used Pm to accelerate the convergence of the series (51). Let N be an odd natural number and let Sm ˆ kˆ1 ck be the sequence of partial sums of (51). We de®ne e-sequence by e0;m ˆ 0, e1;m ˆ sm and en‡1;m ˆ en

1;m‡1

‡

1 ; en;m‡1 en;m

n; m ˆ 1; 2; 3; . . .

…53†

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M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

Fig. 5. Representation of the stress distribution ryy as a function of x on the surface plane.

It can be shown (see e.g. [28]) that the sequence e1;1 ; e3;1 ; . . . ; eN;1 converges to g…t† ‡ ED …c0 =2† faster than the sequence of partial sums sm ; m ˆ 1; 2; . . . The actual procedure used to invert the Laplace transforms consists of using Eq. (52) together with the e-algorithm. The values of c and L are chosen according the criteria outlined in (see e.g. [27]). The last step in the inversion is to evaluate the integral in (48). This was done using Romberg integration with adaptive step size. This method uses the results from successive re®nements of the extended trapezoidal rule followed by extrapolation of the results to the limit when the step size tends to zero. The details can be found in (see e.g. [28]). The whole process was carried out on a 586 personal computer. 6. Numerical results In order to illustrate the above results graphically the source r…x; t† was taken in the following form: r…x; t† ˆ H …x

jaj†H …t† exp… bt†;

M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

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where a and b are ®xed constants and H denotes Heaviside's unit step function. This represents a localized heat source acting in the region a 6 x 6 a starting at t ˆ 0 with a value of unity and exponentially decaying in time. The double transform of r…x; t† is given by r 2 sin…qa†‰1 ‡ iqpd…q†Š r …q; s† ˆ ; p q…s ‡ b† 

where d…q† denotes the Dirac delta function. The copper material was chosen for purposes of numerical evaluations. The numerical technique outlined above was used to invert the iterated transforms in Eq. (39) giving the temperature and (42)±(47) giving the displacement components u, v and the stresses components rxx ; ryy ; rxy ; rzz on the plate …y ˆ 1†. The results are shown in Figs. 1±7, respectively. In these ®gures, the solid and dotted lines represent the solution obtained in the absence and in the presence of the external magnetic ®eld, when the medium is a perfect electric conductor. The phenomenon of ®nite speeds of propagation is manifested in all these ®gures. We note that since the displacement v is an odd function of y, its value on the middle plane is always zero. It can be seen from these ®gures that the magnetic ®eld acts to decrease both the magnitude of the stress

Fig. 6. Representation of the stress distribution rxy as a function of x on the surface plane.

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Fig. 7. Representation of the stress distribution rzz as a function of x on the surface plane.

and the magnitude of the displacement components. This is mainly due to the fact that the e€ect of the magnetic ®eld corresponds to term signifying positive forces that tend to accelerate the metal particles.

7. Concluding remarks The importance of state space analysis is recognized in ®elds where the time behavior of physical process is of interest. The state space approach is more general than the classical Laplace and Fourier transform techniques. Consequently, state space is applicable to all systems that can be analyzed by integral transforms in time, and is applicable to many systems for which transform break down (see e.g. [29]). Owing to the complicated nature of the governing equations for the generalized magneto± thermoelasticity with two relaxation times, few attempts have been made to solve problems in this ®eld. These attempts utilized approximate method valid for only a speci®c range of some parameters. In this work, the method of direct integration by means of the matrix exponential, which is a standard approach in modern control theory and developed in detail in many texts (see e.g. [30]),

M.A. Ezzat et al. / International Journal of Engineering Science 39 (2001) 1383±1404

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is introduced in the ®eld of electromagneto-generalized thermoelasticity with two relaxation times when the medium is a perfect conductor and applied to two-dimensional problem in which the temperature, displacements and electromagnetic ®eld are coupled. This method gives exact solutions in the Laplace transform domain without any assumed restrictions on either the applied magnetic ®eld or the temperature and the displacement distributions. The method used in the present work is applicable to a wide range of problems. It can be applied to problems which are described by the linearized Navier±Stokes equations (see e.g. [31]). 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]

M. Biot, Thermoelasticity and irreversible thermo-dynamics, J. Appl. Phys. 27 (1956) 240±253. H. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solid. 15 (1967) 299±309. R. Dhaliwal, H. Sherief, Generalized thermoelasticity for an isotropic media, Quart. Appl. Math. 33 (1980) 1±8. H. Sherief, Fundamental solution of the generalized thermoelastic problem for small times, J. Therm. Stresses 9 (1986) 151±164. H. Sherief, M. Anwar, Problem in generalized thermoelasticity, J. Therm. Stresses 9 (1986) 165±181. H. Sherief, M. Ezzat, Solution of the generalized problem of thermo-elasticity in the form of series of functions, J. Therm. Stresses 17 (1994) 75±95. H. Sherief, M. Anwar, State space approach to two-dimensional generalized thermoelasticity problems, J. Therm. Stresses 17 (1994) 567±590. I. M uller, The coldness, a universal function in thermo-elastic solids, Arch. Rat. Mech. Anal. 41 (1971) 319±332. A. Green, N. Laws, On the entropy production inequality, Arch. Rat. Mech. Anal. 54 (1872) 47±53. A. Green, K. Lindsay, Thermoelasticity, J. Elast. 2 (1972) 1±7. E. S ß uhubi, Thermoelastic solids, in: A.C. Eringen (Ed.), in: Cont. Phys. II, Academic Press, New York, 1975 (Chapter 2). S. Erbay, E. S ß uhubi, Longitudinal wave propagation in a generalized thermo-elastic cylinder, J. Therm. Stresses 90 (1986) 279±295. J. Ignaczak, A strong discontinuity wave in thermoelastic with relaxation times, J. Therm. Stresses 8 (1985) 25±40. J. Ignaczak, Decomposition theorem for thermoelasticity with ®nite wave speeds, J. Therm. Stresses 1 (1978) 41±52. M. Ezzat, Fundamental solution in thermoelasticity with two relaxation times For cylindrical regions, Int. J. Eng. Sci. 33 (1995) 2011±2020. G. Paria, On magneto-thermoelastic plane waves, Proceeding, Cambridge Phil. Soc. 58 (1962) 527±531. A.J. Wilson, Proceeding, Cambridge Phil. Soc. 59 (1963) 483±488. G. Paria, Magneto-elasticity and magneto-thermoelasticity, Adv. Appl. Mech. 10 (1967) 73±112. A. Nayfeh, S. Nemat-Nasser, Electromagneto-thermoelastic plane waves in solids with thermal relaxation, J. Appl. Mech. E 39 (1972) 108±113. S. Choudhuri, Electro-magneto-thermoelastic plane waves in rotating media with thermal relaxation, Int. J. Eng. Sci. 22 (1984) 519±530. H. Sherief, Problem in electromagneto-thermoelasticity for an in®nity long solid conducting circular cylinder with thermal relaxation, Int. J. Eng. Sci. 32 (1994) 1137±1149. H. Sherief, M. Ezzat, A thermal shock problem in magneto-thermoelasticity with thermal relaxation, Int. J. Solids Struct. 33 (1996) 4449±4459. M. Ezzat, Generation of generalized magneto-thermoelastic waves by thermal shock in a perfectly conducting halfspace, J. Therm. Stresses 20 (1997) 617±633. M. Ezzat, State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity, Int. J. Eng. Sci. 35 (1997) 741±752. H. Sherief, M. Ezzat, A problem in generalized magneto-thermoelasticity for an in®nitely long annular cylinder, J. Eng. Math. 34 (1996) 387±402.

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[26] M. Ezzat, M. Othman, Electromagneto-thermoelastic plane waves with two relaxation times in a medium of perfect conductivity, Int. J. Eng. Sci. 38 (2000) 107±120. [27] G. Honig, H. Hirdes, A method for the numerical inversion of the Laplace transform, J. Comp. Appl. Math. 10 (1984) 113±132. [28] H. William, in: Press al. Numerical Recipes, Cambridge University Press, Cambridge, 1986. [29] D. Wibery, Theory and problems of state space and linear system, in: Schaum's Outline Series in Engineering, McGraw-Hill, New York, 1971. [30] K. Ogata, in: State Space Analysis Control System, Prentice-Hall, Englewood Cli€s, NJ, 1967 (Chapter 6). [31] M. Ezzat, State space approach to unsteady two-dimensional free convection ¯ow through a pours medium, Can. J. Phys. 72 (1994) 311±317.