on the plane problem of orthotropic quasi-static thermoelasticity

JournalofElasticity 41: 161-175, 1995. 161 © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

On the Plane Problem of Orthotropic Quasi-static Thermoelasticity

LIN WEI 1 and ZHAO YU-QIU 2 1Mathematics Department, ZhongShan University, GuangZhou 510275, China 21nstitute of Computer Applications, GuangZhou 510075, China

Received 20 December 1994; in revised form 9 August 1995

Abstract. The plane displacement boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using an accurate estimate and the contractive mapping principle. Representation of solution of the field equation is obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.

Mathematics Subject Classifications (1991): 35K50, 73C03, 73C25.

Key words: thermoelasticity, displacement boundary value problem, orthotropic, decoupling, singular integral equation, contractive mapping principle.

1. Introduction

The investigation of the influence of the temperature distribution in an elastic body on the internal stresses and strains is a main interest of the theory of thermoelasticity. One of the difficulties in solving the problems of thermoelasticity lies in the fact that temperature and stress-strain are generally coupled. Essentially, a non-homogeneous temperature field brings about internal heat stresses, while the stretch and constriction caused by such heat stresses obstruct the heat conduction in the elastic body and consequently produce an effect on the temperature distribution. Hence it is worthwhile to seek an effective approach to decouple the system. Some work has been done on this subject. For example, Shi and Xu [7] and Zhao [9] discuss the homogeneous isotropic plane thermoelasticity problems corresponding to displacement boundary value conditions. In this paper, we consider mainly the orthotropic case. This time there are different elastic material constants along different directions, and the elastic body possesses two orthogonal principal directions. In the discussion that follows, the system is decoupled and a further study of the decoupled heat equation is carded out. The results are of interest both in theoretical analysis and in numerical computation.

Let ft be a bounded simply-connected domain on the plane. For a homogeneous orthotropic elastic body with reference configuration f~, suppose the x- and y-axes are the two principal directions, then the direct stresses try, ay, the shear stress rxu

1 6 2 LIN WEI AND ZHAO YU-QIU

and the displacement u = (u, v) have the following stress-strain relations (see, e.g.

a= - 1 - v12v21 ~ + v12E22~y/ '

( _ 1 v 2 1 E l l ~ + E22 , ~ Y - 1 - ~12~21

r ~ = G ~ + ~ ,

Lekhnitskii [5]):

(1.1)

where El l and E22 are respectively the Young's moduli along x and y, and G12 is the shear modulus, v~2 is a Poission's ratio, which describes the constriction in y-direction caused by stretch in x. v2~ has an analogous meaning only with y and x interchanged. These constants obey the following symmetric property: v21Ell = v12E22. A n d 0 < v~2 < ½,0 < v2~ < ½.

The quasi-static equilibrium equation (the field equation) and the heat equation are

Oa~ ~ ~0 Oy = ~yy' (1.2)

(1.3)

Oa: Or:~ = ~ O0 Or:~ ~ + ~ ~ , o~ +

Oo 0 qAO = p s - ~ + flc~-~ divu,

where 0 is the temperature, ~ denotes the interaction constant, q, p, s and c denote respectively the heat conductivity, the density, the specific heat and the absolute reference temperature. All the above constants are positive.

In this paper we consider (1.2) and (1.3) subject to the following conditions:

uls r = O, vls T = 0, (1.4)

01st = o, 0(x,y;O) = o0(x,y) , (1.5)

where S = 0f~ is the boundary off~, ST = S × [0, T], f/T = ft × [0, T], T > 0. The rest of this paper is organized as follows: In Section 2, we rewrite (1.1),

(1.2) and (1.3) in their equivalent complex forms, and obtain some estimates of norms. In Section 3, by solving a system of singular integral equations, we express the displacements u and v in terms of the temperature 0, hence to show that the system is decoupled. A further study of the decoupled heat equation is presented in Section 4, which leads to some contractivity and solvability results.

2. Complex F o r m and Est imate of Norms

Denoting w = u + iv, z = z + iy, we then have

Ou Ov Ow div(u, ~ ) = ~ + ~ = 2r~e Oz"

ORTHOTROPIC QUASI-STATIC THERMOELASTICITY 163

Hence heat equation (1.3) turns equivalently to

., o{ .o} qAO = ps-~ + 2flc~-~Re ~ z " (2.1)

Now we want to turn (1.1) (1.2) into complex forms. To this end, we utilize some results and techniques from Hua, Lin and Wu [3], Gilbert and Lin [2] and Begehr and Lin [1]. Set

E ~ 2 ( 1 /.,2). K - 2G1-----~ u, L = - - -

Using these notations and substituting stress-strain relation (1.1) into equilibrium equation (1.2), we get

E { [ 62 L] 02 /: --~ L

O0

(2.2) = fl 00

We know that0 < u12 < 1 and0 < u21 < 1, hence62 > u,6 -2 > u , K > - u > -1 . Since the case K = 1 coincides with the isotropic case discussed in Zhao [9], we need only consider the problem corresponding to (2.1), (2.2), (1.4), (1.5) in the following two cases: K > 1 and ]K[ < 1.

We are now in position to deduce the complex form and the norm estimate when K > 1. Letting k = K - ~ - 1, A = kL, carrying out the following transformations (transformations of coordinates and unknowns and linear combination of equations):

v -6 V

and multiplying (2.2) on the left by

1 + k u , (2 .5)

-6

164 LIN WEI AND ZHAO YU-QIU

we have thus turned (2.2) to

1 - v 2

1]o2 A - 1

A-- k 2 k

Ox o---T + k -Y~ v

= /3

v'~ k+ u O0 6 1 + kv OX

- 6 ~ o°°y

(2.6)

Next, multiplying (2.6) by 1 × 2 matrix (1 - i) on the left, one obtains

k - A 02W k + A 02ITI ,r

2k ozoz-------~ + 2-----£- o2oz-----i

Zo~ re-, oo ~oo 1 L ° o -x+ oyj

= _ ~2) (90 ~2 O0 ,~0 4 [(CO 1 ,~"~ + ( c o l + ) ~ - - ~ ] ,

(2.7)

~(1--V 2) l+kv, and where n o = E , c o = k+v

k + l Z k _~- i W = U + i V , Z = X + i Y , Z ~ - 2----k + Z = X + - ~ Y ,

( o) 0 1 0 -z-~-~ o z - ~ U~

o 1(o o) o z - ~ -5Y + '-6V '

( o) 0 l + k 0 1 - k 0 _ 1 0 -zk~-~- Ogl T 0--2 + 2 o z 2 - ~

0 1 - k 0 l + k 0 1 ( 0 . 0 ) O Zl --5-o--2 + 2 o z - ~ -b-~ + ~k-~-f "

This is the complex form we are seeking. From (2.7), one gets

k-~ ,ow k+~ (ow) 2--U o z-S + - -

= 3o-T[ (c o - e2)n~zo + (co 1 + ,~2)o] + ~,(z), (2.8)


where flz is the image of fl under the mapping z --~ Z, and ~o(2) is an analytic function on ~2 with respect to Z. T and 1-[ are Vekua's operators (see, e.g. Vekua [8]):

1/o f(z) daz, (2.9) (TDf)(() = - ~ z - (

l ID f(z) dcrz, (2.10) (IIDf)(() = -- 7 ( ; _--~)2

where daz is the area element. (2.8) implies

A + k, -1 A -- k. 1 _ 62)iin.0] o w _ ~o-~ [(c~ 1 + ~ o + ~x--t~o - ~ . , o + - ~ 021 J

+ - ~ - - ~ + ~ - - ~ o ( Z ) . (2.11)

Hence { .,,~ A + k , -1

W o.1 + + 62)~-~0

- I - - ~ (c~-l--~2)Ilfl~O])

A + k A - k -~ 2A(1 + k) ~(Z) + 2A(1 - k) ~(Z) + ql(Z1). (2.12)

Here ~ ' (Z) = 2~p(Z), 9tz~ is the image of 9t under mapping z ~ Z1, and ~(Z1) is an analytic function on ftz~ with respect to Z1.

In the section that follows, • and • in (2.12) are to be determined (expressed in 0) by using condition (1.4). But by now we restrict our attention to the estimation of norm, using the above equality and (2.11). We notice from transformations (2.3) and (2.4) that

2 R e ~ : = 1 t~x/~[(co - 62) ReWz1 + (co + t52) Re Wgl].

Hence from (2.11) and (2.12), it follows that

2Re~--~-Wz = c~0 + M0,

¢~o ~ = ~(~o + ~ ) ( ~ + ~),

1 [(co - 62) Re WZl M - 6 ~

(2.13)

(2.14)

166

where ReWz~ can be derived from (2.12). Now we claim that the following estimate holds:

IIM 011L2(a) ~< ~ll011L~(a)-

In fact, we first derive the Green's equality

/o {(,,>([,, o, 1 - O-~x2 + [ v + L

+I (5 2 v 0 [

l +(u~v~u~,+v~) v ,5 -2 0 0 0 L uy

[ (xnu + ynv) ds = 0, d8

LIN WEI AND ZHAO YU-QIU

(2.15)

v+ L] 0 2

&rOy

u~ } vu dx dy

+ vx

(2.16)

= 28 £ 0(Re w~) dx dy. (2.17)

where x,~ = nlax + n2rz~,yn = nlrxy + n2ay, and (nl,n2) is the unit external normal vector at the boundary point (z, y) E Oft. Notice that

u~ = Re Wz + Re we,

vy = Re Wz - Re w~,

uy = - I ra Wz + Im We,

v~ = Im Wz + Im we.

From (2.16), it follows that

l - v 2 [(,52 + 2v + ,f-2)(Rewz)2 + 2(,52-,5-2)Rew,,Rew~+

+(6 2 - 2v + ~-e)(Re we) 2 + 4L(Im we) 2] dz dy

f =-28__jnRe \ Ozz] dxdy

= 2~Re [[ Ow o Oz dx dy]


(62-6-1) 2 4( l -v 2) Setting Cl = (6 2 + 2u + 6 - 2 ) - 6~,_2v+6_2 = 62_2v+6_ 2 , one gets

Cl fa(Rewz)E dz dy

[cl(Rewz) 2 + ( .~2 -- ~_2 <. Rew. ~ , \ v / 6 2 - 2 v + 6 - 2

2 (2.18)

+ x /62 -2v+6-2Rew~) +4L(Imw~) 2] d x d y

2flo fn 0(Re w,) dx dy.

Noticing that 62 > v, 6 -2 > v, we are able to show that ~ • a f> rio. Hence 4 (2.18) and (2.14) suggest that

el ( a21[0[[2 + 2 a / n 0MOdtr + [[M0[[2)~< flo/n 0M 0dtr+ aflo[10[[ 2.

This in turn implies

a ( 4 a - riO)[[0[[2 + 4[1M0[12 ~ < - ( ~ ' ~ - r i o ) fn OMOdtr

~< - (-c2-~--flO) [[O[I[[MO H,

i.e.

( 4 ~ ) (a[[OH - HMO[[)2 + -~ HMO[[2 "< fl°HO[[HMOH"

Thus the fact ~t _ ~0~ /> 0 suggests [[M011 ~< ~11011. This is (2.15). Similar discussion is valid in case [K[ < 1. This time we give only a brief

description of the complex form and the estimate of the norm. Setting p -- ( 1 ~_+_z_+_~h vS-di-s-g, k = p2, A = p2 \ l - , ], carrying out the transformation of coor-

dinates [' ] and the transformation of unknowns

1 - v 1

- 6 (1 - u)6 2p(1- L)

(2.19)

A - k 2 k

A - k 2 k

(2.20)


and multiplying (2.2) on the left by

1

1

1 - v 1 - v

A - 1 ( A - 1)t5 2 p ( 1 - L ) 2 p ( 1 - L )

, (2.21)

turns (2.2) to

{[ 1

A 0 2

+

A - 1

.~ - - k 2

k 02 02 }

OXOY + [ A

flO 2

6" OO P -1 6) OO 1 (6_ l + + 1 - v - )0--X 1 - - - ~ (6 OY

k - 1 .6_ 1 6" 00 A - 1 /$. 00 2p-(-l-"L) ( + )O-X + 2(1 L) (6-1 - )O--Y

(2.22)

The left hand side of (2.22) can be rewritten as an elliptic operator D

k--)~ 02W k q - ~ 1921/v/,r

D W = 2k cOZOZ-------7 + 2----~ tgZOZ1 (2.23)

in the same way as discussed before concerning (2.6). Thus ~ and ~ can be

expressed by 0 and by the analytic functions ~(Z) and ~(Z1 ) (the way to determine • (Z) and k~(Z1) is to be shown in the section that follows). So we can write

2Re Wz = aO + M 0,

where M 0 is an expression in term of 0, • and 9. The previous routine (2.16)- (2.18) is still valid, that is to say, in each case, our discussion leads to the estimate (2.15):

IIMOIIL (m llellL=(m.

3. So lu t ion

In the previous section, our discussion was divided into two parts: one for the case K > 1 and the other for IKI < 1; equation (2.2) is then reduced to Standard forms (2.7) and (2.22), respectively. In each case, there is an expression 2Re wz = aO + MO. Moreover, we have also shown that IIMOIIL2(m .< IIOlIL2( ) holds, where M depends on ~(Z) and ~(Z1).


In order to determine M so as to decouple the displacement-temperature system, it is necessary to solve • and q (in 0) by using boundary condition (1.4). For the purpose of succinctness, we unify (2.7) and (2.22) in the following form:

/ 1

A °2 I A - 1

A-- k 2 k

0 2 0 2 }

[Tl 0 ] T2(o)

(3.1)

Similarly to (2.9), (2.11) and (2.12), (refer to Hua, Lin and Wu [3] as well), W = U + iV can be expressed as follows:

A + k A - k W + T(O)+ 2A(1 + k ) O(Z)+ 2A(1 - k) q~(Z) + IIl(Zl), (3.2)

whereT(0) = T•z ' [ ~ T a z ( T l ( O ) - iT2(O)) + ~Tf~z(TI(O ) - iT2(O))].The unknown functions ~ (Z) and • (Zl) are respectively analytic in f~-z and flz,. Then the boundary condition (1.4) reads

Wlzeon = 0. (3.3)

a-k ~+k -T(O) = Uo(O) + iVo(O), where Uo(O) If we denote a = ~ , b = 2~0+k)' and Vo(O) are real, then (3.2) and (3.3) yield, on the boundary

(a + b) Re~(Z) + Re ~(ZI) = (Uo(O))(Z),

(a - b) ImP(Z) + Im q(Z1) = (V0(0))(Z). (3.4)

Assume that (3.4) has a solution of the following form (for background knowl- edge of systems of singular integral equations, see e.g. Muskhelishvilli [6])

¢(z) = a z r - z '

l f0 #2(r) drl L fo III(ZI) = ff-i DZl ~1 -~ Zll + D z # 3 d S ' ~ i Dz#4ds '

(3.5)


where #1, #2, #3 and #4 are unknown real functions, then on the boundaries of £tz and £z~, we have

• (t) =

• ( t l ) =

1 f #1(7-) dr + Jo ' t ~ OS2z, r'l~', 7ri f~Z 7" - t

1 ~ #2(7-) dT-1 p2(t)+ 7r--~ f~zl r l - t l + a z #3ds

+i~o #4ds, tl E OQzl, 12z

(3.6)

according to the Plemlj-Sohochii formula. Hence

a e ~ ( t ) - #l(t)"JC L Re [ ~ 7"t 1 az Lrir - tJ #l(r)ds,

I m ~ ( t ) = f a im[1 r ' ] . z ~-7 7- ------/ #1(7-) ds,

Re~( t l )=#2( t )+ Re ~i7-1 t-----~ #2(7-)ds+ #ads, ~z fG

Img( t l )= f~z Im ~irl t-----~ n~

(3.7a-d)

Here s is the arc length parameter of O~z and C = d~ -a~, r~ = -~. By substituting (3.7a-d) into the boundary condition (3.4) and adding two additional equalities, it follows that

g [1<] ( a + b ) # l ( t ) + # 2 + ( a + b ) Re ; i r - - - t #l(r)ds 12z

(a-b) faf~z Im [ l r _ r-I t] #l ( r )ds+ fa .z Im [1 i rl-r~tl] #2(r)ds

+ [ #4 ds = Vo, Ja f2z f

- c o l # 3 d s = 0, #3 JO f~z

-- C O / #4 ds = O, #4 da f~z

(3.8)


where co fO~z ds = 1. On introducing the operator S,

S(~l , ~2, #3, #4)

a + b 1 0 0

0 0 0 0

0 0 1 0

0 0 0 1

#2

#3

#4

+ f oz

(1 r ' ) (1 r__.~ ) 1 0

( 1 C ) ( 1 r_____~ ) 0 1 ( a - b ) I m (rir----tt Im ~ir l tl

0 0 -co 0

0 0 0 -co

#1

#2

#3

#4

ds,

the above system of singular integral equations has the compact form

i] S(# l , #2, #3, #4) = V0 . (3.9)

The characteristic operator corresponding to S is

B fo #dr S ° ( # ) = A # + r-i ~ z r - t ' (3.10)

where

#2 # = ; A =

#3

#4

a + b 1 0 0

0 0 0 0

0 0 1 0

0 0 0 1

, B =

0 00i] - (a + b)i - i o

0 0 0

0 0 0

Hence the index of the singular integral operator S is

a(S) = 2-~[arg{det((A - B)(A + B)-I)}]o~ = 0. (3.11)


By the introducing of #, we can conclude that if # = (]z1,~2,~3,]~4) is a solution of the homogeneous equation corresponding to (3.9), then ~ and • satisfy the homogeneous condition corresponding to (3.4), that is

z e a .

This implies that ¢ and • are constants. (In fact, differentiating with respect to 21, one gets

), k ¢ ' ( z ) + = o

A A

which implies ¢ ' ( Z ) - O, thus • is a constant and so is ~.) Set O(Z) = C1 + iC2, then ~ ( Z1) = - ( a + b)C1 - i(a - b)C2. (3.6) suggests

L /~I(T) d " " - ~ T -- CI + iV2, ~ri fOnz v - Z

-~Z flZ 1 7" 1 - - Z 1 + flz #3 ds + i nz tz4 ds = - ( a + b)Cl - i(a - b)C2.

Now it is easy to show that C2 = 0. In fact, the first equality indicates that #1 (v) - c, +ic22 = (#1 (v) - -~2 ) - i - ~ is the boundary value of a certain function, analytic outside f~ and vanishing at co. Since the imaginary part of the boundary value coincides with the constant -_~z on 0fl and vanishes at 00, one concludes C2 -- 0, hence ~I(T) ---- ~-L,~ 4 ------ 0, and 2#2 + Co1#3 = --(a + b)C1. A complete set of solutions to the homogeneous equation corresponding to (3.9) (i.e. S# = 0) is

(1, - ( a + b), 0, 0), (1,0, -co(a + b),0). (3.12)

Since the index n = 0, the Notherian theorem says that the real conjugate equation possesses two linearly independent solutions. Notice that

fa ( 0 0 1 0 ) S ( # ) d s = 0, ~z

fa ( 0 0 0 1 ) S ( # ) d s = 0 , 12z

hence a complete set of solutions of the homogeneous conjugate equation is

(0, 0, 1, 0), (0, 0, O, 1). (3.13)

It is then obvious that (U0, V0, 0, 0), the right hand side of (3.9), is orthogonal to both of them. This shows that (3.9) is solvable (for its various solutions, see,

ORTHOTROPIC QUASI-STAT|C THERMOELASTICITY 173

e.g. Muskhelishvilli [6]). Hence ~ and • can be determined through solving # = (#1,#2,#3,#4) from (3.9). This provides us with an expression of W in 0. Thus systems (1.3) and (1.2) are decoupled, so that we are led to solve an integro-differential equation in the temperature 0.

4. Contractivity and Unique Existence of Solution

According to the discussion in the above two sections, we have obtained an expression of W in terms of 0, hence we have

and

Ow 2Re-~- z = c~0 + M 0 (4.1)

IIM011L2(~) ~ ~II011L~(~)- (2.15)

Substituting (4.1) into the field equation (2.1), and using boundary condition (1.5), we obtain the decoupled integro-differential equation for the temperature 0 subject to the initial-boundary value conditions

Ot - a]AO = b l M O t =- MoOt, ]

O(z, 0) = O0(z), / (4.2)

OIsT = O,

_ ___P_z__ q bl = Obviously, Mo = bl M, a linear operator on where a I = ps+[~cc~, ps+~cc~" L2(f~), satisfies the following estimate of the operator norm:

/3ca IIM011op ~ ps +/~c~ - 7 < 1. (4.3)

As a preparation for the discussion of the decoupled heat equation (4.2), we introduce several function spaces as below:

W 2 ' I ( ~ T ) = {WlW; Wx, Wy, Wt; Wxx, Wxy, Wyy E L 2 ( ~ T ) } (4.4)

with norm

Ilwllw~,,(~T)

= [[_ (Iwl 2 + Iwxl z + Iwul 2 + Iwtl z + k d i t T

+lw~yl z + Iwy~l z) dz dydt ] l/Z .


2,1 We denote by W~, 0 (fT) the completion of the family of C°°(fT) functions 2,1 vanishing in a neighbourhood of ST in W~ (fiT). There is a result concerning the

norms of W~,J ( fT ) .

2,1 LEMMA 4.1. (Ladyzhenskaya [4]). Assume Of~ E C 2, then W~, o ( fT ) possesses an equivalent norm

[[WIIW~,~(~T) = [/~T(IwtlE + IAAwIE)dxdydt] ~/2 (4.5)

where the constant A ~ O. ([4], p. 109-113, in fact, this is a combination of the statement (2.7) on p. 109 of [4] and Remark 2.1on p. 113 of the same book.) []

Up to now, we can conclude the following theorem.

THEOREM 4.2. Assume that 19o E Hd( f~ ), then there exists a unique solution in 2,1 W~,o ( f~ T ) to (4.2) and the solution is analytic in the space variables for almost all

fixed t E [0, T]. []

On the basis of the estimate (2.15), the proof of the above theorem can be found in Shi and Xu [7], and Zhao [9]. For the purpose of clarity, we give a brief account of the proof here.

In fact, for 0 E W~,~ (f~T), define 19 = FO by the following relation:

Ot - alAO = MOt, )

o ( z , o ) = o0(z),

Olsr = 0.

(4.6)

Since 0 E W~,'d(fT) is uniquely determined (see, e.g. [4], p. 112-113), the 2,1 above F is a map from W~, 0 ( f T ) to itself. To prove the unique solvability, it

suffices to prove that F is a contractive map of W~'~(fT). It makes no difference if we set Oo to be zero. (4.3) and (2.15) suggest that

T 2 2 < J -4- 7 IlO:llL:(n ),


where

J = 2 a l f n OtAOdcrzdt T

= -2a l fn (OxtOx + O~tO~)dcrz dt r

T 0 d ~ ] = --~1 ~ ~ [ f ~ ( ~ : + ~ ) d~

:

~ 0 .

Hence

2 2 2 .210 t IW, Z't/'-'12 (4.7) " II0elIL (.T) r , . 2,0 k a~T )

Equation (4.7) shows ~at F is a conCactive mapp~g of W ~ (~r). ~ i s proves • e ~ q u e soNabifi~ of (4.2). ~ e fact ~at F is con~active is also of ~terest ~ numefic~ compu~fion.

Acknowledgement

This research is supported in part by Mathematical Tlan Yuan foundation of China.

References

1. H. Begehr and W. Lin, A mixed-contact boundary problem in orthotropic elasticity. In: H. Begehr and A. Jeffrey (eds), Research Notes in Maths, series no. 263. Pitman Publishing Ltd. (1992) pp. 219-239.

2. R. P. Gilbert and W. Lin, Function theoretic solution to problem of orthotropic elasticity. J. Elasticity 15 (1985) 143-154.

3. L. Hua, W. Lin and Z. Wu, Second Order Systems of Equations on the Plane. Pitman Publishing Ltd. (1985).

4. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics. Springer, New York (1985).

5. S. G. LeLhnitskii, Anisotropic Plates. Gordon and Breach, New York (1968). 6. N. I. Musl~elishvilli, Singular Integral Equatio~ Holland, Nordboof (1953). 7. P. Shi and Y. Xu, Decoupling of the quasi-static system of thermoelasticity on the unit disk. J

Elasticity 31 (1993) 209-218. 8. I. N. Vekua, General&edAnalytieFunction. Addison-Wesley (1962). 9. Y. Q. Zhao, On plane displacement boundary value problems of quasi-static linear thermoelas-

ticity. Applicable Analysis 56 (1995) 11%129.

on the plane problem of orthotropic quasi-static thermoelasticity

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