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Some theorems in generalized nonlocal thermoelasticity
Inl. 1. EngngSci. Vol. 32, No. 3, pp. 473-479. 1Y94 Copyright fQ 1YY4 Elscvicr ScicnccLtd Printed in Great Britain. All rightsrcscrvcd 0020-7225/W $6.00 + O.(x)
Pecgamon
SOME THEOREMS IN GENERALIZED THERMOELASTICITY RANJIT
S. DHALIWAL
NONLOCAL
and JUN WANG
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N lN4
Abstract-A
reciprocity theorem in the context of the generalized thermoelasticity is established. A uniqueness theorem for the mixed initial boundary value problem is also proved without assuming the positive definiteness of the elastic constants. 1. INTRODUCTION The nonlocal theory of continuum mechanics is of recent origin. It takes into account the fact that the interacting forces between material points are far reaching in character. From the systematic point of view, it differs from the classical theory by assuming the balance laws to be valid only on the whole of the body. In some cases, such as in phonon dispersion in solids, in electromagnetic solids, in fracture mechanics, when the continuity hypothesis made in classical theory can no longer be supported because of the discrete characteristics of the medium, only a nonlocal theory might provide the right answer while the classical ones would fail. The references to the physical background of nonlocal theory of continuum mechanics can be found in Kunin [l] and Rogula [2]. Eringen [3,4] developed the linear theory of nonlocal elasticity and thermoelasticity in 1972 and 1974, respectively. It is shown in [3] that the elastic wave propagation is dispersive by employing the nonlocal theory while the classical theory gives no dispersion. Also, using nonlocal theory to solve the crack problems in [5], it is shown that the crack tip singularity in stress appearing in classical solution disappears. In recent years physicists have also been giving increasing attention to the so-called “second sound” effect in solids. This effect arises from the possible transport of heat by a wave propagation process rather than diffusion. As is well known, the diffusion equation predicts an infinite velocity of propagation for thermal disturbances, which is contradictory to the physical observations. It seems reasonable to modify the coupled theory of heat conduction to allow time for acceleration of the heat flow in response to an applied temperature gradient. Such an effect has been exhibited in liquid helium and there is speculation concerning its possible existence in solids. In 1977, Balta and Suhubi [6] extended Eringen’s work by including the temperature rate among the constitutive variables and formulated a generalized theory of nonlocal thermoelasticity. A comprehensive discussion of the literature concerned with theories of generalized thermoelasticity can be found in Chandrasekharaiah [7] and Gncti and Moodie
Bl. Owing to the newness of the nonlocal theory, many aspects are open to study. One of the most important questions is whether the initial-boundary value problems defined in the frame of nonlocal thermoelasticity are well-posed. For this reason, we are going to study that under what conditions the solutions of those problems are unique. We prove a reciprocity theorem and a uniqueness theorem for the solution of the mixed initial boundary value problem in generalized nonlocal theory of thermoelasticity, which requires no positive definiteness on the elastic constants. Also, the positive definiteness of interaction kernels assumed in this paper is partial. 2. PRELIMINARIES AND BASIC EQUATIONS In this section we summarize, with slight modifications, the basic equations derived in [6] and postulate certain conditions on nonlocal constants to make a more convenient version of these equations which will be used in later sections. 473
R. S. DHALIWAL and J. WANG
474
For a homogeneous dQ,
the
basic
and isotropic
equations
strain-displacement
of
relations
solid with a configuration
generalized
linear
Q, bounded
nonlocal
by a closed surface
thermoelasticity
consist
of
the
[6] 1 eij
the stress-strain-temperature
=
5
t”i,j
+
relations
UiJ = Ae,, 6iJ + @eiJ + (Cl6 + ‘228) 6ij +
+
IR
(2.1)
uj,i),
[C;e(x’,
t) + c;~(X’,
t)] 6ij
IR
[k’ekk(X’, t) 6ij + Zp’eiJ(X’,
t)]
dS2’
dS2’,
(2.2)
the equations of motion UiJ,j +
the heat conduction
Fi = p&y
Uij
=
(2.3)
Uji,
equations qi = -KO,i -
IR
K'e,i(X', t) da’,
(2.4)
and the energy equation I-
-qi,i + K,e + K,e + l'~ii +
Jn
IY'&ii(X'pt) d&'
-
JR
[K;i)(X', t)+ K;G(X'y t)]d&’
+ Q=O.
(2.5)
In these equations Uij and eij are, respectively, the Cauchy stress and strain tensors, ui is the displacement vector, 8 is the temperature deviation above the initial temperature t&,, qi is the heat flux vector, p is the mass density, Fi and Q are, respectively, the body force and the strength of the internal heat source, each measured per unit volume, K,, K*, K, Y, c,, c2, A and p are material constants, KI, K;, 18, v’, c;, c;, h’ and ,u’ are nonlocal coefficients which are functions of )x - ~‘1. The above equations do not readily yield the uniqueness theorem. With a view of conforming with the coupled nonlocal theories (see, e.g. Eringen [9] and Altan [lo]), we postulate certain conditions on nonlocal coefficients so that the equations (2.2), (2.4) and (2.5) can be rewritten into more convenient forms as Uij =
IR
cU(lX -x’])[hekk(x’,
t) 6iJ + Zpeij(x’,
+ o P(]x -x’l)[c,O(x’, I
t) + c,~(x’,
qi = - n k(l~ -x’])e.i(x’, I -qi,i + o B(lX -X’])eii(X’;), I
t)]
dS2’ + Q =
IR
t)
t)]
dB’
t)] SiJ d&I’,
(2.6)
d&‘,
y(]X -X’])[C,8(X’,
(2.7) t) + $8(X’,
t)] dQ’,
(2.8)
where )c, p, c, and c2 are constants. The functions (Y, /3, y and k appearing in these equations are known as interaction kernels or moduli of nonlocality and are assumed to depend on the distance measured from the point where the independent constitutive variables are evaluated to another point in the body. They express that the values of independent constitutive variables at x’ have influence on their values at x. The solution of a mixed initial-boundary value problem in the dynamic theory of homogeneous, isotropic, linear generalized nonlocal thermoelasticity is a thermoelastic process {ui, eij, uij, 8, qi} that satisfies the basic equations (2.1), (2.3) and (2.6)-(2.8) with the initial
Theorems in generalized nonlocal thermoelasticity
475
conditions pi =O,
Ui= 0,
8 =o,
e =o,
qi=O,
(2.9)
and the boundary conditions Ui=
on dS, x [0, co);
Ui
e=6
aijnj
=
Ui
qini = q
on dt& X [0, m);
on 3s: x [0, w);
(2.10)
on IX& X [0, 00).
(2.11)
In the above boundary conditions, X2,, df& and &?t, ~9s; are arbitrary subsets of d8 and their complements with respect to dS2, ni is the unit outward normal to dSJ and Ui, Ui, 6 and q are prescribed functions in the domains of their definition.
3. A UNIQUENESS
THEOREM
In this section, we shall extend Brun’s method [ll] to nonlocal thermoelasticity and prove a uniqueness theorem for the mixed initial boundary problem without assuming the positive definiteness of the elastic constants. Also, the positive definiteness on interaction kernels assumed is partial. Unfortunately, this method fails in dealing with static problems when the positive definiteness of the total energy becomes important. 3.1 (uniqueness). If cl > 0, c2 > 0 and the interaction kernel k(b -x’j) is positive definite, then there exists at most one solution for the mixed initial-boundary value problem described in Section 2.
THEOREM
PROOF.It is sufficient to show that for Fi = 0, Q = 0 and homogeneous the solution is trivial. Let B(t) =
A(t) =
pliilii dS2 + C,
In
k(lx -x’l)O,\e,j
boundary
d8’ dB,
1f
CY(~X -x’l)[Ae;ke,I + 2pei’jeij] dG’ dS1
+f
If
conditions,
(3.1)
RR y(lx -dl)[c,e
+ &][c,e'
+
0 R R
c2eqdW d&2,
(3.2)
where primed quantities are functions of x’ and t. Let t, and f2 be two instants of time in (0, ~0) and
+
If
y(lX
+
By straightforward
dQ’ dB
k(Ex -W?j(W,~02) If RR
dQ; dQ
+ c&)][cJ?(~~)
pii,(t l)tii(f2) dS2 + Cl
fR
+ c2
+ &(f2)
-x’I)[C18(tt)
RR
If RR
k(l~ -x'l)b,j(t,)e~j(Q dCZ’dS2.
(3.3)
calculation, we find that
B(t)-A(t)=$f[L(r+s,t-s)-L(r-s,t+s)]d.v,
(3.4)
0 and B(t) +A(t) =2i:L(s,
s)d..r -2c,~~bbk(lx-xfl)e:0)e,i(~)dn’
dads.
(3.5)
R. S. DHALIWAL and J. WANG
476
Employing the Green-Gauss that
theorem and using equations (2.1), (2.3) and (2.6)-(2.8),
L(t,, f2) = aR (oij(rl)ui(tz) - ]cle(t~) + Qb(t,)]qj(rz)]nj I Now taking into account the homogeneous
dS*
we find
(3.6)
boundary conditions, we can see that I!+, , f*) = 0.
(3.7)
Substituting from equation (3.7) into equations (3.4) and (3.5), we obtain B(t) -A(t) B(t) + A(r) = -2c,
= 0,
(3.8)
k(Jx -X’])0,‘i(S)0,i(S) dS2’ d&I dr.
(3.9)
We notice that the equation (3.8) represents a counterpart of Brun’s theorem in the classical theory (see [12], Section 60), and the equation (3.9) represents the conservation of total energy. From equations (3.1), (3.8) and (3.9), we find that
In
PQili dB +
~2 Ifon
k(lx
-x’l)O,iO,‘i
dS2’ dQ
f + Cl
111 0 R R
k(Jx -x’l)W~(s)O,i(s)
dQ’ d& d.r = 0.
(3.10)
Since p > 0, c, > 0, Q > 0 and the kernel k((x -x’]) is positive definite, we find that equation (3.10) will be satisfied only if ui=O and Bsi=O. (3.11) Again, taking into account the homogeneous initial and boundary conditions, give Ui=O and 8=0.
equations (3.11)
Now equation (2.1) gives eij = 0 and equations (2.6) and (2.7), respectively, give oij = 0 and qi = 0. This complete the proof. This theorem avoids any restrictions on Lame constants A and p and no positive definiteness is imposed on the interaction kernels CY,/3 and y.
4. A RECIPROCITY
THEOREM
In this section we give a reciprocity theorem for generalized nonlocal thermoelasticity in homogeneous and isotropic media. The parallel result for coupled nonlocal thermoelasticity was derived by Altan [lo]. Let O(X, t) and Y(X, t) be functions defined on Q x [0, to). For 0 < t < tO, the convolution of o and v is defined by I w*v=
III
The notation
w(x, t - t)v(x,
t) dt.
(4.1)
I
l*v=
I0
V(X, r) dt
(4.2)
will also be used in this paper. Let R’= {ui, 0’, o~j, ql, eij} be the solution of the mixed initial boundary value problem stated in Section 2, corresponding to the external data system S’s {Ff, o’, Vi, ai, 6’, q’}, r = 1, 2; then we will prove the following lemma and the reciprocity theorem:
Theorems in generalized nonlocal thermoelasticity
LEMMA~.~.
Fort,,
477
r,SE{1,2},let
tze[O,m),
J(F:(*,)u;(*z)+ [c,@(*,I+ cze’(*,)lQ”(*zN dQ
+
R
IJ If
y[c,&‘(f2)
-
+
c28S’(f2)][c, W(f,) + c2e’(f,)]dC2’ d!2
s-2
-
~[Aer,‘(tl)k~i(t2) + 2&'(fl)&j(f2)]
dS2’ dS1,
(4.3)
R
then
A&,> fz) = As& PROOF. Using
the divergence theorem,
-
r,).
(4.4)
we find that equation (4.3) may be written as
1~y[c,@(fz)
+ c-#‘(f2)][c,
W(f,)
+
c&f,)] dO’ dC2
n
Using equations (2.1), (2.3), (2.6), (2.7) and (2.8) in the above result, we find that
Ad*, *A = 7
11c,kfYi(f,)~~,l(f2)
dS2’ dC!
P =
c,k6f,‘(tl)0~,(t,) d&2’dS2
II n = f
I
c,ke;(f2)Wi)(f,)
dW dP
n
which completes the proof. THEREOM4.1 (reciprocity). Let R’ be the solution of the mixed initial boundary value problem stated in Section 2, corresponding to the external data system S’, r = 1, 2, then
fR =fJR [$*u; fas2 f R
I,,[o:‘uf-~e’*q4ds-l*~*
-qe**q’]dS-
+ JF:*u;+c#**Q’]dP+l*j-
c,e’*q*dS+
l*
[F;*u;+~e’*Q*]d8+1*
fh-2 c,#*Q*dQ
c,e**q’dS
c,t?**Q’dB.
(4.6)
R. S. DHALlWAL
478
and J. WANG
PROOF. Taking t, = f - t, t, = z and r = 1, s = 2 in equation (4.4) and integrating with respect to t, from 0 to t, we get I +j+
II n -II
-II
[c,B’*q2+CLe1*q2]dS-bpiij*BTdn Gkd!i * Sf,’ dS2’ da +
(Ff * tir + c, 8’ * Q2 + c,# * Q2) dSJ
IR
)‘[C, i)*’ -t c,,~~‘] * [c,8’
-I- c-#]
dCJ’ d$2
a cu[Ae$ * Cg(tJ -I-2pe{/ * e$ dB’ dS2 n
(F~*uf+c,e2*Q’+c28*Q1)d~
-~I~k&*fI!~dWdO+ R
Since
o*ir= o*v,
o*v=v*w,
(4-g)
we find that
II
y[c$*
-t
c.#“‘] * [c,8’ + c&l’] dSY dS2 =
y[c,@
$ c$“]
* [c1e2 f c,e”] dS2’ dS&
(4.11)
+ 2pe$’ * e!j] dQ’ dB.
(4.12)
o
Q and
cu[Ae&* ei(r2) + 2pe:/ * e$] dSZ’ d&2=
II o
Substituting from equations (4.9)-(4.12)
I
aa[o;*u; =
-[c10’*q2+q&*q2]dS+
J,[&*a:
II Tt
@e’,; *Eli
into equation (4.7), we find that
IR
(Fi’*u~+cti?*Q2+c2~‘*Q2)dS2
-[c,82*q’+~~i)2*q’ldS+~~(Ff~~~+c,82*Q’+C,jt2*Q’)d~.
(4.13)
In equation (4.13), replacing variable t by t, and integrating from 0 to t with respect to r, we get the desired result. Acknowledgement-We
thank the reviewer for his constructive suggestions which helped to improve the manuscript.
Theorems in generalized nonlocal thermoelasticity
479
REFERENCES [1] I. A. KUNIN, Elastic Media with Microstructures. Springer, Berlin (1983). [2] D. ROGULA (Ed.), Nonfocul Theory of Material Media. Springer, Berlin (1982). [3] A. C. ERINGEN, ht. /. Engng Sci. 10,425 (1972). [4] A. C. ERINGEN, Inr. J. Engng Sci. l2, 1063 (1974). [5] A. C. ERINGEN, C. G. SPEZIALE and B. S. KIM, J. Mech. Phys. Solids 25,339 (1977). [6] F. A. BALTA and E. S. SUHUBI, ht. J. Engng Sci. 15, 579 (1977). [7] D. S. CHANDRASEKHARAIAH, Appl. Me&. Rev. 39, 355 (1986). [8] T. S. ONCU and T. B. MOODIE, Archs Rut. Me& Anal. l21, 87 (1992). [9] A. C. ERINGEN, Cry&f Lattice Defects 7, 109 (1977). [lo] B. ALTAN, J. Therm. Stresses W, 207 (1990). ill] L. BRUN, J. Mhnique 8, 125 (1969). [12] M. E. GURTIN, The linear theory of elasticity. In Hundbunch der Physik (Edited by C. TRUESDELL), a/2. Springer, Berlin (1972). (Revision received 9 February
1993; accepted 14 April 1993)
Vol. VI
Pecgamon
SOME THEOREMS IN GENERALIZED THERMOELASTICITY RANJIT
S. DHALIWAL
NONLOCAL
and JUN WANG
Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N lN4
Abstract-A
reciprocity theorem in the context of the generalized thermoelasticity is established. A uniqueness theorem for the mixed initial boundary value problem is also proved without assuming the positive definiteness of the elastic constants. 1. INTRODUCTION The nonlocal theory of continuum mechanics is of recent origin. It takes into account the fact that the interacting forces between material points are far reaching in character. From the systematic point of view, it differs from the classical theory by assuming the balance laws to be valid only on the whole of the body. In some cases, such as in phonon dispersion in solids, in electromagnetic solids, in fracture mechanics, when the continuity hypothesis made in classical theory can no longer be supported because of the discrete characteristics of the medium, only a nonlocal theory might provide the right answer while the classical ones would fail. The references to the physical background of nonlocal theory of continuum mechanics can be found in Kunin [l] and Rogula [2]. Eringen [3,4] developed the linear theory of nonlocal elasticity and thermoelasticity in 1972 and 1974, respectively. It is shown in [3] that the elastic wave propagation is dispersive by employing the nonlocal theory while the classical theory gives no dispersion. Also, using nonlocal theory to solve the crack problems in [5], it is shown that the crack tip singularity in stress appearing in classical solution disappears. In recent years physicists have also been giving increasing attention to the so-called “second sound” effect in solids. This effect arises from the possible transport of heat by a wave propagation process rather than diffusion. As is well known, the diffusion equation predicts an infinite velocity of propagation for thermal disturbances, which is contradictory to the physical observations. It seems reasonable to modify the coupled theory of heat conduction to allow time for acceleration of the heat flow in response to an applied temperature gradient. Such an effect has been exhibited in liquid helium and there is speculation concerning its possible existence in solids. In 1977, Balta and Suhubi [6] extended Eringen’s work by including the temperature rate among the constitutive variables and formulated a generalized theory of nonlocal thermoelasticity. A comprehensive discussion of the literature concerned with theories of generalized thermoelasticity can be found in Chandrasekharaiah [7] and Gncti and Moodie
Bl. Owing to the newness of the nonlocal theory, many aspects are open to study. One of the most important questions is whether the initial-boundary value problems defined in the frame of nonlocal thermoelasticity are well-posed. For this reason, we are going to study that under what conditions the solutions of those problems are unique. We prove a reciprocity theorem and a uniqueness theorem for the solution of the mixed initial boundary value problem in generalized nonlocal theory of thermoelasticity, which requires no positive definiteness on the elastic constants. Also, the positive definiteness of interaction kernels assumed in this paper is partial. 2. PRELIMINARIES AND BASIC EQUATIONS In this section we summarize, with slight modifications, the basic equations derived in [6] and postulate certain conditions on nonlocal constants to make a more convenient version of these equations which will be used in later sections. 473
R. S. DHALIWAL and J. WANG
474
For a homogeneous dQ,
the
basic
and isotropic
equations
strain-displacement
of
relations
solid with a configuration
generalized
linear
Q, bounded
nonlocal
by a closed surface
thermoelasticity
consist
of
the
[6] 1 eij
the stress-strain-temperature
=
5
t”i,j
+
relations
UiJ = Ae,, 6iJ + @eiJ + (Cl6 + ‘228) 6ij +
+
IR
(2.1)
uj,i),
[C;e(x’,
t) + c;~(X’,
t)] 6ij
IR
[k’ekk(X’, t) 6ij + Zp’eiJ(X’,
t)]
dS2’
dS2’,
(2.2)
the equations of motion UiJ,j +
the heat conduction
Fi = p&y
Uij
=
(2.3)
Uji,
equations qi = -KO,i -
IR
K'e,i(X', t) da’,
(2.4)
and the energy equation I-
-qi,i + K,e + K,e + l'~ii +
Jn
IY'&ii(X'pt) d&'
-
JR
[K;i)(X', t)+ K;G(X'y t)]d&’
+ Q=O.
(2.5)
In these equations Uij and eij are, respectively, the Cauchy stress and strain tensors, ui is the displacement vector, 8 is the temperature deviation above the initial temperature t&,, qi is the heat flux vector, p is the mass density, Fi and Q are, respectively, the body force and the strength of the internal heat source, each measured per unit volume, K,, K*, K, Y, c,, c2, A and p are material constants, KI, K;, 18, v’, c;, c;, h’ and ,u’ are nonlocal coefficients which are functions of )x - ~‘1. The above equations do not readily yield the uniqueness theorem. With a view of conforming with the coupled nonlocal theories (see, e.g. Eringen [9] and Altan [lo]), we postulate certain conditions on nonlocal coefficients so that the equations (2.2), (2.4) and (2.5) can be rewritten into more convenient forms as Uij =
IR
cU(lX -x’])[hekk(x’,
t) 6iJ + Zpeij(x’,
+ o P(]x -x’l)[c,O(x’, I
t) + c,~(x’,
qi = - n k(l~ -x’])e.i(x’, I -qi,i + o B(lX -X’])eii(X’;), I
t)]
dS2’ + Q =
IR
t)
t)]
dB’
t)] SiJ d&I’,
(2.6)
d&‘,
y(]X -X’])[C,8(X’,
(2.7) t) + $8(X’,
t)] dQ’,
(2.8)
where )c, p, c, and c2 are constants. The functions (Y, /3, y and k appearing in these equations are known as interaction kernels or moduli of nonlocality and are assumed to depend on the distance measured from the point where the independent constitutive variables are evaluated to another point in the body. They express that the values of independent constitutive variables at x’ have influence on their values at x. The solution of a mixed initial-boundary value problem in the dynamic theory of homogeneous, isotropic, linear generalized nonlocal thermoelasticity is a thermoelastic process {ui, eij, uij, 8, qi} that satisfies the basic equations (2.1), (2.3) and (2.6)-(2.8) with the initial
Theorems in generalized nonlocal thermoelasticity
475
conditions pi =O,
Ui= 0,
8 =o,
e =o,
qi=O,
(2.9)
and the boundary conditions Ui=
on dS, x [0, co);
Ui
e=6
aijnj
=
Ui
qini = q
on dt& X [0, m);
on 3s: x [0, w);
(2.10)
on IX& X [0, 00).
(2.11)
In the above boundary conditions, X2,, df& and &?t, ~9s; are arbitrary subsets of d8 and their complements with respect to dS2, ni is the unit outward normal to dSJ and Ui, Ui, 6 and q are prescribed functions in the domains of their definition.
3. A UNIQUENESS
THEOREM
In this section, we shall extend Brun’s method [ll] to nonlocal thermoelasticity and prove a uniqueness theorem for the mixed initial boundary problem without assuming the positive definiteness of the elastic constants. Also, the positive definiteness on interaction kernels assumed is partial. Unfortunately, this method fails in dealing with static problems when the positive definiteness of the total energy becomes important. 3.1 (uniqueness). If cl > 0, c2 > 0 and the interaction kernel k(b -x’j) is positive definite, then there exists at most one solution for the mixed initial-boundary value problem described in Section 2.
THEOREM
PROOF.It is sufficient to show that for Fi = 0, Q = 0 and homogeneous the solution is trivial. Let B(t) =
A(t) =
pliilii dS2 + C,
In
k(lx -x’l)O,\e,j
boundary
d8’ dB,
1f
CY(~X -x’l)[Ae;ke,I + 2pei’jeij] dG’ dS1
+f
If
conditions,
(3.1)
RR y(lx -dl)[c,e
+ &][c,e'
+
0 R R
c2eqdW d&2,
(3.2)
where primed quantities are functions of x’ and t. Let t, and f2 be two instants of time in (0, ~0) and
+
If
y(lX
+
By straightforward
dQ’ dB
k(Ex -W?j(W,~02) If RR
dQ; dQ
+ c&)][cJ?(~~)
pii,(t l)tii(f2) dS2 + Cl
fR
+ c2
+ &(f2)
-x’I)[C18(tt)
RR
If RR
k(l~ -x'l)b,j(t,)e~j(Q dCZ’dS2.
(3.3)
calculation, we find that
B(t)-A(t)=$f[L(r+s,t-s)-L(r-s,t+s)]d.v,
(3.4)
0 and B(t) +A(t) =2i:L(s,
s)d..r -2c,~~bbk(lx-xfl)e:0)e,i(~)dn’
dads.
(3.5)
R. S. DHALIWAL and J. WANG
476
Employing the Green-Gauss that
theorem and using equations (2.1), (2.3) and (2.6)-(2.8),
L(t,, f2) = aR (oij(rl)ui(tz) - ]cle(t~) + Qb(t,)]qj(rz)]nj I Now taking into account the homogeneous
dS*
we find
(3.6)
boundary conditions, we can see that I!+, , f*) = 0.
(3.7)
Substituting from equation (3.7) into equations (3.4) and (3.5), we obtain B(t) -A(t) B(t) + A(r) = -2c,
= 0,
(3.8)
k(Jx -X’])0,‘i(S)0,i(S) dS2’ d&I dr.
(3.9)
We notice that the equation (3.8) represents a counterpart of Brun’s theorem in the classical theory (see [12], Section 60), and the equation (3.9) represents the conservation of total energy. From equations (3.1), (3.8) and (3.9), we find that
In
PQili dB +
~2 Ifon
k(lx
-x’l)O,iO,‘i
dS2’ dQ
f + Cl
111 0 R R
k(Jx -x’l)W~(s)O,i(s)
dQ’ d& d.r = 0.
(3.10)
Since p > 0, c, > 0, Q > 0 and the kernel k((x -x’]) is positive definite, we find that equation (3.10) will be satisfied only if ui=O and Bsi=O. (3.11) Again, taking into account the homogeneous initial and boundary conditions, give Ui=O and 8=0.
equations (3.11)
Now equation (2.1) gives eij = 0 and equations (2.6) and (2.7), respectively, give oij = 0 and qi = 0. This complete the proof. This theorem avoids any restrictions on Lame constants A and p and no positive definiteness is imposed on the interaction kernels CY,/3 and y.
4. A RECIPROCITY
THEOREM
In this section we give a reciprocity theorem for generalized nonlocal thermoelasticity in homogeneous and isotropic media. The parallel result for coupled nonlocal thermoelasticity was derived by Altan [lo]. Let O(X, t) and Y(X, t) be functions defined on Q x [0, to). For 0 < t < tO, the convolution of o and v is defined by I w*v=
III
The notation
w(x, t - t)v(x,
t) dt.
(4.1)
I
l*v=
I0
V(X, r) dt
(4.2)
will also be used in this paper. Let R’= {ui, 0’, o~j, ql, eij} be the solution of the mixed initial boundary value problem stated in Section 2, corresponding to the external data system S’s {Ff, o’, Vi, ai, 6’, q’}, r = 1, 2; then we will prove the following lemma and the reciprocity theorem:
Theorems in generalized nonlocal thermoelasticity
LEMMA~.~.
Fort,,
477
r,SE{1,2},let
tze[O,m),
J(F:(*,)u;(*z)+ [c,@(*,I+ cze’(*,)lQ”(*zN dQ
+
R
IJ If
y[c,&‘(f2)
-
+
c28S’(f2)][c, W(f,) + c2e’(f,)]dC2’ d!2
s-2
-
~[Aer,‘(tl)k~i(t2) + 2&'(fl)&j(f2)]
dS2’ dS1,
(4.3)
R
then
A&,> fz) = As& PROOF. Using
the divergence theorem,
-
r,).
(4.4)
we find that equation (4.3) may be written as
1~y[c,@(fz)
+ c-#‘(f2)][c,
W(f,)
+
c&f,)] dO’ dC2
n
Using equations (2.1), (2.3), (2.6), (2.7) and (2.8) in the above result, we find that
Ad*, *A = 7
11c,kfYi(f,)~~,l(f2)
dS2’ dC!
P =
c,k6f,‘(tl)0~,(t,) d&2’dS2
II n = f
I
c,ke;(f2)Wi)(f,)
dW dP
n
which completes the proof. THEREOM4.1 (reciprocity). Let R’ be the solution of the mixed initial boundary value problem stated in Section 2, corresponding to the external data system S’, r = 1, 2, then
fR =fJR [$*u; fas2 f R
I,,[o:‘uf-~e’*q4ds-l*~*
-qe**q’]dS-
+ JF:*u;+c#**Q’]dP+l*j-
c,e’*q*dS+
l*
[F;*u;+~e’*Q*]d8+1*
fh-2 c,#*Q*dQ
c,e**q’dS
c,t?**Q’dB.
(4.6)
R. S. DHALlWAL
478
and J. WANG
PROOF. Taking t, = f - t, t, = z and r = 1, s = 2 in equation (4.4) and integrating with respect to t, from 0 to t, we get I +j+
II n -II
-II
[c,B’*q2+CLe1*q2]dS-bpiij*BTdn Gkd!i * Sf,’ dS2’ da +
(Ff * tir + c, 8’ * Q2 + c,# * Q2) dSJ
IR
)‘[C, i)*’ -t c,,~~‘] * [c,8’
-I- c-#]
dCJ’ d$2
a cu[Ae$ * Cg(tJ -I-2pe{/ * e$ dB’ dS2 n
(F~*uf+c,e2*Q’+c28*Q1)d~
-~I~k&*fI!~dWdO+ R
Since
o*ir= o*v,
o*v=v*w,
(4-g)
we find that
II
y[c$*
-t
c.#“‘] * [c,8’ + c&l’] dSY dS2 =
y[c,@
$ c$“]
* [c1e2 f c,e”] dS2’ dS&
(4.11)
+ 2pe$’ * e!j] dQ’ dB.
(4.12)
o
Q and
cu[Ae&* ei(r2) + 2pe:/ * e$] dSZ’ d&2=
II o
Substituting from equations (4.9)-(4.12)
I
aa[o;*u; =
-[c10’*q2+q&*q2]dS+
J,[&*a:
II Tt
@e’,; *Eli
into equation (4.7), we find that
IR
(Fi’*u~+cti?*Q2+c2~‘*Q2)dS2
-[c,82*q’+~~i)2*q’ldS+~~(Ff~~~+c,82*Q’+C,jt2*Q’)d~.
(4.13)
In equation (4.13), replacing variable t by t, and integrating from 0 to t with respect to r, we get the desired result. Acknowledgement-We
thank the reviewer for his constructive suggestions which helped to improve the manuscript.
Theorems in generalized nonlocal thermoelasticity
479
REFERENCES [1] I. A. KUNIN, Elastic Media with Microstructures. Springer, Berlin (1983). [2] D. ROGULA (Ed.), Nonfocul Theory of Material Media. Springer, Berlin (1982). [3] A. C. ERINGEN, ht. /. Engng Sci. 10,425 (1972). [4] A. C. ERINGEN, Inr. J. Engng Sci. l2, 1063 (1974). [5] A. C. ERINGEN, C. G. SPEZIALE and B. S. KIM, J. Mech. Phys. Solids 25,339 (1977). [6] F. A. BALTA and E. S. SUHUBI, ht. J. Engng Sci. 15, 579 (1977). [7] D. S. CHANDRASEKHARAIAH, Appl. Me&. Rev. 39, 355 (1986). [8] T. S. ONCU and T. B. MOODIE, Archs Rut. Me& Anal. l21, 87 (1992). [9] A. C. ERINGEN, Cry&f Lattice Defects 7, 109 (1977). [lo] B. ALTAN, J. Therm. Stresses W, 207 (1990). ill] L. BRUN, J. Mhnique 8, 125 (1969). [12] M. E. GURTIN, The linear theory of elasticity. In Hundbunch der Physik (Edited by C. TRUESDELL), a/2. Springer, Berlin (1972). (Revision received 9 February
1993; accepted 14 April 1993)
Vol. VI