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Variational principle for linear coupled dynamic theory of viscoelastic thermodiffusion
Inf. J. Engng Sci. Vol. 21, No. 5, pp. 605-607,1989 Printed in Great Britain. Ail righa reserved
0@20-7225/W $3.00+ 0.M) Copyright @ 1989Pergamon Press ptc
AND
LETTERS IN APPLIED
ENG~E~G
SCIENCES
VARIATIONAL PRINCIPLE FOR LINEAR COUPLED DYNAMIC THEORY OF VISCOELASTIC ~E~ODI~SION J. KUBIK Institute of Civil Engineering,
and J. WYRWAL
Engineering College, ui. Katowicka 48,45-951 Opole, Poland
Abstract-The variational principle for linear coupled dynamic theory of viscoelastic thermodiffusion is constructed using variational theory of potential operators. The functional derived herein leads to all the governing equations, including the boundary and initial conditions, given in form of the Euler equations.
1. INTRODUCTION The governing equations in the linear theory of viscoelastic thermodiffusion
are (see Kubik [1])
p* dui,jj + (A + cl) * dUj,ji+ ~4 - YT* dO,i - YW* dM,, = piii, k8,jj - (FFZ *de f I * dM + YT*da,,,)’ = -Pr, KM,, - (i* dO + II * dM + YM* dui,,)’ = -pR,
(1.1)
where Ui denote the components of displacement vector, @-temperature, M-chemical potential, p~~rn~nents of body force vector, pr -rate of internal heat generation, pR-rate of internal mass generation, p, p, h, yT, yns, m, n, 1, k, K-characteristics of the material, *-the Stielties convolution;
To the above field equations we adjoin the boundary conditions pi
s (P * dU(i,j)+ (A* dUk,k-
YT *
d8 - ye * ~)~ii)~j
=fji on A,,
on A,,,
ui = tit
4 E -kO,ini
= 4 on A,,
O=&onA,,
j = -KM,ini
=j on Ai,
M=&onAM,
@&i(O)= iti,
PLii= i?i,
(l-2)
and the initial conditions
S(O) 3 (m * de + I * dM + YT* du,i)(O) = S, c(0) = (1 *dO + n * dM + ym * dui,i)(0) = e,
(1.3)
where pi, Gi, 4, 6,, i, M, L&,I$, 3 and (.? are prescribed functions; ufbi) denotes the symmetric part of Ui,j, $-the Kronecker delta, n,-the components of the unit outward normal to A; A, (a = a, u, q, 0, j, M) denote subsets of A, so that A = A, UA, =A, UAe = Aj UAMM, A~flA~=AqflA~=AjflA~=IZI. The set of basic equations
(l.l)-(1.3)
Au+f=O,
can be put into the operator form (see Reddy [2]) A:E-,E’,
u E E,
fEE’,
(1.4)
where u = [Ui, 0, M; Ui, pi, 0, 4, M, j; dui, tii, S, 8, C, MIT, f= i-P&
P’, PR; -fit lii, -4,
6, -1, M; -PC,
605
-p
diii, 0, S, 0, elT,
(1.3
(1.6)
606
J. KUBIK and J. WYRWAE
and A is a linear matrix (15 x 15) operator
the non-zero elements of which are given by
Aii = -~*d(.),ji-(A+~~*d(.6ij),ji+p(:‘), A13
=
A21
‘YM * d(*),i,
=
(-YT
AKZ=YT*~(*),~,
* W,J’>
A22
=
Wjj
-
Cm
*d(.)>‘,
Ax2 = (-I * d(n))‘,
A23 = (-I * d(n))‘,
A31 = (-‘/M * d(*),i)‘, A33 =ZK(.),ij - (n *d(.))‘, A45 = A67 = Ass = -A54 = -A,6 = -A98 = 1, A(to
= A(i~oo) = P,
4x3)(12)
=
A
=
-
1.
0.7)
Here E denotes the space of ordered arrays of the form given in (1.5); the dual space of E is denoted by E’.
2. THE VARIATIONAL
PRINCIPLE
FOR VISCOELASTIC
THER~ODIFFUSION
We now discuss the central topic of the paper, namely, the construction of the variational principle for viscoelastic thermodiffusion . The necessary and sufficient condition that there exist a variational principle corresponding to the operator equation (1.4) is that A be symmetric. The corresponding variational functional is given by P(u)=;(Au,u)
+ (f,u),
where ( *, -) denotes the bilinear form on E’ x E. A demonstration be found in Vainberg [3]. We define the bilinear form on E’ x E by
(2.1) of this important result can
(f,u)=bXldu,dV+~2b~*u~dV+~~~~~*du,dA +5lf,*u,u+ Z=6
A
and check whether A is symmetric.
2
~~(O)u~dV,
I=10
v
(2.2)
Substituting (1.5) and (1.7) into (2.2) we get U’E E.
(Au, u’) = (Au’, u),
(2.3)
This indicates that the operator A is symmetric. Now, substituting (1.4) and (2.2) into (2.1), using the divergence theorem and property of the convolution (see Gut-tin and Sternberg [4]), we obtain
+ ; * duk,k *dui,i-~kQ,i*~,i_2KMi$M,i_pF;*dui+pr*8+pR*M
+ v (-pii, I -
I
A,
dui + p d(ui(O) - Bi)Eii+ SO + eM) dV
@i*duidA-
I
A~(g-8i)*doidA-~
Q*OdA A,
Thus, F(u) in (2.4) is the ~nctional associated with Au + fin (1.4) and the equation (1.4) is the Euler equation for the functional F(u).
607
Letters in Applied and Engineering Sciences
Then the problem of finding the solution to the equation (1.4) is equivalent critical point of the functional I;(u). This is proved in the following theorem. Theorem. Let II E E, and let I;(u) have a linear Gfiteaux differential the functional defined in (2.4). Then
to finding a
at every II, where F(n) is
6F(u, II’) = 0,
(2.5)
if and only if u is a solution of (1.4). Proof. Let u’ be an arbitrary element in E. Then the G&eaux differential of F(u) is SF(u, a’) =
& F(u +au’)1,,o
= y ((-p * dui,ji - (A + /J) * duj,ji + Yr * d8,i + YM* dM,i + pi& - pJ$) * dui f
+ (k0,jj - (m * dO + L* dM + Yr * du,,i)’ + pr) * 0’ +(K~,li-(Z*dQ+n*dM+y,*dui,i)‘+pR)*M’)dV + v (~(~i(O)
J
-
+ (-S(O) + $0’
6i) du: + p d(ui(O) - iii)li: + (-C(O) + C)M’) dV
+ I~(p.-Pi)*duldA-~“(ui-li,)*d~:dA + .
-
IAM
(M-~)*j’dA=(Aa+f,u’).
We first prove sticiency. Suppose that u E E is a solution of (1.4). Then (2.6) becomes (2.5). To prove the necessity, assume that (2.5) holds. Using the Lemmas established by Gurtin in [5], we see that u satisfies (1.4). This completes the proof of the theorem.
REFERENCES [1] J. KUBM, Acra mech. 50,285 (19&)). [2] J. N. RBDDY, ht. J. Engng Sci. 14,605 (1976). [3] M. M. VAINBERG, VariationalMethod for the Study of Nonlinear Operators. Holden-Day, San Francisco (1964). [4] M. E. GURTIN and E. STERNBERG, Arch. rad. Mech. Anal. 11,201 (1%2). (51 M. E. GURTIN, Arch. rad. Me&. Anal. 16,34 (1964). (Received 23 Augurt 1988)
0@20-7225/W $3.00+ 0.M) Copyright @ 1989Pergamon Press ptc
AND
LETTERS IN APPLIED
ENG~E~G
SCIENCES
VARIATIONAL PRINCIPLE FOR LINEAR COUPLED DYNAMIC THEORY OF VISCOELASTIC ~E~ODI~SION J. KUBIK Institute of Civil Engineering,
and J. WYRWAL
Engineering College, ui. Katowicka 48,45-951 Opole, Poland
Abstract-The variational principle for linear coupled dynamic theory of viscoelastic thermodiffusion is constructed using variational theory of potential operators. The functional derived herein leads to all the governing equations, including the boundary and initial conditions, given in form of the Euler equations.
1. INTRODUCTION The governing equations in the linear theory of viscoelastic thermodiffusion
are (see Kubik [1])
p* dui,jj + (A + cl) * dUj,ji+ ~4 - YT* dO,i - YW* dM,, = piii, k8,jj - (FFZ *de f I * dM + YT*da,,,)’ = -Pr, KM,, - (i* dO + II * dM + YM* dui,,)’ = -pR,
(1.1)
where Ui denote the components of displacement vector, @-temperature, M-chemical potential, p~~rn~nents of body force vector, pr -rate of internal heat generation, pR-rate of internal mass generation, p, p, h, yT, yns, m, n, 1, k, K-characteristics of the material, *-the Stielties convolution;
To the above field equations we adjoin the boundary conditions pi
s (P * dU(i,j)+ (A* dUk,k-
YT *
d8 - ye * ~)~ii)~j
=fji on A,,
on A,,,
ui = tit
4 E -kO,ini
= 4 on A,,
O=&onA,,
j = -KM,ini
=j on Ai,
M=&onAM,
@&i(O)= iti,
PLii= i?i,
(l-2)
and the initial conditions
S(O) 3 (m * de + I * dM + YT* du,i)(O) = S, c(0) = (1 *dO + n * dM + ym * dui,i)(0) = e,
(1.3)
where pi, Gi, 4, 6,, i, M, L&,I$, 3 and (.? are prescribed functions; ufbi) denotes the symmetric part of Ui,j, $-the Kronecker delta, n,-the components of the unit outward normal to A; A, (a = a, u, q, 0, j, M) denote subsets of A, so that A = A, UA, =A, UAe = Aj UAMM, A~flA~=AqflA~=AjflA~=IZI. The set of basic equations
(l.l)-(1.3)
Au+f=O,
can be put into the operator form (see Reddy [2]) A:E-,E’,
u E E,
fEE’,
(1.4)
where u = [Ui, 0, M; Ui, pi, 0, 4, M, j; dui, tii, S, 8, C, MIT, f= i-P&
P’, PR; -fit lii, -4,
6, -1, M; -PC,
605
-p
diii, 0, S, 0, elT,
(1.3
(1.6)
606
J. KUBIK and J. WYRWAE
and A is a linear matrix (15 x 15) operator
the non-zero elements of which are given by
Aii = -~*d(.),ji-(A+~~*d(.6ij),ji+p(:‘), A13
=
A21
‘YM * d(*),i,
=
(-YT
AKZ=YT*~(*),~,
* W,J’>
A22
=
Wjj
-
Cm
*d(.)>‘,
Ax2 = (-I * d(n))‘,
A23 = (-I * d(n))‘,
A31 = (-‘/M * d(*),i)‘, A33 =ZK(.),ij - (n *d(.))‘, A45 = A67 = Ass = -A54 = -A,6 = -A98 = 1, A(to
= A(i~oo) = P,
4x3)(12)
=
A
=
-
1.
0.7)
Here E denotes the space of ordered arrays of the form given in (1.5); the dual space of E is denoted by E’.
2. THE VARIATIONAL
PRINCIPLE
FOR VISCOELASTIC
THER~ODIFFUSION
We now discuss the central topic of the paper, namely, the construction of the variational principle for viscoelastic thermodiffusion . The necessary and sufficient condition that there exist a variational principle corresponding to the operator equation (1.4) is that A be symmetric. The corresponding variational functional is given by P(u)=;(Au,u)
+ (f,u),
where ( *, -) denotes the bilinear form on E’ x E. A demonstration be found in Vainberg [3]. We define the bilinear form on E’ x E by
(2.1) of this important result can
(f,u)=bXldu,dV+~2b~*u~dV+~~~~~*du,dA +5lf,*u,u+ Z=6
A
and check whether A is symmetric.
2
~~(O)u~dV,
I=10
v
(2.2)
Substituting (1.5) and (1.7) into (2.2) we get U’E E.
(Au, u’) = (Au’, u),
(2.3)
This indicates that the operator A is symmetric. Now, substituting (1.4) and (2.2) into (2.1), using the divergence theorem and property of the convolution (see Gut-tin and Sternberg [4]), we obtain
+ ; * duk,k *dui,i-~kQ,i*~,i_2KMi$M,i_pF;*dui+pr*8+pR*M
+ v (-pii, I -
I
A,
dui + p d(ui(O) - Bi)Eii+ SO + eM) dV
@i*duidA-
I
A~(g-8i)*doidA-~
Q*OdA A,
Thus, F(u) in (2.4) is the ~nctional associated with Au + fin (1.4) and the equation (1.4) is the Euler equation for the functional F(u).
607
Letters in Applied and Engineering Sciences
Then the problem of finding the solution to the equation (1.4) is equivalent critical point of the functional I;(u). This is proved in the following theorem. Theorem. Let II E E, and let I;(u) have a linear Gfiteaux differential the functional defined in (2.4). Then
to finding a
at every II, where F(n) is
6F(u, II’) = 0,
(2.5)
if and only if u is a solution of (1.4). Proof. Let u’ be an arbitrary element in E. Then the G&eaux differential of F(u) is SF(u, a’) =
& F(u +au’)1,,o
= y ((-p * dui,ji - (A + /J) * duj,ji + Yr * d8,i + YM* dM,i + pi& - pJ$) * dui f
+ (k0,jj - (m * dO + L* dM + Yr * du,,i)’ + pr) * 0’ +(K~,li-(Z*dQ+n*dM+y,*dui,i)‘+pR)*M’)dV + v (~(~i(O)
J
-
+ (-S(O) + $0’
6i) du: + p d(ui(O) - iii)li: + (-C(O) + C)M’) dV
+ I~(p.-Pi)*duldA-~“(ui-li,)*d~:dA + .
-
IAM
(M-~)*j’dA=(Aa+f,u’).
We first prove sticiency. Suppose that u E E is a solution of (1.4). Then (2.6) becomes (2.5). To prove the necessity, assume that (2.5) holds. Using the Lemmas established by Gurtin in [5], we see that u satisfies (1.4). This completes the proof of the theorem.
REFERENCES [1] J. KUBM, Acra mech. 50,285 (19&)). [2] J. N. RBDDY, ht. J. Engng Sci. 14,605 (1976). [3] M. M. VAINBERG, VariationalMethod for the Study of Nonlinear Operators. Holden-Day, San Francisco (1964). [4] M. E. GURTIN and E. STERNBERG, Arch. rad. Mech. Anal. 11,201 (1%2). (51 M. E. GURTIN, Arch. rad. Me&. Anal. 16,34 (1964). (Received 23 Augurt 1988)