Variational principle for nonsteady multicomponent continuum system

VARIATIONAL

PRINCIPLE

MULTICOMPONENT

FOR

CONTINUUM

NONSTEADY SYSTEM

O. S. L i m a r c h e n k o

UDC532.5:539.3

The equation of motion of a multicomponent continuum s y s t e m written for the p a r t i c u l a r case when t h e r e is no heat t r a n s f e r between the components is shown to be equivalent to some variational principle analogous to the O s t r o g r a d s k i i principle in the m e c h a n i c s of a d i s c r e t e s y s t e m , which is thus conventionally called a v a r i a t i o n a l principle. The method proposed in [3, 8] is used to show that this principle m a y be used to d e s c r i b e dynamic p r o c e s s e s in t h e r m o d y n a m i c a l l y noninteracting i m m i s c i b l e multicomponent continuum s y s t e m s in the g e n e r a l c a s e when internal dissipative f o r c e s a r e acting. In c o n t r a s t to [9], where a general variational principle for the description of mechanical s y s t e m s with complex p r o p e r t i e s is formulated, in the p r e s e n t work the variational principle is not postulated but is der i v e d f r o m the equation of motion on the basis of a method analogous to that used to derive the Ostrogradskii principle in the m e c h a n i c s of d i s c r e t e s y s t e m s . In addition, specific e x p r e s s i o n s a r e obtained for all the quantities appearing in the r e s u l t i n g v a r i a t i o n a l relation for the s y s t e m investigated. (In [9], only the general p r o c e d u r e for their derivation is given.) 1. Following [7], the differential equation of motion and continuity equation for a multicomponent c o n tinuum s y s t e m in an inertial C a r t e s i a n coordinate s y s t e m a r e written in the f o r m -> > = 0; p k ( ~ - - Yh)-Div 2 -Pk

(1.1)

dpk ~ Ohdivvh = 0,

(1.2)

dt - -

w h e r e the s u b s c r i p t k indicates that the quantities appearing in Eqs. (1.1) and (1.2) r e l a t e to the k-oth c o m ponent of the given system. Equations (1.1) and (1.2) a r e valid in the region Vk occupied by the k-th component of the system, bounded by the s u r f a c e Sk. (The total n u m b e r of components in the s y s t e m is N.) In a c c o r d a n c e with [2, 3, 8], the possible displacement 5~k of points of the k-th component of the continuum s y s t e m is d e t e r m i n e d as the a r b i t r a r y small displacement of points of this component consistent with the coupling (internal and e x t e r n a l [4, 8]) at that moment, and satisfying the r e l a t i o n s ->

6u~---O

(t=t~; t-~t2);

(6U,,,-- 6Uk).17 = 0

(MES,,a,; m=/=k),

(1.3) (1.4)

w h e r e t 1 and t 2 a r e the beginning and end of the c o n s i d e r e d time interval; M is a point of the common boundary Smk = Sm D Sk of two different contacting components of the system; ~ is the v e c t o r of the common n o r m a l to the boundary Smk (of a r b i t r a r y direction). The condition in Eq. (1.3) means that at the initial and final t i m e s the t r u e motion and the c o m p a r a t i v e motion (motion o v e r a variational t r a j e c t o r y ) a r e the same. The condition in Eq. (1.4) is the m a t h e m a t i c a l formulation of the r e q u i r e m e n t that t h e r e be no interpenetration or discontinuity of the different components of the system. Multiplying Eq. (1.1) by 5~k, integrating o v e r the region Vk, and summing over all k gives N

.>

->

->

~ [pk (~h -- a'h) + Div~Ph].~u~dY---- O.

(1.5)

k=l V k

Equation (1.5) is the m a t h e m a t i c a l e x p r e s s i o n of a principle analogous to the D e l a m b r e - - L a g r a n g e principle for the given s y s t e m [81 .

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. T r a n s l a t e d f r o m Prikladnaya Mekhanika, Vol. 14, No. 10, pp. 28-34, October, 1978. Original a r t i c l e submitted November 21, 1977.

0038-5298/78/1410-1039S07.50 9

Plenum Publishing Corporation

1059

2. Taking into account that f D i v ~ . 6 ~ d V = !" "~'~'.5*uhdS--.f ~h.grad 6LdV (where

(2.1}

=fP"k" n~ (grad ~Uk}ij = ~(Su }/0xi is a tensor of second rank; x i is a Cartesian coordinate in an

inertial reference frame}, Eqs. (1.5} and (2.1) lead, after a series of transformations, to the result

(SA~" + 8A~') dV + .f t,~'6u,AS -- ~ p,,~,, .~,,~,~V = O. Sk

k=l

(~'.~4

Vk

In Eq. (2.2) the following notation has been introduced: 6A~* = PkFk 96uk is the density of the element a r y work of external bulk forces acting on ~ e u o ~ t s of the__k-th component of the system on the possible displacements of points of this component; 5A~ = - 2 P k- grad 6u k is the density of the elementary work of the internal forces acting on the particles of the k-th component of the system on the possible displacements of points of this component (6A~ is determined by the theological properties of the component considered). The t e r m vJipk~vk 95~kdV will be transformed separately. Consider, in accordance with [9], the time ==~

==~

derivative of the integral of pkVk 96u k taken over the moving volume Vk. Using the continuity relations in Eq. (1.2) it is found that I

PhWh'SukdV = "~d f phv~,'6u~,dV-"> § § § ~ p~,v~.~vhdV. -> §

t) Yk

(2.3)

Yk

~

Substituting Eq. (2.3} into Eq. (2.2), the result is integrated with respect to time from ti to t 2. Taking into account that the variation of the bulk kinetic-energy density of the k-th component of the given system is

8 ~ r;dv= .f 8r;dv, Vk

r

Vk

the following result is obtained

I'~-" [8 S r'~dv + .f (SAt + 6A;)aV + ( p~.~,,~dS I ,~t --

I O"v'~'~u~eVl = O.

Here T~ is the bulk kinetic-energy density of the k-th component of the system. In view of Eq. (1.8) the last integral sum in this relation vanishes. Since the normal components of the pressure forces on the boundaries 8k of the regions Vk that do not belong to S, the boundary of the region Ar V = ~ Vk, are in equilibrium, it follows from Eq. (1.4) that the elementary work done by pressure forces k=

1

on the possible displacement of the internal boundaries of the given system is zero. Then Eq. (2.5) may be written in the final form

.t

8 f T'kdV-k- ~ (SA~'-4-SA~')dV + .( p~.6u~dS dt-~O, vk s'k

(9~.6)

t, Ik=l 1. vk

where S~ = Sk fq S is the "external part ,r of the boundary Sk. Equation (2.6) is the mathematical formulation of a principle analogous to the Ostrogradskii principle for thermodynamically noninteracting noninterpenetrating multicomponent continuum systems. The characteristic feature of the principle is the presence of t e r m s characterizing the work of internal forces of the system (potential and nonpotential) on the possible displacement of points of the system. The principle may be formulated as follows. The true motion of a thermodynamically noninteracting noninterpenetrating multicomponent continuum system is realized when Eq. (2.6) is satisfied. The resulting principle may only conventionally be called an Ostrogradskii principle, since in this case the system is described by Euler variables (in the classical mechanics of discrete systems only Lagrange variables are used) and, in addition, it is impossible to separate the functional in Eq. (2.6) [5].

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Note that no s i m p l i f y i n g a s s u m p t i o n s w e r e m a d e in p a s s i n g f r o m Eqs. (1.1) and (1.2) to Eq. (2.6). All the t r a n s f o r m a t i o n s involved a r e r e v e r s i b l e , since the 5 ~ k a r e independent ( 5 ~ k and 5 ~ m , d e t e r m i n e d in the r e g i o n s Vk and V m , depend only on the s u r f a c e s S m k ) . All this indicates that the v a r i a t i o n a l p r i n c i p l e obtained and the equation of motion a r e equivalent, In g e n e r a l , Eqs. (1.1) and (1.2) - a n d hence Eq. (2.6) - do not constitute a c l o s e d s y s t e m . To f o r m a c l o s e d s y s t e m of equations, it is n e c e s s a r y to add to the given s y s t e m an equation of s t a t e - the e n t r o p y - c o n s e r v a t i o n law, x~itten s e p a r a t e l y for each component of the s y s t e m . In a n u m b e r of c a s e s (an i n c o m p r e s s i b l e liquid with constant v i s c o s i t y , an e l a s t i c body obeying H o o k e ' s law, etc.) t h e r e is no n e e d to use the equation of state, since this law has a l r e a d y been u s e d in deriving the equation of motion. E l i m i n a t i n g the c o m p o n e n t s of the p r e s s u r e t e n s o r 2P"k f r o m Eq. (2.6) as a r e s u l t of the equation of state, it is found that Eq. (2.6), together with a s y s t e m of continuity equations and initial and kinematic bounda r y conditions, f o r m s a c l o s e d m a t h e m a t i c a l p r o b l e m . The p r i n c i p l e obtained m a y be u s e d to d e s c r i b e s y s t e m s of contacting m e d i a in v a r i o u s a g g r e g a t e states and s y s t e m s with a d i s c r e t e - c o n t i n u o u s s t r u c t u r e . The advantage of this p r i n c i p l e is that when it is used t h e r e is no n e e d to d e t e r m i n e the internal interaction f o r c e s between the individual components of the s y s t e m and the dynamic boundary conditions, since t h e s e follow a u t o m a t i c a l l y f r o m the principle. Note that, when the given s y s t e m is in equilibrium, T = 5T = 0 and, if N = 1, Eq. (2.6) in this c a s e coincides, except for the notation, with the L a g r a n g e v a r i a t i o n a l f o r m u l a for static p r o b l e m s of e l a s t i c i t y t h e o r y

[10]. In a n u m b e r of c a s e s when t h e r e is a potential density of all the f o r c e s acting on the s y s t e m and coupling r e a c t i o n s (internal and external} and when N = 1, Eq. (2.6} takes the f o r m t~

6 3' .[ (r* - n*) dVdt =

0.

(2.7}

tlV

H e r e T* and 1I* a r e bulk densities c o r r e s p o n d i n g to the kinetic e n e r g y of the s y s t e m and the potential of the f o r c e s acting on the s y s t e m and the coupling r e a c t i o n s (internal and e x t e r n a l ) . In the f o r m in Eq. (2.7), the H a m i l t o n - O s t r o g r a d s k i i p r i n c i p l e is p o s t u l a t e d in [1, 6]. The condition that t h e r e m u s t be a potential of the f o r c e s acting on the s y s t e m and the coupling r e a c t i o n s r e s t r i c t s the region of application of the p r i n c i p l e in this f o r m , since in g e n e r a l it is not s a t i s f i e d in the m e c h a n i c s of continuous media. Thus, the p r i n c i p l e in the f o r m in Eq. (2.6) includes, as s p e c i a l c a s e s , the p r i n c i p l e s given in [1, 6, 10], and is t h e r e f o r e t h e i r generalization. 3. In c o n t r a s t to the a l r e a d y known p r i n c i p l e s , which a r e only valid for c o n s e r v a t i v e s y s t e m s , the p r e s ent p r i n c i p l e m a y be used to d e s c r i b e the d y n a m i c s of n o n c o n s e r v a t i v e s y s t e m s , including t h o s e with d i s s i p a tion. To c o n f i r m that the p r i n c i p l e is valid, consider its application to the c a s e of a s y s t e m of solid p a r t i c l e s of a r b i t r a r y f o r m moving in a flow of v i s c o u s c o m p r e s s i b l e fluid. It m a y be shown that in this c a s e the p r i n c i ple is equivalent to the equation of motion of the s y s t e m , in combination with the dynamic b o u n d a r y conditions. The v a r i a t i o n a l r e l a t i o n in E~. (2.6) m a y now be written in the f o r m

8

-~- ~n=lE --

+

-> ">

pvz dV~, ~pF.SudV--

t'n(n)~(nJt'2c-i,k=l 2-

:'

"'ri "vk ]

_ aSU: Pu~dV

.~g(n} ~(nJ,~(n)

H e r e p is the density of the fluid occupying region r ; v, v e l o c i t y of the fluid p a r t i c l e s ; F, bulk f o r c e acting on the fluid; P i j , components of the p r e s s u r e t e n s o r [7]

*"; = ' \ a,~ . a-g-,] -

p+ T ~

aT~ ~'J;

(a.2}

m (n) is the m a s s of the n - t h p a r t i c l e ; ~(n ), d i s p l a c e m e n t v e c t o r of the c e n t e r of inertia of the n - t h p a r t i c l e with r e s p e c t to the initial position; J i ~ ), i n e r t i a t e n s o r of the n - t h p a r t i c l e ; -~ (n), v e c t o r c o r r e s p o n d i n g to the angle of rotation of the n - t h p a r t i c l e with r e s p e c t to its initial position; F (n) and ~I ~a>, r e s p e c t i v e l y , the m a i n

1 041

moment of the external forces applied to the particles; 5~, possible displacement of the liquid particles; 5~ (n) ancl 5"~(n), possible displacements of the solid particles, characterizing the translational and rotational motion, respectively. Note that there are no external surface forces. The kinematic conditions of flow continuity lead to the following relations between the possible displacements at the flow boundaries of the fluid 8u-~ 8~(n) + 8~(n) X § (nJ on S~,

6u ---~0 on E;

(3.3)

where z is the flow boundary of the fluid; Sn, external surface of the particles; ~(n), N external unit-vector normal to the surface Sn; ~(n), vector joining the center of inertia of the n-th particle to some point of the s u r f a c e Sn. The t e r m s appearing in Eq. (3.1) are now successively transformed, isolating the coefficients of the independent variations ~j, oc~j ~ .(n),5r . Taking into account the motion of the region T, the following relation may be written e, ,,

1 §

t.

f 1.7

.

.

t~'

.

.

i~ 1

"

§247

">

]

.=,s.

Using Eq. (3.3) and integrating by parts with respect to the time in the first integral, the result obtained is as follows

tl 6

9 t1

dVdt-~

--

9

t, , t1 9

P ~d~ .8~ dVdt +

-~ pvZN(~).(8 r

+ 8~(~) x r(n)) dsdt.

(a.4)

t1Sn

n=l

Integration by parts with respect to the spatial coordinate in the third integral of Eq. (3.1) gives t=

3

t=

-;2"

t 1 7; l , ] = l

3

=o

t I Y. l , l = l

t=

3

n=l tlSni,]=l

t,

3

tx x

i,i=l

Note that, when Eq. (3.3) is taken into account, the first integral vanishes. Without giving the transformations of the remaining t e r m s in Eq. (3.1) and a number of intermediate calculations, it is simply noted here that on the basis of Eqs. (3.2)-(3.5), equating to zero the coefficients of the independent variations 5uj, ~c~j~ .(n) 5q~n), the following relations may be obtained 2

dv~

P-~=P~'J

OP

~

~

3

ox--f,+y~a

I~Z

O~j

1

O~v~ Ox~j '

(3.6)

9

(3.7)

(3.8)

3

Z s',

MI"+Q?.

i,j=l

Here R~n) are the components of the internal forces of the system, the forces of the particle interactions with the fluid, given by 3

.,f

- 2

Sn

l~1

3

+

+

z k~l

and Qcn) are the components of the internal moment of the system, characterizing the particle interactions ] with the fluid, given by 3

3

3

2

Ovk

~Vq V.qjir l .S a

q .t=l

while e i~tj r ~ is the Levi--Civita tensor [9].

1042

l.q ,l=I

k=l

Equation (3.6) gives the equation of motion of the f l u i d - t h e N a v i e r - S t o k e s equation [7]. Equations (3.7) and (3.8), taking into account Eqs. (3.9) and (3.10), may be regarded as the dynamic boundary conditions for the fluid and the equation of motion of the particles. Thus, from the variational principle in the form in Eq. (2.6) for the particular case of a nonconservarive multicomponent system, the equation of motion and dynamic boundary conditions may be obtained. This means, according to [6], that the given principle is valid for the d i s c r e t e - c o n t i n u o u s system investigated. LITERATURE 1. 2. 3. 4.

5. 6. 7. 8. 9. 10.

CITED

G. Goldstein, Classical Mechanics [in Russian], Nautm, Moscow (1975). A . A . Ii'yushin, Mechanics of Continuous Media [in Russian], Moscow State Univ. (1971). N.A. KiPchevskii, Course of Theoretical Mechanics [in Russian], Vol. 2, Nauka, Moscow (1977). N . A . Kil'chevskii, N. E. Tkachenko, and L. M. Shal,da, " P r o b l e m s of the analytic mechanics of s e m i aggregate systems," in: Nonlinear and Therm al Effects in Transient Wave P r o c e s s e s [in Russian], Proceedings of a Symposium, Vol. 1, Gorki (1973)9 pp. 211-234. M.O. Kil'chevskii, G. D. Nechiporenko, and L. M. Shal, da, Principles of Analytic Mechanics [in Ukrainian], Naukova Dumka, Kiev (1975). J . W . Leech, Classical Mechanics, Chapman and Hall, London (1965). L . G . Loitsyanskii, Mechanics of Liquids and Gases [in Russian], Nauka, Moscow (1973). V . V . Rumyantsev, "Some variational principles in the mechanics of continuous media," l>rikl. Mat. Mekh., 36, No. 6, 963-973 (1973). L . I . Sedov, Foundations of the Non-Linear Mechanics of Continua, Pergamon (1965). S . P . Timoshenko, Theory of Elasticity, McGraw-Hill (1970).

DYNAMIC

PROPERTIES

OF STOCHASTIC B.

P.

OF C O M P O S I T E

MATERIALS

STRUCTURE

Maslov

UDC539.4

Methods of calculating the dynamic c ha ract eri st i cs of nonuniform media have been developed in [1, 2, 4-8], etc. In the present work, the propagation of elastic harmonic waves in an infinite macroscopic isotropic medium with random properties is investigated. Analogous problems were considered under various r e s t r i c tions in [1, 2, 4, 5, 7, 8]. The problem is solved using the method of conditional moment functions used e a r l i e r to calculate static c ha r act er i s t i cs [3]. By using the apparatus of conditional averaging, limitations on the fluctuations of the elastic properties of the components may be avoided, and hence it is possible to use the method outlined to predict the properties of building materials of fiberglass type, etc. 1. The equation of motion is written in the form a~ - p ~ = o,

(1.1~

where ~ is the s t r e s s tensor; p, density; U, displacement vector; a dot over a symbol, differentiation with r e s p e c t to the time t; O, operator of differentiation with r e s p e c t to the coordinate; the product of operators and tensors is taken to be their convolution with r e s p e c t to internal indices. Considering wave motion of the form U (x, t) ~-- u (x) exp (iot),

(1.2)

substitution of Hooke's law = k~ -----kau

(1.3)

into Eq. (1.1) gives

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 14, hTO. 10, pp. 35-40, October, 1978. Original article submitted October 18, 1977.

0038-5298/78/1410-1043507.50 9 1979 Plenum Publishing Corporation

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