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A p p l ie d M a t h e m a t i c s a n d M e c h a n i c s(Engl i sh Edi t ion , Vol . 13 , No. 9 , Sep. 1992)

P u b l i s h ed b y S U T,S h an g h a i , C h i n a

S O M E G E N E R A L T H E O R E M S A N D G E N E R A L I Z E D A N D P I E C E W I S E

G E N E R A L I Z E D V A R I A T I O N A L P R I N C I P L E S F O R

L I N E A R E L A S T O D Y N A M I C S *

Xing J ing- t ang ( J ]]~ .' !~ )

( Pek ing Institute o f Aeronautics an d Astronautics, Beijing)

Zh en g Zh ao - ch an g (~ [~ J~~ )

( Tsinghua Un iversity, B eijing )

(Received June 28, 1988; Com mun icated by Chien W ei-zang)

A b s t r a c t

From the concept o f our-dimensional space and under the fou r kinds o f t ime l imit

conditions, some general theorems fo r elastodynamics are developed, such a s the principle

o f possible w ork a ction, the virtual displacemen t principle, the virtual stress-mom entum

principle, the reciprocal theorems and the related theorems of time terminal conditions

derived fro m it. The variational principles o f pote ntial energy a ction a nd complementary

energy action, the H - W principles, the H -R principles an d the,constitutive variational

principles fo r elastodynumics are obtained.'Hamilton's principle, T oupin's work an d the

fo rmu la t ions o f Re f [5],[17]- [24] m ay be regarded as some special cases o f the general

principles given in the paper. B y considering three cases: piecewise space-time dom ain,

piecewise space domain, piecewise time domain, the piecewise variational principles

including the potential, the complementary an d the m ixe d energy action fash ions are given.

Finally, the general form ulatio n o f piecewise Variational principles is derived. If the time

dimension is no t considered, the form ulation s obtained in the paper will become the

corresponding ones fo r elastostatics.

K e y w o r d s variational principle, elastodynamics, general theorem, boundaryvalue problem of four-dimensional domain, dy namics

I . I n t r o d u c t i o n

Th e gene ra l t heor em s and var ious v ar i a t i ona l p r inc ip l es fo r e l as tos ta t i cs have been s tud ied and

sum m ed up sys t emat i ca l ly . Espec i a l ly , ma ny wo rks o n th is t op ic t~-~0] have been p ub l i shed in ou r

co u n t r y . H o w ev e r , it s eems t h a t t h e s am e p r o b l em f o r e l a s to d y n am i cs h a s n o t b een s t u d i ed

exhaus t ive ly .

* Collected in the Proceedings o f the Invitational China-Am erican Workshop on Finite Element

M ethods, Chengde, People's Republic o f China,June 2 - 6, 1986."Supported by the Doctorate Training Fun d

of National Education Comm ission of China.

825

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826 Xing Jing-tang and Zheng Zhao-chang

The reciprocal theorems for dynamic problem were studied by so me authors . In 1873,

Rayleigh0 ~3gave a reciprocal theorem between the amplitude and the phase of a harmonic motion.

The fashion of reciprocal theorem with the inertia forces to be added, which is based on the Betti's

theorernfor elastostatics, may be found in Refs. [1 -2 ]. The reciprocal theorems with convolution

integrals are discussed in Refs. [12 - 15]. The ones given in this paper are different from those o f the

previous works and the formulation given in Ref. [16] may be regarded as a special case of the

reciprocal theorem in the paper.

Studies on some variational principles with the conditions at two time limits t~ and t 2prescribed

are as follows. The famous Hamilton principle usually considers the potential fashion in which some

consistence displacements are used to describe the configuration of a system and the variations o f

the displacements must vanish a t the two time limits t z and t 2. Usingsome momenta that satisfy the

dynamic equilibrium and ~he variation vanishing at the time limits t~ and t 2, Toupin07] and

Crandalit~sJ developed the complementary one o f the Hamilton principle. Green and Zerna derived

the fashion of the Hamil ton principle for elastodynamics. Chen t2~ extended the work done by

Toupin to linear elastodynamics. Under the conditions that the variations of displacement at the

time limits t~ and t 2 vanish, Truesdell and Toupint2Jj, Yut22], BarrtZ3J, and Dean and Plass t241 and

others studied the Hamilton principle for elastodynamics in order to obtain the fashions of the H-W

principle, the H-R principle, etc., for elastodynamigs. Oden and Reddyt51 made an excellent

description about some variational principles for theoretical mechanics. Under the conditions of

both displacement and momentum vanish at the time limits t~ and t2, they discussed some

variational principles for linear elastodynamics in which the constitutive principles are included. In

Refs. [27 -28], the reasonable conditions at the time limits t~ and t 2 are proposed, and some

corresponding general theorems and variational principles are derived, .the formulations of whichgiven by other authors in the previous works may be regarded as some special cases. This paper is a

part of Ref. [28].

I I . Basic Defini t io ns a n d E q u a t i o n s

The following definitions are introduced in this paper.

1. Action

If the variable f(t ) is dependent on the time t, the integral

tzC

I -- -- Jr ~ f ( t ) d t (2.1)

is called the action of the variable f(t) in the interval [t~, t2]. We suppose that integral. (2.1) always

exists in the problems concerned here. The action represents the integral effect of the var iablef(t )

with respect to the time t. Iff( t) is a force, the integral/ is the impulse of the force. But here, we do not

require that the variablef(t) be a force. For the clearness of description we shall call the integral I

"some act ion" by means of the name of the variable fit). The word " act ion " is quoted from the

Hamilton Principle.

2. T h e f u n c t i o n s o f p o t e n t ia l e n e r g y a n d c o m p l e m e n t a r y e n e r g y

Suppose that the elastic body considered in the paper is a hyperelastic body and the forces

related are potential forces.There exist the functions of potential energy and complementary one as

follows, which are calculated by means of the following integrals from the reference configuration

(static and unstreched) to the current configuration.

The function o f strain energy density:

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G e n e r a l T h e o r e m s o f L i n e a r E i a s to d y n a m i c s 8 27

,JA ( E , D = cr , jdE , j

Th e f u n c t i o n o f co m p l em en t a r y s tr a i n en e r g y d en si ty :

rE~J

A * ( E ~ ) - - -- J o E , j d a , j

Th e f u n c t i o n o f k i n ema t i c en e r g y d en s it y :

T ( V , ) = I ~ P , d V ,

Th e f u n c t i o n o f co m p l em en t a r y k i n ema t i c en e r g y d en s it y :

Th e p o t en t ia l o f b o d y f o rce :

i P iT * ( P , ) = V , d P ,

0

G ( U , ) - - - - I U i - F , d U ,

( 2 . 2 a )

( 2 . 2 b )

( 2 . 2 c )

(2.2d)

( 2 . 2 e )

Th e co m p l em en t a r y p o t en t ia l o f b o d y f o r ce :

G * ( P D = - U , d F , ( 2 . 2 f )

Th e p o t en t i a l o f t r ac t i o n o n t h e s u r f ace o f b o d y :

rU~ Ug ( U ~ ) ~ o - T M , ( 2 . 2 g )

Th e co m p l em en t a r y p o t en t ia l o f tr ac t i o n o n t h e s u r f ace o f b o d y :

g . ( , , ) = I ~ ' _ U , d T , ( 2 . 2 h )

I t is easy to o b ta in t he fo l lowing re l a t i ons be tween the func t ions o f po ten t ia l an d the i r

c o m p l e m e n t a r y o n e s:

A ( E , D "b A * ( a i D --- -a ltE ,~ ]

T ( V O + T * ( P O ~ P I V ~

G ( U , ) + G * ( ~ , ) = - - ~ U , ( 2 . 3 )

g ( U , ) + g * (~I',) - T , U ,

I t i s n e c e s s a r y t o p o i n t o u t t h a t t h e f u n c t i o n s i n ( 2 .2 ) r e d e p e n d e n t n o t o n l y o n t h e el as ti c o d y

t o b e co n s i d e r ed b u t a l so o n t h e b o d i e s w h i ch ac t th e f o r ce s o n t h e e l a st ic b o d y . A l t h o u g h s o me t i mes

we do n o t n eed to kn ow the exp l ic i t fo rm s o f t he func t ions i n (2.2) , we suppose tha t t hey a ll ex is tw h e n t h e w h o l e m ech an i ca l s y s t em co n s id e r ed is g iv en . O b v i o u s ly , f o r t h e d ead l o ad s, w e h av e

G ( U , ) = - I ~ , U ~ , G * ( I ~ , ) = 0( 2 . 4 )

g ( U , ) = - T , U , , g * ( T , ) = 0

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8 28 X i n g J i n g - t an g a n d Z h e n g Z h a o - c h a n g

3. T h e a c t i o n o f p o t e n t ia l o f m o m e n t u m a t t h e t i m e l i m i t s a n d i ts c o m p l e m e n t a r y

o n e

L e t u s c o n s i d e r a n el a st ic b o d y a t re s t. P , ( t v ) s t a n d s fo r an i m p u l s e b o d y f o r c e a c t i n g o n t h e

e la s t i c body a t t he moment tp , wh ich may be rep re sen ted by

F~ (x , t) = P , ( t D z l ( t - - tD ( 2 . 5 )

w h e r e z / ( t - - t t ) is D i r a c ' s d e l t a f u n c ti o n . S u b s t i t u t in g th e b o d y f o r c e i n ( 2. 5) i n t o t h e m o m e n t u m

equ a t ion (2.1 l ) and in t eg ra t ing i t f rom t .~ - to t p§ , w e m a y h a v e

P~ (x ,tF ) -----P, tD (2 .6 )

by mea ns o f the l imi t a tion t~_ - -* t ~+ . Thus , g iv ing the m om en tu m P~ ( tp ) a t t he m om en t tpm e a n s t h a t t h e c o r r e s p o n d i n g i m p u l s e b o d y f o r c e ( 2 . 5 ) a c t s o n t h e e l a s t i c b o d y . A c c o r d i n g t o

de f in i tions (2.1 ) and (2.2 ), t he ac t ion o f po ten t i a l o f m om en tum a t t he t ime ~ imi ts and i ts

c o m p l e m e n t a r y o n e m a y b e a s f o l l o w s .

T h e a c t io n o f p o te n t ia l o f m o m e n t u m a t t h e ti m e m o m e n t t :

( U D = ] t,_ I U i - p , ( t , ) , d ( t - t , ) d U , d l

U i"- - - - - - J o P ~ ( t D d U l (2. 7 a )

T h e a c t i on o f c o m p l e m e n t a ry o f m o m e n t u m a t t h e t im e m o m e n t tp:

rtp+ r P i~ * ( P') -- -- Jtp -J * - U ' d P ' ( t D ' d ( t - t ' ) d t

= - - I y ' U , d P , ( 2 . 7 b )

T h e t r a n s f o r m a t i o n r e la t io n b e t w e e n ~ ( U D a n d ~ *(- P~ ) is

( U , ) + ~ * ( P , ) ----- P , U ~ ( 2 . 7 c )

4. T h e g o v e r n i n g e q u a t i o n s

Rec tangu la r Ca r t e s i an coord ina te s z ( x , , i -- --1 , 2 , 3 ) w i ll b e e m p l o y e d f o r d e f in i n g t h e

t h r e e- d i m e n s i o n a l s p a c e ~ c o n t a in i n g th e b o d y . T h e i n te r i o r d o m a i n o f ~ is d e n o t e d b y r a n d t h e

su r face o f the bo dy can be d iv ided in to two pa r t s : pa r t S r ove r wh ich the ex te rna l fo rce s ~ a re

p re sc r ibed and pa r t Sv ove r wh ich the d i sp laceneb t s U~ a re p re sc r ibed . Ob v iou s ly S-- -- S r I.J S t ] ,

Sc r 17 S t ] = qb o D eno te the s t r ess t enso r by c r~ , t he s t r a in t enso r by E~ j , t he bo dy fo rce s by

P~. The U~ , P '~ and P~ a re the com po nen t s o f d i sp lacem en t f ie ld , t he ones o f ve loc i ty f i eld and

t h e o n e s o f m o m e n t a o f t h e d o m a i n , r e s pe c ti v e ly , v ~is t h e d i r e c ti o n c o n s i n e s o f t h e u n i t n o r m a l

d r a w n o u t w a r d s o n t h e b o u n d a r y S , p t h e m a s s d e n s i t y o f t h e b o d y , a ~ j k ~ a n d b t j k , t h e t e n s o r

o f el a s ti c cons tan t s a nd i ts r eve rse t enso r . Th e l inea ri zed gove rn ing eq ua t ion s o f de sc r ib ing the

m o v e m e n t 0 f t h e b o d y d u r i n g t h e t im e i n te r v a l I t I , t_ ,] E ~ 0 , o o ) m a y b e s u m m a r i z e d a sf o l l o w s .

The s t r a in -d i sp lacemen t r e l a t ions and ve loc i ty -d i sp lacem en t r e l a tions :

1E i j ~ . . ~ U i , j . ~ U j , ~ ) ( ~ , t ) r ( 2 . 8 a )

V , =U , , t ( 2 : 8 b )

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G e n e r a l T h e o r e m s o f L i n e a r E l a s t o dy n a m i c s 8 29

Th e s t r e ss - s tr a in r e l a t i o n s an d th e m o m en tu m - v e lo c i t y r e la t io n s :

E , ~ = b ~ a , t , ( x ,t ) {r X ( t ,, t, )P , = p V , ,

V , - - - - P , / p .

U s in g f o r m u la t i o n ( 2.2 ), w e h av e

1A(E,D =-[a,~,E,sE~,,

1T ( V , ) = ~ p V , V , ,

T h e m o m e n t u m e q u a t i o n s o f e q u i l i b r i u m :

~,~,~+F,=P,,, ( ~ _ , t ) ~ ( t , , t , )

T h e b o u n d a r y co n d i t i o n s :

1

1T * ( P ; ) = 2---~-P,P,

( 2 . 9 a )

( 2 . 9 b )

( 2 . 9 c )

( 2 . 9 d )

( 2 . 1 0 )

(2 11)

a l ~ y $ = T ; ( _x ,t )E S ~ • [ t l , t 2 ] ( 2 . 1 2 a )

U~-----U, . (_x , t )E Sv x [ t~ , t~ ] (2 .1 2 b )

Th e co n d i t i o n s a t t h e t im e l im i ts :

U , ( $ ~ ) - - - - U , ( ~ v ) , z _ E ~ v ( 2 . i 3 a )

P , ( ~ , ) = P , ( ~ , ) . z _ E ~ , ( 2 . 1 3 b )

w h e r e ~ , ~ , a r e t h e s e t o f t h e t i m e li m it s t w h i c h t h e d i s p l a c e m e n t s a r e p r e s c r i b e d a n d t h e s e t

o f t h e t i m e l i m i t s a t w h i c h t h e m o m e n t a a r e p r e s c r i b e d , r e s p e c ti v e l y . h e s e t { t ,, t , } -- - -

~ vU ~ , ,Zv n ~ , = do ; 9 v -- -- ~ X ~ v, u 1 6 5 tp .O b v io u s ly , t h e r ea r e f o u r ca s e s t o b e i n c lu d ed in

condit ions (2 .13) , that is

a ) | t r = { t , , ta }~ | t- -- - do : the d i spla ce m en t f ie lds at th e l imits t, an d t 2 are prescr ibe d;

b ) t u = dO, t , = { t , , t ~ } r t h e m o m e n t u m f ie ld s a t t h e l im i ts t, a n d t ~ a r e p r e ~ r i b e d ;

c) t v = { t , } , t , - - - ~ t ~ } : t h e d i s p l acem e n t a t t o e l im i t t, an d t h e m o m en tu m a t t h e l im i t t2 a r e

p rescr ibed ;

d ) . /v- -- -{ t2} , t t = { t~ } : the m om en tu m a t the l imi t t~ an d the d isp lace me n t a t the l imi t t2 a re

p rescr ibed .

I I I . S o m e G e n e r a l T h e o r e m s f o r L i n e a r E l a s t o d y n a m i c s

F i r s t, w e i n t r o d u c e s o m e d e f in i ti o n s a s f o ll o w s .

a ) A d m is s ib le d i s p l acem en t f ie lds U ~ ( A D F ) an d g en e r a li z ed ad m is s ib le d i s p l acem en t f ie lds

U~* * * ( G A D F ) . Th e A D F a r e d e f in ed a s a s y s t em o f d i s p l acem en t s s a t i s fy in g re l a t i o n s ( 2.8 ), ( 2 . ; 2 b )

an d ( 2 .1 3 a ) a s w e l l a s b e in g r e s t r i c t ed t o b e in g t r i p ly d i f f e r en t i ab l e . Th e G A D F a r e t h e A D F

r e lea s in g t h e r e s t r i c t i o n t o b e in g t r i p ly d i f fe r en t iab l e . O b v io u s ly , a ll A D F a r e i n c lu d ed in a l l G A D F .

b ) A d m i s s i b l e s t r e s s - m o m e n t u m f i e l d s a ~ * * a n d P y * ( A S F ) a n d g e n e r a l i z e d a d m i s s i b l e

s t r e s s - m o m e n t u m f i e l d s c r y # * * * , P ~ * * * , / ? , * * * a n d ~ , * * * ( G A S F ) . T h e G A S F a r c d e f i n e d

a s a s y s t e m o f s t r e s s e s r ~ * * * , m o m e n t a P ~ * * * , b o d y f o r c e s / ~ * * * a n d t r a c t i o n s T ~ * * * s a t is f y in g

e q u a t i o n s ( 2 .1 ) , ( 2 . 1 2 a ) a n d ( 2 . 1 3 b ) . T h e A S F a r e t h e G A S F r e s t r i c t i n g F i * * * - = F ~ p r e s c r i b e d

a n d T ~ * * * ~ T , p r e s c r i b e d . S i m i l a r l y , a l l A S F a r e a l s o i n c l u d e d i n a l l G A S F .

1 . T h e p r i n c i p l e o f a d m i s s i b l e w o r k a c t i o n

F o r a n a r b i t r a r y A S F c r y # * * a n d P , * * , w e h a v e t h e i n t e g r a l r e l at i o n

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8 3 0 X i n g J i n g - ta n g a n d Z h e n g Z h a o - c h a n g

Obvious ly , we a l so have the in t eg ra l r e l a t ion

I l l { I ' [ c r "* * * E " * - P ' * ' * V ' l d r ) d t = ~ ; : { ~ * . " ' ' ' ' U ' ~ d S + J s . a ' ' ' ' ' ~ ' ' U ' d S

+ I p , * ' * U , d r } d t - I , ~ , , " " U , d r - ~ , u ~ P , ' " O , d r ( 3 . l b )

f o r a n a r b it ra r y A D F U ~' a n d a n a r bi tr a ry G A S F cr ~ * ** , F # * * , ~ # * * a n d P # * * , w h e re

s t ands fo r the un i t ou twa rds no rma l a t t he t ime l imi ts . Obv iou s ly , fo r the t ime in t e rva l E tu , t , ] ,

w e h a v e ~ ( t e ) - - - - - - 1 , s . F o r m u l a t i o n s ( 3 .1 ) m a y b e d e m o n s t r a t e d e a s il y a n d a r e

neglec ted here .

2. T h e p r i n c ip l e o f v i r t u a l d i s p l a c e m e n t s

I f w e t a k e th e ac t u a l s t re s s - m o m e n t u m a ~ j a n d P ~ a s t h e A S F a n d t h e s u m o f t h e v i r tu a l

d i s p la c e m e n t s ~ U ~ a n d t h e a c t u a l d i s p la c e m e n t s U ~ a s t h e A D F i n ( 3. 1a ), w e m a y o b t a i n t h e

p r inc ip le o f v i r tua l d i sp lacem en t s

- - I~ r ~ P l3 U ~ d r (3 t 2 )

3. T h e p r i n c i p le o f v i r t u a l s t r e s s - m o m e n t a

I n r e l at i o n (3 .1 a ), w e c a n t a k e t h e ac t u a l d i sp l a c e m e n t s U ~ a s th e A D F a n d t h e s u m o f th e

a c tu a l s t re s s - m o m e n t a a ~ j , P ~ a n d t h e v i rt u al s t re s s - m o m e n t a ~ a ~ , ~ P , a s t h e A S F . T h e n w e

m ay de r ive the p r inc ip le o f v i r tua l s t r e ss -mom enta

S i m i l a r l y . f o r t h e G A S F a n d A D F w e h a v e

N ow , l e t u s exp la in the phys ica l me an ing o f the in t eg ra ls on ~ , o r ~ u in r e l a t ions (3 .1 ) - (3.3 ). By

u s e o f f o r m u l a t i o n s ( 2 .5 ) a n d ( 2 .6 ), t h e a c t i o n o f t h e v i r tu a l w o r k c o m p l e t e d t h r o u g h t h e v i r tu a l

d i s p la c e m e n t s ~ U ~ b y t h e i m p u l s e b o d y f o r c es P ~ ( _ z, t) c o r r e s p o n d i n g t o t h e m o m e n t a P ~

p r e s c r i b e d a t t h e t i m e m o m e n t t , E [ t l , t 2 ] m a y b e c a l c u la t e d b y t h e i n te g r al

i t ,~ ~ F , ( z , t ) , U , d r d t = i t , , f P , ( t , ) , d ( t - - t , ) 3 U , d r d tt z - J * d t z - d ~

= I , (P ,c ~ U ,) tp d r ( 3 . 4 a )

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G en er a l Th e o r em s o f L i n ea r E l a s t o d y n ami cs 83 1

C o n s i d e r i n g t h e ca s e o f t ~ = { t , ,t ~ } an d t~----~b , the last integ ral in f or m u la t io n (3.2) is

- ~ , p ~ ' , d U , d r = - { I [ ( ~ , , d U , ) , t + ( ~ ' , ~ U , ) , ~ ] d r }

= I ( P ~ d U ~ ) , ~ d r - f ( P ~ J U O , ~ d r ( 3 . 4 b )

wh ere the def in i t ion o f ~ is u sed . Phys i ca lly , i n t eg ra l (3 .4b) mean s the sum of t he ac t ions o f the

v i rt u a l w o r k co m p l e t ed t h r o u g h t h e d i s p lacemen t s J U ~ b y t h e i mp u l se b o d y f o rce s ~ ( _ x , t )

co r r e s p o n d i n g t o t h e m o m en t a P ~ ( t t ) an d P , ( t D prescr ibed a t t he t ime l imi t s t~ an d t r Th e m inus

befo re t he i n t eg ra l a t t he t ime t 2 m eans tha t t h is pa r t o f ac t ion is expor t ed f rom the t ime in t e rva l [ tp

/ 2 ] "4. T h e t h e o r e m o f a c t i o n o f s t r a i n a n d k i n e m a t i c e n e r g y

Thi s t he orem i s co r re spon d ing to t he t heo rem o f s t r a in energy fo r e l as tost a t ics . I t s t a tes t ha t fo r

an a rb i t r a ry e las t ic bod y in dyna m ic equ i l i b r ium in the space ~, • t t , t 2 J t h e ac t ion o f t he d i f f e renceo f t h e s t ra i n en e r g y an d t h e k i n em a t i c en e r g y eq u a ls t h e s u m o f t h e ac t io n s o f t h e w o r k co mp l e t ed b y

t h e b o d y f o r ce s , t h e t r ac t i o n s an d t h e mo men t a a t t h e t i me l i m i t s t h r o u g h t h e co r r e s p o n d i n g

displacements , i .e .

Rela t ion (3.5 ) m ay be de r ived f rom (3.1a).

5 . q~he r e c i p r o c i t y t h e o r e m

S u p p o s e t h a t a l i n ea r e la s ti c b o d y can b e i n t w o d i f f e r en t s t at e s o f d y n am i c equ i li b ri u m w h i ch

are i den t i f i ed by m eans o f t he super scr ip t s (2.1) a nd (2.2) . O n acc oun t o f l inear e l as ti c it y , t he

rec ip roc i ty t heorem

It~ Y J U ~ d S +

- , " - 'i ~ .~ ~ jU [ ~ d S + U~l~dr d t~ , t l 8 0 S t : r

c e' , ,' v : , , a s ( 3 . 6 )

m ay be eas i ly ob ta ined f rom (3. l a ) .

6 . T h e r e c i p r o c i ty t h e o r e m o f w o r k a c t i o n

Let t he two s t a tes o f equ i l i b r ium in (3.6) be t he eases i n which on ly a fo rce is app l i ed

respectively, i .e.

S t a t e (1) : ~ [z~(~x , t) - -- -~~ ( .x~ , t ) ,d (_x - -_x m )

State (2): ~ 2 ) ( Z , t ) ~ - - - F < 2 , ~ ( X ( 2 ) , t ) z ~ ( z - - x t ~ ) ( 3 . 7 )

Based on (3.6) , we have the r ec ip roc i ty t heo rem o f w ork ac t ion

J " ' ' ot~ h

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8 3 2 X i n g J i n g - ta n g a n d Z h e n g Z h a o - c h a n g

a n d i t s t at e s t h a t t h e a ct i o n o f th e w o r k c o m p l e t e d b y th e f o r c e p ~ t~ t ~ o u g h t h e d i s p l a c em e n t U [ ~

e q u a l s t h e a c t i o n o f th e w o r k c o m p l e t e d b y t h e f o r c e J ~ t~ t h r o u g h t h e d i s p la c e m e n t U ~ t ).

7 . T h e r e c i p r o c i t y t h e o r e m o f d i s p l a c e m e n t s

L e t ~ ( x o ~ , t ) = , d ( t - t o ~ ) a n d l ~ ( ~ _ ~ , t ) - - - - - , d ( t - t e z ~ ) , ( t t < ~ t v ~ t ~ < ~ t ~ ) in

re l a tions (3 .7 ) and (3.8 ). Then we ma y ge t t he rec ip roc i ty theo rem o f d i sp lacemen t s

U , "~ (~_ ~ , t ~ ) - - U [ ~ ( x~ t~ , t " ~ ) ( 3 . 9 )

I t s t a t e s tha t t he d i sp lacemen t a t t he po in t x r z~ and the t ime m om en t t (z; p r o d u c e d b y t h e u n it

fo rce ac t ing on the po in t ~o~ and the t ime t (~ w i ll equa l t he d i sp lacem en t a t t he po in t " ~o~ and

the t ime t o~ wh ich shou ld be p re sc r ibed fo r the equ i l ib r ium o f the un i t impu l se fo rce ac t ing on th~

po int x~ z~ an d the t im e t (~ ' :

S imi la r ly , t he rec ip roc i ty theo rem o f the t r a c t ions on the Sv m ay be de r ived and i s neg lec ted here .

8 . T h e t h e o r e m s o f t h e c o n d i t i o n s a t t i m e l i m i t s t~ a n d t~

I f a ll var iables e xcep t for the va r iables a t the t im e l imi ts t~ and t~ in re la t ion (3 .6) vanish , we hav e

the theo rem s o f the cond i t ions a t t he t ime l imi t s t , and tz

o r

I r r p ~ ) r r c 2 ~ , _ / p ~ , ~ U ~ z , ) t z ] d r ~~ E(pl - -U~x' ) t L _ ( p i " - , U ~ , ' ) t t ] d r ( 3 1 1 )1" L k ~ ( ~ ( I l l k J ( r

which con ta ins the fou r c a se s a s fo l lows .

a ) S t a te (1 ): P ~ t ~ ( t t ) = O , U [ l ) ( t t ) - - - - M ( x - x ~ t ~ ) ;

Sta te (2) : p~t~(t2)=O, U ~( t~ ) - -_ - J ( x - x~ z~ ) ; p~2~ (x ~t) , t ~ ) _ _ p ~ ( ~ r t z) .

( 3 . 1 2 )

I t s t a te s t h a t t h e m o m e n t u m a t t h e p o i n t x c ~ a n d t h e t im e t 2 p r o d u c e d b y t h e u n i t d is p l a c e m e n t

impu l se ac ting on the po in t x o~ an d the t ime t~ w i ll equa l the minus o f the mo m en tum which shou ld

a c t o n t h e p o i n t ~ o ; a n d t h e t im e t~ i n o r d e r t o p r o d u c e a u n i t d i s p l a ce m e n t i m p u l s e a t t h e p o i n t

x ~z~ an d the tim e tr

b ) S t at e (1): U ~ ( t ~ ) ~ 0 , p ~ ' ~ (t ~ ) -- - -- z /( x - x ~ ~ ) ;

Sta te (2) : U~Z~( t~ )=O, P~2~(t2)-- -- ,d($-_x~2-~) ; U~Z~(xCa~, t~) -- - - - U ~ ' ( g ~ , tz ) .

( 3 . 1 3 )

I t s t a t e s tha t t he d i sp lacem en t a t the po in t x ~ and the time t~ p ro du ced by the un i t m om en t umimpul se ac t ed a t t he po in t x ca~ and the t ime t~ w i ll equa l the minus o f the d i sp lacem en t w h ich sho u ld

b e i m p o s e d a t th e p o i n t g ~u a n d t h e t im e t~ i n o r d e r t o p r o d u c e a u n i t m o m e n t u m i m p u l s e a t t h e

po int x ~ and the t ime tz ,

c) S t a te ( I) : P ~ ( t ~ ) - - - - 0 , U ~ ( t ~ ) - - - - ' , d ( g - x _ ~ ) ;

Sta te (2) : P ~ > ( t ~ ) - - - - d ( x _ - x _ ~ ) , U ~ z ~ ( t ~ ) = O ; ' P ~ ( ~ t ~ , t t ) - - -- -U~' (x_ ~ , t~) .

( 3 . 1 4 )

I t s t a t e s tha t t he d i sp lacemen t a t the po in t x<2~ and the t ime t2 p ro du ced by the un i t d i sp lacem en t

impu l se ac t ing on the po in t $ ~t ~ and the t ime t~ w il l num er ica l ly equ a l the m om en t um which s hou ld

ac t on the po in t go~ and the time tt i n o rde r to p rod uce a un i t m om en t um impu l se a t t he po in t x ~z~

and the t ime t r

d) Sta te (1) : P ~ t ~ ( t O = , d ( x _ - x . ~ u ) , U ~ i ~ ( t , ) = o ;

S ta te ( 2 ): P ~ )( tD - - -- O , U ~ Z ~ ( t ~ ) = ~ / ( x - ~ ' ~ ) ; P ~ ( x . (z~ , t~)- - - -U~:)(~ ~t) , t , ) .( 3 . 1 5 )

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G e n e r a l T h e o r e m s o f L i n e a r E l a s to d y n a m i c s 83 3

I t s ta t es t h a t t h e mo m en t u m a t t h e p o i n t ; ca) an d t h e t ime t 2 p r o d u ced b y t h e u n i t m o m en t u m

impul se ac t ing on the po in t x~ l> and the t ime t , wi ll nume r i ca l ly equa l t he d i sp l acem ent wh ich

shou ld be imp osed a t t he po in t z (1) an d the t ime t~ in o rd er t o p rodu ce a un i t d i sp l aceme nt impu l se

at th e poin t z,<z~ an d the t ime t :.

Th e rec ip roca l t heo rem s g iven in Ref . [16 ] a re con ta ined in t heo rem (3 .11).

I V . T h e G e n e r n l l z e d V a r i a t i o n a l P r i n c i p l e s f o r L i n e a r E l a s t o d y n a m i c s

1. T h e p r i n c i p l e o f p o t e n t i a l e n e r g y a c t i o n

By us ing the p r inc ip l e o f v i r t ua l d i sp l acem ent s (3.2 ), t he fu nc t iona l o f t he p r inc ip l e o f po ten t i a l

en e r g y ac t i o n

H , ,E U ,] - - - - I t t ' ( I [ A ( E , j ) - T ( V , ) - ' , U , 3 d r - I s " , U , d S ~ d t + I , ~ ' ,U , d r( 4 . 1 )

m ay be der ived . Th e co ns t ra in t cond i t i ons fo r t h i s func t iona l a re r e l a t ions (2 .8 ), (2.12b), (2 .13a) and

(2 .9) , and i t s Euler equat ions are the (2 .11) , (2 .12a) and (2 .13b) . The Hamil ton pr inciple for

e l as todyn am ics r equ i r ing the var i a t ions o f t he d i sp l acem ent s a t t he t ime l imi t s t~ and t 2 van i sh is a

special ease o f pr inciple (4 .1) .

2. T h e p r i n ci p le o f c o m p l e m e n t a r y e n e r g y a c t i o n

May be eas i ly der ived by use o f t he p r inc ip l e o f v i r t ua l s t r ess -momentum (3 .3a) as fo l lows:

/ -/x ~ [a ,~ , P~]-- -- [ A * ( c r ~ s ) - T * ( P ~ ) ] d r

- I S V ,C r ~ j~ ,,d S d r + I ,o ~ U i P ,d r ( 4 . 2 a )

wh ose cons t rain t con di t ion s consis t of (3 .11), (3 .12a), (3 .13b) an d (2 .9) , an d i t s Euler equa t ions a re

(3.8) , (3 .12b) a nd (3 .13a). T he T ou pin 's pr inc iple (3.4) (3 .5) requ i r ing the var iat ions of the m om en ta

(or veloci t ies) at the t ime l imi ts t~ an d t~ van ish i s al so a sp ecial case o f the g ener i l ized one (4.2a) .

F o r t h e G A S F , t h e f u n ct io n a l m a y b e w r i t te n a s

t l q

+ I n # ~ ( ' , ) d S - J s u U ,c r ,j~ ) ,d S d t 't - l , o ~ U , P ,d r - I , p ~ , * ( P , ) d r (4.2b)i n wh ich the var i a t i ons 3 /~ , , ~ and c~P~ do no t van i sh , so t ha t t he i ndepe nden ce o f

JScr~s an d c3P~ m ay be perm i t ted .

ft. T h e d y n a m i c f a s h i o n s o f th e H - W p r i n c i p l e

B y use of Lag ran gia n mul t ip l iers releasing co nst ra in ts (2 .8), (2 .12b) a nd (2 .13a) f rom

funct iona l (4 .1 ), w e ma y have o ne fash ion o f t he H -W pr inc ip le .

I " Sl 6 [ c r ~ s ,P i , U ~ , E ~ s , V ~ 3 = [ A ( E ~ D - T ( V ~ ) - ~ U ~[1 *

1 1 [ (U ~ -U ~ )C r~ T ~ d S

- I n , , U , d S ) d r + I , ~ J , ~ ( U , - U , ) P , d r ( , 1 . 3 a )

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834 Xing J ing-tang and Zhen g Zh ao-chan g

R e p l a c i n g t h e A ~ * ( ~ ,~ ) a n d T*~(P,) n (4 .2 a) w i t h A ( E , ~ ) a n d T ( F , ) b y m e a n s o f ( 2 . 3 )

a n d u s i n g t h e L a g r a n g i a n m u l t ip l ie r s, w e m a y o b t a i n a n o t h e r d y n a m i c f as h i o n o f t h e H - W p r in c i p le

+ I , v ZT ,P ,r r ( 4 . 3 b )

T h e r e is a t ra n s f o r m a t i o n b e~ w e en H i 6 a n d / '/ x 5

H ls+ Fime=O ( 4 . 3 c )

4 . T h e d y n a m i c f a s h i o n s o f t h e H - R p r i n c i p l e

S i m i l a rl y , B y u s e o f r e l a t io n ( 2 . 3 ), w e m a y o b t a i n e d t h e t w o f a s h i o n s o f th e H - R p r i n c i p l e f r o m

(4 .3a ) and (4 .3b ) , i . e .

I " { fI s [a~ ,P , ,U ~]= [T * t (P , ) -A **(a~D -P ~U I -P ~U ~, ,$1 .r

1 d v - ~ s ( U , - lT , ) a , ,v , d S - I s ff,U , d S } d ,l ' ( :Y , j ( ~ U , , $ 'g i -~ U s . , ) ]

§ ; p , u , d , : + f , (4.4a)

a n d

/ / ~ 8 [ a , ~ , P , , U , ] = [ A * ( a , ~ ) - T * ( P D § 2 4 7 D ] d v1 v

- -~ , U ~ , $ ~ ) J U , ~ S - - I S ( ~ ,J ~ ) J -- ~ J v i)U d S § I , U l T , P ,d = +Iep , (P , - -P , )U ,dT

( 4 . 4 b )A l s o , t h e r e i s a t r a n s f o r m a t i o n r e l a t i o n

H I s § = 0 ( 4 . 4 C )

F u n c t i o n a l s ( 4 .3 a ), ( 4 .3 b ) , ( 4 .4 a ) a n d ( 4 .4 b ) a r e t h e g e n e r a l f a s h io n s o f th e H - W p r i n c i p l e a n d

t h e H - R p r i n c ip l e fo r e l a s t o d y n a m i c s . T h e r e su l ts i n R e f s . ( 3 . 7 - 3 .1 D m a y b e s o m e s p e c ia l c a s es o f

t h e m .

5 . T h e c o n s t i t u t i v e v a r i a t i o n a l p r i n c i p le s

C o n s i d e r in g th e G A S F a n d t h e G A D F , w e h a v e t h e tw o c o n s t i t u ti v e v a r i a t io n a l p r i n c ip l e s

H ls[a ,~ ,P i , U i , ( P , , ~ l , P D

a n d

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G e n e r a l T h e o r e m s o f L i n e a r E l a s to d y n a m i c s 8 35

/ T e , E a , j , P , , U , ( g , , T , , P , ) ]

f ' S E a , ~ E , j - A ( E ~ D - P , V , + T ( V , ) - P , U I - G ( U , ) ] d ~

- I e ( , ,U , + g ( U , ) ) d S - I s u f f , a , , , , d S } d t+ I% C O ,P ,d r

"l- J~p ~ ( P , U , + g ( U , ) ) d r ( 4 . 5 b )

Th e con s t r a in t s fo r t hese two func t iona l s a re (2 .8 ), (2 .12b) and (2.13a) fo r t he G A D F and (2.11),

(2 .12a) and (2 .13b) fo r t he GASF. The Eu lcr equa t ions a re t he cons t i t u t i ve , equa t ion (2 .9b) fo r

func t iona l (4 .5a) and the cons t i t u t ive equ a t ion (2 .9a) fo r func t iona l (4 .5b) . I t i s no t d i f f i cu l t t o p rove

t h a t

H s s ' t- // ~ s = 0 ( 4 . 5 c )Th e const i tu t ive va r ia t iona l p r inc ip les fo r e las tici ty we re proposed by Oden and R eddy [~ .

Som e remarks a bou t the Oden 's const i tu t ive var ia t iona l p r inc ip les were g iven in [28 ] . I t shou ld be

po in ted ou t t ha t the G AS F to be used in (4 .5a ) fo r i ndependence o f 3c r j j and 3P~ in ~"

X ( t l , t 2 ) , an d t h e G A D F t o be u s ed i n (4 .5 b) f o r i n d ep en d en ce o f ~ E~ j an d 3 V= i n r

x ( t~ , f2 ) 9I f t h e G A S F a n d t h e G A D F a r c re p la c e d b y t h e A S F a n d t h e A D F , r es p ec ti ve ly , t he s e

indcpen dence s m ay n o t ex i st i n t he gen era l cases. Som e de t a i ls on t h i s p rob l em wi ll be g iven in

an o t h e r p ap e r .

I n t r o d u c i n g s o m e co n s t r a in t s o r r e l ea s in g s o me co n s t r a i n ts b y m ean s o f th e Lag r an g i an

mu l t ip l i e r m etho d in to o r f rom the abov e l i st ed var i a t i ona l p r inc ip l es , r espec t ive ly , we m ay d er ive

som e o the r f ash ions o f t he var i a t iona l p r inc ip l es fo r e l as todynam ics , wh ich a re a ll neg l ec t ed here .

V . T h e P i e c e w i s e V a r i a t i o n a l P r i n c i p l e s f o r E l a s t o d y n a m i c s

F o r s imp l ic i ty , s o me co n v en t i o n s an d n o t a t i o n s a r e u s ed . Le t ~ ,. d en o t e a s u b d o m a i n o f t h e

an d it s s u f f ace b e S = , S u . = S . f ] S v , S ~ . = S ~ n S . ; Eta, t .]8 b e a s u b in te r v al * ~ " o f

the t ime in t e rva l [t~,tz] and ~v~---- ' .U~, t , } ~ , ~ ~ l , a = { f q , f ~ ' } '~ l , . F u r t h e r m o r e , w e

define.

~u~ (x_,t) lx_E~.,tE~v6, ( x _, t) E~ = x ~u 6} ~ t ~ ( x _ ,t ) I x E ~ . , t E ~ , , , ( x _ , t ) E ~ x 5 ,, }

( 5 . 1 )

H B ~ E ~ , • t , ~ 6 3 = I t ' { I 1r, ~o E - d ( E , D - P , U , - a , ~ ( E , ~ - 1 U , ,~ - -2 U ~ , ~ )] d r

- I s ~ . o ( U , - ~ , ) a , ,y ~ d S - I s ~ " , U , d S }d r ( 5 . 2 )

[ a ~ E , ~ - A ( E ~ ~ )+ ( a ,~ , ~ + ~ , ) U ,] drt , . r a

- - i S u o a i J ' J ~ i d S - yS ,~ a ( r i f ~ J - " ) U ' d S } d ' (5 .4 )

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836 Xing J ing-tang and Zhen g Zha o-chang

H ~ [ + . X [ t r [ T ( V D - P , V ~ - P , , , U , ] d r } d t

+ J % . , (5.5>

1. T h e 5 - a r g u m e n t p i e c e w i s e v a r i a t i o n a l p r i n c ip l e o f m i x e d e n e r g y a c t i o n

a ) T h e c a s e o f n o t d iv i d i n g t h e t i m e i n t e r v a l [ t ~ , t~ ] . T h e s p a c e d o m a i n + is d i v i d e d i n t o

t h e t w o s u b d o m a i n s # . a n d ~ ?b , ~ -- -- ~ oI.J "?~ . Th e i n t e r f ace be tw een ~o an d ~ is de no t ed by

S . b . W e r e g a r d + o a s a d o m a i n i n w h i c h t h e a c t i o n o f p o t e n t i a l e n e r g y is e m p l o y e d a n d t ~ a s a

d o m a i n i n w h i c h t h e a c ti o n o f c o m p l e m e n t a r y e n e r g y is u s ed . I n t h e s u b d o m a i n v . • (t~ , t ~ ) , W e

( a ) ~< a) U (,~) ~-, (a>a y c h o o s e t h e f iv e a r g u m e n t s ..cr,~ , , ~ , , ~ ( ~ a n d V ~ ~ i n d e p e n d e n t o f e a c h

o t h e r a n d s o d o i n t h e s u b d o m a i n T b X ( t j , t ~ ) . I f t h e re i s n o c o n f u s i o n , t h e s u p e r s c r ip s ( a ) o r ( b ) w ill

b e n e g l e c t e d i n t h e f o l l o w i n g .

T h e f u n c t i o n a l :~ - I ~ ' ~ = H ~ ~ - - [I~ ~' + J-l~n~ ( 5 . 6 )

T h e E u i e r e q u a ti o n s : E q u a t i o n s ( 2 . 8 ). - ( 2.1 3 ) f o r e v e ry s u b d o m a i n a n d t h e c o n t i n u o u s c o n d i t io n s

o n t h e i n t e r f a c e

w h e r e

T~*) +T~a>----0 ( ~ , t ) E S . ~ x [ t t , t ~ ]

{ U ' ~ = U ~ ~ '( 5 . 7 )

w h e r e

9 f(- ~= H ~ *> H ~ ~ - / la.~

H ~ ~ x [ t i , t q ] o ] + H 6 ~ [ ~ x [ l~ , tq]o]

/-/~ a) = / 7 5 B [ ~ x [ t q , t z] a ] + 1 7 5 K [ ~ X [ t q , t z ] a ]

9 l~ nJ r= I+:d P~")s

W e c a l l t h e d- luax t h e a c t i o n o f m i x e d e n e r g y o n t h e i n t e r f a c e u .

T h e E u l e r e q u a t i o n s : E q u a t i o n s ( 2 .8 ) - ( 2 .1 3 ) f o r e v e r y s u b d o m a i n ;

o n t h e i n t e r f a c e

( 5 . 9 )

( 5 . 1 0 )

t h e c o n t i n u o u s c o n d i t io n s

/ ' -/ ~ * ) - - - - /' /~ s [ ~ , X E l , , / 2 ]- I + / - / 5 ~ [ ~ ' o x E t a, t . . ] ]

/75(~ ) = / 7 ~ . [ ~ . • [ t~ , t = ] - l+ / 7 5 x E ~ b x [ t , , t = ] ] ( 5 . 8 )

d-InnB= T~ b>U ~ "> d S d t = ~ b>,~ ~,jb>U ~ , > d S d tt~ S.b ti S,b

a n d w e c a ll d T n a ~ t h e a c t i o n o f m i x e d e n e r g y o n t h e i n t e r fa c e S o y x l-t I , t ~ ] .

b ) T h e c a s e o f n o t d i v i d i n g t h e s p a c e d o m a i n . T h e t i m e i n t e r v a l l - t i, t2 ] i s d i v i d e d i n t o t w o

s u b i n t e r v a l s c a n d d , a n d t h e t i m e m o m e n t t q ( t ~ < t q < t z ) i s t h e d i v i d i n g p o i n t . W e r e g a r d t h e

s u b d o m a i n ~, X [ t j , tq-1 o a s a d o m a i n i n w h i c h t h e a c t i o n o f p o t e n t i a l e n e r g y is e m p l o y e d a n d t h e

a c t io n o f c o m p l e m e n t a r y e n e r g y is u s e d in t h e d o m a i n + • [-tq,t2]n . L e t ~ d = + • ~ tq ~ . T h e

v a r i a t i o n a l p r i n c i p l e f o r t h i s c a s e i s a s f o l l o w s .

T h e f u n c t i o n a l :

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G e n e r a l T h e o r e m s o f L i n e a r E i a s t o d y n a m i c s 8 3 7

p~.~ ~ .~ + p 1 4 ~ ( ~ _ 0(~,~)E~o4 ( 5 . 1 1 )

TT(o) ~TT(d)

c ) T h e c a s e o f b o t h s p a c e a n d t im e d o m a i n s d i v i d ed . L e t ~ ----~ o U ~ , S ~ = ? . N ~ b ; E / t , t2 3

- - [ t t , t q ] . U Etq~ t ,qa, ~ tq }~ E tt , tq ]. 17 Etq,t2]a a n d w e d e f i n e th e f o l l o w i n g s u b d o m a i n s :

(a t ) : i . • E t't, t q ] . , e l a s t i c p o ten t i a l ene rgy an d k inem a t i c ene rgy employ ed ;

(ad ) : 7 . X Etq, t2qa, e l as ti c p o t e n t i a l e n e r g y a n d c o m p l e m e n t a r y k i n e m a t ic e n e r g y u s e d ;

(be ) : ~ b • E t ~ , t ~ ~ com plem en ta ry e l a s ti c ene rgy and k inem a t i c ene rgy em ployed ;

Cod) : ib • [tq,12]a, c o m p l e m e n t a r y e l a s t i c e n e r g y a n d c o m p l e m e n t a r y k i n e m a t i c e n e r g y

e m p l o y e d .

T h e f i v e a r g u m e n t s i n d e p e n d e n t o f e a c h o t h e r m a y b e c h o s e n f o r e v e r y s u b d o m a i n . T h e

cor re spond ing va r i a t iona l p r inc ip le i s a s fo l lows .

T h e F u n c t i o n a l :

g-15~ H , I E ( a c ) ]+ H s~[ ( a d ) 3 + HBxE ( a c ) 3 + H s~[ (b e ) 3 -1 7 5 sE (b e ) ]

- - I I , sE (bd) ] - I I , z C (a d) ] - ] -I sx[ (bd) ] + J-Inns - ,FInnz ( 5 . 1 2 )

T h e E L de r e q u a t i o n s : E q u a t i o n s ( 2 . 8 ) - ( 2 .1 3 ) f o r e v e ry s u b d o m a i n ; t h e c o n t i n u o u s c o n d i t i o n s o n

in te rfaces (5 .7) and (5 .11) .

2 . T h e t r a n s f o r m a t i o n r e l a t i o n s b e t w e e n c o m p l e m e n t a r y e n e r g y - a c t i o n a n d

p o t e n t i a l e n e r g y - a c t i o n

I t i s n o t d i f f i c u l t t o p r o v e t h a t t h e re a r e t r a n s f o r m a t i o n r e la t io n s b e t w e e n c o m p l e m e n t a r y

ene rgy-ac t ion and po ten t i a l ene rgy-~c t ion a s fo l lows :

I " I,aE ~ ox Ete , t . ]a3q ' I - l sJET. • [ tq , te]8] = T~ *~ U~~ (5.13)tq S.b

EYaxEtq,t,]~]=~_P ~ a) C(~'U~~)dr ( 5 . 1 4 )3 17

T h e s e t r a n s f o r m a t i o n r e la t io n s m a y b e u s e d t o t r a n s f e r f r o m o n e o f th e t h r e e k in d s o f 5 - a r g u m e n t

p iecewise va r i a t iona l p r inc ip le : the mixed one , the po ten t i a l o ne and the com plem en ta ry one , to

a n o t h e r .

3 . T h e p i e e e w i s e g e n e r a l iz e d v a r i a t i o n a l p r i n c ip l e s o f p o te n t i a l e n e r g y -a c t i o n a n dc o m p l e m e n t a r y e n e r g y - a c t i o n

By use o f the t r ans fo rm a t ion re l a t ions (5 .13 ) and (5 .14 ) , we m ay t rans fe r f ro m the ac t ion o f

com plem en ta ry en e rgy to tha t o f po ten t i a l ene rgy o r d o the reve rse. Then , the p iecewise gene ra li z ed

va r i a t iona l p r inc ip le o f po ten t i a l ene rgy -ac t ion and i t s com plem en ta r y on e a re ea s ily ob ta in ed a s

fo l lows :

, l - l ~ : H ~ [ (a c) ] + H ~ [ ( ad ) ] + H ~ E (b c) ] + H ~ E ( bd ) ] + H ,~ r[ (a c) ]

+ H ~ x [ ( a d ) ] + H ~ r [ ( b c ) ] + H ~ r [ ( b d ) ] + g T . . s - - f l H e ~ ( 5 . 1 5 )

w h e r e

9-1.~s = f T~ b~ ( U~ *~ -- U J b~ )d Sd tt t d Sob

( 5 . 1 6 )

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8 38 X i n g J i n g - ta n g a n d Z h e n g Z h a o - c h a n g

a n d

w h e r e

J 1 R n x = I -,~162176 ~ , -- U ~ a )) d r"?e d

J - l + n : " I I 6 s [ ( a c ) ] - I I ~ s [ ( a d ) ] - I I s s [ ( b e ) ] - I I s s [ ( b d ) ] - I I 6 x [ ( a c ) ]

- [ I s x [ ( a d ) ] - 17 5 x[ ( b c ) ] - - l - l b r [ ( b d ) ] + , l- In n j - f l n n x ( 5 . 1 7 )

J l n n i = (T, (*>-I T ,<b>)U,(*JdSdth S.h

s ----f (P,<~176 (5.18)J Cod

If some constraints arc imposed into the picccwise variational principles given above, the

corresponding principles for 4 arguments, 3 arguments, 2 arguments and ] argument may be

obtained and are neglected here.4 . T h e g e n e r a l f a s h i o n o f p i e c e w i s e v a r i a t i o n a l p r i n c i p l e s

The space dom a in ~" is d iv ided in to seve ra l sub dom a ins an d i a a nd f% s t and f o r a subd om a in

o f p o te n t i a l e n e r g y - a c t io n a n d a s u b d o m a i n o f c o m p l e m e n t a r y e n e r g y - a c ti o n , r e s p ec t iv e ly . T h e

t im e i n te r v al F t , , t 2 ] is d i v i d ed i n t o s e v er a l s u b - i n te r v a ls a n d t h e [ t q , / . ] ~ a n d t h e

I -t~ , t . 34 deno te a sub - in te rva l o f k inema t i c ene rg y-ac t ion an d a sub - in te rva l o f com plem en ta r y

one , r e spec tive ly . The in t e r face s be tw een the space sub dom a ins a re den o ted b y S . n , S B n a n d

S n n , t he t ime d iv id ing po in t s a re r ep re sen ted a s { t q } n n , { tq } n . a n d { t q } n n 9 T h e s u b s c r i p t s H

a n d / 7 m e a n p o t e n t i a l a n d c o m p l e m e n t a r y s u b d o m a i n s , r e s p e c t i v e l y , a n d a s t h e s u b s c r i p } H H

m e a n s t h a t t h e i n t e r f a c e S H n i s t h e i n t e r f a c e b e t w e e n t w o a d j a c e n t s u b d o m a i n s i n w h i c h t h e

po ten t i a l ene rgy ac t ion is em ploy ed , e t c. Le t ~?HR =~ • { t . } z n , ~ . U = ~ • { t , }H a a n d

Hn = ~ • { t~ } o n . Th e gene ra l f a sh ion o f p i ecewise va r i a t iona l p r inc ip le s i s a s fo l lows:

, / 7 = ~ ] H B F ~ . x E t , , t z ] q + Y ~ H x E ~ x E l q . t . 3 o 3a o

- ~ / 7 ~ E ~ , x [ t t , ~ 2 - 1 l - 3 - -] ,/ -/ xE ~ x l - t q , t . l a 3b d

+ ESnn S.~ S,,,

(5.19)

w h e r e t h e n u m b e r o f f ie ld a r g u m e n t s m a y b e f r o m s in g le t o f iv e . I t is e a s y t o p r o v e t h a t t h e

s t a t i o n a r y c o n d i t i o n s d e ri v e d b y m e a n s o f ~ u = 0 c o n s i s t o f t h e s y s t e m o f d y n a m i c e q u a t i o n s a n d

t h e c o n t i n u o u s c o n d i t i o n s o n t h e i n te r fa c es .

I t is no t d i f f i cu l t t o de r ive the gene ra l f a sh ion o f po n ten t i a l ene rgy-ac t ion a nd tha t o f

c o m p l e m e n t a r y e n e r g y - a c t i o n .

The gene ra l i t i e s o f the func t iona l (5 .19 ) a re a s fo l lows .

a ) T h e s p a c e d o m a i n a n d t h e ti m e i n te r v al c a n b e d i v i d e d i n t o s o m e s u b d o m a i n s a n d s u b -

in te rva ls a nyw ay . D iv id ing e i the r o f the space dom a in o r the t ime in t e rva l i s a spec ial c a se , and no

d iv id ing i s a more pa r t i cu la r c a se .

b ) C h o o s i n g p o t e n t ia l o r c o m p l e m e n t a r y s u b d o m a i n i s a r b i t ra r y , t h e t r a n s f o r m a t i o n r e l a ti o n

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G e n e r a l T h e o r e m s o f L i n e a r E l a s t o d y n a m i c s 8 39

p r o v i d e s a w a y o f t r an s f e r ri n g b e t w e e n p o t e n t ia l o n e s a n d c o m p l e m e n t a r y o n e s ; th e p i ec e w is e m i x e d

o n e , t h e p i e ce w i se p o t e n ti a l o n e a n d t h e p i ec e w is e c o m p l e m e n t a r y o n e a r e t h e t h r e e f a s h i o n s o f t h e

gene ra l f a sh ion (5 .19 ) .

c ) I n e v e r y s u b d o m a i n t h e i n d e P e n d e n t a r g u m e n t s t o b e s u p p o s e d m a y v a r y f r o m s in g le t o f iv e ,

a n d , a s a p a r t i c u l a r c a s e , t h e a n a l y ti c a l, s o l u t i o n m a y . b e u s e d i n s o m e s u b d o m a i n s , a l s o .

d ) T h e t r a c t io n s , t h e d i sp l a c e m e n t s a n d t h e m o m e n t a o n t h e in t e rf a c e s b e tw e e n t w o a d j a c e n t

s u b d o m a i n s m a y b e i n d e p e n d e n t o f e a c h o t h e r a n d n o c o n t i n u o u s c o n d i t i o n s o n t h e in t e rf a c es a re t o

b e s a t i s f i e d p r e v i o u s l y .

T h e p ie c e w is e v a ri a t io n a l p r in c i p le s f o r e l a s to d y n a m i c s g iv e n a b o v e m a y b e u s e d t o f o r m u l a [ e

s o m e m o d e l s o f fi n it e e l e m e n t m e t h o d s f o r d y n a m i c an a ly s is . T h e a u t h o r s h a v e r e s e a rc h e d t h e ~

a p p l i c a ti o n s in m o d e s y n t h e s is m e t h o d s a n d t i m e e le m e n t m e t h o d s f o r d y n a m i c a n a ly s is o f

s tru ctu re st2 5-2 91 . T h e r e s e a r c h f o r a p p l i c a t i o n t o f o u r - d i m e n s i o n a l e l e m e n t m e t h o d s r e m a i n s t o b e

d o n e .

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