(1)journal of elasticity volume 41 issue 1 1995 xiaoguang zhong -- dynamic behavior of the interface...

8/13/2019 (1)Journal of Elasticity Volume 41 Issue 1 1995 Xiaoguang Zhong -- Dynamic Behavior of the Interface Between Tw

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JournalofElas t ic i ty 41 : 39-72 , 1995 . 39 1995 Klu we rAc ad em ic Publishers. Printed in the Netherlands.

D yn am ic ehav ior of the Interface e tw een Tw oSo l id Ph ases in an Elast ic arX I A O G U A N G ZHONGDivision o f Engineering a nd Applied Science California Institute o f TechnologyPasadena CA 91125 USARe c e iv e d 2 4 J u ly 1 99 5Abs t rac t Us in g t h e c o n t i n u u m m e c h a n i c a l m o d e l o f s o l i d - s o l i d p h a s e t r a n si t io n s o f A b e y a r a tn e a n dKn o wle s , t h i s p a p e r e x a m in e s t h e l a r g e t im e d y n a m ic a l b e h a v io r o f a p h a s e b o u n d a r y . T h e p r o b l e ms tu d i e d c o n c e r n s a s e m i - in f in i t e e l a s t i c b a r i n i t i a l ly in a n e q u i l i b ri u m s t a t e t h a t i n v o lv e s two m a te r i a lp h a s e s s e p a r a t e d b y a p h a s e b o u n d a r y a t a g iv e n l o c a t i o n . I n t e ra c t io n b e twe e n t h e p h a s e b o u n d a r y a n dth e e l a s t i c wa v e s g e n e r a t e d b y a n im p a c t a t t h e e n d o f t h e b a r a n d s u b s e q u e n t r e fl e c ti o n s i s s t u d i e d i nd e t a i l, a n d a n e x a c t s o lu t i o n o f t h e d y n a m ic a l p r o b l e m , wh ic h i s g o v e r n e d b y a n o n l i n e a r r e. cu rs iv ef o r m u la , i s o b t a in e d . I t i s s h o wn th a t t h e p h a s e b o u n d a r y r e a c h e s a n e w e q u i l i b ri u m s t a t e fo r l a r g et im e . N u m e r i c a l c a l c u l a t i o n s b a s e d o n t h e r e c u r s iv e f o rm u la a r e c a r r ie d o u t t o i l lu s t r a te a n a ly t i c a lresults .

1. Introduct ionEr icksen [14] p ion eered the r esea rch on the use o f f in i te the rmoe las t i c i ty theory tom ode l phase t r ansfo rmat ions in so l ids by cons ide r ing the equ i l ib r ium of a ba r o fm ate r ia l t ha t can cha nge phase . N ow i t is w e l l e s t ab l ished tha t a no n l inea r e las t icmate r i a l capab le o f phase t r ansfo rmat ion has a no nconv ex f ree energy func tion ; seefo r exam ple Ab eyara tne [1 ] Rosak i s [21]. The nonconvex i ty o r m ore genera l lythe loss o f s t rong e l l ip t i c ity o f the f r ee energy func t ion l eads to the nonun iquen esso f the so lu t ions to bo unda ry va lue p rob lems o r in i t ia l -boundary va lue p rob lemsin f in it e e l as t ic i ty even though en t ropy cond i t ions a r e im posed a t s t ra in d i scon t in -uit ies [2 4] .

Recen t ly the re has bee n s ign i f ican t p rogress in the m ode l ing o f mar tens i t ict r ansfo rmat ions wi th the h e lp o f non l inea r the rmo e las t ic i ty theory . Th e m ainapproaches in th is ca t egory inc lude the min im iza t ion o f energy by w hich som emicros t ruc tu res a r e success fu l ly p red ic t ed by Ba l l and James [ 10] and B ha t t acharya[11] and the A beya ra tne -K now les m ode l wh ich qua l i t a tive ly p red ic ts hys te r es i s[2] and the sh ape m em ory ef fect [9] in m ar tensi t ic t ransformat ions and i s consis tentw i th v iscosi ty- reg ular ized theory [5] .

In the se t t ing o f con t inu um mech an ics the nonun iquen ess o f so lu t ions o f bas i cboun dary - o r in i t ia l -va lue p rob lems i s due to the l ack o f cons t i tu t ive in fo rmat ion

Ad d r e s s a f t e r Au g u s t 1 5, 1 99 5: D e p a r tm e n t o f E n g in e e r in g S c i e n c e a n d M e c h a n i c s , V i r gin i aP o ly t e c h n i c I n s t i tu t e a n d S t a t e Un iv e r s i ty , B I a c k s b u rg , VA2 4 0 6 1 , US A .


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4 0 X IA O G U A N G ZI ION Gabou t the phase t r ans fo rm at ion p roces s ; s ee [2 4 ] . In o rde r to r esto re un iq uen es si n s u c h p r o b l e m s v a r i o u s c o n s t i t u t iv e p o s t u l a te s h a v e b e e n u s e d s u c h a s t h em a x i m u m e n e r g y d i s s i p a t io n c r i te r i o n [ 13 ] a b s o l u t e m i n i m i z a t i o n o f e n e r g y [ 1 014] v i s cos i ty r egu la r iza t ion [15 ] [22 23 ] and pos tu la tes o f a cons t i tu t ive na tu rea t a p h a s e b o u n d a r y [ 2 4 ] . H e r e w e w i l l c o n c e n t r a te o n th e m o d e l p r o p o s e d b yA b e y a r a t n e a n d K n o w l e s .

B o r r o w i n g f r o m a v i e w p o i n t w i d e l y u s e d in m a t e r ia l s s c i en c e A b e y a r a t n e a n dK n o w l e s p o s t u l a t e d t w o a d d i t i o n a l c o n s t i t u t iv e re l a ti o n s a t a p h a s e b o u n d a r y . O n ei s t h e n u c l e a t i o n c r i te r io n w h i c h d e t e r m i n e s w h e n a n d w h e r e a n e w p h a s e w i l li n i ti a te f r o m a p a r e n t p h a s e a n d t h e o t h e r i s a k in e t i c r e l a ti o n w h i c h d e t e r m i n e st h e r at e o f p h a s e t r a n s f o r m a t i o n o r i n o t h e r w o r d s t h e p r o p a g a t i o n s p e e d o f p h a s eb o u n d a ri e s. I n th i s m o d e l t h e m o t i o n o f t h e p h a s e b o u n d a r y is a m e c h a n i s m fo rd i s s ipa t ion .

A b e y a r a t n e a n d K n o w l e s s h o w e d t h a t f o r a fi n it e b a r w i t h a n o n m o n o t o n i cs t r es s- s t ra in re la t ion the bou nda ry -va lue p rob lem as soc ia ted w i th a quas i - s t a ti cl o a d i n g p r o c e s s h a s a n i n f in i te n u m b e r o f s o l u t i o n s [ 2]. I n o r d e r t o s e l e c t a u n i q u es o l u t i o n t h e y i n v o k e d t h e k i n e t ic r e l a t io n a n d n u c l e a t i o n c r it e ri o n d i s c u s s e d a b o v e .I n t h i s w a y t h e y s e l e c t e d a u n i q u e s o l u t i o n w h i c h p r ed i c t s m a t e r ia l re s p o n s e t h a ti s q u a l i ta t i v e ly i n a g r e e m e n t w i t h e x p e r i m e n t a l o b s e rv a t i o n s i n u n i - a x ia l t e n s i o nt e st s. L a t e r t h e y e x t e n d e d t h e i d e a t o a f u l ly d y n a m i c a l m e c h a n i c a l th e o r y a g a i ns e l e c t in g a u n i q u e s o l u t i o n f r o m m a n y p o s s i b l e s o l u t i o n s [4 ] b y m e a n s o f t h e k i n e t -i c r e la t ion and nuc lea t ion c r i t e r ion . Th is approach has a l so been app l ied to thet h e r m o m e c h a n i c a l t h e o r y f o r a q u a s i- s ta t ic p r o c e ss [6 1 6] a n d t o d y n a m i c a l p r o -ces ses w i th hea t co ndu c t ion [7] o r to ad iaba t i c dyn am ica l p roces ses [8] . J i ang [17 ]h a s g e n e r a l i z e d t h e A b e y a r a t n e - K n o w l e s m o d e l to e l e c t r o - th e r m o - m e c h a n i c a l p r o -c e s s es . F o r s i m p l i c it y w e c o n s i d e r o n l y th e p u r e l y m e c h a n i c a l t h e o r y o f A b e y a r a t n eand K now les [4] .

I t h a s b e e n s h o w n b y A b e y a r a t n e a n d K n o w l e s [ 4 ] a n d J a m e s [ 1 5 ] t h a t t h em o t i o n o f a p h a s e b o u n d a r y i s d i s s i p a t i v e i n a n o n l i n e a r e l a s t o d y n a m i c t h e o r yf o r s o l i d -s o l i d p h a s e t r a n s f o r m a t i o n s . T h u s o n e e x p e c t s t h a t th e d y n a m i c s y s t e ma s s o c i a t e d w i t h p h a s e t r a n s f o r m a t i o n s m a y r e a c h a n e q u i l i b r i u m s t a t e a t l a r g et i m e o r f r o m a n e n e r g y p e r s p e c ti v e t h e d y n a m i c a l s y s t e m w i l l e v e n t u a l l y r e a c h am i n i m u m e n e r g y s ta te .

R e c e n t l y L i n a n d P e n c e [ 18 ] [1 9] i n v e s t i g a t e d t h e la r g e t i m e d y n a m i c a l b e h a v -i o r o f a p h a s e b o u n d a r y i n a o n e - d i m e n s i o n a l s e t ti n g in v o l v i n g a b a r o f f i ni teleng th . Due to the fo rmidab le ana ly t i ca l d i f f i cu l t i e s encoun te red in the i r fu l lyd y n a m i c p ro b l e m t h e y u s e d a n a p p r o x i m a t e e n e rg y a p p ro a c h . T h e y s h o w e d t h a tt o ta l e n e r g y d i s s i p a t e d is t h e e n e r g y n e c e s s a ry t o s e tt le t h e d y n a m i c a l s y s t e m i n t oa n e w m i n i m u m e n e rg y s t at e.H e r e a n a t t e m p t is m a d e t o o b t a i n a n e x a c t a n a l y t ic a l s o l u t i o n f o r t h e l a rg et i m e d y n a m i c a l b e h a v i o r o f a p h a s e b o u n d a r y i n t h e f r a m e w o r k o f A b e y a r a t n ea n d K n o w l e s [ 4]. I n o r d e r to a s s e ss t h e p h a s e b o u n d a r y m o t i o n f o r la r ge t im e w ec h o o s e t h e s i m p l e s t p o s s i b l e p r o b l e m . A s e m i - i n fi n i te b a r i n it i a ll y in a n e q u i l i b r iu m


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DYN MIC L BEH VIOR OF PH SE BOUND RY 41s t a te w i t h a s i n g l e p h a s e b o u n d a r y i s s u b j e c t t o i m p a c t a t i ts e n d . T h e m a t e r i a l o f t h eb a r i s a s s u m e d t o b e t r il in e a r , w i t h c o m m o n e l a s ti c m o d u l i i n t h e t w o m e t a s t a b l ep h a s e s . W i t h i n th i s s e tt in g , w e m a k e n o a p p r o x i m a t i o n s i n o b t a i n i n g o u r a n a l y t i c a lso lu t ion .U n f o r t u n a t e l y t h e a p p r o a c h e m p l o y e d h e r e c a n n o t b e u s e d to s o l v e th e p r o b l e mo f t h e f in i te b a r c o n s i d e r e d b y L i n a n d P e n c e [ 18 ] , [1 9 ]. A n u m e r i c a l a p p r o a c h i sn e c e s s a r y f o r t h e f in i te b a r p r o b l e m . W e h a v e i n v e s t i g a t e d t h e f in i te b a r p r o b l e mb y n u m e r i c a l a n a l y s i s [2 4 ].

I n S e c t i o n 2 , t h e m o d e l t o b e u s e d i s s u m m a r i z e d b r ie f ly . T h e i n i t ia l - b o u n d a r y -v a l u e p r o b l e m t o b e s o l v e d i s f o r m u l a t e d in S e c t i o n 3 . W e a s s u m e th a t t h e m a t e r i a li s th e s p e c i a l t r i li n e a r m a t e r i a l d e f i n e d i n S e c t i o n 2 . T h e k i n e t i c r e la t i o n i s a s s u m e dt o b e t h e s i m p l e s t p o s s i b l e o n e , f = ~ h , w h e r e ~ i s a c o n s ta n t , f is t h e d r i v i n gt r a c t io n o n t h e p h a s e b o u n d a r y , a n d h i s t h e p h a s e b o u n d a r y v e l o c it y . T h e r e i s ap r e e x i s t i n g p h a s e b o u n d a r y . I n S e c t i o n 4 , w e a n a l y z e t h e s h o r t t i m e b e h a v i o r o f t h ep h a s e b o u n d a r y d u e t o a n e x t e r n a l i m p a c t d is t u r b a n c e . W e d e t e r m i n e r e s t r ic t i o n so n t h e e x t e r n a l d i s t u r b a n c e a n d m a t e r i a l p a r a m e t e r s t h a t p r e v e n t t h e i n i t i a t i o no f n e w p h a s e b o u n d a r i e s . W e s o l v e t h e p r o b l e m f o r l r ge t ime i n S ec t i on 5 ,r e l y i n g o n t h e f a c t t h a t t h e r e i s o n l y o n e p h a s e b o u n d a r y a t a n y t i m e . S i m p l en o n l i n e a r r e c u r s i v e fo r m u l a e a r e o b t a i n ed f o r t h e m o t i o n o f th e p h a s e b o u n d a r ya n d t h e d e f o r m a t i o n i n t h e ba r. I n S e c t i o n 6 w e c o n s i d e r t h e p r o b l e m f o r m u l a t e di n S e c t i o n 3 f o r g e n e r a l m o n o t o n i c a l l y i n c r e a s i n g k i n e t i c f u n c t i o n 4 h ) . I n S e c -t i on 7 , num er i c a l ca l cu l a t i on s tha t supp or t ou r ana ly t i ca l r e su lt s a re ca r r i ed ou t .

2 P r e l i m i n a r i e sT h e m a t e r i a ls i n t h e A b e y a r a t n e - K n o w l e s m o d e l a re c h a r a c te r i z e d b y n o n c o n v e xs t ra i n e n e r g y f u n c t i o n s . T h e s e m a t e r i a l s c a n d e s c r i b e s ta b l e d e f o r m a t i o n s as w e l la s m e t a s t a b l e o n e s . I t i s a s s u m e d t h a t a d e f o r m a t i o n w i ll j u m p f r o m o n e s t a b l eo r m e t a s t a b l e p h a s e t o a n o t h e r w h e n c e r t a i n c r i ti c a l c o n d i t io n s a r e s a t is f ie d . T h eproc es s l eads t o a s t r a i n d i s con t i nu i t y ; i f t he s t r a in s on e i t he r s i de o f the d i s con t i nu i t ya r e i n d i f f e r e n t p h a s e s , t h e d i s c o n t i n u i t y i s c a ll e d a p h a s e b o u n d a r y ; o t h e r w i s e it isa s h o c k w a v e .

T h e b a s ic a s s u m p t i o n s i n t h e A b e y a r a t n e - K n o w l e s m o d e l a re :1 ) T h e d e f o r m a t i o n is in s m o o t h n e s s c la s s C 2 a w a y f r o m s h o c k w a v e s o r p h a s e

b o u n d a r i e s; d e f o r m a t i o n g r ad i e n ts a r e d i s c o n t in u o u s a c r o ss s h o c k w a v e s o rp h a s e b o u n d a r i e s , b u t th e d i s p l a c e m e n t is c o n t i n u o u s .

2 ) T w o s u p p l e m e n t a r y c o n s t i tu t i v e re l a ti o n s a r e p o s t u la t e d : a k i n e t ic r e l a ti o n a ta p h a s e b o u n d a r y a n d a n u c l e a t i o n c r i te r i o n .T h e a s s u m p t i o n s i m p l y t ha t t h e d e f o r m a t i o n at a p h a se b o u n d a r y i s c o h e r e n t

a n d p h a s e b o u n d a r i e s a r e k i n e t i c a l l y d r i v e n . U n l i k e t h e s i tu a t i o n i n c o n v e n t i o n a ls h o c k w a v e t h e o r i e s , a p h a s e b o u n d a r y i s n o t t h e r e s u lt o f t h e o v e r l a p p i n g o f


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4 2 X I A O G U A N G Z H O N Gc h a r a c t e ri s ti c s . I n t h e f o l l o w i n g , o n l y t h e o n e - d i m e n s i o n a l v e r s i o n o f th e m o d e l i sp r e s e n t e d .

2 .1 . GOVERNINGEQU TIONSC o n s i d e r a o n e - d i m e n s i o n a l b a r w i th u n i f o r m c r o s s s e c t io n A t h a t o c c u p i e s t h ei n t e rv a l [ 0 , L ] i n a n u n s t r e s s e d r e f e r e n c e c o n f i g u r a t i o n . I n a l o n g i t u d i n a l m o t i o no f t h e b a r, t h e p a r t i c le a t x i s c a r d e d t o t h e p o i n t x + u x , t ) a t t i m e t , w h e r e t h ed i s p l a c e m e n t u i s r e q u i r e d to b e c o n t i n u o u s w i th p i e c e w i s e c o n t i n u o u s f ir st a n ds e c o n d o r d e r d e r i v a t i v e s o n [ 0, L ] f o r t > 0 . L e t

7 = u x , v = u t 2 .1 )d e n o t e s t ra i n a n d p a r ti c l e v e l o c i t y r e s p e c t iv e l y . I t i s a s s u m e d th a t 7 x , t ) > - 1 s ot h at th e m a p p i n g x ~ z + u x , t) i s i n v e r ti b l e a t e a c h t i m e t . T h e s tr e s s a x , t ) i sr e l a te d to t h e s tr a in b y a = 5 7 ) .

T h e e q u a t i o n o f m o t i o n a n d t h e c o m p a t i b i l i ty e q u a t io n a r e5 ( 7 ) 7 ~ = p v t = O , 2 .2 )v ~ - 7 t = 0 . 2 . 3 )

T h e c h a r a c t e r i s ti c s f o r t h i s s y s t e m a r ed~c~ - = = k c 7 ) , 2 . 4 )

_/ 277_.xw h e r e c = ~ / - - . T h e c o r r e s p o n d i n g R i e m a n n i n va r ia n t s a l o n g e a c h c h a r a c t e r i s -t i c a re :

v - / c d 7 = co n s t a n t , 2 .5 )a l o n g d x / d t = c 7 ) ;

v + f c d 7 = c o n s t a n t , 2 .6 )a l o n g d x / d t = - c 7 ).

I f th e r e i s a s t ra i n d i s c o n t i n u i t y a t x = s t ) , t he f o l l o w i n g j u m p c o n d i t i o n s m u s th o l d :

~ 7 + - 7 - ) = - v + - v _ ) , 2 .7 )~ 7 + ) - ~ 7 - ) = - p ~ v + - v _ ) , 2 . 8 )


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DYNAMICALBEHAVIOROF A PHASEBOUNDARY 43w h e r e ( ) + , ( ) _ d e n o t e q u a n ti ti es to t h e r i g h t a n d l ef t o f t h e d is c on ti nu i ty , r es pe c -t ively .

L e t

fO W ( 7 ) = o ( 7 ) d 7 (2.9)b e t h e s t r a in e n e r g y p e r u n i t v o l u m e fo r t h e m a t er ia l . C o n s i d e r t h e r e st r ic t i o n o ft h e m o t i o n t o t h e t i m e i n t e rv a l [ t l, tz ] a n d t o t h e p i e c e o f th e b a r t h a t o c c u p i e sthe in te rva l [X l, x2] in the r e fe rence s t a te . Su ppo se tha t 7 and v a re sm oo th on[x l , x2] in the t im e in te rva l [ t l , t2 ] excep t a t the m ov ing d i s co n t inu i ty x = s t ) .L e t E t ) b e t h e t o t al m e c h a n i c a l e n e r g y a t t im e t f o r t h e p i e c e o f th e b a r u n d e rc o n s i d e r a t i o n :

f x a:2t ) = [ W 7 x , t ) ) + P V 2 x , t ) ] A d x . (2 .10)A d i r e c t c a l c u l a t i o n e s t a b li s h e s t h e f o l l o w i n g w o r k - e n e r g y i d e n t it y :

a x 2 ) A v x 2 , t ) - a x l ) A V X l , t ) - J ~ t ) = f t ) h t ) A , (2 .11)

w h e r e t h e d r i v i n g t r a c t i o n f t ) a c t i n g o n t h e d i s c o n t i n u i t y i s d e f i n e d b yf = ] 7 - , 7 + ) = f v + _ a 7 ) d T - l [ a 7 + + a 7 - ) ] 7 + _ 7 - ) . (2 .12)

T h e a d m i s s i b i li t y c o n d i t i o n i m p o s e d o n t h e p h a s e b o u n d a r y i sf t ) h t ) >_ O . (2 .13)

U n d e r i s o t h e r m a l c o n d i t i o n s t h e a d m i s s ib i l i ty c o n d i t i o n i s a c o n s e q u e n c e o f t h es e c o n d la w o f th e r m o d y n a m i c s .2 2 S U P P L E M E N T A R Y CON STITUTIVE RELATIONSB e s i d e s t h e e q u a t i o n s o f m o t i o n , t h e s t re s s- s tr a in r e la t io n , j u m p c o n d i t i o n s a n d t h ea d m i s s i b i li t y c o n d i t i o n , t w o s u p p l e m e n t a r y c o n s t it u t i v e r e l at i o n s m u s t b e s p e c i fi e di n o r d e r t o u n i q u e l y d e t e r m i n e a s o l u t i o n o f th e p h a s e t r a n s f o r m a t i o n p r o b l e m . T h et w o s u p p l e m e n t a r y c o n s t i t u t i v e r e la t io n s a r e t h e k i n e t i c r e la t io n a n d t h e n u c l e a t i o ncr i t e r ion . The k ine t i c r e l a t ion r e la tes the phase boundary p ropaga t ion speed h tot h e d r i v i n g t r a c t io n f t ) a c t i n g o n t h e p h a s e b o u n d a r y . T h e n u c l e a t i o n c r it e ri o nd e t e r m i n e s w h e n a n e w p h a s e w i l l b e n u c l e a t e d f r o m t h e p a r e n t p h a s e . T h e s e t w or e l a ti o n s a r e m a t e r i a l -d e p e n d e n t o n ly . I n t h is w o r k , w e u s e t h e f o l l o w i n g k in e t i cr e la t ion an d n uc lea t ion c r i te r ion :


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4 4 XIAOGUANGZHONG1 K ine t i c r e l a t ion :

f -- ~b(h) (2.14 )w h e r e ~(h ) i s a m o n o t o n i c a l l y in c r e a s i n g f u n c t i o n th a t may b e d i s c o n t in u o u sa t ~ = 0 , b u t is a l w a y s c o n t i n u o u s e ls e w h e r e .

2 . Nu c l ea t i on c r i t e r i on :A s s u m e t h a t t h e r e i s a t w o - p h a s e m a t e r i a l , w i t h l o w - s t r a i n - p h a s e a n d h i g h -s t ra i n - p h a s e . T h e n f o r t h is m a t e r i a l w e h a v e

f _< fcr (2.1 5)f o r a l o w - s t r a i n - p h a s e t o h i g h - s t r a i n - p h a s e t r a n s f o r m a t i o n , a n d

f -> fcr (2.16 )f o r a h i g h - s t r a i n - p h a s e t o l o w - s t r a in - p h a s e t r a n s f o r m a t i o n ; h e r e f i r , f e r a r ec r i t i ca l d r i v ing t r ac t i ons .

2.3. T R I L I N E A R M A T E RIA L ST h e s i m p l e s t e la s t i c m a t e r i a l m o d e l c a p a b l e o f d e s c r i b i n g p h a s e t r a n s f o r m a t i o n s ist h e s o c a l l e d trilinear material The t r i l inea r s t r e s s -s t r a in r e la t i on can be exp res seda s

13', - 1 < 3 ' < 3 ' m ,t r (3 ) = ( t rm - ~rM)(3 - - 3 M )/ (Tm -- 3 M) + a M , 3 ,n < 3 < 3 M, (2 .17)

g 2 ( 3 ' - - 3 ' T ) , 3 ' M < 7 < O 0 ,w h e r e a m = ~ 1 3 ' m , OrM - ~ 2 ( 3 ' M - - 3 ' r ) .

W e c a l l t h e s t ra i n i n t e rv a l - 1 < 3 < 3 m p h a s e 1 o r t h e l o w - s t r a i n p h a s e ,w h i l e t h e i n t e rv a l 3 ,~ < 3 < 7 M i s p h a s e 2 o r t h e u n s t a b l e p h a s e a n d 3 M 3i s p h a s e 3 o r t h e h i g h - s tr a i n p h a s e . T h e lo w - s t r a i n p h a s e a n d t h e h i g h - s t r a inp h a s e a r e m e t a s t a b l e p h a s e s . I f w e c o n s i d e r t h e m a r t e n s i ti c t r a n s f o r m a t i o n a s o u rp r o t o t y p e , w e c a n i d e n t i f y t h e l o w - s t r a i n p h a s e a s a u s t e n i t e a n d t h e h i g h - s t r a i np h a s e a s m a r t e n s i te . L e t ci = V/W , i = 1 , 2 be t he sound speeds i n t he l ow-s t ra inp h a s e a n d t h e h i g h - s t r a i n p h a s e .

I t i s e a s y t o s e e t h a t o n e m u s t h a v e~ 1 > 0 , # 2 > 0 , 3 'm < 3 ' M .

T o g u a r a n t e e t h e n o n c o n v e x i t y o f t h e c o r r e s p o n d i n g s tr a in e n e r g y f u n c t io n , w erequ i re t ha t~ 1 3 ' m > ~ 2 ( 3 ' M - - 3 ' T ) -


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DYNAMICAL BEHAVIOR OF A PHASE BOUNDARY 45F o r t h i s m a t e r i a l t h e d r i v i n g t r a c ti o n a c t in g o n a p h a s e b o u n d a r y w i t h th e h i g h -

s t ra i n p h a s e o n t h e l e f t a n d t h e l o w - s t r a i n p h a s e o n t h e r i g h t is , f r o m ( 2 . 1 2 ) a n d(2.17),

- 2 7 T ,f ( 7 - , 7 + ) - ( 1 2 ~ 2 ) ( 7 + 7 - - 7 M 7 ra ) + T [ 7 + + 7 - - - 7 M - - 7 ra ). (2 .18)I f t h e l o w - s t r a i n p h a s e i s o n t h e l e ft , th e d r i v i n g t r a c t io n c h a n g e s s ig n .

I n a n equil ibr ium s t a t e , on e mus t hav e a+ = ~r_ = ~r so t ha t t he d r i v ing t r ac t i onm a y b e e x p r e s s e d i n t e r m s o f st re s s a :

~1 -- 2 a2 1 -- 2 27 T r- + 2 m 7 M - -- tT m + 7 M - 2 . 1 9 )

I t is e a s y t o s h o w t h a t f a ) i s m o n o t o n i c a l l y i n c r e a s i n g , a n d f a ~ ) < O, f a M ) >0 . T h e M a x w e l l s tr e ss a 0 o f t h e m a t e r i a l i s t h e u n i q u e s tr e ss f o r w h i c h f ( a ) = 0 .

I f w e l e t ~ l = 2 = t h e n t h e M a x w e l l s t re s s i s g i v e n b ycr0 = (am + aM ) . (2 . 20 )

T h e l o w - s t ra i n a n d h i g h - s tr a i n v a l u e s o f s t ra i n c o r r e s p o n d i n g t o t h e M a x w e l l s t re s sa r e r e s p e c t i v e l y g i v e n b y

7~ ---- l ( T m q- 7M -- 7T ), (2 .21)70 = (Tin + 7M + 7T ) . (2 . 22 )

T h e j u m p c o n d i t i o n s ( 2 .7 ) a n d ( 2 .8 ) b e c o m ec 2 ( 7 + - 7 - ) + h ( v + - v _ ) = c 2 7 T , ( 2 .2 3 )( v + - - v _ ) + s ( 7 + - - 7 - ) = 0 . ( 2 .2 4 )

T h e R i e m a n n i n v a r i a n t s (2 . 5) a n d ( 2 .6 ) a r ev + c7 = cons t an t . (2 . 25 )

T h e j u m p c o n d i t i o n s (2 .7 ) , ( 2 . 8 ) a n d t h e s t re s s - st r a in r e l a t io n i m p l y t h a t< c (2 .26)

f o r a p h a s e b o u n d a r y p r o p a g a t i o n s p e e d .I n t h e f o l l o w i n g s e c t io n s , t h e s p e c i a l t r i li n e a r m a t e r i a l w i t h 1 = 2 = i su s e d . F o r t h i s m a t e r i a l, w e c a n r e f o r m u l a t e t h e n u c l e a t i o n c r it e r i o n ( 2 .1 5 ) a n d( 2 .1 6 ) i n t e r m s o f s t ra i n s f o r c o n v e n i e n c e ,

7 -> % r > 7o (2 .27)


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4 6 X I A O G U A N G Z H O N G

low strain phaseF i g . 1. T h e i n i t i a l s t a t e

high strain phase

f o r t h e l o w - s t r a i n p h a s e t o t h e h i g h - s t r a i n p h a s e t r a n s f o r m a t i o n , a n d7 < 7or < 70 2.28)

f o r t h e h i g h - s t r a i n p h a s e t o t h e l o w - s t ra i n p h a s e t r a n s f o r m a t i o n .3 F o r m u l a t i o n o f a n I n it ia l V a l u e B o u n d a r y V a l u e P r o b l e mC o n s i d e r a s e m i - i n f i n i te b a r o f t h e s p e c i a l t r il i n e a r m a t e r i a l d e f i n e d in S e c t i o n 2 .T h e m o t i o n o f a p h a s e b o u n d a r y is g o v e r n e d b y k i n e t i c r e l a ti o n f = ~ b ~ ), w i t h~ b h) = w h . A s d e f i n e d in S e c t i o n 2, f i s t h e d y n a m i c a l d r i v i n g tr a c t i o n o n a p h a s eb o u n d a r y , a n d h i s th e p h a s e b o u n d a r y p r o p a g a t i o n s p e e d . T h e n u c l e a t i o n c r it e r i o ni s t he on e g iven in S ec t i on 2 ; s ee 2 . 27 ) o r 2 . 28 ).

In i t ia l l y , t he ba r is i n t he l ow -s t r a in pha se fo r 0 < x < so , and i n t he h igh - s t r a inp h a s e f o r 80 < x < o c . T h e b a r is i n a n e q u i l i b r i u m s ta t e. T h u s th e p h a s e b o u n d a r ya t 80 i s in i t i a l l y s t a t ionary . F ro m the k ine t i c r e la t i on , f = wh , w e con c lu de tha tf = 0 , so t ha t t he ba r is in i t i a l ly i n t he M ax w el l s ta t e

v x ,O)=vf f = 0 , 3 . 1 )1- ~ ( x , 0 ) = 7 o = ~ ( T m + 7 M - ' r r ) 3.2)

f o r 0 < x < a 0 ;v x,O) = v ~ = O , 3 .3 )- ~ ( ~ , o ) = - t o = ( - r . .k 71- / U ~ - ~ T ) 3.4)

fo r 80 < x < oo see F ig ure 1).A t t = 0 , t h e e n d o f t h e b a r i s s u d d e n l y s u b j e c t e d t o a v e l o c i t y v0 > 0 w h i c h is

m a i n t a i n e d c o n s t a n t u n t il t i m e t * a n d t h e n r e d u c e d i n s ta n t l y t o z e r o . T h e r e s u l t i n gb o u n d a r y c o n d i ti o n s c a n b e d e s c r i b e d a s

v O , t ) = v o , o ___ t < t * , 3 . 5 )


9/34

D Y N M I C L B E H V I O R O F P H S E B O U N D R Y

V4 7

v 0

~gt tFig 2 The incident square wave

v 0 , t ) = 0 , t > t * . 3 .6 )T h i s b o u n d a r y c o n d i t i o n i s e q u i v a le n t to t h e f o ll o w i n g d i s p l a c e m e n t b o u n d a r y

c o n d i t i o n : s e e F i g u r e 2 )u O , t ) = u o t , O < t < t * , 3.7)u O , t ) = v o t * , t > t * , 3.8)

w h e r e t h e i n i ti a l d i s p l a c e m e n t o f t h e p a r t ic l e a t x = 0 w i t h r e s p e c t t o a r e f e r e n c ec o n f i g u r a t i o n i s t a k e n t o b e z e r o .4 A S h o r t T i m e A n a l y s isI n th i s s e c t i o n w e w i l l s o l v e th e p r o b l e m f o r m u l a t e d in S e c t i o n 3 f o r s h o r t t i m e . W ea s s u m e t h a t t h e re a r e n o n e w p h a s e b o u n d a r i e s n u c l e a t e d d u e to t h e i n t e r ac t io n o ft h e o r i g in a l p h a s e b o u n d a r y a n d t h e i n c i d e n t w a v e o r d u e t o t h e b o u n d a r y c o n d i ti o n .W e w i l l d e t e r m i n e r e s t r ic t i o n s o n t h e v 0 a n d o t h e r p a r a m e t e r s t h a t a s s u r e t h a t th i sa s s u m p t i o n i s v a l i d .A s w e h a v e p i e c e w i s e c o n s t a n t in it ia l d a ta a n d b o u n d a r y v a l u es , w e c a n s o l v et h e p r o b l e m b y d e a l i n g w i t h s u i t a b l e R i e m a n n p r o b l e m s . I t i s e a s y t o s h o w t h a tt h e s o l u t i o n i s p i e c e w i s e c o n s t a n t ; s e e F i g u r e 3 . T h e g o v e r n i n g e q u a t i o n s a r e t h e nr e d u c e d t o j u m p c o n d i t io n s o n s h o c k w a v e s a n d j u m p c o n d i t i o n s a n d t h e k i n e ti cr e l a t io n o n t h e p h a s e b o u n d a r y .

A t f ir s t t h e i n c i d e n t w a v e p r o p a g a t e s t o w a r d t h e s t a ti o n a r y p h a s e b o u n d a r y . T h es tr a in 7 0 s e e F i g u r e 3 ) c a n b e d e t e r m i n e d b y t h e j u m p c o n d i t i o n o n th e s h o c kA B

S O

v 0 - V o + c 7 0 - 7 0 ) = 0 , 4.1)

v07 0 = - - - + 7 0 < 7 0 . 4 .2 )c


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8 X IA O G U A N G Z t tO N G

F

Fig. 3 .

H- - -

' 7

2 B

A s c h e m a t i c fi g u r e f o r t h e s h o r t ti m e a n a l y s i s

O n s h o c k C D ( s e e F i g u r e 3 ), t h e j u m p c o n d i t i o n is- v 0 + c ~ - 7 0 ) = 0 .

D u e t o t h e b o u n d a r y c o n d i t io n , w e h a v e( 4 . 3 )

W h e n t h e in c i d e n t w a v e r e a c h e s t h e p h a s e b o u n d a r y , t h e i n c i d e n t w a v e i s b o t hr e f l e c te d b y t h e p h a s e b o u n d a r y a n d t r a n s m i t te d t h r o u g h i t . T h e j u m p c o n d i t i o n so n t h e p h a s e b o u n d a r y a n d s h o c k w a v e s a r e

( v + - v + ) + c ( 7 1 - 3 + ) = 0 , ( 4 . 6 )/ z ( 7 1 t - - 7 T ) - - ~ 7 1 -]- Phl V+l - V l ) ---- 0 , ( 4 . 7 )( v + - V l ) + h l ( 7 + - 7 1 - ) = O , ( 4 . 8 )

( v i - - v o ) - c ( T i - - 3 0 ) = 0 . ( 4 . 9 )S o l v i n g ( 4 . 6 ) - ( 4 .9 ) , w e f i n d t h a t

C817Tv 1 = v o -F 2 ( c -F s l ) ( 4 . 1 0 )

= 7 0 , ( 4 .4 )= O . ( 4 . 5 )


11/34

DYN AM ICAL BEHAVIOR OF A PHASE BOUNDARY 4 93 1 = - - V O 3 - 3 0 q - 8 1 7 T ( 4 .11 )c 2 ( c + ~ 1 )

CSl3 T ( 4 .12 )V = V 0 2 ( C - - ~ 1 ) '

7+ = _ vo + 70 + - 617T ( 4 .13 )c 2 ( c - ~ , )

H e r e ~1 i s de t e r m ine d by the k ine t i c r e la t i on , f = wh, i .e .. 3 ' T + 3 ' ~ - 3 ' + - 3 ' ? ) = ~ i .3'M ( 4 .14 )

T h e v e l o c i t y v a n d s t r ai n 3' a r e d e t e r m i n e d f r o m j u m p c o n d i t i o n s o n th e s h o c k sF D a n d D E ( s ee F i g u r e 3 ):

T h u s

v - v ~ - + c ( 3 ' - 3 ' ? ) = o ,v - ~ - c ( 3 ' - 9 ) = o .

c&13 TV 2 ( c + ~ l )

3 m + 3 M C3 T3 ' - 2 2( c + h i )

W e d e t e r m i n e v l a n d 3' f r o m t h e j u m p c o n d i ti o nv - vl + c(3' - 3'1) = 0

a n d t h e b o u n d a r y c o n d i t i o n . W e fi n d t h at

( 4 .15 )( 4 .16 )

( 4 .17 )

( 4 .18 )

( 4 .19 )

, ( 3 ' 2 + - 3 ' ~ ) - . 3 ' ; + 5 2 ( v ~+ - v ; ) = o ,( v ~ + - v ; ) + 5 2 ( 3 ~ - 3 ; ) = o ,v ; - v - c ( 3 ~ - - 3 ) = o ,

( 4 .22 )( 4 .23 )( 4 .24 )

V + _=t :N o w w e d e t e r m i n e 2 , 7 2 f o rm a l ly . T h e c o r r es p o n d i n g j u m p c o n d i t io n s a re

v~ = 0, (4.2 0)3 'm + 3 'M (c - - ~1)7T (4 .21 )3'1 - - 2 2 ( c + ~ 1 )


12/34

5 0v - v , + c 7 2 + - 7 ,+ ) = o .

F r o m e q u a t i o n s ( 4 . 2 2 ) - ( 4 . 2 5 ) ,( 7 r a C T T7 2 - = 2 , + m ) 2 ( c + ~ 2 ) '

C T T7 2 = ( T m~ + 7 M ) + 2 ( c - h 2 )

XIAOGUANG ZHONG(4 .25/

(4 .26)

(4 .27)

T h e r e f o r e , t h e d r i v i n g tr a c t io n f o n t h e p h a s e b o u n d a r y E H w i t h ~ 2 c a n b e e x p r e s s e da s

~ ' ) ' T rf - ~ ~ T M + T m - - 7 2 - - 7 + )

2( c 2 _ ~2) (4 .28)I t i s e a s y t o s h o w t h a t t h i s f i s m o n o t o n i c a l l y d e c r e a s i n g w i t h r e s p e c t t o ~ 2.

F r o m t h e k i n e t i c c o n d i t i o n f - - w ~, w e h a v e

7 2 C 8 2 - - 0 dh 2. (4 .29)2 ( c 2 _ ~ 2 )T h e r e i s o n l y o n e s o l u t i o n o f ( 4 .2 9 ) , n a m e l y

~2 --- 0. (4 .30 )T h u s f r o m ( 4 .2 2 ) - (4 . 2 5 ), w e h a v e

v 2 = 0, (4.31 )7 2 = 7 0 , ( 4 .3 2 )v+ = 0 , (4 .33)7 + = 7 + . (4 .34)

T h e r e f o r e , a f t e r t h e p a s s a g e o f t h e i n c i d e n t w a v e , t h e p h a s e b o u n d a r y w i l l b es t a t i onary fo r a wh i l e .

C e r t a i n r e s t r ic t i o n s m u s t n o w b e i m p o s e d t o a s s u r e t h a t t h e r e is n o n e w p h a s eb o u n d a r y n u c l e a te d .


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DYN AM ICAL BEHAVIOR OF A PHASE BOUNDARY 51

c i c

- /

R

F i g . 4 . F o r m o f s o l u t i o n t o a R i e m a n n p r o b l e m w i t h in i t ia l d a t a i n d i f f e r e n t p h a s e s

4 . 1 . R E S T R I C T I O N S O N T H E S H O R T T I M E A N A L Y S I SF i r s t, w e r e q u i r e t h a t

2 s o- - > t * . ( 4 .3 5 )c

T h i s is b e c a u s e i t i s r e q u i r e d i n t h e a b o v e a n a l y s i s th a t t h e s h o c k w a v e r e f l e c t e d b yt h e p h a s e b o u n d a r y w i l l n o t re a c h x = 0 u n t i l a t i m e t > t * .

S e c o n d l y , w e r e q u i r e t h a t 7 1 b e l o n g to a p p r o p r i a t e p h a s e s . T o c h e c k th i s , i tis n e c e s s a r y t o c h e c k w h e t h e r a R i e m a n n p r o b l e m w i t h in i ti a l d a t a 7 L a n d 7 R i nt h e l o w - s t r a i n a n d h i g h - s t r a i n p h a s e s r e s p e c t i v e l y h a s a s o l u t i o n t h a t s a ti sf ie s t h ep h a s e s e g r e g a t i o n c o n d i t io n s . F o r t h e R i e m a n n p r o b l e m w i t h in i ti a l d a t a VL, 7 L )a n d ( v R , 7 R ) , w h e r e 7 L b e l o n g s t o t h e l o w - s tr a in p h a s e a n d 7 R b e l o n g s t o th eh i g h - s tr a i n p h a s e ( s e e F i g u r e 4 ) , w e h a v e

)' - --'-- h

w h e r e

7 + = h + CTT ( 4 . 3 6 )2 ( e - a )C T T ( 4 . 3 7 )2 ( e + ~ )

h = ~ c ( V R - [ - C T R - - V L { - CTL). ( 4 . 3 8 )T h e p h a s e s e g r e g a t i o n c o n d i t i o n r e q u ir e s t h a t 7 + > 7 M a n d 7 - < 7 m - T h u s

w e h a v eCTT CTT (4 .3 9)7 M 2 c - < h < 7 m + 2 c +

T h e e x p r e s s i o n ( 4 . 3 9 ) c a n b e e x p r e s s e d i n t e r m o f th e d r iv i n g tr a c t i o n f . F r o m( 2 .1 8 ) , a s t h e l o w s t ra i n p h a s e o n t h e le f t, w e h a v e

~ T T /f = 2 t T M + 7 m - - 7 - - - 7 + ) , ( 4 . 4 0 )


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52 XIAOOUANOZHONG

F i g . 5 . A d m i s s i b l e r e g i o n f o r th e k i n e t i c r e la t i o n

w e m a y r e w r i t e 4 .3 9 ) a s

2 7 M T i n c f f ~ 2 ] < f < T 7 m - - T M + c ~ - _ - - ~ 2 4 .4 1)Ineq ua l i t y 4 . 41 ) can be i l lu s t r a t ed i n t he f - ~ p l ane as i n F igu re 5. Th e shad ed

reg ion i n F igu re 5 is t he p l ac e wh ere 4 . 41 ) i s sa t is f ied .I n o r d e r t o h a v e a s o l u t i o n w h i c h s a t is f ie s t h e p h a s e s e g r e g a t i o n c o n d i ti o n s , t h e

c u r v e c o r r e s p o n d i n g to k i n e t i c r e l a ti o n f = a ;~ m u s t l ie in t h e s h a d e d r e g i o n inF i g u r e 5 . T h i s w i ll b e t h e c a s e i f w e f u r t h e r r e q u i r e th a t

0 < a~ < w0, 4.4 2)w h e r e

I T y0 . ~ . ~ y

and y sa t i s f ies4 . 43 )

y 2 _ ~ y + 7 M - - 7 m _ 0 . 4 . 4 4 )2 7 T

T h i r d l y , w e r e q u i r e th a t 7 , 7 1 a r e in t h e l o w - s t r a i n p h a s e . It c a n b e s h o w n t h a tt h is r e q u i r e m e n t is e q u i v a l e n t t o t h e i n e q u a l i t y

) 'M 4- ) 'm C - - .~2 2 c + h ) 7T < 7m , 4 . 45 )


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DYN AM ICAL BEHAVIOR OF A PHASE BOUNDARY 5 3w h e r e h i s d e t e r m i n e d b y 4 . 1 4) . E q u a t i o n 4 . 1 4 ) c a n b e re w r i t t e n a s

c T T c 2 - ~--------~ h .T h e i n e q u a l i ty 4 . 45 ) c a n n o w b e s i m p l i f ie d t o r e a d

4 . 4 6 )

1 - - ~ .h < c 4 .4 7 )1 + c . ~t h u s t h e r e s t r ic t i o n o n v 0 i s

4 . 4 8 )

w h e r e s 0 = c 1 - c . ~ ) / 1 + c -y ) a n d c.y = T M - - 7 m ) / T T .W h e n t h e t h r e e c o n d i t i o n s 4 . 3 5) , 4 . 4 2 ) a n d 4 . 4 8 ) a r e s a ti s fi e d , t h e r e i s n on e w p h a s e b o u n d a r y n u c l e a t e d , a n d a f t e r t h e i n it ia l i n te r a c t io n b e t w e e n t h e p h a s eb o u n d a r y a n d t h e i n c i d e n t w a v e , t h e p h a s e b o u n d a r y w i ll re m a i n s t a t io n a r y f o r aw h i l e .

I f w > w 0 w e c a n n o t f i n d s o l u t io n s o f R i e m a n n p r o b l e m s t h a t s at is f y t h e p h a s es e g r e g a t i o n c o n d i t i o n . I f v 0 i s t o o la r g e , t h e n 4 . 4 8 ) w i l l b e v i o l a t e d a n d n e w p h a s eb o u n d a r i e s w i ll b e n u c l e a t e d .

T h e r e s u lt o b t a i n e d h e r e t h a t t h e p h a s e b o u n d a r y r e m a i n s s t a t io n a r y f o r a w h i l ea f te r th e e n c o u n t e r o f t h e i n c i d e n t w a v e a n d t h e p h a s e b o u n d a r y w a s postulatedb yP e n c e [ 20 ]. B y u s i n g t h e a p p r o a c h o f A b e y a r a t n e a n d K n o w l e s , w e o b t a i n t h e re s u ltr i g o r o u s l y a n d e x h i b i t t h e e x p l i c i t r e s t r i c t i o n s o n i n i t i a l d a t a , b o u n d a r y d a t a a n dm a t e r ia l p a r a m e t e r s n e c e s s a r y t o a s s u m e t h e r e s u lt . T h i s re s u l t c a n b e g e n e r a l i z e dt o s i t u a t io n s i n w h i c h t h e i n c i d e n t s q u a r e w a v e i s r e p l a c e d b y a n a r b it ra r y p u l s e , t h ek i n e t i c r e l a t i o n f = w ~ is r e p l a c e d b y f = ~ b ~ ), a n d t h e m a t e r i a l i s a n arbitraryt r i l in e a r m a t e r i a l .5 . L a r g e T i m e P h a s e B o u n d a r y B e h a v io r : M o t io n o f t h e P h a s e B o u n d a r y5 .1 . R E C U R S I V E F O R M U L A E F O R v , 7 )B e c a u s e o f th e a s s u m p t i o n th a t n o n e w p h a s e b o u n d a r y i s n u c l e a te d , th e f o r m o ft h e s o l u t i o n t o b e f o u n d c a n b e i ll u s t ra t e d b y t h e z - t p l o t s h o w n in F i g u r e 6 . A si n S e c t i o n 4 , w e c a n s h o w t h a t t h e s o l u t i o n i s p i e c e w i s e C o n st an t . T h u s t h e p h a s eb o u n d a r y p r o p a g a t i o n s p e e d h t ) is p ie c e w i s e c o n s t a n t in t i m e a s w e l l . T h e p a r to f t h e tr a j e c t o r y o f t h e p h a s e b o u n d a r y t h a t s e p a r a t e s re g i o n s P f f s e e F i g u r e 6 )c o r r e s p o n d s t o t h e p h a s e b o u n d a r y p r o p a g a t i o n s p e e d h k. T h e p a r t o f th e tr a je c t o ryt h a t s e p a r a t e s r e g i o n s S ~ c o r r e s p o n d s t o th e p h a s e b o u n d a r y p r o p a g a t i o n s p e e d4- -4 -~ k. W e d e n o t e p a r t ic l e v e l o c i t i e s a n d s t r a in s i n r e g i o n s P f f b y v k a n d 7 k , i n r e g i o n s


16/34

54 X I A O G U A N G Z H O N G

B

BI

3 J

S P o S P oF i g . 6 . A s c h e m a t i c x - t p l o t

S P~ b y v -+ and ~ f f . De no te t he s t r a in an d pa r t i c l e ve loc i t y i n r eg ion Bk by vk , 7k ,a n d i n r e g i o n S k by vk , 7k .F r o m t h e b o u n d a r y c o n d i t i o n s, w e h a v e vi = 0 and ~ i = 0 fo r i = 1 , 2 , 3 , . . . .F r o m r e s u lt s o b t a i n e d i n S e c t i o n 4, w e k n o w t h a t

~ = o , 5 . 1 )e i - = o , 5 . 2 )2 i - = 7 0 , 5 . 3 )~+ = 0 , 5 .4)9 , + = . ~ o . 5 . 5 )

N o w w e w i l l s h o w b y i n d u c t io n th a t~ i = 0 , 5 . 6 )~ / - = o , 5 . 7 )


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D Y N A M I C A L B E H A V IO R O F A P H A S E B O U N D A R Y 5 5

~ - = 7 0 , 5 . 8 )~+ = 0 , ( 5 . 9 )9 + = 7 + , ( 5 . 10 )7 i = 7 0 , ( 5 . 1 1 )

f o r i = 1 , 2 , . . . .A s s u m e th a t

~k = 0 , (5 .12 )~ - = 0 , ( 5 . 1 3 )7 k- = 7 0 , ( 5 . 1 4 )-+ = 0, (5 .15)k- + 7 + ( 5 . 16 )k =

~k = 7 0 , ( 5 . 17 )i n r e g i o n s SP~. T h e n i n r e g i o n s SP~ , f r o m j u m p c o n d i t i o n s o n t h e p h a s eb o u n d a r y a n d s h o c k w a v e s, w e h a v e

- + - +Vk l -- V k + l -1 - 8 k + l ( 7 k - t - 1 - - ' ~ k + l ) ---- 0 , ( 5 . 1 8 )

~ ( '~ k - t -+ l - - 7 T ) - - # ' ~ k + l hI- P ~ k + l ( ' O k + + l - - 0 k + l ) - -- - 0 , ( 5 . 19 )

0 + c - +k + l - v + + T k + l - 7 + ) = o , 5 . 2 0 )0 k + l - - V k -t-1 - - C ( ' ~ k + l - - ' ~ k - I- 1 ) ---- 0 , ( 5 . 2 1 )

V k + 1 - - O k - - C ( ' ~ k + 1 - - ' ~ k + l ) = 0 . ( 5 . 2 2 )F r o m ( 5 . 1 8 ) - ( 5 . 2 2 ) , w e f i n d t h a t

+ 1 = 7 k = 7 o ,7k+1 - 7m -t- 7M CTT2 2 ( c + ~ k + l ) '

(5 .23)(5.24)

- [m + 7M 7T 5 .25 )7 k + l - - 2 + 2 ( c - ~ k + l )


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DYNAMICALBEHAVIOROF A PHASE BOUNDARY 5 71 C T T

[/~-1 ---- 2 C 3- 8 i + 1_ + .~ o ++ v ~ - + c ~ 7 ; ) (5.39)+ { C 2 ) ' T- - + ~ .r o - v . + c - r ? ) ) , (5 .40)

1 C T T + 7 0 + + v i - + c T / - . )Y h , = ~ c - ~ ,+ -----S ~ (5 .41)

F r o m t h e s e r e s u lt s , w e i n f e r t h e f o l l o w i n g s im p l e fo r m u l a :v~- + cT i- - c27-------LT c7 + . (5 .42)c + h i

Fro m the d e r iva t i o n w e see t ha t (5 .42) h o lds fo r i _> 2 , bu t i t i s e a sy t o showd i r e c t l y t h a t i t al s o h o l d s f o r i = 1 . T h u s w e h a v e t h e f o l l o w i n g f o r m u l a e f o r v , 7f o r a ll r e g i o n s P f f ; s e t ti n g

h ix i = - - , (5 .43)Ct h e n

_ C T T ( 1 + 1 ) (5 .44)V i+ I ---- T 1 3 - Z i+ l 1 + X---~' '1 ( 7T )'T + 2 7 + ) , (5 .45)7 ~ 1 = ~ l + x i + l l + x i

C Y T ( 1 + 1 ) ( 5 .4 6 )Vi + l - - 2 1 - - X i + 1 1 + x i '1 ( /T 7_.__Z _._T 2 7 + ) . ( 5 . 4 7 )7i+ +1 = 2 1 - - x i + l l + x l

H e r e x i i s d e t e r m i n e d b y t h e k i n e t i c r e l at i o n t h r o u g h t h e r e c u r s iv e r e l a t io n

l ~ x l 1 2 q - a X i + l i = 1 , 2 , 3 , . . . ,- - X i + I

(5.48)a n d

2 v o ( 1 )C T T - - l - - x 2 1 + a X l , (5 .49)


20/34


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DYNAMICALBEHAVIOROF A PHASEBOUNDARY 5 9N o t i c e t h a t

- x 2 2 a > 0 ,s o t h a t

x 2 < 0 .A s s u m e t h a t k = i , X 2 k_ 1 > 0 , X 2 k < 0 ; t h e n

0

1 )1 X 2 i 1 2 - 4- a x 2 i l ,- - X 2 i + lx21 1 )1 + x 2 i + l - - 1 2 + a x2 i+2 ,- - X2i+2

5 . 5 7 )

5.58)

5 . 5 9 )

5 . 6 0 )

X 2 i +I > O , X 2i +2 < 0 . 5 .6 1 )W h u s k = i + l , X 2 k - 1 ~> 0 , X 2k < 0 . S o X 2 k - 1 > 0 , X 2k < 0 f o r a l l k .F r o m

X 2 k - I 1 . ~ 2 + a 5 . 6 2 )1 - ~ X E k _ 1 1 - - z k X 2 k ~a n d t h e f a c t s t h a t x 2 k - 1 > O , X 2k O , w e h a v e

X 2 k - 1 > 1 - - - - - - k -[- a - x 2 k ) > 1 + a ) ( - - X 2 k ) , 5 . 6 3 )

i .e .X 2 k - 1l + a

T h i s c o m p l e t e s t h e p r o o f o f L e m m a 1.P R O P O S I T I O N 1 . W h e n t - - . ~ , h ~ O .

P r o o f . F r o m 5 .4 8 ) , w e h a v e

5 .6 4 )

x2kl 1 )_1 + X 2 k _ l 1 - X 2 k + a X 2 k ,X2k -- 1 )1 q- X2k 1 2 -~ a X2k +l ,-- ~g2kT1 5 . 6 5 )


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6 0SO

XIAOGUANGZI-IONG

- - - - X 2 k + a X 2 k 2 + a Z 2 k + l . 5.66)1 - - X2k+l

I t f o l l o w s t h a t

X 2 k - I 1> 1 + X 2 k - l ) 1- ~ X 2 k 1 + a x 2 k + l> ( 1 + a ) z 2 ~ + l , ( 5 . 6 7 )

i e0 < X2k+l 0f o r a l l n o r a,~ < O , f o r a l l n ) , t h e n e c e s s a r y a n d s u f f i c ie n t c o n d i t i o n f o r I I ~ = l q nt o c o n v e r g e i s t h a t ~ = l a n c o n v e r g e s .L E M M A 4 . T h e s e r i e s ~ = l X 2 k _ l / 1 - - X2k-1 a n d ~ = l X 2 k / 1 - - X2k) con-ver g e .

P r o o f . B y ( 5 .6 8 ) , i t c a n b e s h o w n th a tX2k+l Xl< ( 5 .7 3 )

1 - X 2 k l ( 1 + a ) k + l - - 1


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DYNAM ICAL BEHAVIOR OF A PHASE BOUNDARY 61

S k + l

x k + l

Fig . 7 . Def in i t ions o f S k , t k , t ks k

t

A st i m ( a ) ~ + I ) / ( z , ) 1k ~ ( 1 + - 1 ( l + a ) k - 1 - l + a < 1 , (5 .74)

~ k C ~= lX l/(( 1 d - a ) k + l - 1 ) c o n v e r g e s , s o d o e s ~ = l Z 2 k _ l / ( 1 - Z 2 k _ l .By (5 .6 8 ) an d x 2 k < 0 ,X2k x 1< - - Z 2 k < (5 .7 5 )1 - Z 2 k (1 q- a) k

A s ~ z = l X l / 1 + a ) k c o n v e r g e s , s o d o e s ~ n ~ = l X 2 k / 1 - - X 2 k ) , a n d L e m m a 4 i sp r o v e d .P R O P O S I T I O N 2 . F o r a l l k , s k > so , a n d l i m k ~ o o s k < -M h l (1 + I / a ) , w h e r el~I is som e f i n i t e num ber .

Proo f . D e n o t e t h e p e r io d o f t i m e d u r i n g w h i c h h = h k b y t k , th e k t h i n c i d e n tw a v e ' s t i m e i n t e r v a l b y { k, t h e p o s i t i o n o f t h e p h a s e b o u n d a r y w h e n i t s e p a ra t e sr e g i o n s S P ~ ( i t i s s ta t i o n a ry a t t h i s t i me t o o ) b y s k , s ee F i g u re 7 .

F r o m t h e d e f in i ti o n , w e h a v e 5 1 = t * , 8 1 = 80 , andtk = sk + ({k - ~ ) s k + {k - s__~k (5 .7 6 )

C - - S k cs o t h a t

t k -~ 1 - zk (5 .77)


24/34


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DYNAMICALBEHAVIOROF A PHASEBOUNDARY 63w e h a v e

tk = t * l - I l + 1 - x i / < l + 1 - - X li = lTh ere fo re , the re ex i s t s M such tha t fo r a l l k

(5 .87)

{k < M 1. (5 .88)B y ( 5 .7 9 ) a n d t h e f a c t t h a t x k d e c a y s e x p o n e n t i a l ly w i t h k , t h e r e e x i s ts a c o n s t a n tM s u c h t h a t

tk < /~t , for a l l k . (5 .89)N o w l e t u s e s t i m a t e s oo . W e h a v e

S k+ l = 81 q- ~ l t l ~- 82t2 Jr- ' -] - sk tk (5 .90)B u t

1)S 2 k - t 2 k - 1 < l~/1C ~ X 2 k - 1 < / ~ 8 1 1 + < c , ( 5 . 9 1 )k=l k---1o o o o- ~ t 2 ~ < M c ~ ~ < M ~ 1

k = l k = l

so tha t

SO ~ k = l - - 8 2 k t2 k c o n v e r g e s . T h u s

Soo < / ~ t ,~ l ( l q - 1 ) < o o .

(5 .92)

(5 .93)N o w w e v e r if y t h a t Sk > sO.

82k+1 x 2k t2 k= 82k ~1 x2k

= s l + \ \ l - z t + l - z 2 / + ' 'X 2 k - , 2 k - 1 +

q- \ i ~- X2k_----- 1 - - X 2k ] ] (5 .94)


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6 4 XI AOGUANGZ H O NX 2 i - l t - 2 i - 1 Z 2 i t 2 i X 2 i - I + Z 2 i+ - - - - t 2 i - 1 > 0, (5.95)1 - x 2 i _ 1 1 - x 2 i 1 - x 2 i

so that8 2 k + 1 > .S l , (5.96)

~ 2 k + l t 2 k + l8 2 k + 2 = 8 2 k + 1 + > 8 2 k + 1 . (5.97)1 - - X 2 k + lThis l eads to

8 k > 81 -~- 80 . 5.98)

5 3 R E S T R I C T I O N S O N P A R A M ET ER SI t has been assum ed tha t the re i s no ne w phase b oundary in i t ia t ed by the impac t .N o w w e v e r i fy a po steriori that this is the case.

I t is ob viou s that the s t ra ins ~ff in regions S P ~ and the s t ra in 7k in region SBkbe long to appropr i a t e phases. W e check w he the r 7 i f , 7k ( in reg ion B k) , 7 Ik and7Jk be lon g to appropr i a te phases fo r a l l k . In the fo l lowing , we sha l l show tha tthese s t ra ins belon g to appropria te phas es w hen the res t ric t ions (4 .35), (4 .42) and(4.48) are satisfied. T he p ro of is by indu ction.

W hen k = 1 , we kno w tha t 7~ , 7k , 71k and 7Jk be long to appropr i at e phasesf rom Sec t ion 4 .

As sum e 7k , 7k , 71k and 7dk be long to appropri a te phases when k = i .S ince V~m and 7 /~1 a re ob ta ined by so lv ing a Riemann p rob lem wi th in i t i a l

data V d i , 7 a i ) a n d (~ + , ~+ ) , a n d s in c e 7di belong s to the low-st ra in phase , 9 +be longs to the h igh-s t ra in phase , t hen i+1 belon g to appropria te phase s i f (4 .35)is satisfied.

F rom the recurs ion fo rm ulae (5 .44) - (5 .47) and the jum p con d i t ions (2 .7 ), (2 .8 )on sho ck w aves , we f ind tha t

7m + 7M 7T 5 . 99 )71i 1 = 2 2(1 + zi+l)I f / + 1 is e v en , t h e n T T / ( 2 ( 1 + z i + l ) ) > 7T /2 > T M - - T m ) / 2 , S O 7 I i + 1 < / rn .

I f i + 1 i s o d d , t h e n - -7 T / (2 (1 + X i - 1 ) ) > - - 7 T / ( 2 ( 1 + X i + I ) ) , S O 71i 1 < 7 I i - 1


27/34

D Y N MIC L BEH V IO R O F P H S E BO U N D RY 6 5A s 7 ~ 1 < 7 m a n d 7 i+ 1 < 7 m , 7Si+l < 7 m , w e c o n c l u d e t h a t 7 i f , 7k , 7 t k a n d 7 d kb e l o n g to a p p r o p r i a t e p h a s e s w h e n k = i + 1 .

W e h a v e s h o w n t h a t n o n e w p h a s e b o u n d a r y i s n u c l e a t e d if t h e re s t ri c ti o n s4 . 3 5 ) , 4 . 4 2 ) a n d 4 . 4 8 ) a r e s a t is f ie d .

5 4 I N T E R P R E T T I O N O F T H E N L Y S I SF o r a s e m i - in f i n i t e b a r i n i t ia l l y in a t w o - p h a s e e q u i l i b r i u m s ta t e w i t h a s i n g l e p h a s eb o u n d a r y , w e s e e t h at , d u e t o a n i n c i d e n t s q u a r e w a v e , t h e m o t i o n o f t h e p h a s eb o u n d a r y h a s t h e f o l l o w i n g fe a tu r e s:

1. T h e p h a s e b o u n d a r y m o v e s f o r w a r d w h e n t h e s q u a re w a v e h i ts it fo r th e f ir stt im e . T h e p h a s e b o u n d a r y m o v e s in th e f o l l o w i n g w a y :

f o r w a r d ~ s t at io n a r y ~ b a c k w a r d ~ s ta t io n a r y ~ f o r w a r d . . .T h e p h a 6 e b o u n d a r y m o v e s m o r e s l o w l y w h e n m o v i n g b a c k w a r d t h a n i t d i dt h e l a s t t i m e i t m o v e d f o rw a r d .

2 . T h e p h a s e b o u n d a r y s p e e d d e c a y s e x p o n e n t i a l l y t o z e r o . T h e r e i s a s i n g l ee f f e c t i v e m a t e ri a l c o n s t a n t t h a t d e t e r m i n e s t h e r a te o f d e c a y o f p h a s e b o u n d a r yp r o p a g a t i o n s p e e d : i t i s a = 2w c/( 7~) > O.

3 . T h e p h a s e b o u n d a r y p o s i t i o n is a l w a y s t o t h e r i g h t o f i ts i n it ia l p o s i t i o n , a n di ts f i n a l p o s i t i o n i s a f i n i te d i s t a n c e a w a y f r o m i ts i n i ti a l p o s i t i o n .

6 R e s u l t s f o r O t h e r K i n e t ic R e l a t i o n sA n i m p o r t a n t g e n e r a l i z a t i o n i n v o l v e s r e p l a c i n g t h e k i n e t i c r e l a ti o n f = ~ b y t h ef o l l o w i n g k i n e t i c r e l a t io n :

f - f . ) / w , f > f . ,~ = 0 , - f . < f _ _ _ f .

( f + f . ) / w , f < - f . ,6 .1 )

w h e r e f . a n d w a re p o s i t i v e m a t e r ia l c o n s t a n t s s e e F i g u r e 8 ) . T h i s k i n e t ic r e l a ti o ni s o f in t e r e s t b e c a u s e i t i n v o l v e s a b a r r ie r a g a i n s t p h a s e b o u n d a r y m o t i o n .

F o r t h e k i n e t i c r e l a t i o n 6 . 1) , th e i n it ia l s ta t e g i v e n i n S e c t i o n 3 n e e d n o l o n g e rb e a M a x w e l l s t a t e :

v x , O ) = O , 6 . 2 )7 z , 0 ) = 7 o 6 .3 )

f o r 0 < x < s o ;v z , 0 ) = 0 , 6 .4 )


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66 XtAOCtJANOZHONC

f

- f

F ig . 8 . A k ine t i c r e la t ion tha t i s d i s co n t inu ous a t ~ = 0

7 ( x , O ) = 7 + ( 6 .5 )f o r s o < x < ~ .

B u t w e r e q u i r e a ( S o , 0 ) = a ( So + ,0 ) , i.e . 7 + - 7 o = 7Tand-f < f o < f , ,w h e r e f 0 = T T ( T m + 7 M - - 7 + - - 7 0 ) / 2 .I t is c l e a r t h a t w e c a n s o l v e t h e i n i ti a l - b o u n d a r y v a l u e p r o b l e m f o r t h e k i n e t i cr e l a t i o n ( 6 . 1 ) u s i n g a p r o c e d u r e w h i c h i s e x a c t l y t h e s a m e a s t h a t e x p l a i n e d i nS e c t i o n s 4 a n d 5 . W e w i l l n o t r e p e a t t h e p r o c e d u r e . I n f a c t t h e e x p r e s s i o n s f o rp a r t i c l e v e l o c i t y a n d s t r a i n r e m a i n t h e s a m e a s t h o s e g i v e n i n S e c t i o n s 5 . 2 a n d5 .3 ; t h e r e c u r s i v e f o r m u l a f o r p h a s e b o u n d a r y p r o p a g a t i o n s p e e d i s, h o w e v e r ,d i f f e ren t .6 . 1 . S H O R T T I M E A N A L Y S ISA s t h e r e i s a b a r r i e r a g a i n s t t h e p h a s e b o u n d a r y m o t i o n , t h e p h a s e b o u n d a r y w i l lr e m a i n s t a t i o n a r y w h e n i t i n t e r a c ts w i t h a n i n c i d e n t w a v e , p r o v i d e d t h a t v 0 i s s m a l le n o u g h . I f th i s is t h e c a s e ( F i g u r e 3 ), f r o m ( 4 . 1 0 ) - ( 4 . 1 3 ) , w e h a v e

vi- = vo, (6.6)7 7 = - v o + 7 o , ( 6 .7 )cv + = vo, (6.8)


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Fig. 9.7 -

r

t ime

De cay o f the phase boundary p ropaga t ion speed

O O Q ~ a ~ O O I m O O I O I D Q O O O O ~ O ~ O I D O O t O Q O O O Q O Q O l e O m O . . . ..

Q I O O I Q O O Q I O Q I O O D D O O g Q O O ~ O ~ O Q I O I D U I O O O O J O . . . . . Q ~ O~

Q U Q m v A I

DYNAMICALBEHAVIOROF A PHASE BOUNDARY 6 7

0.75t

Fig. 10.

2.25

Variation o f strains o n both sides o f the ph ase IJoundary

. y + = _ v o + + .

A c c o r d i n g l y , t h e d r i v i n g t ra c t io n o n t h e p h a s e b o u n d a r y i s(6 .9 )

' ) ' T ,f - ~ - t ' r m + ' y M - 7 - - ' r + )

= f0 -4- / tTTv0 < f . ,ci .e . v o


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68 XIAOGUANGZHONGv

i m

v +

Fig. 11. Variationof particle velocities on both sides of the phase boundaryt

r - - - ' -

Fig. 12. Trajecto ryof the phase boundary

I f v 0 < c ( f , - f o ) / ( T T ) , t h e n t h e i n c i d e n t w a v e is t o t a l ly t r a n s m i t t e d t h r o u g ht h e p h a s e b o u n d a r y a n d t h e p h a s e b o u n d a r y r e m a i n s s t a t io n a r y f o r t > 0 .

I f v0 > c ( f , - f o ) / ( # 7 7 ) , then ~1 ~ 0 T h e s o l u t i o n i s a g a i n g i v e n b y ( 4 . 1 0 ) -( 4 .1 3 ) , b u t t h e p h a s e b o u n d a r y p r o p a g a t i o n s p e e d w i l l b e d e t e rm i n e d b y f =f . ~ , i .e .

1 ~ ) 2 c f ~ / 0 ) )1 S x , , + a X l = v 0C T T ( 6 .11 )T h e n o t a t i o n h e r e i s th e s a m e a s t h a t i n S e c t io n 4 .


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DYNAMICALBEHAVIOROF A PHASEBOUNDARYInitial state

6 9

State at t=m

~ phasel ~ phase3 ~ disturbedphase 3Fig 13 The deformed state of the bar at f = oo

A s i n S e c t i o n 4 , w e c a n s h o w t h a t h 2 = 0 a n d w e c a n a l s o d e r i v e t h e r e st r ic -t i o n s o n v 0 a n d o t h e r p a r a m e t e r s t h a t a s s u r e t h a t t h e r e i s n o n e w p h a s e b o u n d a r yn u c l e a t e d .6.2. LARGETIME ANALYSIST h e s t r ai n a n d p a r t ic l e v e l o c i t y i n t h e b a r ar e g i v e n b y 5 . 6 ) - 5 . 1 1 ) a n d 5 . 4 4 ) -5 . 4 7 ); s e e F i g u r e 6 . T h e d r i v i n g t r a c ti o n o n t h e p h a s e b o u n d a r y t h a t s e p a r a t e s

r e gi o n s P ~ I i s

/= /0 - -5 -- i7 7 + l_xL, 6 .12 )W e c ons ide r a t2 fi rs t. F r o m t h e s h o r t t i m e a n a l y s is , w e k n o w x 1 > O. I t can be

s h o w n t h a tXl

l + x ll )

72 = 1 - x 2 + a z 2 , 6 .13 )

s o t h a ta:_______L_l< 2 f , + fo )

1 + X l - / . t / ' 2 --+ ~2 = O; 6 .1 4 )

X l 2 f , + f 0 )- - >1 + X l # 7 ~

--+ hz < O. 6 .1 5 )

I t f o l l o w s t h a t x 2 < O.


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7 0 XIAOGUANGZ H O NF u r t h e r m o r e b y i n d u c t i o n w e c a n s h o w t h a t x 2 i - 1 > _ 0 a n d x 2 i < _ O T h erecursive fo rmu lae fo r the phase bound ary speed a re

1 + X2i_ 1 '7 2 = 1 -~ X2i + a X2i ,( 2 ( f . ~ f 0 ) ~ ( 1 )2 i =

1 7 ~ 2 i ~ T ] 1 -- X2i+1-2 + a X2i+l .A cco rdin g to the k inet ic re la t ion (6 .1),

X 2 i _ 1 2 ( f . + f0 ) ---+ k2 i+ l = O .1 + X i , . [ 2

(6.16)

(6.17)

(6.18)

(6.19)W e n o w sh o w tha t x 2 i - 1 a n d - x 2 i decay exponentia l ly as i ~ oo. Le t ~i be thesolut ion to (6 .16) and (6.17) w hen f . = 0 (subsequent ly f0 = 0) ; then w e kn ow

t h a t x 2 i - I a n d - ' 2 i decay exponen t ia l ly f rom resu lt s ob ta ined in Sec t ion 5 . For agive n v0, i t is easy to sh ow tha t X > X ~ 0.Ifx 2 < 0 , then from (6 .13),

1 + ~1 1 :g-----~ + a '2, (6.20)1 )l + x l 1 - - ~ + a i~ 2, ( 6 .2 1 )1 + X1 ~ T ] = 1 - x 2 + a x2. (6.22)

O ne can s how that 0 _> x2 > &2 > ~2. B y induct ion, w e can fur ther show thatX2i--I > X 2 i - I >_ 0 a n d - T , 2 i > - x 2 i >_ O . S o x 2 i _ 1 and --X2i decay expo nentia l ly .Thus i f x 1 # 0 , i t only takes a f in i te t im e for x i to reach zero; se e (6.18) o r (6.19).A d i f fe ren t genera l iza tion o f the k ine t ic re la tion o f Sec t ion 3 i s

f = ( ~ ), (6.23)wh ere i s con t inuous fo r a l l ~ and mo noton ica l ly inc reas ing , and

d e > 0 , (6 .24)d h -


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DYNAMICALBEHAVIOROF A PHASE BOUNDARY 71> 6 . 2 5 )

~b'(0) > 0, (6.2 6)i f( h ) = - 0 ( - ~ ) . ( 6. 27 )

F o r th i s k i n e t i c r el a ti o n , w e c a n o b t a i n r e s u l ts o n p h a s e b o u n d a r y m o t i o n s i m i l a r t otho se in S e c t ion s 4 o r 5 . S imi l a r ly , i f ~b(h ) i s a m on o to n ic a l ly inc r e a s ing f unc t ionw i t h a d i s c o n t i n u i t y a t h = 0 , th e n w e c a n o b t a i n r e s u lt s s i m i l a r t o t h o s e f o r k i n e ti cre la t ion (6 .1) .

F o r t he spe c ia l t r i l ine a r m a te r i a l c ons id e r e d he r e , a nd f o r a la r ge c l a s s o f k ine t i cr e l a t i o n s , w e h a v e s h o w n t h e p h a s e b o u n d a r y w i l l r e t u r n t o a n e q u i l i b r i u m s t a t ea f t e r a l ong t ime .7 A N u m e r i c a l C a l c u l a t i o nT o g a i n a q u a n t i t a ti v e i m p r e s s i o n o f t h e a n a ly t i c a l re s u l ts w e h a v e o b t a i n e d s o f ar ,a n u m e r i c a l c a l c u l a t i o n b a s e d u p o n t h e r e c u r s iv e f o r m u l a s ( 5 . 4 4 ) - ( 5 . 4 9 ) i s c a r ri e do u t . A s t h e p u r p o s e o f t h e c a l c u l a t i o n i s t o v i s u a l i z e t h e a n a l y ti c a l r e s u lt s , t h ep a r a m e t e r s c h o s e n a r e n o t r ea l p h y s i c a l d a t a a n d t h e i r u n i t s a re n o t s p e c if i ed . T h ep a r a m e t e r s u s e d a r e a s f o l l o w s :

1 . = 1 ,p = 12 . 7T = 1 .5 , 7M 2 , 7m = 13. c = V/-~pZ= 14 . 7 o = (T M + 7 m -- 7 T ) = 0 . 7 55 . 7 0 = I ( T M Jr )' m q - ) ' T ) - - -- 2 . 2 56 . v0 = 0 .25 , s0 = 1 , t* = 1 ,w = 0 .5

I t i s e a sy to c he c k tha t t h i s s e t o f pa r a m e te r s s a t i s fi e s ( 4 .35 ) , ( 4 .42 ) a n d ( 4 .48 ) . Th en u m e r i c a l r e s u l t s a r e s h o w n i n F i g u r e s 9 - 1 3 .

F r o m F i g u r e 9 w e s e e t h a t t h e p h a s e b o u n d a r y p r o p a g a t i o n s p e e d o s c i l l a t e sa s t i m e i n c r e a s e s a n d d e c a y s v e r y f a s t e v e n w i t h a < 1 . A f t e r o n l y a b o u t t e ni n t e r a ct i o n s o f t h e p h a s e b o u n d a r y a n d t h e r e fl e c te d w a v e , t h e p h a s e b o u n d a r yp r o p a g a t i o n s p e e d i s t o o s m a l l to o b s e r v e . S o a r e t h e v a r i a ti o n s o f p a r t ic l e v e l o c i t ya nd s t r a in ; s e e F igu r e s 10 a nd 11 . F r o m the num e r i c a l r e su l t s we se e tha t so

(1)journal of elasticity volume 41 issue 1 1995 xiaoguang zhong -- dynamic behavior of the interface...

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