Nesterenko Journal of Applied Mechanics

8/12/2019 Nesterenko Journal of Applied Mechanics

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4. V. Ya. Arsenin, Math emat ical Physics [in Russian], Nauka, Mos cow (1966).5 . N . S . K o s h l y a k o v , E. B . G l i n e r, a n d M . M . S mi r n o v , E q u a t i o n s i n P a r t i a l D e r i v a t i v e s

i n M a t h e m a t i c a l P h y s i c s [ i n R u s s i a n ] , V y s s h a y a S h k o la , M o s c o w ( 1 97 0 ).6 . S . K . G o d u n o v a n d V . S . R y a b e n ' k i i , D i f f e r e n c e S c h e m e s [ in R u s s i a n ] , N a u k a , M o s c o w

(1973).7. V . M . K o l o b a s h k i n a n d V . I . S e l y a k ov , " F l u i d i n f i l t r a t i o n i n a s t r a t i f i e d m e d i u m , "

Gaz. Prom., No. 6 (1980).

P R O P A G A T I O N O F N O N L I N E A R C O M P R E S S I O N P U L S E S I N G R A N U L A R M E D I AV. F. Neste renko UDC 624.131 + 532.215 + 534.22

T h e s t u d y o f m e c h a n i c s o f a g r a n u l a r m e d i u m i s o f s u b s t a n t i a l i n t e r e st , b o t h s c i e n t i f -ically and for the soluti on of applie d problems. Such mater ials are, for example, goodbuffe rs for shock loads. Their, study is important for the devel opmen t of proce sses of thep u l s e d e f o r m a t i o n o f s e v e r a l p o r o u s m a t er i a l s . A r e v i e w o f s t u d i e s o f s m a l l d e f o r m a t i o n sa n d e l a s t i c w a v e p r o p a g a t i o n i n t h e s e m e d i a w a s c a r r i e d o u t i n [ i] o n t h e b a s i s o f d i s c r e t emodels. The structure of a stationa ry shock wave was analyzed in [2] as a function of itsamplitude.

i . S t a t e m e n t o f t h e P r o b l e m. T h e p r o b l e m o f n o n s t a t i o n a r y , n o n l i n e a r p e r t u r b a t i o n si n o n e - d i m e n s i o n a l g r a n u l a r m e d i a i s s t a t e d i n t h e p r e s e n t p a p e r o n t h e b a s i s o f t h e w e l l -k n o w n i n t e r a c t i o n b e t w e e n n e i g h b o r i n g g r a n u l e s .

As an intera ction l aw we choo se the Hertz law [3]

3 i : r

where F is the compress ion force of granules, E is the Youn g modul us of their material, R,and R2 are radii, ~ is the Poisson co efficient, and x, and x2 are the coordinat es of spheri-cal granules (x2 > x,).

It is neces sary to point out that a depend ence of the form ~s/~, where 6 is the closestapproa ch of parti cle centers, is valid not only for spheres, but also for contacts of otherf i n i t e b o d i e s [ 3 ] . I n t e r e s t i n g l y , i t is o n l y d u e t o t h e f i n i t e p a r t i c l e s i z e s o f a l i n e a r l ye l a s t i c m a t e r i a l c o n s t i t u t i n g t h e g r a n u l a r m e d i u m t h a t i t s b e h a v i o r h a s a n o n l i n e a r l y e l a s t i ccharacter.

T h e u s e o f t h e s t a t i c H e r t z l a w i n s o l v i n g d y n a m i c p r o b l e m s i m p l i e s t h e f o l l o w i n g r e -strictions: I) the maxi mum stress achieved at the center of the contact must be less thanthe elastic limit; 2) the sizes of the contact surface are much smal ler than the radii ofc u r v a t u r e o f e a c h p a r t i c l e; a n d 3 ) t h e c h a r a c t e r i s t i c t i m e s o f t h e p r o b l e m r a r e m u c h l o n g e rt h a n t h e o s c i l l a t i o n p e r i o d o f t h e b a s i c s h a p e f o r t h e e l a s t i c s p h e r e T

~ > > T ~ 2 . 5 R ~ z ,w h e r e c t i s t h e v e l o c i t y o f s o u n d i n t h e s p h e r e m a t e r i a l .

C o n d i t i o n s 1 - 3 r e s t r i c t t h e m a s s v e l o c i t i e s o f t h e m e d i u m t o q u a n t i t i e s o f t h e o r d e ro f s e v e r a l m e t e r s p e r s e c o n d f o r m e t a l l i c p a r t i c l e s w i t h r a d i i i n t h e i n t e r v a l 1 - 5 m m.D i s s i p a t i o n p r o c e s s e s a r e n o t t a k e n i n t o a c c o u n t a t t h e p r e s e n t s t a g e o f t h e s t u d y .

F o r n u m e r i c a l s t u d y o f p e r t u r b a t i o n p r o p a g a t i o n p r o c e s s e s i n a o n e - d i m e n s i o n a l c h a i no f s p h e r i c a l p a r t i c l e s w i t h a r b i t r a r y r a d i i R t h e s e c o n d o r d e r e q u a t i o n s o f m o t i o n w e r e r e -d u c e d t o a f i r s t o r d e r s y s t e m o f e q u a t i o n s :

x i = F ~ ~ , x = ~ , x ~ . . . , ~ N ) , i = 1 . . . . . 2 N , 1 . 2 )F ~ x ) = ~ x ) - - ~ x ) , ~ = t , . . . , N ,

N o v o s i b i r s k . T r a n s l a t e d f r o m Zh u r n a l P r i k l a d n o i M e k h a n i k i i T e k h n i c h e s k o i F i z i k i , N o .5, pp. 136-148, September- Octob er, 1983. Origin al artic le submit ted April 12, 1982.

0 0 2 1 - 8 9 4 4 / 8 3 / 2 4 0 5 - 0 7 3 3 5 0 7 . 5 0 9 1 9 8 4 P l e n u m P u b l i s h i n g C o r p o r a t i o n 7 3 3


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F l ( x ) ~ x t - x , t = N t . . . . , 2 5 ./ 1 ' 2 E - - - ' - - ' -~ 8 ~ 2 1 , i f 0 ~ - 1 > 0 , t 2 , ,

~ ( x ) = 0 , i f 8 t - ~ < 0 , t = 2 . . . . N ~ ( 1 . 2 )[ R ~ R ~ 1 ~ i , ~ E

~ x ) =0 i f 8 t ~- -< O , i = i . . . . . N - - t ,~ - z = B ~ + B ~ _ ~ - ~ + ~ x ~ + ~ - z ) , ~ = 2 . . . . . N ,

z ~ + ~ ( t = O ) = x ~ + ~ _ ~ + R ~ + R ~ _ ~ , ~ = 2 . . . . N , x ~ + ~ ( t = O ) = O .It is clear from this form of writin g that for N / > i ~ l the quantities x~ are the

velocity values of the i-th particle, while for 2 N ~ i> N they are the coordinate values.The initial conditions for the velocitie s on ~ and ~N are specified below for each separatecase.

2. Analysis of the Anharmoni c and Long-Wave Approximations. Consider a one-dimens ionalchain of identical spherical granules. We assume that it is subject to constant compressionforces F~, applied to ~he chain edges and securing the initial closest approach 6o. As willbe clear from the following, ~ is conveniently introduced into the equation explicitly.With this purpose in mind we replace the coordinate x i by the displaceme nt of the given par-ticle from its equilibr ium position u i. Using the express ion for the force i . i ) , the par-ticle equation of motion becomes

~ l = A ( 8 o - - u l + u i - i ) a /2 - - A ( 8 o - - u ~ + i + u i ) 8 / z, ( 2 . 1 )A = E ( 2 R ) t ; 2 / [ 3 ( t - - v ' ) m ] , N - - I > / t . >I 2 ,

where m is the mass of the particle. It is assumed here that the distance betw een the par-ticle~canters does not exceed 2R.

The equation provided will describe the propagat ion of one-dlmensional perturbatio nsin a three-dimensional simple cubic packing of spheres if the front plane is parallel to thecube boundaries. Differences occur only in the numerica l coefficient in A.

Equation (2.1) can be tr ansformed to a well-in vestiga ted sys tem of nonlinear oscill a-tors under the assumption of small deformatio n in the medium by compariso n with the initialclosest approach 6a, i.e., putting

l U l - ~ - u t I /6 o < < t .In the anharmonic ap proximati on Eq. (2.1 ) is then

u ~ = = ( u ~ + l - 2 u l + u , - 1 ) + ~ ( u i + l - 2 u ~ + u , - 0 xx ( u i - 1 - u ~ + l ), ( 2 . 2 )

3 1= y A 6 o , ~ = A 6 ~ 1 /2 , N - - 1 > t ~ > t 2 .Equations of the type (2 .2) with a quadratic nonlinear ity were solved nume rically in a num-ber of studie s (se e, e.g., [4, 5]), and a brie f revi ew can be found in [6]. It was show nthat in the problem of a piston moving with a constant veloci ty vo there exists an oscillat-ing nonstatio nary structure of a shock wave, in whose head is formed a soliton with thevelocity in the maxim um equal to 2vo. The oscillations are damped behind the front even inthe absence of dissipation. A similar equation can be obtained for small perturbations andfor a different choice of interaction potentials. Oscillations undamped in time were ob-served near the piston for sufficiently high values of its velocity in numerical calcula-tions of a Toda chain with a particle in teraction determin ed by the Morse potential [6, 7].The existence of solitons in a chain with a Morse potential was shown numeri cally in [7],

In the long-wave approxima tion (L >> u = 2R, where L is a characterist ic spatial sizeof the perturbation) and with the usual replacement

u i = u (x ) , u i - 1 = ~ - ~ ( x ) , u i + i = : ( z ) ,D - ~ O / a x

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o n e c a n o b t a i n f r o m E q. ( 2. 2 ) t h e n o n l i n e a r w a v e e q u a t i o nu ~ t c ~ o u 4 2 c 0 ~ u = = - - ~ u ~ u ~ ,

co = Af~/~6R~, ~ = c o R ~ / 6 , e = c ~ R / 5 o . (2 .3 )

l ~ \ I a \2 U \23A l l te s o f o r d e r l a n d h i g h e r w e r e o m i t t e d i n d e r i v i n g 2 . 3 ) .Solu t ions o f th i s equa t ion sa t i s fy the Korteweg--de Vr ies (KdV) equa t ion , whose p rop er t i esa r e qu i t e we l l known [8 ] , accu ra t e ly up to t e rms quad ra t i c in the non l inea r i ty and d i sp er -s ion coef f i c i en t s . Not: dwel l i ng on the de ta i l s o f d i sc uss i ng thes e p rop er t i es , we no te tha tin a g r an u la r med ium of p roper s t ruc tu re , compressed by an ex te rna l f o r ce , t he ex i s t e nce i sposs ib l e o f so l i t a ry waves , pe r i od ic waves, and shock waves wi th an osc i l l a t ing s t ru c tu r e .The qua l i t a t ive beha v io r o f t he so lu t ion s depends on the t ime char ac te r i s t i c s o f t he load in gp u l s e [ 8 ] .

We are in t ere ste d in the cas e 6o -~ 0 . Then, as seen f rom Eq. (2 . 2) , the qu ant i ty 8 |and the anharmonic approx imat io n becomes inap p l i c ab le . For ~o = 0 , i n the long-wave appr ox i -mat ion compress ion waves o f smal l ampl i tude canno t be desc r ibe d by the s t andar d wave appr ox i -mat ion , which i s a d i r e c t consequence o f the anharmonic approx imat io n . I n the case l u ~ - i - - u i l/80 ~ i one can ob ta in f rom (2 ,1) on ly an equa t ion cor r espo nd in g to the long-wave approx ima-t ion (L >> a ) by a r ep lacem ent s i mi l a r t o the p rec eed i ng one . The equa t ion i s( = = )

. 8 ( - - ' u . , ) ' / ~ 64 (_ t t , ) ' J ' ( 2 . 4 )- - u ~ > 0 , c z = 2E

~ P o ( i - 4 ' )I n t h e e q u a t i o n g i v e n t h e d i s p l a c e m e n t u i n c l u d e s t h e i n i t i a l d i s p l a c e m e n t r e s p o n s i b l e f o rthe c lose s t approach ~o. The th r e e l as t t e rms in Eq. (2 .4 ) a r e Of o rder ( a /L ) 2 in compar i -s o n w i t h t h e f i r s t . H i g h e r - o r d e r t e rm s h a ve b e e n o m i t t e d . I n t h e p r e s e n c e o f a c o n s t a n tcompress ion fo rce , guara n tee ing an in i t i a l c los es t approach ~o, Eq. (2 .4 ) r educes to then o n l i n e a r e q u a t i o n ( 2 .3 ) i n t h e s a m e a p p r o x i m a t i o n . T h i s i s e a s i l y d o n e by r e p r e s e n t i n g t h ef u l l d e f o r m a t i o n i n th e f o r m

u ~ = 6 o / 2 R A u xa n d c a r r y i n g o u t a n e x p a n s i o n i n t he s m al l q u a n t i t y A u o f t he t e r ms i n t h e r i g h t - h a n d s i d eo f t h e e q u a t i o n . S t a n d a r d s o l u t i o n s o f E q . ( 2 . 4) c a n b e f o u n d i n t h e f o r m u ( x -- V t ) . I n -t r o d u c i n g t h e n e w v a r i a b l e ~ = - u x , w e o b t a i n

V ~ 3 ~ 1 2 ~ x a2 (~)~ ~ ( 2 . 5 )~- ~ = ~ + 8 ~v~ ~ / ~

E q u a t i o n ( 2. 5) c a n be i n t e g r a t ed , p e r f o r m i n g t h e v a r i a b l e r e p l a c e m e n t 5 = z~ / ~:

( Cx i s a n i n t e g r a t i o n c o n s t a n t ) . T h e l as t e q u a t i o n i s c o n v e n i e n t l y w r i t t e n i n d i m e n s i o n l e s sf o r m, w i t h v a r i a b l e r e p l a c e m e n t :

z = y, x = y- '~ -U ] ~ (2 .6 )y~,5 = y~5 y l :Synn C2

( C 2 i s a c o n s t a n t ) .If C2 = 0, Eq.

s o l u t i o n i s( 2 .6 ) i s e a s i l y i n t e g r a t e d . I n a s y s t e m m o v i n g w i t h v e l o c i t y V t h e

s v 2 \ 2 V ~ x ( 2 . 7 )

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For x = ~ - ~ t T n = 0 ~ . . ~ vanishes, This is found to contr adlc t the condi tion ~ > 0,under which Eq. (2.4) was derived, Therefore it is necessary to study the behavior of thesolution for C= # 0. With thls purpose in mind we rewrite Eq. (2.6 ) in th~ following form,making it possible to use ~he analogy with particle motion in the potential field [8]:

O t y i s ~ l l ~y ~ = - ~ W ( y ) , W ( y ) = - y~ t ~ + - i +It can be noted from the equat ion for W(y) that for 0 < C= < 5/27 the func tion W(y)

has the form of a curve with two extreme, shown in Fig. 1 for the values Cs - 4 / 2 7 (curvei) and 0.i (curve 2). The extrem e yl and Ya, mark ed on Fig. i, corre spond to the caseCs - 0.i. For Cs * 0 the value of the maxim um W(yz) also t ends to zero, as does the value ofits coordinate Yl. For Ca - 0 we obtain a solution of (2.7) in the form of periodic waves.The deviation of Cs from zero changes the nature of the solution qualitatively. From theshape of the function W y ) for 0 < C , < 5 / 2 7 it can be conclud ed that Eq. 2 . 4 ) admits inthis case :the existence of stationary solutions of the type of periodic functions, whilesolitary waves are also possible under certain conditions. Zndeed, near the maxi mum y~ W(y)can be expanded in powers of (y - yi):

w v ) ~ w ( ~ ) ~ v - y , ) ' .This form of potential energy leads within the mechanic al analogy to an infinite time ofparticle fall with total energy W(yz) at the point Yz, which c orresponds to the formationof~a solitary wave [8 ]. The dependence of the potential energy on coordina te near yz cor-responds to the stationary case for the KdV equation near the analog point. Therefore, thebehavior of the solution for y y: will coincide asymptotica lly with the soliton soluti onof the KdV equat ion.

The restriction Cs < 5 / 2 7 guarantees a value of the phase velocity of a solitary com-pression wave larger than the initial velocity of sound. Indeed, the phase velocity of asoliton V and the velocit y of sound co equal, respectively:

- I . / 4 / 3 ~ 1 / 9v = ~ | l ~ u i - ~ /~ , C o = c ~ o ~ , - ~ - ] ,where ~o is the initial deformation. By comparing the expressions above it is seen that theinequ ality V > co is satisfie d for yl < (2/3) /a. The given cond itio n is satisfi ed for Cs y~ is near the behavior of the solution (2.7), corre sponding to the caseCs = 0 and to vanishing total energy. Consequently, the maxim um deformatio n value in asolitary wave is near the amplitude value of the periodic wave (2.7):

~ . = i T T )The chara cteristic spatial size of a soliton is determined in this case by the period of thesolution (2.7), which equals

Sa

The solitary waves found differ fro m solitons of the KdV equati on by the dependence of thephase ve lo ci ty on the wave ~p l i tu de ~o th er important dif fer ence is the independence ofthe characteris tic spatial size of the soliton of Eq. (2.4) L = 5u on the wave amplitude.As already noted, for a small deformation in the w a v e in comparison with ~o the solution ofEq. (2.4 ) in the form of a solitary wave will be close to the analo solution of the KdVequation.

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w 16 ~

-41 1

Fig. i

f

2

-

~ 25

15

J ~ S I

I

2 5 0Fig. 2

__

55 ~ O ~ ~ec

In the presenc e of dissi pation in the medium, for Cs = 0 the deformation behind thefront of the stationar y sh ock wave cor respondin g to the mini mum W(yz) equals the value ~c'coinciding wi th that found from the conservation laws for a chain with o = 0:

B r = V 4 .By comparing the quantities ~m and ~c it is seen that for wea k dissipa tion the ratio

of maxi mum d eformatio n at the fron t of the stationary shock wave to its establish ed val uebehind the front reaches the value

~ m / ~ c = 5 / 4 ) ~ .~ 1 . 5 6 .The given difference in deformat ions leads to the circumsta nce that the ratio of the

maxi mum pres sure at the front of the stationary shock wave Pm to the establish ed pressurebehind it Pc can achieve the value

P~p~ = ( ~ , , , / ~ )~ / ~ = ( 5 / 4 p ~ i . 9 5 .F u r t h e r u s e o f th e o n e - d i m e n s i o n a l n o n l i n e a r c h a i n o f o s c i l l a to r s , i n t e r a c t i n g c o m p r e s s i v e l yaccordin g to the Hertz law, was carried out numerically.

3. Analysi s of Computat ional Results. For numeri cal solution of the system of nonlin-ear equatio ns (1.2) we used the Eammin g and fourth order Runge- Kutta methods. The controlsolution was obtained by the mome ntum and energy co nserva tion laws and comparis on of the re-s u l t s o b t a i n e d b y v a r i o u s m et h o d s . D u r i n g t h e c a lc u l a t i o n t h e m o m e n t u m c o n s e r v a t i o n l a w iss a t i s f i e d w i t h a n a c c u r a c y n o t w o r s e t h a n 1 0- 5 % , a n d t o t a l e n e r g y c o n s e r v a t i o n l a w w a ssatisfied within i0-s-I0-2%. An estimate of the relative error in determi ning the particlevelocit y gives the value 10-2%. With the purpos e of verifyin g the correct ness of the calcu-l a t i o n w e a l s o c o m p a r e d t h e n u m e r i c a l s o l u t i o n o f t h e p r o b l e m o f i m pa c t o f a s i n g l e p a r t i c l ewith a chain of i00 identical particles, arranged with initial gaps, with its obvious exactsolution. This comparison showed coincidence of the numerica l and exact solutions with in thelimits of relative errors mentioned above.

We make one more comment. For 6o = 0 and identical R i Eq. (2.1) with variabl e replace-m e n twi u i / v o , 9 t ( A ~ V o ) / 5

is transformed to dimensionl ess form

[ 5 I i 3 . 1 )Clearly, for a chain with free ends (~i = N = 0) identical ~ values correspond to

identical w i values if the problem is solved with initial conditionsw ~ ( t = O ) = t , i = 1 . . . . k , w ~ ( t = O ) = O , k < l < N

A similar conclus ion is also valid for solving the problem of a piston moving with constantv e l o c it y . T h e r e fo r e , o n e f u r t h e r c o n t r o l m e t h o d w a s v e r i f i c a t i o n o f t h e o b s e r v e d e q u a l i t yof dimension less particle velocity w i at identical momen ts T for problems with d ifferent v al-ues of initial velocities. For convenience of comparison with experiment the proble m was

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solved in real time. The choice of the initial step At in time was done by starting fromt h e n a tu r a l p h y s i c a l c o n d i t i o n

At 2 , N =I00.

The time depende nces obtain ed for xi/vo are shown in Fig. 2 (curves a). It is seent h a t f o r t h e s e i n i t i a l d a t a t w o s o l u t i o n s a r e f o r m e d w i t h p r a c t i c a l l y r e s t i n g m a t e r i a l b e t w e e nthem. The formation of an initial perturba tion of two stati onary solitar y waves is alreadyconcluded on the twentie th particle. They further propag ate as stationary. As a consequen ceof formation of solitar y waves the final velocity of the last partic le x differs fr om vo.For example, x~o/~ = 1.2338, xbo/~ = 1.2342, xlo,/~ = 1.2342. The equality of veloc ities xso =x , o o s e r v es a s f u r t h e r v e r i f i c a t i o n o f t h e s t a t i o n a r y n a t u r e o f a l a r g e - a m p l i t u d e s o l i t o n .

S i n c e t h e s o l i t o n a m p l i t u d e s v a r y , t h e d i s t a n c e b e t w e e n t h e m increases continuously.A c o m p l e t e l y s i m i l a r p a t t e r n i s a l s o o b s e r v e d f o r o t he r v a l u e s o f t h e i n i t i a l v e l o c i t i e s o fthe first two partic les with varyi ng time scale. It is neces sary to note t h a t for free endsof the chain (~, = $ N = 0) there occurs a recoil of the first partic les and a destruc tionof the system. For example, at the moment of time t = 2-10 ~ sec~ whe n the second solitonof lesser amplitude is located near the 2 7 t h particle, the particl es from 1 to 7 have a nega-tive velocity, and those fr om 1 to 8 are removed fr om each other by a distanc e larger than2R. The veloc ity values of the fl rs ts ev en pa rticle s equal --0AI:--0.048,--0.013,--0.0068,~-0.0029, and --0.0004 m/set. The re moved fraction al pulse is approx imate ly 10% of theinitial value, Clearly, the destruc tion of the syste m at a certai n moment of time willnot affect the sollton shape, but the very fact of destructi on renders it impossible to ac-c u r a t e l y d e t e r m i n e t h e s o l i t o n a m p l i t u d e f r o m t he i n i t i a l c o n d i t i o n s i n a n y c o n t i n u u m d e -scription.

It follows from Eq. (3.1) and the fact of existence of stati onary so litary waves thatt h e t e m p o r a l h a l f - w i d t h o f a s o l i t o n % a nd i t s ph a s e v e l o c i t y V m u s t d e p e n d o n t h e p a r t i c l evelocity at the m a x i m u m of the solit ary wave v as follows:

~ . . v ~ ~ / ~ V N v ~ 5 .T h e o b v i o u s d i f f e r e n c e b e t w e e n s o l i t o n s i n t h e g i v e n s y s t e m a n d s o l i t o n s o l u t i o n s o f

t h e K d V e q u a t i o n i s t h e n o n l i n e a r d e p e n d e n c e o f t h e p r o p a g a t i o n v e l o c i t y a n d o f t h e s q u a r eo f t h e r e c i p r o c a l h a l f - w i d t h % - * o n t he v e l o c i t y a m p l i t u d e v . I t i s a l s o i n t e r e s t i n g thatthe spatial sol iton size is in the given case indepe ndent of the~,velocity amplit ude Vm, ande q u a l s f i v e p a r t i c l e d ia m e t e r s . W e re c a l l that solutio ns of the cont inuum equati on (2.4)i n t he f o r m of s o l i t a r y w a v e s e x i s t o n l y f o r n o n ~ a n i s h i n g i n i t i a l d e f o r m a t i o n s ~ o . I t s f i n alvalue gu arante es the required asymp totic behavi or of the solut ion for ~ + ~o, leading to so-liton formation. In a numer ical calculati on solitar y aves also exist for ~o = 0. In spe-c i f i c n u m e r i c a l c a l c u l a t i o n s i t w a s c l a r i f i e d that i n t h e p r e s e n c e o f i n i t i a l d e f o r m a t i o n sto


7/11

~ ~ ? 2 R O ~

29 t

0 t " O ~ -s.-ecFig. 3

5 /

2

i

0 40 50 80 t 9 On, seeFig. 4

60 and 61 (curve i), obtai ned wi th a time step 0. 25.10 -s sec, as wel l as the fu nctio n (2.7)(curve 2), in whic h the value of the phase velo city was taken equal to its value in the nu-merica l account. The moment of achieving maximum deformati on is reached later on. It isseen from Fig. 3 that the continu um approxi mation is in satisfactory agreement wi th the nu-merical calcula tion with t ~ 1 0 -5 sec. At the same time the difference is large for ~ + 0,which is natural, since the functio n (2.7) does not give the correct asym ptotic beha vior inthis region.

A l s o i n v e s t i g a t e d w a s t h e c a se i n w h i c h t h e i n i t i a l v e l o c i t y i s c o m m u n ic a t e d b y t h efirst 4 particles. In this case the pertu rbatio n was decomposed into 4 solitary waves. Ifthe initial velocity is communicated by the single first particle, a single soliton is formedin the system.

3.2. Soliton Interaction. Two cases were investigated.A. Colllsion 'of solitons movin g toward each other. The initial veloci ty conditions are:

x l t = O ) = z ~ t = O ) = 5 m / ~ c - z l o . t = O ) = z , 9 t = 0 = - - 5 m l s e cx ~ t = 0 ) = 0 , ~ = f= i , 2 , t 0 0 , 9 9 , N = i 0 0 .

The ends of the chain are assumed free. The result of the collision is shown in Fig. 2,where it is seen that initially there is a decompo sition into 4 solitons, m oving toward eachother. Until the collisio n each of the pairs is displaced si milarly to the case 3.1. Fol-lowing the interaction, the solitons do not change their shape. Only phase changes occur.The differe nce in the arrival time of the first veloc ity maxi mum at the 75th particl e be-tween case A and the correspo nding case 3.1 is around i0 ~sec (see Fig. 2). A similar soli-ton interaction wit hout a change in their shape was obse rved by us during reflectio n from arigid wall of a sequence of 6 solitary waves, g enerated by impact with a chain of particles(N ffi 80) b y a pi st on wi th mas s 5m.

B. Overta king of a lower amplitude sol iton by a larger amplitude soliton. The initialconditions for the velocit y are

x l t = O ) = x ~ t = O ) = t 0 m / s e c , x l s t = O ) - -- -- l s ( t = O ) = 5 m / s e cx ~ t = O ) = 0 , ~ = g = t , 2 , t 8 , 1 9 , N = 1 5 0 .

The ends of the chain are free: ~i = ~ = 0 In particular, the leading group of the soli-tary waves I-3, shown in Fig. 4, were formed for the choice of initial condi tions in thesystem. The time interval 2.~0-4 < t 8. 10 -4 sec -- to the veloci ty of the 145th particle. In the latter casesoliton 2 is not shown, since even for t > 5 10 4 sec it does not partici pate in the inter-action betwee n solitons of a given group. It is seen from Fig. 4 that soliton 3 is subse-quently passed by solitons 2 and i. The amplitude and shape of the solitary waves did notchange in this case. It is interesting that for these initial conditions 6 solitary wavesformed in the system, desp ite the fact that the initial perturbati on, consisti ng only of 2moving particles, decomposes into 2 solitons (Section 3.1). The fact given is a consequen ceof the nonlin ear nature of the partic le interaction.

3.3. Actio n of Externa l Force of Triangu lar Profile. The initial conditions for thevelocity are:

7 3 9


8/11

F i g 5

i [

Fig, 6

x , t = 0 ) = 0 , N > ~ i > i , N = i 0 0 .The force acting on =he left end of the chain was gi ven as follows:~ i = 2 ' 1 0 e (i - - 1 0 4 0 m / s e ~ , 0 ~ < t < 1 0 ~ s e c ~ i = 0, t > i 0 - ~ sec

The right end of the chain Is assum ed free (~N-- 0),It is seen from Fig. 5, where the velocities of the 35-th and 60-th par ticles are shown,

that the perturbat ion decomposed into 7 solitons. For decreasing time of force action downto 10 ~ sec the number of solitons decreased to one. The pulse durations of 10-100 ~sec aretypical of blast experiments. Therefore the discrete structure of the granular medium appearsto have a substantial effect on the formation of pulse compression in this case.

3.4. The Piston Problem. The initial conditions for the velocitie s are:x l t = 0 ) = 5 m / s ac , x ~ t = 0 ) = 0 , N ~ > ~ > I , N = 2 0 0 .

The right end of the cha in is assume d free ( N = 0), while the left end is subject to thecondition ~I =~ 2, guarantee ing constancy of the first particle velocity.The values of all particle velocities at m o m n t of time t = 5.45,i0 ~ sec are give n in

~ig. 6. For convenience of perception the velocity values of neighbori ng par=icles were in=er-preted as a union of straight lines. The real velocity values correspond only =o integers. The amp-li=ude of the first maxim um v approaches wi=h =ime twice =he value of the pis=on velocity vo.This result is similar to those ob=ained for a Toda chain [6], for particles mutually inter-acting according to the Morse potential [7], and for the nonlinea r KdV equation [8]. It isalso seen from Fig. 6 that a soliton with a velcoity am plitude equal to 2vo is formed withthe flow of time at the head of the wave. The motion of the mediu m does not merge on thestationary regime, despite the practical constancy, startin g with i = 20, the velocit y ofthe leading front equal to the soliton velocity with an amplitude 2vo, and the achievementof a stationary state by particles near the piston. The nonstationarity of moti on consistsof a continuous increase in the number of particles pa rticipat ing in the oscillatory motion.It is necessary to note that the velocity of the leading front gets near the velocity of theestablished so liton with an accuracy of 1% already at the first ten particles. Therefore,fixing in the experiment only the velocity of the leading front, one can reach erroneousconclusions concerni ng process stationarity.

It is interesting to compare the velocity of the leading front of a nonstati onary waveV wi=h the velocity DI of a stationary im pact wave, pro pagatin g in the given case in thepresence of dissipation. Starting from the equations of stats of the mediu m and the massand momen tum cons ervation equations, we find that

2E J . . , , 5 .l : 1 ' 5Dl (i -- ~ - ) ~PoUsing the numerical results, we obtain that the soliton phase velocity V with maximum veloc-i=y v equals

I = 5EV ~ 0 . 9 1 5 ( i ~ ~ ) ~r ./ l ~ i ~ .For v = 2vo we have a ratio V/DI : 1.05.

Thus the difference between the established velocity of the leading front of a nonsta-tionary wave and a stationary wave is *5%. The given difference renders the possibi lity of

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e s t i m a t i n g a n e x p e r i m e n t o n l y f r o m t h e l e a d i n g f r o n t v e l o c i t y u n r e a l u n l e s s w e a r e i n th es t a t i o n a r y r e g i m e .

I f t h e re a r e d i s s i p a t i v e l o s s e s i n t h e m e d i u m , t h e n i f t h e y a r e s m a l l i t i s n a t u r a l t oexpect by analog y with the KdV equation) that the shape of the leadin g front of the wavei s n e a r a s o l i t o n s h a p e . I t s a m p l i t u d e V m , m u s t g u a r a n t e e t h e f r o n t v e l o c i t y , e q u a l t o D , .U s i n g t h e e x p r e s s i o n f o r t h e s o l i t o n v e l o c i t y , w e f i n d f r o m t hi s r e q u i r e m e n t

v ~ i ~ i . 5 6 % .T h u s , f o r s m a l l d i s s i p a t i o n t h e m a s s v e l o c i t y a t t h e s t a t i o n a r y f r o n t c a n e x c e e d t h e p i s t o nv e l o c i t y b y 1 , 5 6 t i m e s . I t i s i n t e r e s t i n g t h a t t h e r a t i o f o u n d V m ,/V o c o i n c i d e s w i t h t h ev a l u e s o f ~ m / ~ C , o b t a i n e d f r o m t h e c o n t i n u u m t r e a t m e n t ( s e c . 2 ) . I t w a s f o u n d i n [ 6 , 7]t h a t f o r a s u f f i c i e n t l y l a r g e i n d e x o f n o n l i n e a r i t y t h e p a r t i c l e s n e a r t h e p i s t o n c a r r y o u tu n da m p ed o s c i l l a t i o n s . U n l i k e t h e s e t w o c a s e s , i t w a s f o u n d i n o u r n u m e r i c a l e x p e r i m e n t t h a tt h e v e l o c i t y p r o f i l e s i n t h e c o o r d i n a t e s w - - i a r e i d e n t i c a l f o r t h e v a l u e s v o / V = 0 . 9 5 an d5 . 1 0 - a t t h e c o r r e s p o n d i n g m o m e nt s o f t i m e , w h i c h i s i n a g r e e m e n t w i t h E q . ( 3 . 1 ) . I t f o l -l o w s f r o m t h e d e p e n d e n c e o f t h e s h o c k w a v e v e l o c i t y O n v o t h a t t h e i m p a c t a d i a b a t o f t h e g r a nu l a r m e d iu m i s i n d e p e n d e n t o f t h e s i z e o f t h e p o w d e r p a r t i c l e s , a s i s t h e f r o n t v e l o c i t y f o rn o n s t a t i o n a r y w a v e m o t i o n i n t h e a b s e n c e o f d i s s i p a t i o n .

W e n ow t u r n o u r a t t e n t i o n t o t h e e n e r g y d i s t r i b u t i o n i n t h e s y s t e m b e t w e e n k i n e t i c Eka n d p o t e n t i a l E p e n e r g i e s . T h e r a t i o E k / E p = 1 . 2 4 9 i s e s t a b l i s h e d b y t h e t i m e t ~ 3 1 0 - 5s e c , w h en p r i m a r i l y 6 p a r t i c l e s p a r t i c i p a t e i n t h e m o t i o n . I t f o l l o w s t h a t E k / E p i s p r a c -t i c a l l y c o n s t a n t . T h e m a xim u m d e v i a t i o n f r o m t h i s v a l u e d o e s n o t e x c e e d 0 . 8% . T h e g i v e nv a l u e o f E k / E p i s n e a r t h e e x p e c t e d o n e b y t h e v i r i a l t h e o r e m f o r p a r t i c l e s i n t e r a c t i n g a c -c o r d i n g t o t he H e r t z l a w a n d p e r f o r m i n g f i n i t e o s c i l l a t o r y m o t i o n :

= = i 2 5

I t d o e s n o t f o l l o w f r o m t h e v i r i a l t h e o r e m , h o w e v e r , t h a t d e v i a t i o n s f r o m t h e m e an v a l u e w i l lb e s m a l l , a s i s o b s e r v e d i n t h e n u m e r i c a l c a l c u l a t io n . T h e s m a l l n e s s o f d e v i a t i o n s f r o m m e anv a l u e s i s , o b v i o u s l y , e x p l a i n e d b y t h e l a r g e n u m b er o f p a r t i c l e s i n t h e s y s t e m . A s i m i l a re n e r g y d i s t r i b u t io n b e t w e e n Ek a n d E p w a s a l s o o b s e r v e d i n c a s e s 3 . 1 a n d 3 . 2 .

W e a l s o n o t e a f e a t u r e o b s e r v e d i n t h e n u m e r i c a l c a l c u l a t i o n . S i n c e t h e v e l o c i t y a m p l i -t u d e o f t h e s o l i t o n p e a k i n c r e a s e s , a p p r o a c h i n g 2 v o , th e v a l u e o f t h e m a xim u m v e l o c i t y o f t h el a s t p a r t i c l e V N w i l l e x c e e d 2 V o . T h e v a l u e s o f V N /V o f o r N = 2 0 , 5 0 , 1 0 0 , a n d 2 0 0 e q u a l ,r e s p e c t i v e l y , 2 . 7 0 8 , 2 . 8 3 6 , 2 . 8 7 6 , a n d 2 . 8 9 5 . A s a l r e a d y n o t e d , t h e r a t i o V N/V o i s i n d e p e n -d e n t o f v o .

T h e r e s u l t s o f t h e c a l c u l a t i o n s ho w t h a t i n t h e p r o b l e m u n d e r c o n s i d e r a t i o n t h e p r e s s u r ei n t h e m a xim u m o f t h e f i r s t o s c i l l a t o r Pm s u b s t a n t i a l l y e x c e e d s t h e p r e s s u r e e s t a b l i s h e d n e a rt h e p i s t o n P c F o r e x a m p l e , f o r a p i s t o n v e l o c i t y v o = 5 m / s e c a t t h e m o m e nt o f t im e t h e w a vf r o n t i s f o u n d a t t h e 5 0 t h p a r t i c l e , p m / P c = 2 . 2 4 . W e r e c a l l t h a t o n t h e b a s i s o f t h e c o n -t i n u u m a p p r o x i m a t i o n f o r t h e s t a t i o n a r y s h o c k w a v e t h e P r e s s u r e r a t i o m ay r e a c h t h e v a l u ep m / P c = 1 . 9 5 . T h u s , t h e p r e s s u r e i n t h e sh o c k w a v e a t t h e n o n s t a t i o n a r y p o r t i o n c a n e x c e e dt h e p r e s s u r e a t t h e f r o n t o f t h e s t a t i o n a r y w a v e .

~ . 5 . P e r t u r b a t i o n P r o p a g a t i o n i n a C h a i n o f V a r y i n g R a d i u s G r a n u l e s . T h e p a r t i c l e r a d iwere a ssign ed by us by using a sequence of random numbers in the interval 0, i), found by

m e a n s o f t h e s t a n d a r d p r o g r a m R A N D :R ~ = { R o + R A N D ( i , O ) } I O - 8 m .

A . T h e P i s t o n P r ob l e m . T h e i n i t i a l a nd b o u n d a r y c o n d i t i o n s a r e s i mi l a r t o t h e c a s e3.4. The veloc ity p rofil e for Ro = 0.5 at the momen t of time t = 3.8.10 -4 sea), corre-s p o n d i n g t o t h e p i s t o n v e l o c i t y v o = 5 m / s e a , i s g i v e n i n F i g. 7 a. F i g u r e 7 b s h o w s t h e v e l o c -ity of the 25th part icle as a funct ion of tim for Ro = 0.5 and for the same pisto n velo c-ity. The radius of this parti cle is 7.824 65-10 -~ m. The valu es of the consta nts equalE = 2 l i 0 n N / m 2 P0 = 7. 8 X i 0 S k g / m a w = 0 . 2 9 . T h e m o s t i m p o r t a n t d i f f e r e n c e s b e t w e e n t h e s y s t e mw i t h d i f f e r e n t p a r t i c l e s i z e s a nd t h e c a s e w h e n t h e i r s i z e s ar e i d e n t i c a l s e e F i g . 6) a rethe following.

I. In a syste m wit h c haotic p artic le sizes Fig. 7a) no unif orm state in veloc ities isreached ev en near the piston, unlike the case of ident ical size s see Fig. 6) for the number

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a

5 ~OO 150 ;Fig. 7

b

_ L _ ItO 15 ~ t ~ sec

~

-O,S, ~ 0 4 0 5 0 6 0 t, . f O ~ se

Fig. 8of particles investiga ted by us N = 200. The given feature is also characteristic of thecases Ro = 2 and 3 at the same piston ve locity of 5 m/sec.

2. The velocity amplitude at the front can be lower than its value behind the front(Fig. 7a, b), and does not tend monot onica lly to the value of half the piston velocity.The value of the particle vel ocity in the first leading oscilla tion vari es with the propa-gation into the material in a nonmonot onic way, and can be both smaller than Vo and sub-stantially excee d this value. In this case the smallest particle mass does not necessa rilycorrespond to a high velocity in the first maximum, and conversely. Behind the wave frontdifferent particles can reach veloc ity values larger than 2vo (see Fig. 7a, b). Particlevelociti es larger than 2vo were determ ined in all three investigate d cases with Ro = 0.5s2, 3 for a piston velocity v, - 5 m/sec. The'excess of individual par ticle veloci ties over2Vo increased with decreasin g Ro, i.e., with the increase in spread in values of particl esizes and masses. The maxim um velocity value behind the front 11.76 m/sec was fixed at the25th particle with Ro - 0.5, Vo - 5 m/sec. The wave front is at this moment at the 170thparticle. A feature of the problem with chaotic particle sizes is also the appearance ofnegative velocities. For example, in the problem with Ro = 0.5s Vo = 5 m/sec the same 25thparticle had a velocity o f--2.142 m/sec. The wave front was at this time at the 148th par-ticle.

When the pertur bation emerges from the free ends of the chains the final velocit y forthe last particl e is for Ro = 2 less than in the case of identical particles. For example,Vxoo/Vo = 2.366, distinctly from the value 2.876 for the case of identical particles. Atthe same time, for Ro = 0.5 Vxoo/Vo = 2.969, which is larger than the similar velocit y ratioin the case of identical particles.

Thus, although the chaotical ly varying particle size causes disordered oscil lations oftheir velociti es behi nd the fronts the nonlinear ity of the interaction can also lead in thiscase to substantial excess of the velocity amplitude at the front and behind it in compari-son with the piston velocity.

3. The ratio of kinetic to potential e nergies in a system with varying partic le sizealso contains oscillatio ns near the mean values as is the case for a system with identicalparticle sizes. The differences between these two cases are that the maxim um deviationsfrom the mean value are substantia lly larger in the first cases and the mean values of Ek/E p differby more from the value 1.25, following from the virial theorem. For examples the values ofEk/Ep, averaged over the time interval during which the leading front of the wave t r a v e r s e sthe 8ista nce from the 70th to the 90th parti cle for the cases Ro = 0.5s 2, and 3, equalsrespectively, 1.178, 1.236, and 1.232. In this case the maxim um relative deviations fromthese mean values equals respectivelys 9, 7, and 6 .

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B . D e c o m p o s i t i o n o f a n I n i t i a l P e r t u r b a t i o n i n a C h a i n of G r a n u l e s w i t h F r e e E n ds .The initial conditions are similar to the case 3.1.

T h e t i m e d e p e n d e n c e s o f t h e v e l o c i t i e s f o r t h e 61 s t a n d f i r s t o s c i l l a t i o n s o f v e l o c i t yfor the 96th parti cle are given in Fig. 8 (Ro = 2). Compare d with the case of identi calp a r t i c l e s , t h e p e r t u r b a t i o n d o e s n o t d e c o m p o s e . n o w i n t o 2 s o l i t o n s, b u t h a s a s i g n i f i c a n tamount Of rando m character. At the same time it is chara cteri stic that the leading pertur-b a t i o n c a n b e n e a r i n s h a p e t o a s o l i t o n, a n d d e c a y i n a m p l i t u d e e v e n i n t h e a b s e n c e of 9dissi pative losses. It is seen from Fig. 8 that this dampin g is relate d to scatte ring ofe n e r g y b y a l a r g e n u m b e r o f p a r t i c l e s . T h e v e l o c i t y a m p l i t u d e d a m p i n g i s c o n v e n i e n t l yt r a c e d b y t h e m a x i m u m v e l o c i t y v o f t h e l a s t p a r t i c l e . T h u s , f o r R o = 0 . 5 v i o o / v = 0.892 ,for R 0 = 2 v100/u = 0.507, and for R 0 = 3 V~o[V, = ~.003 . It is se en tha t the se va lu es are sub -s t a n t i a l l y s m a l l e r t h a n 1 . 2 34 2 , c h a r a c t e r i s t i c o f a s y s t e m o f i d e n t i c a l p a r t i c l e s ( s ee S ec .3 . 1 ) . I t i s i n t e r e s t i n g t h a t h e r e , a s i n th e p i s t o n p r o b l e m c o n s i d e r e d i n t h e p r e c e d i n gs e c t i o n , a s i g n i f i c a n t i n c r e a s e i n t h e a m o u n t o f p a r t i c l e c h a o t i z a t i o n i n s i z e d o e s n o t l e a din the trans ition fro m Ro = 2 to Ro = 0.5 to an enhan ced dampin g of the velo city a m p l i t u d eO n t h e c o n t r a ry , d a m p i n g i s s m a l l e r i n th e c as e w i t h R o = 0. 5 . I n c o n c l u s i o n w e e n u m e r a t ethe basic results: ~

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

2 . T h e r e a c t i o n o f th e g i v e n s y s t e m t o v a r i o u s p e r t u r b a t i o n s w a s i n v e s t i g a t e d . T h ei n t e r a c t i o n o f s o l i t o n s a nd t h e i r b a s i c p r o p e r t i e s a n d f e a t u r e s w e r e s t u d i ed .

3 . I t w a s f o u n d t h a t t h e c o n t i n u u m e q u a t i o n o f a n o n l i n e a r c h a i n o f o s c i l l a t o r s , b e i n ga l o n g - w a v e a p p r o x i m a t i o n , h a s s t a t i o n a r y s o l u t i o n s i n s a t i s f a c t o r y a g r e e m e n t w i t h n u m e r i c a lc a l c u l a t i o n s .

4 . T h e b a s i c f e a t u r e s o f p e r t u r b a t i o n p r o p a g a t i o n i n s y s t e m s w i t h c h a o t i c a l l y v a r y i n gg r a n u l e s i z e s w e r e i n v e s t i g a te d .

The author is grateful to L. V. Ovsyanni kov, A. A. Deribas, and R. M. Garipo v for dis-cussing the results, and to V. V. Deineko, L. N. Shcheglov, and N. G. Ann ikov for assist ancei n p e r f o r m i n g t h e n u m e r i c a l c a l c u l a t i o n s .

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tices ," J. Geophy s. Res. , 85, No. BI2 (1980).7. D . F . S t r e n z w ik , " S h o c k p r o f i l e s c a u s e d b y d i f f e r e n t e n d c o n d i t i o n s i n o n e - d i m e n s i o n a l

quiesce nt latt ices," J. Appl. Phys., 50, No. Ii (1979).8 . I . A . K u n i n , T h e o r y o f E l a s t i c M e d i a w i t h a M i c r o s t r u c t u r e [ i n R u s s i a n ] , N au k a , M o s c o w

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