Zhang and Whiten_1996

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E L S E V IE R Powder Technology88 (19 96) 59--64

he calcula t ion o f contact forces be twee n par t ic les u s ing spr ing andd mping models

D . Z h a n g , W . J . W h i t e n *Julius Kruttschnitt Mineral Research Centre Universityof Queensland Isles Road Indooroopilly Brisbane. QId 4068 Australia

Received I Decem ber1995; evised20 January 1996

A b s t r a c t

Discrete e lemen t simulations can be used to model the behaviour of bo th fluids and contacting particles in a ve ry flexible manner. Thesesimulations ha ve wide ap plications in both m ining and mineral processing. They require the calculation of forces between th e discrete elementsused in the simulation typically by assuming spring, dashpot and slider components at the contact points. The accuracy of the simulaticrtsdepends on the assumptions m ade in the calculation of interelement forces. The different m ethods that can be used to calculate the forces havebeen examined and unrealistic he haviour found for most ot methods commonly used. The non-linear force formula of Tsuji et al . ,P o w d e rTechnol. 71(199 2) 239, w ith the particles separating w hen the force returns to zero rather han w hen the distance between th e centres exceedsthe sum o ftb e radii is found to give realistic results.

Keywords:Discreteelementsimulations;Springcomponent;Dashpotcomponent;Contactpoint

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

Owing to t he r ap id p rog res s i n compute r ha rdware , i t i sb e c o m i n g m o r e a n d m o r e r e a l i s ti c t o s im u l a t e t h e b e h a v i o u ro f f l u id and so l id s a s an a s semblage o f d i sc re t e e l emen t s .Sma l l e l emen t s , such a s pa r t ic l e s , can r ep resen t sma l l so l id so r a f l u id . La rge e l em en t s a r e u sed a s bound a r i e s t ha t may bef ixed , such a s a con ta ine r, o r mob i l e .

T h e d i s c r e t e e l e m e n t m e t h o d s i m u l a t e s t h e m e c h a n i c a lr e sponse o f sys t ems by us ing d i sc re t e e l emen t s . In t h i sme tho d , t he fo rces be tween a s su med o r ac tua l d i sc re t e com-p o n e n t s a r e c a l c u l a t e d a n d u s e d t o d e t e r m i n e t h e m o t i o n o ft h e d i s c re t e c o m p o n e n t s t h u s g i v i n g a d y n a m i c s i m u l a ti o n .Dur ing the s im u la t ion p rocess , t he s imu la t ion t ime i s d i sc re -t i zed in to sma l l t ime in t e rva l s . The mo t ion o f each pa r t i c l eand bo unda ry in each t im e in t e rva l i s ca l cu la t ed . The pos i -t i ons o f t hese pa r t i c l e s and bounda r i e s a r e upda ted a t eachsma l l t ime in t e rva l .

T h e d i s c re t e e l e m e n t m e t h o d t h a t d e s c r i b es t h e m o t i o n o fa s s e m b l i e s o f p a r t ic l e s w a s p r o p o s e d b y G r e e n s p a n [ 2 ] a n dCund a l l and S t r aek [ 3 ] an d o the r s [ 1 ,4 -7 ] hav e made ex ten -s ive use o f t he t echn ique .

C u n d a l l a n d S t r a c k [ 3 ] p r o p o s e d a m o d e l b y a s s u m i n gsp r ing , dash po t and s l i de r com pone n t s a t t he con tac t po in t s

* Correspondingauthor.

0032-5910/96/ 15.00 © 1996ElsevierScienceS.A. All fights eserved

Fig. I. Particlecontact mod el for normal orce.

o f ad j acen t pa r t i c l e s (F ig . 1 ) . Th i s mode l i s com mo nly usedfo r ca l cu la t ing the pa r t ic l e impac t fo rce w h ich i s t hen usedfo r upda t ing the pa r t ic l e pos i t ion . Seve ra l a s sum p~o ns abou tthe sp r ing and dam ping fo rces u sed in the imp ac t ca l cu la t ionsa re , ma de by d i f f e ren t r e sea rche r s . The accw acy o f t he s im-u la t ions depen d on the ca l cu la t ion o f i n t e re l em en t fo rces . Inth i s pape r, t he d i f f e ren t me thods tha t can be used to ca l cu la t ethe impac t fo rces a r e examined .



D. Zhang, W.J. Whiten Powder Technology 88 1996) 59-64

2 L i n e a r d a m p i n g a n d sp r i n g m o d e l

The mos t comm only u sed me thod [3 ] a s sumes a l i nea rsp r ing and damper. The fo rce ca l cu l a t ed f rom the l i nea ra s sumpt ions i s app l i ed when the pa r t i c l e s ove r l ap , t ha t i swhen the d i s t ance be tween cen t r e i s l es s t han t he sum o f t hepar t ic le radi i .

Fo r a l i nea r sp r ing and dam per, t he equa t ion o f mo t ion o fimpac t ing pa r t i c l e s can be exp re s sed a s :

d2x dx, ,~ - 7 + q -~ ; + k ~ ( t ) = 0 ( 1 )

To s im p l i fy t he ca l cu l a t i on , Eq . I ) c an be no rm a l i zed a sfol lows:

d2.f eLf ^ ~ ,-~+ 2a - ~+ x( t )= 0 ( 2 )

whe re

x= /q,t= t(m /q) ~/2

and

ka 2 ( m q ) l / z

Th e con tact cond i t ion fo r two par t ic les i s x > O. The in i t ia lcond i t i ons fo r t h i s equa t ion a t t=0 can be s e t to x (O) =0 ,x'(O)= 1 w i thou t l o s s o f gene ra l it y. The so lu t i on o f Eq . (2 )g ives t he r e l a t i ve pos i t i on o f o ne o f t he pa r t i c l e s a t t ime twhen a > , a s show n be low:

2 ( / ' ) =exp { - [ a - ( a 2 - 1 ) , / 2 ] ~}2 ( a 2 - 1 ) I /2

e xp { - [ a + ( a 2 - l ) l / 2 ] t }2 ( a ? - 1 ) 1 / 2 ( 3 )

For the cases a < l and a = l , the pos i t ion is g iven by:

g (~ ) s i n [ ( - a 2 + l ) l / 2 t "] e x p ( - a t ' ) w h e n a < l( - - a 2 + l ) t / 2

i 4 )

g ( t ) = e x p ( t ) w h e n a--=l

The no rma l i zed ve loc i t y and acce l e r at i on can be ca l cu l a t ed

f rom these and a r e g iven i n Append ix A .To inves t i ga t e t he cha rac t e r i st i c s o f p os i t i on , ve loc it y, and

fo rce du r ing t he who le p roces s o f co l l i s i on , t he g raphs o fthe se va r i ab l e s ve r sus t ime were p lo t t ed w i th s eve ra l chosenva lues o f a .

The g raph o f pos i t i on ve r sus t ime i s shown in F ig . 2 ( a )and looks r ea sonab le . The d i sp l acemen t i nc rea se s f rom ze roun t i l a max im um i s r eached , a f t e rward r educes t o ze ro aga in .

A g raph o f ve loc i t y ve r sus t ime was p lo t t ed fo r s eve ralva lues o f a , show n in F ig . 2 (b ) . The v e loc i ty dec rea ses f romthe in i t ia l ve loci ty unt i l i t reaches zero . The di rec t ion ofveloci ty then reverses and decreases unt i l i t reaches a mini -

.... a= 0

a ] ~ a = 0 . 3

0 . ~ - 7 , _ - - ~ ~0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

-o. 2 a)

V ~-~. . . . . a=4~ k \ ' \ ~ .a = 1 . 5f i \ X ~ ' ~ \ . . . . a = 0 7 q

o . s l ~ \ \ . . . ~ . . . ~ . . . :f i :0 . , ...... ......a=0

. . . . .. . . _ . .

- 1 - t b )

/ o o.s I 1.5 2 2 .s ~ s

,.. .. ... ..

_ 1 .5 1 / ~'... ... ..... -..2 1 , a = l 5 ~ , ~ - , -.

- - - • = = v , . a - 0 3

- 2 . ~ i - " a = 0]- 3 1 c )

Fi l ' 2 .. f a ) Norma l i zed pos i t i on v s . no rma l i zed t ime us ing l i nea r Eq . (2 ) ;(b) normalized velocity vs. normalized ime using linear Eq. (2 ); (c)normalized force vs. normalized ime using inearEq. (2).

m u m w h i c h , c o m p a r i n g F ig . 2 ( a ) a n d ( b ) , i s b e f o r e t h ed i sp l acemen t r e tu rns t o ze ro .

The g raph o f fo r ce ve r sus t ime i s p lo t ted i n F ig . 2 ( c ) , andind i ca t e s t ha t Eq. ( 1 i s i nco r r ec t s i nce t he magn i tude o f t helb rce s ( excep t a= 0 ) s t a rt s f rom non-ze ro and r educes toze ro du r ing t he co l l i s i on p roces s .

The equa t ion o f i n i t i al f o r ce i n t e rm o f a can be ca l cu l a t edby subs t i t u t ing t= 0 i n to t he fo rce equa t ions (A ppen d ix A) .

These equa t ions can t hen be s imp l i f ied t o :i 0 ) = - 2 a 5 )

From the expe r imen ta l r e su l t s and t he mechan i sminvo lved , i t i s t hough t t ha t t he fo rce shou ld i nc rea se f romze ro un t i l i t r e aches a max imum then r educe t o ze ro . Theshape o f t he se fo rce ve r sus t ime cu rves shou ld b e c lo se r t othe cu rve when a = 0 , t ha t i s, no damping , and wou ld beexpec t ed t o become a sym met r i c a s t he damping inc rea sed .Bourgeo i s e t a l . [ 8 ] g ive an examp le o f such an expe r imen ta lcu rve . I t c an a l so be s een f rom F ig . 2 ( c ) t ha t t he fo rce changess ign , t ha t i s i t becomes a t t rac t ive , be fo re t he d i sp l acem en t



D. Zhang, W.J. Whiten/Powder Technology 88 1996) 59- 64 61

(F ig . 2 ( a ) ) r e tu rns t o ze ro . Th i s r eve r se co r r e sponds t o theinc rea se i n ve loc i t y a ft e r the min im um in F ig . 2 (b ) .

The a l t e rna t i ve t o t he l i nea r mode l i s t o u se t he H er t z i anconta ct theory wh ich g ives th e e las t ic force . Tsuj i e t a l . [ 1 ]

added a damping t e rm to t h i s t ha t g ives an i n i t i a l f o r ce o fzero.

3 . N o n - l i n e a r s p r i n g a n d n o n - l i n e a r d a m p i n g m o d e l s

Tsuj i e t a l . [ 1 ] u sed a H e r t z i an non - l i nea r con t ac t mode lin one o f h i s pape r s . He then u sed t he l i nea r con t ac t mode l i nh i s l a t er pape r [ 7 ] . As t he l i nea r mode l does no t s eem rea l -i s ti c , i t i s wor th i nves t i ga t i ng t he no n - l i nea r mode l . I n t h i sequa t ion , t he dam ping t e rm i s a func t ion o f d i sp l acemen t andve loc i t y, and d i f f e r s f rom the damping t e rm in t he l i nea requa t ion w h ich i s on ly a func t ion o f ve loc i ty. Th i s t ype o fdamp ing ensu re s t ha t t he i n i t ia l f o r ce s t a r t s f rom ze ro . Tsu j is t a t e s t he damping t e rm was found heu r i s t i c a l l y, and t hedam ping coe ff i c i en t a i s an empi r i ca l cons t an t r e l a t ed to t he

....a = O. . . . . . . ., - a = 0 .5

z t ~ a = l

-o .J ( .) 4 .^

V . . a = O

1 ~ . ~ . . . . . . . , a = 0 . 5

I \ \ <~ a = l

° Vo 2 _

- 1 - 1 ( b) . . . . . . . . . . '

x , : :; , / . . ..

- 0 . , , . . / . .

a = 1 . 5Fig. 3. (a) Normalizeddisplacementvs. normalized im e using non-linearEq. (9); (b ) normalizedvelocityvs. normalized ime usingnon-linearEq.(9); (c) normalized orce vs normalized ime usingnon-linearEq. (9).

coe ff ic i en t o f r e s t i tu t i on . Howeve r, t he d amp ing t e rm can b ede t e rmined by d imens iona l ana lys i s .

The equa t ion o f mo t ion fo r t he sys t em o f two pa r t i c l e sa s suming n on - l i nea r sp r ing and dam ping i s :

m ~+ a m k)l/2x t)l/4~ -~+6 )

To m a k e e a c h t e r m n o n - d i m e n s i o n a l, d is p l a c em e n t x ( t ) a n dt ime t a re replaced by:

x = m , : 2 1 k ) 2 / s £ 7 )

t = [ m u o / k ) 2 / 5 / Vo ] t ( 8 )

Eq. (6 ) i s rewri t ten as:

d2.~(t) . ^ . ^ i / 4 d x l ) ^ . ^ . 3 / , ~ ^- - - d - - ~ , a x t t ) - - - - ~ + x t ) - = u ( 9 )

The p lo t s o f ve loc i ty, d i sp l acemen t and fo rce ve r sus t imea re ob t a ined a f t e r i n t eg ra t ing Eq . ( 9 ) w i th t he i n i t i a l cond i -t i on x (O) = 0 , x ( 0 ) = 1 and a r e show n in F ig . 3 .

These p lo t s show tha t t he equa t ion o f mo t ion w i th non -l i nea r sp r ing and damping i s r ea sonab le . S imi l a r ly t o t hel inear case , the force reverses and the veloci ty s tar ts toinc rea se be fo re t he d i sp l acemen t r e tu rns t o ze ro .

4 . E x p e r i m e n t a l r e s u l t s

Expe r imen t s t o compare t he two mode l s we re conduc t edus ing t he Hopk inson ba r equ ipmen t de sc r ibed i n Re f. [ 9 ] . Ashor t s t ee l ba r was impac t ed on to a d i sk wh ich was he ld

aga ins t a l onge r s t ee l ba r. S t r a in gauges o n t he ba r s we re u sedto r eco rd t he im pac t fo r ce a s a func t ion o f t ime .Seve ra l t ypes o f rocks such a s s ands tone , g r an i t e and b asa l t

we re u sed . F ig . 4 show s the comp ar i son o f expe r im en ta lr e su l t and n on - l i nea r mode l u s ing basa l t . The expe r imen t sus ing o the r t ype o f rocks gave s imi l a r r esu l t s. These exp e r i -men ta l r e su l t s show tha t t he l i nea r mode l i s no t su i t ab l e fo r

f o r c e

o2~ . 0 o . z 0 . 4 0 . 6 0 . 8 1 . 0 . 2

0. 0 . • - I , , , i , , ,

-O.4

0.6

eo re t i ca l-o.8- ~ .o - ~ = e x p er im en ta l d a t a

.l.z

F i g . 4 . C o m p a r i s o n o f e x p er i m e n t al r e su l t i t h n o n - l in e a r o d e l a = 0 . 3

using basaltstone.



D. Z.hang, W.J. Whiten/Powder Technology 88 1996) 59-64

t h i s t ype o f s imu la t ion . The non- l inea r mode l i s m uch c lose rto the exper imental resul ts . However, a bet ter form of thedamping t e rm m igh t g ive a be t t e r f i t t o t he cu rve . More inves -t i ga tion o f t he impac t m echan i sms i s neces sary.

5 . S e p a r a t i o n t i m e

Correct ly determining impact ing par t ic le separat ion t imeis cr i t ica l in the calcula t ion. Most papers a ssum e the par t ic lesseparate a t the t ime tx=o, which is when the displacem ent xre turns to zero . Ho wev er, the impact ing force changes di rec-t i on a t t he t imet :~o ( the t ime a t which the force re turns tozero) both for the l inear and non- l in ear equat ions . Therefore ,a t t he time txffio ( t x=o>t /=o , s ee F igs . 2 (a ) and ( c ) , 3 (a )aad ( c ) ) , t he fo rce has changed d i r ec tion . Th i s fo rce i s t henpul l ing the two par t ic les back towards each other ins tead of

forcing them apart . T his i s not real is t ic and hence two par t i -c les must separate when the impact ing force reaches zero ,that i s , a t t imety=o

The t ime tx=o fo r a l i nea r mode l can be ca l cu la t ed us ingEqs . (3 ) and (4 ) g iv ing :

71t.~ =o ( _ a 2 + l ) , / 2 w h e n a < l ( 1 0)

The fo rce a t th is t ime is :

i x = o = 2 e x - ( ' _ a 2 + l ) t / w h e n a < 1 ( 11 )

For a > 1 the pa r ti c l e s neve r reach ze ro d i sp l acemen t aga inand hencet~=obecomes inf ini te .

From E q. (1 0) , i t can be seen that a = is the cr i t ica ldam ping va lue of th is l inear sys tem. Wh en a < 1 , the sys temosci l la tes , but wh en a > 1 , the sys tem is overdamp ed.

In Tsu j i ' s e t a l . [ 1 ] paper, ha l f of the osci l la t ion per iod isused as separat ion t ime, whic h is the t ime when the displace-men t equa l s ze ro . In t he non- l inea r mode l , t he fo rce i s equa lto ze ro when d i sp l acemen t x= 0 , t ha t i s t he po in t t~=o inFig. 5 . This f igure shows that the force changes s ign a ttf= 0be fo re becoming a t t rac t ive . The fo rce d rops ve ry r ap id ly backto ze ro a t t imet~ffioi f the calcula t ion is cont inued, assum ingthe calcula ted forc e s t i l l ac ts on the par t ic le .

F ig . 6 (a ) and (b ) shows the d i f fe rence in no rma l i zed con-t ac t t ime w hen the fo rce d rops to ze ro and when the d i sp l ace -men t r e turns to ze ro a s suming the fo rce equa tion con t inuesto hold .

Both the l inear equat ion and the no n- l inear equat ion in t ro-duce a a s t he damping coe ff i c i en t o r t he no rma l i zed dampingcoeff ic ient . This dam ping co eff ic ient a i s usual ly determine dfrom th e coeff ic ient of res ti tu t ion e of par t ic les . The coeff i -c ient of res t i tu t ion is the ra t io o f the veloci t ies af ter impact ingand be fo re impac t ing . The re fo re , a wrong sepa ra t ion t im eresul ts in a wro ng dam ping c oeff ic ient a . The graphs in Fig .6 (c ) and (d ) compa re the d i f f e ren t r e sul t s. Th e f - - O p lo t sare found by plot t ing e versus a , assuming the separat ion t ime

^

~ L . . • . 1 . , , , o . . . . ~ . ' . ~ , , ~

0 i\

F i g . 5. N o r m a l i z e d f o r c e v s . n o r m a l i z e d t i m e u s i n g n o n - l i n e a r e q u a t i o n

(a=0.5).

a t t i r e Similar ly, the x- - 0 plots are plot ted e versus a assum -ing the separat ion t ime a t tx= 0 . For low values o f e , i t can beseen the two cu rves g ive v e ry d i f f e ren t va lues o f a .

6 . C o n c l u s i o n s

The l i nea r mode l g ives t he g raph shown in F ig 2 (c ) , byp lo t ti ng fo rce ve r sus t ime and shows the p rob lem wi th th i smode l . Excep t fo r t he p lo t when the damping fo rce i s ze ro ,a l l o ther p lots s tar t f rom negat ive values ins tead of zero . Fromthe expe r imen ta l r e su l t s and the phys i ca l mechan i sm i t i sunders tood that the forces should increase f rom zero. Theshape o f t he expe r imen ta l fo rce cu rves i s s imi l a r to t ha t shownin F ig . 2 (c ) fo r t he fo rce cu rve wi thou t a damping fo rce andobtained by plot t ing force versu s t ime. This indicates that thedamping fo rmula i s i nco r rec t. The d amping coe ff i c i en t qshould not be constant . R is expected to be a funct ion ofd i sp l acemen t s ince the con tac t d i sp l acemen t i nc reases f romthe zero .

A second p rob lem i s t hat t he fo rces change d i r ec t ion be fo rethe par t ic les separate , based on thei r radi i , which m eans thatthe force is pul l ing the par t ic les together befo re they separate .

The Tsu j i e t a l . [ 1 ] mode l u sed non- l inea r sp r ing and non-l inear damping . The damping fo rce i s t he func t ion o f d i s -placem ent and veloci ty. This form ula ensures that the in i t ia lfo rces inc rease f rom a ze ro va lue . F ig . 3 (c ) ( and a l so F ig . 3f rom Ref . [ 1 ] ) shows the fo rce become s a t t r ac tive be fo re thepar t ic les separate based on thei r radi i . The force betwe en thecontact par t ic les changes di rect ion before the par t ic les sepa-ra te . This i s not real is t ic , the par t ic les should separate a t thet ime o f t he fo rce go ing to ze ro and be fo re the fo rce s t a r ts t opul l the par t ic les together. After the force has reached zero ,the par t ic les are separat ing fas ter than they are recover ingtheir or ig inal shape.

A l inea r damping fo rce fo rmula wi l l a l so cause the ca l cu -la t ion of unreal is t ica l ly large forces between the par t ic les ,which wi l l then be t ransmit ted to other par t ic les . The non-l inear damping form su ggested by Tsuj i e t a l . [ 1 ] g ives a



D. Zhang W.J. Whiten/Powder Technology 88 1996)59--64

i

5

~ ~ f=O. . . . . . , , , v , , , , | . . . . , .-: ---~ .

0 0 2 0 4 0 6 0 8 0 9

1. c)

, 2 o . 2 . 4 . 6 . 8 1 1 ~ : 1 .4 1 .6 1 .8

a

'1 (b)

o .2 . 4 .G ,8 1 1 ,2 : . 4 ' 1 .6 . 8

a

e

11 - ~ d )

.~ ~.~.

- .2 O .2 .4 .G .8 1 1 Z 1 .4 1 f i ] .8a

F i g . 6 . ( a ) N o r m a l i z e d t i m e v s . a a t t , = o u s i n g l in e a r e q u a t i o n ( b ) n o r m a l i z e d t i m e v s . a a t t ,~ o u s i n g n o n - l i n e a r e q u a t i o n : ( c ) c o e f f i c i e n t o f r e s t i t m i o n e v s.

a using inearequation: (d) coefficientof restitution e vs, a using non-linearequation•

qual i ta t ively acceptable form for the in ter-par t ic le force .How ever, more r e sea rch shou ld be done on the bes t fo rm fo rthe damping fo rce .

The p ar t ic le separat ion cr i ter ion that uses the p ar t ic le radi iresul ts in a t t ract ive forces between par t ic les i f damping ispresent . The corr ect ca lcula t ion separates the par t ic les wh enthe force (w hich is negat ive as def ined in th is paper) r i ses tozero. Afte r th is t ime the par t ic le surfaces re turn to thei r or ig-inal shape s low er than the veloci ty of separat ion of theparticles.

In a d iscre te e le me nt s imulat ion, the calcula t ion o f instan-taneous forc e or the var ia t ion in forces could be ser iously iperror unless careful considerat ion is g iven to the assumptkm smade in de t e rmin ing the magn i tude o f t he in t e r-pa r t i c l eforces .

7 . L i s tof symbols

a damping coe ff i c i en te the coeff ic ien t of res t i tu t ionf contact force

i In l i n e a r e q u a t i o n s , i t i s a n o r m = . i z e d c o e f f i c i e n t . I n n o n - E n e a r e q u a t i o n ,

it is an em piricalconstant hat can be related o 6~e coefficient f restitution.

no rma l i zed con tac t fo rcek s t i ffness coeff ic ientm par t ic le massq damping coe ff i c i en tt t imet no rma l i zed t imev par t ic le veloci ty0 no rma l i zed ve loc i tyv in i t ia l veloci ty

: x par t ic le d isplacem ent£ norma l ized par t ic le d isplacem ent

cknowledgements

Author s wou ld l i ke to t hank M r Cameron Br iggs fo r sup -p ly ing the expe rimen ta l da t a i n F ig . 4 and the JKM RC m an-agem ent for supper t of th is project .

A p p e n d i x A

The ve loci ty and normal ized force a t t ime t us ing a l inearequa t ion can be de r ived a s shown:



D. Zhang. W .J. Whiten/Powder Technology 88 (1996) 59-64

d .f ( t ) [ - a + ( a 2 - 1 ) ~ / 2 ] e x p l - [ a - ( a - l ) J ~ 2 ] t }6 ( t ) = d ~: 2 ( a 2 - 1 ) 1 /2

[ - a - ( a 2 - I ) i /2 ] e x p l - [ a + ( a 2 - 1 ) 1 / 2 ] t }w h e n a > 12 ( a 2 - 1 ) I /2

v ( t ) = - - - d - ~ = c o s [ ( - a 2 + 1 ) e x p ( - a t ' )

s in [ ( - a z + 1 ) i / 2~ ] a ex p ( - a t )w h e n a < 1( - a 2 + l ) ~ / 2

t ~ (1 ) = d . f ( h ; ld l = - e x p ( t ' ) + ~ w h en a = 1

^ . ~ . d 0 ( D [ - - a + ( a 2 - - 1 ) l / 2 ] 2 e x p { - - j a - - ( a 2 - - 1 ) l / 2 ] t }f ( t ) = d [ 2 ( a 2 - 1 ) I /2

[ - a - ( a 2 - 1 ) i / 2 ] exp { - [ a + ( a 2 - 1 ) ~ /2] ~ i

2 ( a 2 - l )U / 2 w h e n a > 1^

= ~ = - s i n [ ( - a 2 + I ) I /2 t' ] ( - a 2 + 1 ) 1 /2 e x p ( - a t ' ) - 2 c o s [ ( - a 2 + l ) I / 2t '] a e x p ( - a t )

s in [ ( - a 2 + 1 ) ,/ 2~ ] a 2 ex p( - a t ' )-4 w h e n a < 1( - a 2 + l ) ~ / 2

~ , ^ . d tT ( i ) t 2t ) = d t = e x p ( . ; ) e x p ( t ) w h e n a = 1

( A l )

A2)

A3)

( A 4 )

( A S )

( A 6 )

e f e r e n c e s

[ I ] Y . Tsuj i , T. Tan aka a nd T. Ishida,Powder TechnoL, 71 (1992)239.[2 ] D . Greenspan ,Discrete NumericaIMethods in Physicsand Enginee iug,

Aca dem ic Press. Ne~, , York. 1974.[3] P.A. Cundal l and O.D.L. Strack,Geotechnique. 29 (1979)4 1.[4] R. E Barbosa-Carr i llo ,Pi~.D Thes;,. ,'he.ln:versity of Illinois, Urba na,

IL. 1985.

[5] B.K. Mishra,Ph.D Th esis,The Unive r s i ty o f Utah , Logan . UT. 1991.[6} B.K. Mishra,Appl. Math. Modelling, 61 (1992)598.[7 ] Y . Tsu j i , T. Kaw aguch i and T. Tanaka ,Powdrr Technal., 77 (1993)

79.[8 ] F. Bourgeo i s , R .P. K ing and J .A . H erbs t in S .K . Kaw ata ( ed . ) ,

Comminution Theory andP raclice,AIM E, Lit lle ton, 1992, Ch. 8, p . 99.[9 ] C . Br iggs and R . Bearman ,The Assessment of Rock Breakage and

Damag e in Crushing Machinery. Proc. EXPLO'95.1995,p. 167.

Zhang and Whiten_1996

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