hydrodynamic model of the impact of a solid on ice

8/2/2019 Hydrodynamic Model of the Impact of a Solid on Ice

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HYDRODYNAMIC MODEL OF THE IMPACTSOLID ON ICE

V. A. Kurdyumov and D. E. Khei sin

O F A

UDC 539.3

Expe rime nts on impact on ice have shown that the insert ion of a solid in ice occ urs be cause of localshatt ering of the ice surface [6]. The proce ss of impac t of steel hem isp her ica l 156- and 300-kg casting s in a1-6 m/ se e vel ocity range on the surface of ice cover was invest igated. The durati on of the impact was 10 -2sec. Plastic strains did not succeed in developing. The elastic strains were also insignificant. A compara-tively thin intermediate layer of finely dispersed construction was formed between the surface of the bodybeing introduced and the mass of unruptured ice. The boundary between the shattered substance of the layerand the main mass of the coarse crystalline ice was sufficiently clear without transition regions [6]. Theshattered ice from the intermediate layer is displaced to the free surface during the insertion. A certainquantity of water is apparently also present in the shattered material under pressure.

Depending on the quantity of liquid phase, the intermediate layer can be represented as a pasty orpowdery substance. Such a substance may possess both viscous and plastic properties, which permits usingthe Hencke system for a viscoplastic body [2]. In the case of the axisymmetric problem, this system has thefollowing form in cylindrical coordinates (Fig. 1):

ap = (~ + )~ ) ( O ~ 'u I c )u u 02u ~ O E . .{_ 9 a )~--Or ~ -+ r Or r2 ~- az 2 ] + 2"e -~ - y - ~ ;1 1 O w a ~ O ~ w ) . 9 a~ , . a~ ,aza--s= ( ~ + ~ ) ~ T . _ ~ + o __ ~_ + . ~ - . 2 ~ , - - & - + v - ~ ;

. . . 1

z = [2 (g~ + ~ + ~=,)+ v~ l -~ ;

(1)(2 )

(3)

a__~u-k ~ (4)O r r O z - -where u is the radia l velocity component; w is the ver tic al compon ent; p is the coefficient of internal fri ctio nin the layer; k is the plastic factor;

9 O u . u ; O w 9 a u O w ( 5 )

The conta ct between the solid and the ice is elastic in the initial stage of impact. Then local shatte ringof the ice surface and the formation of the intermediate layer o ccur. Furth er insertion occur s in the presenc eof the developed intermediate layer. This stage is dominant for sufficiently intense impacts, so that the initi aland final elastic phases can be neglected in the description of the collis ion process.

The inelasti c nature of the impact is verified prin cipall y by the results of expe rime nts on the magnitudeof the rec ov ery fa ctor e0 which de cr ea se s rapidly with the incr ease in the initial impact veloc ity v 0 (Fig. 2).As is seen, the lower boundary of the veloci ty for the applicability of this model is at the 1-1.5 m /s ec level.Here the energy of the reflec ted motio n does not exceed 2-3% of the total ener gy of impact, which decr eas esrapidly with the incre ase in velocity. At lower velocities the inelastic phase of the impact does not pred omi-nate because of too small a volume of shattered ice. This boundary will apparently be lowered for bodies oflarge mass .

An analytic solution of the nonlinear system (1)-(5) can be obtained by assuming the intermediate layerto be thin. To this end, let us introduce a sma lln ess pa ra me te r ~ = h/ r 0


2/6

F i g . 1

~ o

4z 8

#,z#

o,2o

0 , / 5a, e

2 J Vo, m/se cF i g . 2

a n d r 0 i s t h e r a d i u s o f t h e i m p r e s s i o n ( s e e F i g . 1 ) . S u c h a n a s s u m p t i o n i s i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a lr e s u l t s .

L e t u s c o n s i d e r t h e p r o b l e m i n a c o o r d i n a t e s y s t e m c o u p l e d t o t h e s o l i d b e i n g i n s e r t e d i n to i c e a t a v e l o c -i ty v . T h e b o dy p o s s e s s e s a x i a l s y m m e t r y . F r o m t he k i n e m a t i c c o n d i t i o n o n t he b o d y s u r f a c e , w e h a v e w ~ v ,w h i l e th e i n c o m p r e s s i b i l i t y c o n d i t i o n ( 4) y i e l d s t h e e s t i m a t e

Vu ~ ~ - - ~ . (6 )

iU s i n g (6 ) a n d o m i t t i n g h i g h e r o r d e r i n f i n i t e s i m a l s , w e f i n d f r o m (3 )_ {o . V , ( 7 )- ~ ~ \ a z ] "

L i n e a r i z i n g (1 ) i n t h e s m a l l n e s s p a r a m e t e r ~ , w e o b t a i nO p _ ~ - - 0 2 u ( 8 )O r O zz

A s w e s e e , t h e m e m b e r s o f (2 ) w i t h th e f a c t o r ta t u r n o u t to b e a n o r d e r o f m a g n i t u d e l e s s t h a n th e r i g h ts i d e o f ( 8) . T h e p l a s t i c c o m p o n e n t s w i l l h a v e th e s a m e o r d e r i f

~, ~, ~, ---- 4- -- .--kr~ (9 )V3 vH o w e v e r , v ~ 1 0 2 c m / s e c f o r i m p a c t s a t m o d e r a t e v e l o c i t i e s . T h e r a d i u s o f t he i m p r e s s i o n w a s r 0 ~

1 0 c m i n t h e e x p e r i m e n t s c o n d u c t e d [ 3] , a n d t h e t h i c k n e s s o f t h e i n t e r m e d i a t e l a y e r w a s o n t h e o r d e r o f 1 c m ,i . e . , ~ ~ 1 0 - 1 . T h e e x p e r i m e n t a l v a l u e o f t h e i n t e r n a l f r i c t i o n c o e f f i c i e n t w a s t~ ~ 1 - 1 0 - 1 k g . s e c / c m 2 . T h e r e -f o r e , t h e p l a s t i c f a c t o r k s h o u l d b e o n t h e o r d e r o f 102-103k g / c m 2. S u c h v a l u e s o f t h e f a c t o r k a r e c h a r a c -t e r i s t i c f o r m e t a l s , b u t no t f o r s h a t t e r e d i c e .

A f t e r l i n e a r i z i n g t h e s e c o n d e q u a t i o n , w e o b t a i nO____pp= O. (1 0 )O z

T h e r e f o r e , t h e p r e s e n c e o f p l a s t i c p r o p e r t i e s i n t h e s u b s t a n c e o f t h e t h i n i n t e r m e d i a t e l a y e r e x e r t s n oi n f lu e n c e o n t h e p r e s s u r e d i s t r i b u t i o n i n t he l a y e r . T h e c o n t i n u i ty e q u a t i o n is w r i t t e n a p p r o x i m a t e l y a s

aw 2 a " = o . ( l l )O z F O rE q u a t i o n s ( 8) , ( 10 ), a n d ( 11 ) a r e a s i m p l i f i e d R e y n o l d s s y s t e m f o r th e q u a s i s t a t i c s q u e e z i n g o f t h e b o d y

b e i n g i n s e r t e d , a n d t h e o t h e r i s th e r u p t u r e s u r f a c e . I n e r t i a l f o r c e s a r e n o t t a k e n i n t o a c c o u n t h e r e .T h e o b v i ou s k i n e m a t i c c o n d i t io n o f c o n t i n u i ty o f th e v e r t i c a l v e l o c i t y c o m p o n e n t

W I z = 0 - - = U

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

1 0 6 4


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A l i q u i d l u b r i c a n t l a y e r i s f o r m e d o n t he s u r f a c e o f t he s o l i d b e c a u s e o f t he w o r k o f t h e b o u n d a r y f r i c t i o nf o r c e g o i n g o v e r i n t o h e a t a n d b e c a u s e o f t h e t h a w i n g o f t h e i c e . I n t h i s c a s e , t h e t a n g e n t i a l s t r e s s e s( 0 o9 = ~ - -~ + - ~s h o u l d e q u a l z e r o h e r e . S i n c e ~ u / a z >> a w / a r , w e s h o u l d h a v e O u / a Z l z = o = O.

I t i s m o r e c o m p l i c a t e d t o d e t e r m i n e t he b o u n d a r y c o n d i t i o n s o n t he r u p t u r e s u r f a c e , w h i c h c a n b e i n t e r -p r e t e d a s a m o v i n g c o n t a c t d i s c o n t i n u i t y . A s i s k n o w n , e l a s t i c v i b r a t i o n s e n t r a i n i n g a c e r t a i n p a r t o f t h e i m -p a c t e n e r g y a r e e x c i t e d d u r i n g t h e m o t i o n o f s u c h a d i s c o n t i n u i t y in a so l i d . A n a p p r o x i m a t e e s t i m a t i o n o ft h is p a r t c a n b e c a r r i e d o u t b y c o n s i d e r i n g t he i m p a c t o f a n a b s o l u t e l y s o l i d s p h e r e o n a n e l a s t i c h a l f - s p a c e .T h e f r a c t i o n o f i m p a c t e n e r g y b e i n g r a d i a t e d w i t h e l a s t i c v i b r a t i o n s i s d e t e r m i n e d i n t h is c a s e b y t he f o r m u l a[1 ]

w h e r e e is t h e s p e e d o f s o u n d i n t h e h a l f - s p a e e a M v 0 i s th e i m p a c t v e l o c i t y .T h e s p e e d o f s o u n d i n l e e i s e = 3 0 0 0 - 3 5 0 0 m / s e e . I n t h e v 0 = 1 - 6 m / s e e r a n g e u n d e r e o n s i d e r a t i o n ,

t h e f r a c t i o n o f e l a s t i e a l l y r a d i a t e d e n e r g y i n t h e t o t a l e n e r g y b a l a n e e d o e s n o t e x c e e d 2 - 3 % . T h e r e f o r e ,t h e e l a s t i c s t r a i n s o f t h e i c e ca n b e n e g l e c t e d .

[ f a j u m p i n d e n s i t y f r o m P0 t o o l o c c u r s o n th e r u p t u r e s u r f a c e , t h e n w e h av e f r o m t h e c o n d i t i o n o fc o n s e r v a t i o n o f m o m e n t u m

d z , p i w i - - Polo-dT = Pl -- Po

H e r e d z s / d t i s t h e r a t e o f d i s p l a c e m e n t o f t h e r u p t u r e s u r f a c e i n t h e z d i r e c t i o n , a n d w t an d w 0 a r e t h ev e l o c i t i e s o f p a r t i c l e m o t i o n o n t he l e f t a n d t h e r i g h t o f t h e s u r f a c e a s z ~ z s -

N e g l e c t i n g t h e e l a s t i c s t r a i n s o f t h e i c e , l e t u s s e t w 0 = 0 . I f t h e d e n s i t y o f t h e i c e d o e s n o t c h a n g ed u r i n g c r u s h i n g , t h e n f o r t h e v e l o c i t y Z s t o r e m a i n f i n i te , t h e a b s o l u t e v e l o c i t y o f p a r t i c l e m o t i o n n e a r t h er u p t u r e s u r f a c e m u s t b e s e t e q u a l t o z e r o . T h e n

w Jz=h = 0( h i s t h e l a y e r t h i c k n e s s ) .

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

H e r e t h e r u p t u r e s u r f a c e i s n o t a t a n g e n t i a l d i s c o n t i n u i t y . H e n c e , i t i s n a t u r a l to c o n s i d e r t h a t th et a n g e n t i a l c o m p o n e n t o f t h e p a r t i c l e v e l o c i t y i n th e l a y e r , e q u a l t o z e r o ( ul z = h = 0 ), a s t h e s e e o n d b o u n d a r yc o n d i t i o n o n t h e r u p t u r e s u r f a c e .

S o l v i n g t he s i m p l i f i e d s y s t e m o f m o t i o n e q u a t i o n s a n d u s i n g t he d e r i v e d b o u n d a r y c o n d i t i o n s , w e o b t a i na n e q u a t i o n c o n n e c t i n g tw o u n k n o w n q u a n t i t i e s - t h e p r e s s u r e p a n d t h e l a y e r t h i c k n e s s h :

d~P h 3 - }- 3 dp dh ha 3-~r -~ ------ -~ By (12)Setting h = eonst, we obtain the known solution for a thin layer where the pressure is inversely propor-

tional to the cube of the layer thickness [4]. The assumption h = eonst does not correspond to the actualpicture, so that an additional condition must be involved in order to determine the unknowns p and h. To thise n d , s o m e p h e n o m e n o l o g i c a l r u p t u r e c r i t e r i o n , w h i c h i s s a t i s f i e d on t he r u p t u r e s u r f a c e , c o u l d b e u s e d . A tt h i s t i m e , n o s u c h c r i t e r i o n h a s b e e n e s t a b l i s h e d f o r i ce . H e n c e , l e t u s ta k e th e l i n e a r r e l a t i o n s h i p

P - - P c = k p h (i 3)as the additional condition.

This expression is obtained if the displacement of the rupture surface is assumed proportional to thepressure at a given point. Shtaerman [7] used an analogous hypothesis to take account of local surface strainsin the contact problem of elasticity theory.

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The quantity P0 = coast corresponds to static pressure in the absence of an intermediate layer in theinitial stage of impact, which can be identified with the yield point of ice at a local crumpling. For a developedintermediate layer P0 = 0 should be assumed, since there is no direct contact between the solid and the unshat-tered ice. The empirical factor kp depends on the physicomechanical properties of the ice and can be con-sidered constant in a sufficiently narrow range of velocities.

Substituting (13) into (12), we obtain a differential equation in h whose solution isl

I

when ax ia l symm etry i s taken in to accoun t . Hence1 ! 1p = ~ ( 3 ~ k ~ ) ~ ( c , - r ~ .

We have p = 0 for r = r 0 on the edges of the co nta ct zon e, so that Ct = r 2. Let us d et er mi ne the ta n-g e n t i a l s t r e s s o n the r u p tu r e su r f a c e :

1 12 1_ _~ ( ~ k , ) ~ r" c - -= ( r o - - F " ) "

The tangen t ia l s t res ses g row wi thou t l imi t as r ~ r 0. In fac t , the edges o f the con tac t zone are sp l i to ff , so tha t th is depen dence wi l l be va l id on ly in the reg ion

t"] ' ~ - - ,CL

wher e the coeff ic ient a > 1 takes accou nt of chips on the edge of the cru mpl e zon e.Using the no ta t ion r = :a( r / r0 ) , we ob ta in

1 I [ 1- ~ / r o \ ~ - - ( 15 )

T h e ch ip s a re u su a l ly sm a l l . He n c e, i t c a n b e a s su m e d th at a = c o n s t . Ac c o r d in g to p r e l im in a r y e s t i -mate s a = 1 .05-1 .08 .

As an i l lus t ra t i on , le t us compare the theo re t ic a l dependences ob ta ined wi th the resu l ts o f fu l l - sca leexp er ime nts on the impact of a so l id on ice [3 ]. In these exp er ime nts , s tee l hemis phe r ica l cas t in gs weree lec ted on ice f rom d i f fe r en t a l t i tud es . The maxi mum dep th o f penet ra t ion ~max was hence cons idera b ly lessthan the rad ius o f the cas t in g R, which perm i ts neg lec t ing the curv a tu r e o f the in te rmed ia te layer and so lv ingth e p r o b le m in c y l in d r i c a l c o o r d in a t e s .

Let us exa mine the in ser t ion o f a so l id hemi sph ere o f mass M in ice a t an in i t ia l ve loc i ty v0 .o f m o t io n d u r in g im p a c t w i l l b e

I 1 5 1 1d v , i 7 1 3 _ k 3 J { r o ~ S ( (z ,. _ x ) ~ d x (16)M v - - ~ = : - - p d F = - - , i , , o ' ~ - ~ )F Q

(F is the area of the contact zone) .Fr om the co ndi tio n ~max


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5

gA

0.5

a,2

9~ .,. I

~

F i g . 30,~ g,8 [

o , 8

~ 6

I . " ] /

# z

9 " \

# # ~ # ( 8 fF i g . 4

4 1 2 IPma~ = 0. 79 p v~A/~ (31~k~) (2 R)~ . (1 9 )

T h e t o t a l c o n t a c t f o r c e P r e a c h e s t h e m a x i m u m a t t = t 2 (t1 < t 2 < T) :I1 5 1 5

Pm~x = 1.18 v~M6 3~k~)~ (2R) 6 . (2 0)T h e f a c t o r 3 g k~ i n a l l th e c o m p u t a t i o n a l f o r m u l a s d e p e n d s o n t h e p h y s i c o m e c h a n i c a l p r o p e r t i e s o f t he

i c e a n d c a n b e d e t e r m i n e d f r o m e x p e r i m e n t . T h i s c a n n o t b e do n e f o r th e c o m p a r i s o n o f t he e x p e r i m e n t a l a n dt h e o r e t i c a l r e s u l t s , s i n c e i t i s s i m p l e r t o r e d u e e a l l t h e e x p r e s s i o n s t o d i m e n s i o n l e s s f o r m 9 T o th i s e n d , l e tu s i n tr o d u c e t h e d i m e n s i o n l e s s v e l o c i t y o f i n s e r t i o n , t h e p r e s s u r e , a n d t h e t o t al f o r c e :

94 91 1v = v _ = ( 1 - - ~ 4 ) 7 ; ? = p _ _ = 1 .1 9 ( 1 - - ~ 4 ) 7 ~ 4 ;v0 pmax91 5

P = ~ m ax = 1 ,4 2 (1 - - ~ ) ~ ,w h e r e ~ = ~ / ~ m a x "

T a k i n g i n t o a c c o u n t t h a t

w e c a n o b t a i n th e d e p e n d e n c e o f t h e d i m e n s i o n l e s s d e p t h o f i n s e r t i o n ~ o n t h e d i m e n s i o n l e s s t i m e ~ = t / T .T h e n v a l u e s o f ~r ~ , a n d P c a n be c o n s t r u c t e d a s a f u n c ti o n o f t h e d i m e n s i o n l e s s p a r a m e t e r s ~ o r ~ .

T h e s o l i d l i n e s i n F i g s . 3 a n d 4 r e p r e s e n t t h e t h e o r e t i c a l d e p e n d e n c e s v ( t ) a n d P (t ) , w h i l e th e p o i n tsr e p r e s e n t t he e x p e r im e n t a l d a t a . , T h e g o o d a g r e e m e n t b e t w e e n th e e x p e r i m e n t a [ a n d t h e o r e t i ca l r e s u l t ss h o u l d b e n o te d . T h e c u r v e o f t h e v e l o c i t y v ( t) h a s a n i n f l e c t i o n a t t h e en d o f t h e i m p a c t , w h i c h i s d e s c r i b e db y t h e t h e o r y a n d i s a l s o n o t ed e x p e r i m e n t a l l y . P h y s i c a l l y , t h is c o r r e s p o n d s t o " p r e s s i n g " o f t h e s h a t t e r e ds u b s t a n c e w h e n r u p t u r e o f t h e i c e d o e s n o t o c c u r i n p r a c t i c e 9

A n i m p o r t a n t f e a t u r e o f t he i m p a c t i s th e m a g n i t u d e o f t h e s p e c i f i c r u p t u r e e n e r g y :Mv~e v - 2 V ( 21 )

(V i s t he v o l u m e o f t h e c r a t e r s b e i n g f o r m e d ) .C o n s i d e r i n g t h e c r a t e r a s a s e g m e n t o f a p a r a b o l o i d o f r e v o l u t io n , w e h a v e

2

U s i n g ( 1 7) a n d ( 2 1) , w e o b t a in a l i n e a r r e l a t i o n s h i p b e t w e e n t h e s p e c i f i c r u p t u r e e n e r g y a n d t h e m a x i m u mc o n t a c t p r e s s u r e a v e r a g e d o v e r t h e w h o l e c o n t a c t a r e a : e V = 0 . 9 2 6D m a x .

1 0 6 7


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hydrodynamic model of the impact of a solid on ice

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