erosion over time

8/2/2019 Erosion Over Time

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Theoretical and Applied Fracture Mechanics 6 (1986) 207-215 207North-Holland

O N T H E T I M E - D E PE N D E N C E O F T H E E R O S I O N R A T E - A P R O B A B IL I ST I C

A P P R O A C H T O E R O S I O N

H .W . B A R G M A N N

Ecole Polytechnique F~d~rale de Lausanne, CH-1015 Lausanne, Switzerland

A highly simplified mod el is presented o f the erosion process under liquid impact. The theory ou tlined allows themathematical expectation of the erosion rate as a function of time (erosion curve) to be predicted.

This simplified theory of erosion is based on the s tatistical nature of the repe titive lo adin g due to liquid impac t. Itemploy s a classification of individual impacts according to their qu ality of o r effectiveness in rem oving material. T hespecific loss of m aterial (p er impact) is considered to be a random variable, w hich assumes different, discrete valuesdepend ing on the order of repetitions of effective impacts at the sam e location on the target (su rface). Th e probabilities ofproducing an y of these repetitions are calculated. For particular applications, the theory req uires characteristic data for the

specific loss of material as well as typical data concerning the hydrodynamic loading conditions.For the cas e of a ductile erosion process, a very approx imate solution is presented. N evertheless, it exhibits the typical

features of erosion curves as presented in the literature.

" A s l o n g a s th e s u r f a c e i s s m o o t h , i t o f f e r s n o h o l d f o r t h e i m p i n g i n g d r o p s o f w a t e r a n d t h e

w a t e r f lo w s o f f o n a ll s i de s . T h e r e f o r e e r o s i o n d o e s n o t o c c u r f o r s o m e t i m e . H o w e v e r , a s s o o n

a s a n y r o u g h n e s s f o r m s , e r o s i o n d e v e l o p s r a p i d ly b e c a u s e t h e w a t e r p e n e t r a t e s t h e u n e v e n n e s s

o f t h e s u r f a c e a t a h ig h p r e s s u r e d u e t o th e i m p a c t , a n d a c t s v e r y v i o le n t l y . F i n a ll y , w h e n t h e

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

r o u g h e n e d s u r fa c e . T h i s w a t e r d a m p e n s t h e i m p a c t o f s u b s e q u e n t d r o p s s o t h a t t h e ir

d e s t r u c t i v e a c t i o n i s d i m i n i s h e d . T h e s p e c if i c e r o s i o n c o n s e q u e n t l y d e c r e a s e s a f t e r a c e r t a i n

d e p t h h a s b e e n r e a c h e d . "

E . H o n e g g e r (1 9 2 7 )

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

S i n c e H o n e g g e r ' s e x p e r i m e n t s i n 1 92 7 i t h a s

b e e n k n o w n t h a t e r o s i o n b y l iq u id i m p a c t d o e s

n o t d e v e l o p a t a c o n s t a n t r a t e . H o n e g g e r [ 1 ]

p r e s e n t e d e r o s i o n c u r v e s w h i c h e x h i b i t e d , a f t e ra n i n i ti a l in c u b a t i o n p e r i o d , a p h a s e o f in c r e a s i n g

e r o s i o n r a t e ( a c c e l e r a t i o n ) , f o l l o w e d b y a p h a s e

o f d e c r e a s i n g r a t e ( d e c e l e r a t io n ) . H i s e x p l a n a -

t i o n i s e s s e n t ia l l y th e s a m e a s t h a t p u t f o r w a r d i n

r e c e n t p u b l i c a t i o n s o n e r o s i o n u n d e r d r o p i m -

p i n g e m e n t [ 2 , 3 ] , F i g . 1 . E v e n t o d a y , o p i n i o n s

d i f f e r a b o u t w h i c h s t a g e i n t h e e r o s i o n p r o c e s s i s

t h e m o s t i m p o r t a n t . T h e q u a n t i ta t i v e p r e d i c ti o n

o f th e e r o s i o n c u r v e c o u l d d e f i n i te l y b e a n i m -

p o r t a n t s t e p t o w a r d s f in a ll y e s t a b l is h i n g n o t

o v e r l y c o n s e r v a t i v e , a d m i s s i b l e h y d r o d y n a m i c

l o a d i n g c o n d i t i o n s .

T h e p a r t i c u l a r s h a p e o f t h e e r o s i o n c u r v e d e -

p e n d s , o f c o u r s e, o n t h e m a t e r i a l a n d ( g e o m e t -

r i c) s u r f a c e c o n d i t i o n s o f t h e s o l i d s t r u c t u r e , a s

w e l l a s o n t h e t y p e o f ( t h e r m o - a n d ) h y d r o -d y n a m i c i m p a c t l o a d i n g . I t is o b v i o u s t h a t r e a li s -

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

e x p e c t e d a n d a n a p p r o x i m a t e n u m e r i c a l s o l u t i o n

i s e x t r e m e l y t i m e - c o n s u m i n g a n d e x p e n s i v e ,

e v e n w i t h t h e s i m p l e s t o f a s s u m e d m a t e r i a l b e -

h a v i o r a n d f a i l u r e m e c h a n i s m . E x p e r i m e n t a l

s t u d i e s a l s o a r e d i f f i c u l t , t i m e - c o n s u m i n g a n d

e x p e n s i v e .

W h a t r e m a i n s i s t h e n t h e h o p e t h a t, o n t h e

b a s is o f p r e l im i n a r y e x p e r i m e n t a l e v i d e n c e , a

016%8442/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-H olland)



208 H . W . B a r g m a n n / O n t h e t i m e - d e p e n d e n c e o f t h e e r o s i o n r a te

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T i m e o f E x p o s u r e T ime o f Exposure

F i g 1 . T y p i c a l e r o s i o n r a t e v s . t i m e c u r v e s a c c o r d i n g t o

d i f f e r e n t i n v e s t i g a t o r s [ 2 ] .

h igh ly s impl i f i e d a ppr oa c h w i l l c on t a in t he e s se n-

t i al f e a tu r e s a nd p r o v ide so m e ba s i c i ns igh t, a nd

a n u n d e r s t a n d i n g o f th e c o n t r o l li n g p a r a m e t e r s .

Wi th a c ons i s t e n t s e t o f c onc e p t s i t i s t he n poss -

i b l e t o g u i d e t h e e x p e r i m e n t a l w o r k n e e d e d ; t o

c he c k , i n p r i nc ip l e , t he r e su l t s o f c om pu te r c a l -

c u l a t i ons ; t o a dv i se on t he c ho i c e o f a ppr opr i a t e

ma te r i a l s a nd f i na l l y t o de f ine t he a dmis s ib l eh y d r o d y n a m i c l o a d i n g c o n d i t i o n s .

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

i nvo lve s a w ide va r i e ty o f ma te r i a l , so l i d me c h -

a n i c s a n d f l u i d d y n a m i c s p a r a m e t e r s . W h e n

f a t i gue a nd b r i t t l e f a i l u r e c a n be e xc lude d , t he

e r os ion p r oc e s s i s duc t i l e a nd dur ing a n i nc uba -

t io n p e r i o d s h o w s c o m p l e x d e t a i l s o f g r a in b o u n -

da r y de l i ne a t i on a nd p l a s t i c de pr e s s ion o f i n-

d iv idua l g r a ins be low the o r i g ina l su r f a c e l e ve l ,

f i na l l y l e a d ing t o a ge ne r a l , f a i r l y un i f o r m undu-

l a ti o n o f th e s u r f a c e a n d t h e f o r m a t i o n o f s m a ll ,

s m o o t h - e d g e d p i t s . T h i s h a s b e e n d e s c r i b e d i nd e t a il b y V y a s a n d P r e e c e [4 ]. T o w a r d s t h e e n d

o f th e i n c u b a t i o n p e r i o d , t h e g e n e r a l u n d u l a t i o n s

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

c o m b i n e d p r e s s u r e p u l s e s o f a l a r g e n u m b e r o f

c o l l a p s i n g b u b b l e s , o r f r o m i m p i n g i n g d r o p s ,

de ve lop i n to c r a t e r - l i ke de pr e s s ions w i th l a r ge

s m o o t h l i p s .

M a t e r i a l l o s s i s c o n s i d e r e d t o o c c u r f r o m t h e

l i ps o f t he c r a t e r s by duc t i l e r up tu r e a nd a ga in

those p r e s su r e pu l se s ( e f f e c t i ve impa c t s ) ma y be

c o n s i d e r e d r e s p o n s i b l e f o r it . A m a r k e d t im e -

de p e nd e nc e i n the m a te r i a l lo s s r a t e i s obs e r ve d .

T h e c h a n g e o f s u r f a c e t o p o g r a p h y a s e r o s i o n

p r o c e e d s a n d i t s f e e d b a c k t o t h e h y d r o d y n a m i c

loa d ing ( e .g . t r a ppe d ga s a n d / o r l i qu id a t t he

b o t t o m o f d e e p c r a t e rs ; c h a n g e s i n i m p a c t a n g l e)a r e c o n s i d e r e d t o b e m a j o r f a c t o r s . A n o t h e r

f a c t o r m a y b e t h e c h a n g e i n m a t e r i a l b e h a v i o r

due t o r e pe t i t i ve impa c t l oa d ing , i . e . , w or k ha r d -

e n i n g m a y p l a y a r o l e . F o r e x a m p l e , a n a v e r a g e

inc r e a se i n su r f a c e ha r dne ss by a f a c to r o f tw o

h a s b e e n r e p o r t e d a n d a l m o s t a c u b i c d e p e n d -

e nc e o f e r os ion r e s i s t a nc e on ha r dne ss [ 3] .

A l l t h e s e c h a n g e s a r e l i k e l y t o m a k e r e m o v a l

o f ma te r i a l f a r mor e e f f e c t i ve w hi l e e r os ion p r o -

c e e ds t h r ough t he l a ye r s ve r y c lose t o t he su r -

f a c e , i . e . , a t t he e a r l y s t a ge ( a f t e r a poss ib l e

i n c u b a t i o n p e r i o d ) .

T h e p u r p o s e o f th i s p a p e r i s t o p r e s e n t a

h igh ly s impl i f i ed mo de l o f t he e r os ion p r o c e s s

unde r l i qu id impa c t . A t he or y i s ou t l i ne d w hic h

a l lo w s t h e m a t h e m a t i c a l e x p e c t a t i o n o f t h e e r o -

s ion r a t e a s a f unc t i on o f t ime ( e r os ion c u r ve ) t o

be p r e d i c t e d . T h i s t he or y i s ba se d on t he s t a t i s t i -

c a l na tu r e o f the r e pe t i t i ve l oa d ing b y li qu id

impa c t . I t e m ploy s a c l a ss i f ic a t ion o f i nd iv idua l

impa c t s a c c or d ing t o t he i r q ua l i t y o f o r e f f e c t i ve -

ne s s i n r e moving ma te r i a l . T o t h i s e nd , t he

spe c i f i c l os s o f ma te r i a l ( pe r impa c t ) i s c ons id -e r e d t o b e a r a n d o m v a r i a b l e w h i c h a s s u m e s

d i f f e r e n t v a l u e s d e p e n d i n g o n t h e o r d e r o f r e p e -

t i t ion o f impa c t s a t t he s a m e loc a t i on on t he

t a r ge t ( su r fa c e ) . I n t he c a se o f a duc t i l e e r os ion

p r o c e s s , w e a s s u m e t h a t t h e e a r l y r e p e t i t i o n s o f

t he impa c t l oa d ing a r e mos t e f f e c t i ve f o r t he

r e m o v a l o f m a t er i a l. N e v e r t h e l e s s , t h e v e r y a p -

p r ox ima te so lu t i on e xh ib i t s t he t yp i c a l f e a tu r e s

o f e r o s i o n c u r v e s a s p r e s e n t e d i n t h e l i t e r a t u r e [

T h e t h e o r e t i c a l a p p r o a c h e s a d v a n c e d s o f a r f o r t h e q u a n -t i ta t i v e p r e d i c t io n o f t h e e r o s i o n r a t e t i m e - b e h a v i o r a r e

e s s e n t ia l l y b a s e d o n a d h o c a s s u m p t i o n s c h a r a c t e r i s i n g o v e r -

a l l e f f e c ts . N o n e o f t h e m e x a m i n e s o r m o d e l s t h e a c t u a l

p h y s i c a l o c c u r r e n c e s i n a m o r e s p e c i f ic w a y , s o t h e y d o n o t

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

H e y m a n n [ 5] c o n s i d e r s th e l i f e t i m e s o f t h e t o p s u r f a c e

a n d s u b s u r f a c e l a y e r s a s r a n d o m v a r i a b l e s w i t h a s s u m e d

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

a ll s t a ti s ti c a l a s p e c ts o f t h e e r o s i o n p r o b l e m . T h e a p p r o a c h

i s s e lf - c o n s i s te n t ; t h e r e i s , h o w e v e r , n o d i r e c t w a y o f m a k -

i n g a p h y s i c a l r e f i n e m e n t ,

C o n t i n u e d o n p . 2 0 9



H.W. Bargmann / On the time-dependenceof the erosion rate 209

2 . T h e s p e c i fi c lo s s o f m a t e r i a l

W e c o n s i d e r t h e S p ec if ic l o ss o f m a t e r i a l ( p e r

i m p a c t ) t o b e a d i sc r e te r a n d o m v a r i a b le m ,

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

T h i s p r o b a b il i ty m a y c h a n g e w i th t h e n u m b e r No f i m p a c t s , p~(N )= pN. A t t h e N t h i m p a c t , t h e

m e a n o r e x p e c t e d v a l u e o f t h e s p e c if ic lo s s o f

m a t e r i a l i s t h e n g i v e n b y

N = _ _ e N { m } = E m , p N , p N = _ e U { m = r n ,} ,i

N = 1 , 2 , 3 , . . . (i)

From the physical point of view, we need a

classification of the individual impacts according

to their quality of or effectiveness in removing

material. Various types of classifications are con-

ceivable. We shall adopt a particularly simple

one which appears to be supported by ex-

perimental evidence. Moreover, this approach

should lend itself to straightforward extensions

and refinements.

3 . T h e r e p e t it i v e n a t u r e o f e r o s i o n

A s i m p l e , v e r y a p p r o x i m a t e c l a s s if i ca t io n is

s u g g e s t e d , t a k i n g i n t o a c c o u n t t h e s t a t i s t i c a l n a -

t u r e o f r e p e t i t i v e l o a d i n g b y l iq u i d i m p a c t . L e t u sc o n s i d e r t h e p r o b l e m o f p ro d u c i n g c r a te r s in a

s o l id t a r g e t b y r a n d o m p r o j e c t i o n o f l iq u i d j et s

Footnote 1 Cont.

The fundamental reasoning in Thiruvengadam's theory[6] is, in a sense, a vicious circle. His 'efficiency' of erosion ,introduced as a premise, is very closely connected with his' intensity' (or rate) of erosion for w hich he dra w s hisconclusion. He thus g ives as proo f the assumption fromwhich he starts.

Despite references to concepts such as fatigue andWeibull distribution, Springer [7] assumes from the outsetthat the erosion rate is constant with time. His model comesdown to the assumption that shorter incubation periodscorrespond to higher subsequent erosion rates. This may,for certain empirical data, be an approximation for theacceleration phase if an incubation period exists.

The mathematical approach recently advanced by Nos-kievic [8] definitely takes us outside the discipline otphysics. Notwithstanding numerous allusions to concepts ofrigid body dyn amics, viscoelasticity and dam ped v ibrations,the w hole approa ch is still a curve fitting exercise in elemen -tary geometry.

o r d r o p s ( l i q u i d i m p a c t ) . T h e a c t u a l s i t u a t i o n i s

e x t r e m e l y c o m p l e x . O v e r l a p p i n g c r a t e r s o f v a r i-

o u s s i z e s m a y c o n t i n u o u s l y c o v e r t h e e n t i r e s u r -

f a c e ( F ig . 2 ) . W e a d o p t a s i m p l if i e d m o d e l w h e r e

o n l y n o n - o v e r l a p p i n g c r a t e r s o f t h e s a m e s i z e

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

e i t h e r n o t t o o v e r l a p a t a ll o r t o o v e r l a p c o m -

p l e t e l y , a n d p a r t i a l o v e r l a p p i n g i s e x c l u d e d .

L e t t h e p r o b a b i l i t y o f h i t ti n g a n e x i s t i n g c r a t e r

b y l iq u id i m p a c t b e d e n o t e d b y p . A s s u m e t h i s

p r o b a b i l i t y t o b e c o n s t a n t a n d e q u a l t o t h e r a t i o

o f th e a r e a o f a c r a t e r t o t h e a r e a o f t h e t o t a l

s u r f a c e e x p o s e d t o li q u id i m p a c t . T h e s m a l l e r t h e

i m p i n g i n g j e t o r d r o p c o m p a r e d t o t h e s iz e o f t h e

c r a t e r , t h e b e t t e r t h i s a s s u m p t i o n i s f u l fi l le d . I t

f u r t h e r c o n t a i n s t h e i d e a t h a t w h e r e v e r t h e c r a t e r

i s o n t h e s u r f a c e , t h e c h a n c e s o f h i tt i n g i t a r e t h e

s a m e .

L e t u s c o n s i d e r , a t t h e N t h i m p a c t , t h e N

s e p a r a t e e v e n ts : " p r o d u c i n g a n e w c r a t e r " , " p r o -

d u c i n g a f ir st r e p e t i t i o n " , " p r o d u c i n g a s e c o n d

r e p e t i t i o n " , . . . . " p r o d u c i n g a ( N - 1 ) th r e p e ti -

t i o n " . T h e s e e v e n t s to g e t h e r f o r m t h e c e r t a in

e v e n t . I t i s c o n v e n i e n t t o d e f i n e t h e p r o d u c t i o n

o f a 0 th r e p e t i t i o n a s th e f o r m a t i o n o f a n e w

c r a t e r .

W e m a y t h e n b a s e t h e c l a ss i fi c a ti o n o f t h e

i n d i v id u a l i m p a c t s o n t h e r e p e t i t i v e n a t u r e o (i m p a c t l o a d i n g a n d w e m a y c h o o s e t h e p r o b a b i l i -

t y p U e q u a l t o t h e p r o b a b i l i t y o f p r o d u c i n g a n i t h

r e p e t i t i o n , a t t h e N t h i m p a c t ,

N N

P i = P r e v i, i = O , 1 , 2 , . . . , ( N - 1 ) , ( 2 )

N N

w h e r e P re p 0 - P ne w " W e t h u s h a v e t o d e t e r m i n eN

t h e p r o b a b i l i t i e s P r e p i" L e t u s c o n s i d e r a p r e l i m i -

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

Fig. 2. S urfa ce under repetitive impacts. The surface at thecenter corresponds to the superposition of major craters(left) corresponding to effective impacts and weaker craters(right). In the simplified mod el, craters o f one size only coverthe surface, and we assu me that the successive craters areeither not at all or totally overlapping. Weak impacts affect,first of all, the material behavior.



210 H.W . Bargm ann / On the time-dependence of the erosion rate

3.1. The configurations o f the erosion process

F i r s t l y , w e w i l l c o n s i d e r a n o t h e r s e t o f e v e n t s .

W e r e c o r d , after t h e N t h i m p a c t , t h e t o t a l a r -

r a n g e m e n t o f n e w c r a t er s a n d t h e d i f f e r e n t r e p e -

t i t i o n s o b t a i n e d u p t o a n d i n c l u d i n g t h e N t hi m p a c t . W e c o n s i d e r s u c h a n a r r a n g e m e n t t o b e

u n i q u e l y d e f i n e d b y t h e t o t a l n u m b e r s o f t h e

d i f f e r en t r ep e t i t i o n s . W e s h a l l c a l l s u ch an a r -

r a n g e m e n t a " c o n f i g u r a t i o n C iN a f t e r t h e N t h

i m p a c t " o f t h e e r o s i o n p r o c e s s , a n d c o n s i d e r it a s

a n e v e n t . A g a i n , t h e s e e v e n t s t o g e t h e r , i - -

l , 2 . . . . , r e ( N ) , f o r m t h e c e r t a i n e v e n t ( b u t in a

d i f f e r e n t p r o b a b i l i t y s p a c e f r o m b e f o r e ) .

W e a d o p t a s p e c i a l n o t a t i o n f o r t h e p r o b a b i l i t y

o f h a v i n g o b t a i n e d a p a r t i c u l a r c o n f i g u r a t i o n C iN.N

F o r e x a m p l e , F i g . 3 , w e w r i t eP3,2,2

f o r t h e

p r o b a b i l i t y o f h a v i n g o b t a i n e d , a f t e r t h e N t h

N c o n t a i n i n gm p a c t , t h e c o n f i g u r a t i o n C 3 , 2 , 2 , 1 . . . . . 1

t h e f o l l o w i n g e l e m e n t s : ( i ) o n e s e c o n d r e p e t i t i o n

(3 o v e r l ap p i n g c r a t e r s ) , ( i i) t w o f i r s t r ep e t i t i o n s

(2 o v e r l ap p i n g c r a t e r s e ach ) , an d ( i ii ) N - 7 n ew

c r a t e r s. N o t e t h a t t h e o c c u r r e n c e o f n e w c r a te r s

i s n o t ex p l i c i tl y i n d i ca t ed ; c f . F i g . 3 . F o r t h e

p r o b a b i l i t y o f h a v i n g o b t a i n e d , a f t e r t h e N t h

i m p a c t , a c o n f i g u r a t i o n w i th n e w c r a t e r s o n l y , w e

w r i t e p N . O b v i o u s l y , s u c h a c o n f i g u r a t i o n

e x h i b i ts e x a c t l y N n e w c r a t e r s . W e n o t e h e r e t h e

v a r i o u s p r o b a b i l i t i e s f o r t h e f i r s t t h r e e i m p a c t s .A f t e r t h e f i r st i m p a c t , N = 1 , w e o b v i o u s l y

h a v e

Px --- 1. (3)

A f t e r t h e s e c o n d , N = 2 , F i g . 4 ,

P~ = (1 - P ) P I = (1 - p ) , p22 = P P I = P .( 4 )

A f t e r t h e t h i r d i m p a c t , N = 3 , a c c o r d i n g t o t h e

1 2 3 4 . N - 4

( I ) 0 © 0 - 03 2 2 1 1

pN3 ; 2 , 2

F i g , 3 . C o n f i g u r a t i o n c 3 N , 2 , 2 ,1 . . . . . l w i t h o n e s e c o n d r e p e t i t i o n

( 3 o v e r l a p p i n g c r a t e r s ) a n d t w o f i r s t r e p e t i t i o n s ( 2 o v e r l a p p i n g

c r a t e r s e a c h ) a f te r t h e N t h i m p a c t . T h e p r o b a b i l i ty o f h a v i n g

o b t a i n e d t h i s c o n f i g u r a t io n a f t e r t h e N t h i m p a c t i s d e n o t e d

b y p N3,2,2"

O O O2 p 2

P 2 1

F i g . 4 . P r o b a b i l i t i e s P ~ a n d P~2 o f h a v i n g o b t a i n e d , a f t e r t h e

s e c o n d i m p a c t o n e f ir s t r e p e t i t i o n C 22 , a n d t w o n e w c r a t e r s

C~.,, r e s p e c t i v e l y .

t h e o r e m o n t o t a l p r o b a b i l i t y ,

P~ = (1 - 2p)P~ = ( 1 - 2 p ) ( 1 - p ) ,

P3 = 2pP~ + ( 1 - p ) P ~ = 3 p( 1 - p ) , ( 5)

p~ = p p 2 = p 2 .

T h e f o r m u l a t i o n f o r h i g h e r v a l u e s o f N i ss t r a ig h t f o r w a r d . F o r N = 4 - 7 t h e c o r r e s p o n d i n g

p r o b a b i l i t i e s a r e g i v e n i n A p p e n d i x 1 .

3.2. Dif fer ent order repetit ions

H a v i n g f o u n d t h e p r o b a b i l i t i e s u p t o o r d e r

( N - 1 ) f o r th e v a r i o u s c o n f i g u r a t i o n s , w e c a nN

n e x t d e t e r m i n e t h e p r o b a b i l i ti e s P n e w o f p r o d u c -

i n g a new crater a t th e N t h i m p a c t . W e h a v e , f o r

N = 1 , 2 , 3 , . . . ,

P '~w = P~ = 1 , p 2 ew = P~ = l - p ,(6 )

3 (1 2 p )P : + ( 1 p ) P ~ = ( 1 p )2P ne w ~-- - - -

a n d s o o n . S e e A p p e n d i x 2 f o r t h e c o r r e s p o n d i n g

p r o b a b i l i t i e s a t i m p a c t s N = 4 - 7 .

S i m i l a r l y , w e c a n d e t e r m i n e t h e p r o b a b i l i t i e sN

P re p1 o f p r o d u c i n g , a t t h e N t h i m p a c t , a f i rs trepetition. T h u s , f o r N = 1 , 2 , 3 . . . . .

2 2P ) e p 1 = 0 , P r e p 1 = P 2 = P ,

(7 )P~ep, = 2pP~ = 2p(1 - p )

a n d s o o n . S e e A p p e n d i x 2 fo r t h e c o r r e s p o n d in g

p r o b a b i l i t i e s a t i m p a c t s N = 4 - 7 .N

W e n o t e t h e p r o b a b i l i t i e s P r e p 2 o f p r o d u c i n g ,

a t t h e N t h i m p a c t , a second repeti t ion, a g a i n f o r

N = 1 , 2 , 3 ,

3 2 2P~ep 2 = Prep2 2 = 0 , Prep 2 = PP2 = P (8 )

S e e A p p e n d i x 2 f o r t h e c o r r e s p o n d i n g p r o b -

ab i l i t i e s a t i mp ac t s N = 4 -7 .N N N f o rT h e p r o b a b i l i t i e s P r e p 3 , P r e p 4 a n d P r e p 5

p r o d u c i n g , a t t h e N t h i m p a c t , a t h i r d , a f o u r t h



H.W . Bargm ann / On the t ime-dependence of the erosion rate 2 1 1

a n d a f i f th r e p e t i t i o n , r e s p e c t i v e l y , a r e g i v e n in

A p p e n d i x 2 .

L e t u s f i n a ll y n o t e t h e p r o b a b i l i t ie s p N p ~ 2 o f

p r o d u c i n g , a t t h e N t h i m p a c t , a se c ond o r h i gher

repetition. I t f o l lo w s i m m e d i a t e l y

N

N __ p SP r e p s > 2 - 1 - P r e p I n e w •

T h u s w e h a v e

P ~ e p ~ > 2 2 3 3 p 2: P r e p s 2 = 0 , P r e p s > 2 = P r e p 2 =

(9 )

( lO)

a n d s o o n .

4 . T h e e x p e c t a t i o n o f t h e e r o s io n c u r v e

T o b r i n g o u t t h e e s s e n t i a l p i c t u r e o f t h e d e -

p e n d e n c e o f t h e p r o b a b i l i t y o f a f i r s t r e p e t i t i o no n t h e n u m b e r o f im p a c t s , a n d f o r t h is p u r p o s e

o n l y , l e t u s c h o o s e f o r t h e p r o b a b i l i t y o f h i t t i n g

a n e x i s t i n g c r a t e r t h e v a l u e p = 0 . 5. T h e c o r r e s -

p o n d i n ~ v a l u e s fo r t h e p ro b ab i l i t i e s P~n ew , prN epl

an d P~ep ~2 fo r t h e f i r st 7 i mp ac t s , a r e g i v en i n

Fig . 5 .

A t t h i s p o i n t i t i s i n t e r e s t i n g t o o b s e r v e t h a t aN

w e i g h t e d s u p e r p o s i t i o n o f prNpl and Preps2 re-

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

t i m e c u r v e i f t h e n u m b e r N o f e f fe c t iv e im p a c t s

is ta k e n a s a m e a s u r e o f ti m e ( w h ic h s h o u l d b e a

r e a s o n a b l e a s s u m p t i o n ) , F i g . 6 , c u r v e ( a ) .

i i i.0 l ' ' ' j . ~

P r ~ p X N f Ni X P n e w / / P r e p > 2

o.s i ~

0 .00 1 2 3 4 S 6 7 - ~ i ~ ' - N

Fig. 5. Probab ilities or prod ucing, at the N th impact, a newcrater, a first repetition, or a second or higher repetition. Forthe purp ose of illustration, the probability of hitting anexisting crater has b ee n assum ed to be p = 0.5.

1,0

0.5

0.0 l

~ ( a ) s t a t io n a r y v a lu e

~ " N f o r N " - " o o/ I ~ e p > 2

/ //

/ / / P r N 3/ ep>_./ /

~ r ' r ep 2 / ~ /

w3 /,, 5 6 7 ,'~.m.--N

2.0

; / m 6

t

1.0

Fig. 6 . Eros ion ra te vs . t im e (specif ic loss of mate r ia l vs .

number N o f impac t s ) r e l a t ive to i t s s t a t iona ry va lue . The

d i f f e ren t cu rves (a ) , (b ) , and ( c) co r re spond to th ree d i f f e ren t

in f luences o f ea r ly r epe t i t ions , c f . Tab le 1 .

T h i s o b s e r v a t i o n i s n o s u r p r i s e . I n f a c t t h e

e x p e c t e d v a l u e o f t h e s p e c i f i c l o s s o f m a t e r i a l , a tt h e N t h i m p a c t , c a n b e w r i t t e n , e q . ( 1 ) ,

N N N~ 1 ~ E~{m} m l P r e p 1 "~ m 2 P r e p ~ > 2 ,

N = 1 , 2 , 3 , . . . ( 1 1 )

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

l o ss o f m a t e r i a l m 1 f o r a f i r s t r e p e t i t i o n i s m u c h

g r e a t e r t h a n t h e a v e r a g e s p e c if i c l o ss m 2 f o r t h e

s e c o n d a n d a l l h i g h e r r e p e t i t i o n s ; t h i s i s a m p l y

j u st if i ed t h r o u g h o u t t h e l i t e r a tu r e . W e n o t e t h a t

cu rv e ( a ) i n F i g . 6 ex h i b i t s a l l t h e s t ag es u s u a l l y

f o u n d i n e r o s i o n c u r v e s: a n i n c u b a t i o n p e r i o d , a na c c e l e r a t i o n p h a s e , a d e c e l e r a t i o n p h a s e a n d a

f i n a l s t a t i o n a r y s t a g e w i t h c o n s t a n t e r o s i o n r a t e .

I t is n a t u r a l t o a s k h o w t h e s h a p e o f t h e

e r o s i o n c u r v e w i l l b e i n f l u e n c e d b y t h e h i g h e r

o r d e r r e p e t i t i o n p r o b a b i l i t i e s . T h e v a l u e s f o r t h e• . • N N N N

p r o b a b l l m e s P r e p 2 , P r e p 3 , P r e p 4 a n d P r e p 5 a r e

g i v e n i n F i g . 6 . A g a i n , w e o b s e r v e t h a t w e i g h t e d

s u p e r p o s i t i o n s o f v a r i o u s P r e p / r e s u l t i n s h a p e s

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

a s r e p o r t e d i n t h e l i t e r a t u r e . N o t e t h a t t h e p a r -



212 H . W . B a r g m a n n / O n t h e t i m e - d e p e n d e n c e O f t h e e r o s i o n ra te

t icu lar curves (a) , (b ) and (c) , Fig . 6 , represen t

t h e r e l at i v e e x p e c t e d v a l u e ( i .e . , m e a s u r e d i n

u n i ts o f t h e co r r e sp o n d in g s t a t io n a r y ex p ec te d

v a lu e ) o f t h e sp ec i f i c l o ss o f m a te r i a l , a t t h e N th

impact , hence a t t ime t~¢ ,

kN__ N Nk~k = Ek {m} ~] N: m i P r e p i + l n k + i P r e p ~ k + 1 ,i - 1

N = 1 , 2 , 3 . . . . , ( 1 2 )

wh er e m i d en o te s t h e sp ec if i c l o ss o f m a te r i a l f o r

an i t h r ep e t i t i o n , an d m ,+~ d en o te s t h e av e r ag e

loss fo r the (k + 1) th and a ll h igher re pet i t ion s;

c f . eq . ( 1 ) . Th e cu r v es in F ig . 6 we r e g e n e r a t e d

b y u s in g th e n u m er i ca l v a lu es f o r t h e r e l a t iv e

e f f ec t s o f ea r ly r ep e t i t i o n s a s g iv en in Tab le 1 .

F in a l ly , we n o te th a t eq s . ( 1 ) an d ( 2 ) a l so

a l lo w f o r t h e g en e r a l c a se wh e r e m o ~ O , a n d m o

d en o te s t h e sp ec if i c l o ss o f m a te r i a l w h en a n ewcr a t e r i s p r o d u ced . I t h a s b ee n o b s e r v e d in ce r -

t a in , l e ss f r eq u e n t , c a se s t h a t t h e e r o s io n cu r v e

s t a r t s w i th a m ax im u m v a lu e , w i th o u t an acce l e -

ra t ion phase ; th is cou ld cor respond to a s ign i f ic -

an t m a te r i a l l o ss ev en wh en a n ew c r a t e r i s

f o r m e d [ 5 ] .

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

I t ap p ea r s t h a t t h e f o l lo win g co n c lu s io n s can

n o w b e d r a w n .( 1 ) O n th e b as i s o f t h e s t a t is t i c a l n a tu r e o f t h e

r ep e t i t i v e lo ad in g b y l iq u id im p ac t a t h eo r y h as

b e e n o u t l i n e d w h i c h a l l o w s t h e m a t h e m a t i c a l

ex p ec ta t io n o f t h e e r o s io n r a t e a s a f u n c t io n o f

t im e ( e r o s io n cu r v e ) t o b e p r ed ic t ed . Th e sp ec i f i c

lo ss o f m a te r i a l ( p e r im p ac t ) m ay a ssu m e d i f f er -

en t d i sc r e t e v a lu es , w i th p r o b ab i l i t i e s co r r e s -

p o n d in g to : t h e f o r m a t io n o f a n ew c r a t e r ; a f i r s t

r ep e t i t i o n o f im p ac t s ; a seco n d s r ep e t i t i o n ; o r a

r ep e t i t i o n o f ev en h ig h e r o r d e r , a t t h e sam e

Table 1

Relative effect of early repetitions (normalized by stationary

e f f e c t o f l a t e r r e p e t i t i o n s ) , cf. eq. (12) and Fig. 6

m I m 2 m 3 m4 m5

m 6 m6 m6 m6 m 6

Erosion r a t e

curve

(a) 3 1 1 1 1

(b) 3 0 3 0 2

(C) 3 0.5 1 2 3

l o ca t io n o n th e t a r g e t ( su r f ace ) . Fo r a d u c t i l e

e r o s io n p r o cess an d o n th e a ssu m p t io n th a t t h e

ea r ly r ep e t i t i o n s a r e t h e m o s t e f f ec t iv e f o r r e -

m o v in g m a te r i a l , a so lu t io n i s p r e sen ted .

Al th o u g h th is i s a v e r y ap p r o x im a te so lu t io n i t

ex h ib i t s t h e ty p ica l sh ap e o f e r o s io n cu r v es u su -a l ly f o u n d in th e l i t e r a tu r e : an in cu b a t io n p e r io d ,

th e p h ases o f acce l e r a t io n an d d ece le r a t io n , a s

wel l as the f ina l s ta t ionary s tage of a constan t

e r o s io n r a t e .

( 2 ) W e n o te th a t t h e sp ec i f i c l o ss o f m a te r i a l

en te r s t h e e x p r e ss io n f o r t h e ex p ec ta t io n o f t h e

e r o s io n cu r v e o n ly in t h e f o r m o f a d i sc r e t e

we ig h t in g f u n c t io n , i n a we ig h ted su m o f p r o b -

ab i l i t i e s . W e m ig h t t h e r e f o r e th in k th a t t h e ty p i -

ca l sh ap es o f e r o s io n cu r v es a r e t o a l a rg e ex ten t

s t a t i s t i c a l i n n a tu r e . Th ey sh o u ld th en b e r e l a -

t i v e ly in sen s i ti v e to p a r t i cu l a r l o ad in g co n d i t io n s ,m a te r i a l b eh av io r , an d f a i lu r e m ech an i sm s .

( 3 ) Th e ab o v e sh o u ld b e t r u e a l th o u g h th e

n u m b e r N o f i m p a c t s m a y b e i n t e rp r e t e d i n

sev e r a l way s . I n p r in c ip l e , N m ay r e f e r t o t h e

v e r y l a r g e n u m b er o f i n d iv id u a l im p ac t s ( o f av e r -

ag e in t en s i ty ) . I t m ay e q u a l ly we l l r e f e r o n ly to a

sm a l l n u m b er o f e f f ec t iv e im p ac t s ( a l so o f av e r -

ag e in t en s i ty ) . C o n seq u en t ly th e sp ec i f i c l o ss o f

m a te r i a l v a lu es f o r e i th e r i n t e r p r e t a t io n m u s t b e

a d j u s t e d .

( 4 ) Fo r p a r t i cu l a r ap p l i ca t io n s , t h e th eo r y r e -q u i r e s ch a r ac t e r i s t i c d a t a f o r t h e sp ec i f i c l o ss o f

m a te r i a l a s we l l a s t y p ica l d a t a f o r h y d r o d y n am ic

lo ad in g . Fu r th e r sy s t em a t i c e f f o r t i s n ee d ed to

c l a ssi f y th e w id e r an g es o f m a te r i a l b eh av io r an d

f a i lu r e m ech an i sm s an d th e v a r io u s h y d -

r o d y n am ic lo ad in g co n d i t io n s . I n p a r t i cu l a r , i t i s

r e c o m m e n d e d t h a t a p p r o x i m a t e c a l c u l a t i o n s c o n -

ce r n in g th e f o r m a t io n o f c r a t e r s an d d u c t i l e f a i-

lu re , as wel l as fo r b r i t t le fa i lu re and fa t igue , a l l

c a u s e d b y l i q u i d i m p a c t , b e p e r f o r m e d .

( 5 ) Th e s im p l i fi ed th eo r y p r e sen te d a l lo ws

s t r a ig h t f o r war d in - d ep th r e f in em en t s an d d i r ec tg en e r a l i za t io n s o n v a r io u s l ev e l s t o b e m ad e .

Th e co n cep t s p r e sen ted in th i s p ap e r f o r a d u c -

t i le p r o cess o f e r o s io n can a l so b e ap p l i ed , w i th

m in o r m o d i f i ca t io n s , t o t h e ca se o f e r o s io n b y

fa t igue and br i t t le fa i lu re .

A p p e n d i x 1 : P r o b a b i l i t i e s f o r t h e c o n f i g u r a t i o n s

I n Sec t io n 3 .1 th e p r o b ab i l i t i e s o f t h e co n f ig u -



H.W . B argm ann / On the time-dependence of the erosion rate 213

r a t io n s o f th e e r o s i o n p r o c e s s h a v e b e e n p r e s e n -

t ed , f o r t h e f ir s t 3 im p ac t s , N = 1 - 3 , eq s ( 3 ) - ( 5 ) .

W e m a y w r i t e , i n g e n e r a l ,

p N = p N p ~ .- ~ , N = 1 , 2 , 3 , . . . ( A. 1)

N

w h e r e t h e c o l u m n p N = ( p N . , p 2 N , . . . ,P r o ( N ) ' }

i s f o r m ed o f t h e ( u n c o n d i t io n a l ) p r o b ab i l i t i e s

t h a t , a t t h e N t h i m p a c t , t h e e r o s i o n p r o c e s s

wi l l p a ss t o t h e co n f ig u r a t io n s C N , i=

1 , 2 . . . . , r e ( N ) , r e sp ec t iv e ly .

W e d e n o t e b y P t h e m a t r ix o f t h e tr a n s it i o n

p r o b ab i l i t i e s p q ( N ) = - p N , p N = IIp~ll. The t rans-

i t i o n p r o b ab i l i t y p N i s t h e co n d i t io n a l p r o b ab i l i -

t y , a t t h e N th im p ac t ( o r a t im e t N ) , t h a t a f t e r

t h e N t h i m p a c t C N i s o b t a i n e d u n d e r t h e a s-

s u m p t i o n t h a t c N . ~ w a s o b t a i n e d a t t h e ( N -

1 ) th im p ac t .

W e n o te th a t i n each l i n e o f t h e m a t r ix p Nt h e r e i s a t l e a s t o n e e l e m e n t d i f f e r e n t f r o m z e r o ,

Na n d t h e t r an s i ti o n p r o b a b i l it i e s p q , f o r a n y N ,

m(N)

p N = I , j = l , 2 , . . . , m ( N - 1 ) .

i = 1 ( A . 2 )

W e f in a l ly h av e

p ~ = ( p 2 p 3 . . . p N ) T p ~ , N = 1 , 2 , 3 . . . .

( A . 3 )w h e r e T d e n o t e s t h e t r a n s p o s e d m a t r i x , a n d

P ~ - { P ~ } = 0 } .W e p r e sen t i n t h e f o l lo win g th e p r o b ab i l i t i e s

p N o f t h e c o n f ig u r a ti o n s C N , i = 1 , 2 . . . . .

r e ( N ) , f o r N = 4 - 7 .

N = 4

P31 P32 p3

p l

e~

P ~

P•

e ~ ,2

( 1 - 3p)

3p ( 1 - 2p)( A . 4 )

p (1 -p )

P

Psa t i s f y th e r e l a t io n

N = 5

e~ e~ e~ P~ eLP 51 ( 1 - 4 p )

P s 4 p ( 1 - 3 p )

P53 p (1 - 2p )

p54 p (1 - p )

P~ pP~.2 2p (1 - 2p)

P ~ .2 p 2 p

N = 6

Ps P~ ts pS p5 p52,2 P53,

(A.5)

P ~

e~

P ~

e ~

e ~

P ~

e ~ .2

P~ .2

P~ ,2

P~,2.2

P~,3

( 1 - 5p )

5p (1 -4 p)

P

3p

( 1 - 3p )

P

2p

( 1 - 2p )

P

P

(1 -p )

P

( 1 - 3p)

2 p ( i - 2 p )

P

P

P

( A . 6 )



2 1 4

N = 7

H . W . B a r g m a n n / O n t h e t i m e - d e p e n d en c e o f t h e e r os io n r a te

p ~ p ~ p 6 6 p 6 6 . 6 p . . . .- 3 P4 5 P6 P2 .2 P3 ,2 4 ,2 P2,2,2 P3,2

e ;

e ;

e ;

e~

e~

P2

P;

P~,2

P~,2

P~.2

P~,2

P~,2,2P~,2,2

P~,3

P~,3

(1 - 6p )

6p ( i - 5 p )

p

4p

(1 - 4p )

P

3p

( 1 - 3p)

P ( 1 - 2p )

P (1 - p )

P

( 1 - 4p)

2 p ( 1 - 3 p )

2 p p ( 1 - 2 p )

p p

2p (1 - 3p)p 3p

P ( 1 - 2 p )

2 p

A p p e n d i x 2 : P r o b a b i l i t i e s f o r t h e r e p e t i t i o n s

I n S e c t i o n 3 . 2 t h e p r o b a b i l i t i e s f o r r e p e t i t i o n s

o f v a ri o u s o r d e r h a v e b e e n g i v e n , f o r t h e fi rs t

t h r e e i m p a c t s , N = 1 - 3 , e q s . ( 6 ) - ( 8 ) .N

F o r N = 4 - 7 , t h e p r o b a b i l i t i e s P ,e ,~ o f p r o d u c -

i n g , a t t h e N t h i m p a c t , a n e w c r a t e r a r e a s

f o l l o w s :

4Pnew = (1 - 3 p ) P ~ + ( 1 - 2p)P3~ + ( 1 - p ) p 3 ,

5Pnew = (1 - 4 p ) P ~ + ( 1 - 3 p ) e 4 + ( 1 - 2 p ) e ~

+ (1 - p ) P ~ + ( 1 - 2 p ) P ~ , 2 ,

6Pnew = ( 1 - - 5p)eS~ + (1 - 4 p ) e ~ + (1 - 3p )P53

+ ( 1 - 2 p ) P ~ + ( 1 - p ) P ~

+ ( 1 - 3 p ) P S 2 , : + ( 1 - 2 p ) P ~ . : , ( A . 8 )

7e .ew = ( 1 -- 6 p ) P 6 + ( 1 - 5 p ) P~ + ( 1 - 4 p ) P 6

+ (1 - 3 p ) P ~ + ( 1 - 2 p ) P 6 + (1 - p ) e ~

+ ( 1 - 4 p ) p 6 2 + ( 1 - 3p)P~,2

+ ( 1 - 2 p ) p 6 , 2

- I - ( 1 6 _ _3 p ) P 2 , 2 , z + ( 1 2 p ) p 6 , 3

( A . 7 )

N o f p r o d u c i n g , a t t h eh e p r o b a b i l i t i e s P r e p l

N t h i m p a c t , a f i r s t r e p e t i t i o n a r e , f o r N = 4 - 7 , a s

f o l l o w s :

4 = 3 p e ~ + p e 3P r e p 1

5 4 2 p P ~ + P P 3 ,rep 1 = 4 P P 1 + 4

P~eol = 5pp51 + 3pP52 + 2pP53 + PPS2,2 + P P' , ,

( A . 9 )

P ;e p 1 = 6 p e ~ + 4 p P 6 + 3 p P 6 + 2 p P 6 + p P ~

+ 2 p p 6 ,2 + P P ~ , 2 .

W e n o t e

N N u ( A . 1 0 )P r e p s 2 = V r e p ~ l - P r e p 1

NT h e p r o b a b i l i t i e s P r ep 2 o f p r o d u c i n g , a t t h e

N t h i m p a c t , a s e c o n d r e p e t i t i o n a r e , f o r N = 4 - 7 ,

a s f o l l o w s :

p r 4 p 2 = - p P ~ ,

P ~ep 2 = P P ~ + 2 P P ~ 2 ,

P ~ e p z = p P S + 2 p P ~, z + pP53, 2 , (A . 1 1)

6P~ep ~ = PP ~ + 2PP~ ,2 + PP~ ,2 + pp64 , z + 3PP 2,2 ,2 .



H. W . Bargm ann / O n the t ime-dependence of the erosion rate 2 1 5

W e n o t e

N N N ( A . 12)P r e p > ~ 3 = P r e p s > 2 - P r e p 2 •

N o f p r o d u c in g , a t t h eh e p r o b ab i l i t i e s P r ep 3

N t h i m p a c t , a third repeti t ion a r e , f o r N = 1 - 7 ,

as fo l lows:

P ~ e p 3 2 3 = 0= P r e p 3 = P r e p 3

4 4 3P r e p 3 -~- P 4 = P ,

P ~ ep 3 = P P ~ ,

p 6 o . 3 = p p 5 + p P L ,

7 - - 2pP63,3 + 6 6Prep 3 - P P 3 + P P 3 , 2 •

W e n o t e

N N N

P r e p s > 4 - -" P r e p s > 3 - P r e p 3 •

( A . 1 3 )

( A . 1 4 )

T h e p r o b a b i l i t i e s Pr~p4 o f p r o d u c in g , a t th e

N t h i m p a c t , a fourth repet i t ion a r e , f o r N = 1 - 7 ,

as fo l lows:

P ~ e p 4 2 3 : 4= P r e p 4 = P r e p 4 P r e p 4 = 0 ,

e ~ e p 4 ~ - P55 = p 4 ,

p 6 e p 4 = p p 5 4 ,

7 6 6Prep 4 -~ - P P 4 + P P4 , 2 •

W e n o t eN N N

e r e p > ~ 5 = P r e p s > 4 - P r e p 4 .

( A . 1 5 )

( A . 1 6 )

NTh e p r o b ab i l i t i e s P r ep 5 o f p r o d u c in g , a t t h e

N t h i m p a c t , af i f th repet i t ion a r e , f o r N = 1 - 7 , a s

f o l lo ws :

P ~ e p 5 2 3 4 5 ~ 0= P r e p 5 = P r e p 5 = P r e p 5 : -- P r e p 5 ,

p r6 p5 = p 6 = p 5 , ( A . 1 7 )

7 6P r e p 5 = PP5 •

W e n o t e

N : p N NP r e p s > 6 r e p ~ 5 - - P r e p 5 . ( A . 1 8 )

A c k n o w l e d g m e n t

I t i s a p le a s u r e t o a c k n o w l e d g e v a l u a b l e d i s c u s -

s i o n s w i th P r o f e s s o r I . L . R y h m i n g , a n d h i s v e r y

e n c o u r a g i n g i n t e r e s t t h r o u g h o u t t h e p r e p a r a t i o n o f

t h i s w o r k .

R e f e r e n c e s

[ 1] E . H o n e g g e r , " E s s a i s d ' 6 r o s i o n d e s a i le t t e s d e t u r b i n e sv a p e u r " , R e v u e B B C 1 4 ( 4 ) 9 5 - 1 0 4 ( 1 9 2 7 ) .

[2 ] C . M . P r e e c e , " C a v i t a t i o n E r o s i o n " , i n : C . M . P r e e c e ,

e d . , Erosion, Treatise on Materials Science and Technolo-

g y , V o l . 1 6 , A c a d e m i c P r e s s , N e w Y o r k ( 1 9 7 9 ) .

[3 ] J . H . B r u n t o n a n d M . C . R o c h e s t e r , " E r o s i o n o f S o li d

S u r f a ce s b y th e I m p a c t o f L i q u i d D r o p s " , i n : C . M .

P r e e c e , e d . , Erosion, Treatise on Materials Science and

Technology , V o l . 1 6 , A c a d e m i c P r e s s , N e w Y o r k ( 1 9 7 9 ) .

[4 ] B , V y a s a n d C . M . P r e e c e , " C a v i t a t i o n - I n d u c e d D e f o r m a -

t i o n o f A l u m i n i u m " , i n : Erosion, Wear, and Interfaces

with Corrosion, A S T M S T P 5 6 7 , P h il a d e l p h ia ( 1 9 7 4 ).

[5 ] F . J . H e y m a n n , " O n t h e T i m e D e p e n d e n c e o f t h e R a t e o f

E r o s i o n D u e t o I m p i n g e m e n t o r C a v i t a t i o n " , i n : Eros ion

by Cavitat ion o r Imp ingem ent, A S T M S T P 4 0 8, P h i la d e l -p h i a ( 1 9 6 7 ) .

[6 ] A . T h i r u v e n g a d a m a n d S . L . R u d y , " E x p e r i m e n t a l a n d

A n a l y t i c a l I n v e s t i g a t i o n s o n M u l t i p l e L i q u i d I m p a c t E r o -

s i o n " , N A S A C R - 1 2 8 8 , W a s h i n g t o n , D C ( 1 9 6 9) .

[ 7] G . S p r i n g e r , Eros ion by L iqu id Impac t , W i l e y , N e w Y o r k

( 1 9 7 6 ) .

[8 ] J . N o s k i e v i c , " T h e e x t e n d e d m a t h e m a t i c a l m o d e l o f c a v i -

t a t io n a n d e r o s i o n w e a r " , i n : Proc . 6 th In t . Con f . on

Eros ion by L iqu id and So l id Impac t , C a m b r i d g e , E n g -

l a n d ( 1 9 8 3 ) .

erosion over time

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