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Cavitation phenomena in soil-projectile interaction
InZ.J. Impao~ En~ng. Vol.3, No.3, pp.167-178, 1985 Printed in Great Britain
O734-743X/85 $3.00+0.00 © 1985 Pergamon Press Ltd.
C A V I T A T I O N P H E N O M E N A IN SOIL-PROJECTILE INTERACTION David Z. Yankelevsky Department of Civil Engineering, Technion - - Israel Institute of Technology, Haifa, Israel 32000
(Received 12 September 1984; in revised form 12 July 1985) Stlmm~--~ - An analytical criterion is developed for the contact area of a projectile nose penetrating through soil. The cavitation criterion is based on t~he disc penetration model developed earlier and defines the critical velocity at which separation starts as well as the relationship between any higher velocity and the area of contact. At very high velocities the contact area is found to approach an asymptotic value. Some examples are studied from which both nose shape and soil failure criterion are found to have a pronounced effect on separation.
NOTATION A a
B
b CRH
D flZ)_ Fi(Z) m n
Pst p(z,t) R (z ,t) c
(t)
w(t) z z z cr Z
material parameter geometry constant material parameter geometry constant caliber radius head projectile diameter nose shape function functions of nose geometry (i = 1,2,3,4) projectile mass constant material parameter interaction pressure on internal boundary radial displacement of internal boundary radius of curvature critical velocity penetration depth separation distance local coordinate critical separation distance distance from top surface
angle 8 Am
Po T qJ
angle added mass volumetric locking strain initial mass density principal stress difference at failure angle angle
INTRODUCTION D u r i n g the last few y e a r s m u c h t h e o r e t i c a l w o r k has b e e n d i r e c t e d t o w a r d s the d e v e l o p m e n t of c o m p u t e r c o d e s for p r o j e c t i l e p e n e t r a t i o n a n a l y s i s [I-3]. These c o d e s are b a s e d on the f i e l d e q u a t i o n s and on m e d i u m and p r o j e c t i l e p r o p e r t i e s 167
L68
David
Z. Y a n k e l e v s k y
and enable the d e t a i l e d analysis of the responses of p e n e t r a t o r s and targets. It was found that a few c o m p u t e r hours were r e q u i r e d to c o m p l e t e the calculations of a single p e n e t r a t i o n event; thus c u r r e n t use and p a r a m e t e r analysis are p r a c t i c a l l y prevented. In those cases that were studied, very good agreement was found with test data, and further u n d e r s t a n d i n g of the p e n e t r a t i o n p h e n o m e n o n has been achieved. In most of these cases, it is shown that full c o n t a c t does not always exist b e t w e e n the p e n e t r a t o r and the s u r r o u n d i n g soil, s e p a r a t i o n may occur and partial c o n t a c t then exists. A few s i m p l i f i e d models have been p r o p o s e d for p e n e t r a t i o n c a l c u l a t i o n s , based on the s p h e r i c a l or the c y l i n d r i c a l cavity e x p a n s i o n m o d e l s [4,5]. The models do not p r o p e r l y r e p r e s e n t the s o i l - p e n e t r a t o r interaction, since it is assumed that full c o n t a c t exists b e t w e e n the p e n e t r a t o r nose and the soil, and that the i n t e r a c t i o n p r e s s u r e d i s t r i b u t i o n is known in a d v a n c e [4]. These a s s u m p t i o n s are not s u p p o r t e d by c o m p u t e r code results. When the s i m p l i f i e d m o d e l s were c h e c k e d with respect to test data, d i c t i o n s were found to c o r r e s p o n d very p o o r l y with test data [6].
pre-
R e c e n t l y a new a n a l y t i c a l m o d e l has been developed, in w h i c h the m e d i u m is r e p r e s e n t e d by discs of small thickness [7]. The internal b o u n d a r y radius of each disc equals to the local radius of the p r o j e c t i l e nose as long as comp r e s s i v e i n t e r a c t i o n p r e s s u r e exists. With the aid of the discs model, the c r i t e r i a for s e p a r a t i o n and the p r e s s u r e d i s t r i b u t i o n may n a t u r a l l y be found. Earlier c o m p a r i s o n s of the m o d e l ' s p r e d i c t i o n s with test data and with twod i m e n s i o n a l c o m p u t e r code c a l c u l a t i o n s show good a g r e e m e n t [7,8] and encourage further studies of p e n e t r a t i o n p h e n o m e n a with the aid of the m o d e l ~ This paper is d e v o t e d to studying the c a v i t a t i o n phenomena. A separation c r i t e r i o n is d e v e l o p e d for a p e n e t r a t o r having an a r b i t r a r y shape, and the effect of p e n e t r a t o r and m e d i u m p r o p e r t i e s on the size of the contact zone are studied.
MATHEMATICAL FORMULATION C o n s i d e r a p r o j e c t i l e p e n e t r a t i n g into a s e m i - i n f i n i t e m e d i u m (Fig. I). It is assumed that the p r o j e c t i l e impacts normal to the top surface and its d i r e c t i o n is not c h a n g e d d u r i n g the p e n e t r a t i o n event. A s i m p l i f i e d r e p r e s e n t a t i o n of the m e d i u m by discs has been p r o p o s e d [7], where the b a s i c a s s u m p t i o n s of this model are: I. 2. 3. 4. 5. 6. 7.
Heat t r a n s f e r and d i s s i p a t i o n b e t w e e n p r o j e c t i l e and soil is neglected; The p r o j e c t i l e is a rigid body and only soil d e f o r m a t i o n s are considered; The p r i n c i p a l m e c h a n i s m g o v e r n i n g p e n e t r a t i o n is assumed to be that of high local volume changes and only plastic strains are considered; F r i c t i o n b e t w e e n the soil and the nose surface is a s s u m e d to be negligible; The p r o j e c t i l e nose is slender; Only the p r o j e c t i l e nose, or part of it, is in c o n t a c t with the soil; Shear stresses at the i n t e r f a c e s b e t w e e n discs are n e g l e c t e d c o m p a r e d to the radial stresses~
D e r i v a t i o n of the g o v e r n i n g e q u a t i o n s s u b j e c t e d to the above the e x p r e s s i o n for the i n t e r a c t i o n p r e s su r e [7]: p(z,t) Pst
= Pst
+ A'R2 (z,t)
= 1 . (~) 2 T • £n I
I
B = ~ p-Zn(
)
yields
(I) (l-a)
1
A = ~ P0 + 2 p. Zn(
+ B'R(z,t)-R(z,t)
assumptions
)
(l-b) ( 1 -c)
Cavitation
where
p(z,t)
= the radial c o m p o n e n t (considered p o s i t i v e
phenomena
169
of the i n t e r a c t i o n in compression);
pressure,
R(z,t),R(z,t),R(z,t) = the i n t e r n a l b o u n d a r y radial values ment, v e l o c i t y and a c c e l e r a t i o n c o r r e s p o n d i n g l y ; T
= principal
g
= the v o l u m e t r i c
P0 = the p A local
coordinate =
difference
locking
free field mass
= the mass
z
stress
w(t)
density system
-
at depth
z
of d i s p l a c e -
at failure;
strain;
density;
behind
the plastic
z is a t t a c h e d
shock
to the nose
front. tip,
where:
z.
(2)
w(t) is the p e n e t r a t i o n depth of the nose tip, at time t. The nose shape funct i o n f(z) is s u p p o s e d c o n v e x with c o n t i n u o u s first and second d e r i v a t i v e s , and is equal to the p r o j e c t i l e local radius R(z,t) : R(Z,t) Differentiating
= f(z).
(3)
equation
(3) with
~(z,t)
=
~(t)
~f(z) ~z
~(z,t)
=
~'
•
(t)
•
to time and using
÷
and
w(t) (4-b)
~f(z) ~z
(2) yields: (4-a)
-
w2 (t)-f"(z)
into e q u a t i o n
+ w(t)
(I) y i e l d s
~v(t)
dZ
f(z)
FIGURE I.
equation
_ w(t).f' (z)
~z---~r--. ~2f(z)
S u b s t i t u t i n g e x p r e s s i o n s (4-a) for the i n t e r a c t i o n pressure:
respect
Projectile penetrating into a semli-infinite medi~.
• f'
(z).(4-b)
the e x p r e s s i o n
LTO
David
p(Z,t)
= Pst
Z. Y a n k e l e v s k y
+ {A[f' (z)] 2 + B . f ( z ) f " (z) }w 2 (t)
+ B ' f ( z ) . f ' (z).w(t) . (5)
Thus the i n t e r a c t i o n p r e s s u r e on a disc at d e p t h Z d e p e n d s on the i n s t a n t a n e o u s p r o j e c t i l e m o t i o n , on the n o s e s h a p e f u n c t i o n and on the m e d i u m p r o p e r t i e s . C o n t a c t b e t w e e n the p r o j e c t i l e n o s e and the m e d i u m e x i s t s as long as p(Z,t) > 0. The c r i t e r i o n for s e p a r a t i o n at d i s t a n c e z f r o m the n o s e tip therefore: Pst
+ {A[f' (~)]2
+ Bf(~).f.,(~)}~2
The c o n t r i b u t i o n s of all d i s c s y i e l d to t o t a l r e s i s t i v e force: P(t)
(t)
in c o n t a c t
+ Bf(z) f' (z)w(t) with
the nose
is
= 0.
are
(6)
integrated
to
= f 2~.f(z)-f' (z).p(Z,t)-dz. 0
(7)
S u b s t i t u t i n g e q u a t i o n (5) into e q u a t i o n (7), a n d a s s u m i n g that the m e d i u m p a r a m e t e r s have the same v a l u e s for all d i s c s in c o n t a c t w i t h the p r o j e c t i l e at time t, yields: P(t)
= 2~{Ps t / f(z).f' (z)dz 0 • f " ( z ) d z ] . w 2 (t)
By u s i n g
equations
(l-b)
and
+
+
[A f f(z).(f' (z))3dz 0
[B / f2 (z) . (f, (z) ) 2dz]~(t) }. 0
(I-c)
(8)
we obtain: z
P(t)
+ B I f2 (z)'f' (z) 0
= 2Z{Ps t / f(z)f' (z)dz 0
z
+ B[/ f(z)-(f' (z))3dz 0
+ [ f2 (z).f' (z) 0
z
I • f " ( z ) d z ] w 2 (t) + ~ p0[/ f ( z ) - ( f ' ( z ) ) 3 d z ] w 2 0
z
(t) + B[/ f2 (z) 0
(f' (z))2dz]w(t) }. The
expression P(t)
for the r e s i s t i v e = Pst
(9) ~
force
m a y be w r i t t e n
I
" FI (z)
+ B . F 2 ( z ) w 2 (t) + ~
P0 F3(z)
as follows:
.~2
(t)
+ B.F4(z)'Q(t) (10)
where
Fi(z)
are
of s e p a r a t i o n
functions
of the n o s e
geometry,
depending
on the
coordinate
z: z
F I (z)
= 2z / f(z)'f' (z)dz 0
= 2~'f 2 (z)
(10-a)
F2(Z)
= 27 /
F3(z)
= 2~ I f(z)" (f' (z))3dz 0
(I0-c)
F4(z)
= 27 / f2 (z)" (f' (z))2dz. 0
(10-d)
z
[f(z)" (f' (z)) 3 + f2 (z)f' ( z ) f " ( z ) ] d z
0
= 2 ~ [ f ( z ) ' f ' (z)] 2 (10-b)
Z
It is a s s u m e d f (o)
that = 0
the nose
is pointed: (11)
Cavitation
Using Newton's
second
m~(t)
law,
= - PstF1lz)
w h e r e m = the p r o j e c t i l e The
the e q u a t i o n
instantaneous -
The t e r m B'F4(z) (13)
Pst [I -
is t h e r e f o r e : I (~) + ~ P0F3
B F2(z)
F4(~)
is: (12)
w 2 (t).
(13)
m+B F4(~)
m a y be c o n s i d e r e d
into e q u a t i o n
of m o t i o n
mass.
deceleration
m+B
171
I [B F 2 ( z ) - + ~ 0 0 F3lz)]w2 (t) - B F4(z)Q(t)
-
Pst.F1(z) W(t)
phenomena
as the a d d e d m a s s
Am.
Substituting
equation
(6) y i e l d s :
B-f(z) f' (z).F I (~) B F2(z) + I P0F3 (z) m + Am ] - {B.f(z) f' (z) m + Am
I
2 P0 [f' (z)]2
- B[(f' (z)) 2 + f(z) f"(z°)]} ~2 (t) = 0.
In e q u a t i o n (14) the s e p a r a t i o n d e p e n d i n g on the m e d i u m p r o p e r t i e s , o r d i n a t e ~.
(14)
v e l o c i t y may be found for e v e r y nose shape, the shape f u n c t i o n a n d the s e p a r a t i o n cob
W h e n the s e p a r a t i o n v e l o c i t y is d e t e r m i n e d i n t e r a c t i o n p r e s s u r e d i s t r i b u t i o n m a y be found p(z,t)
= Pst[1
for a g i v e n c o o r d i n a t e from e q u a t i o n s (5) and
z, the (13):
I B'F 2(z)+ ~ P0F3(z) - {B.f(z)f' (z) re+Am
F I (z) - B.f(z)f' (z) m - ~ ]
I - ~p0[f' (z) ]2 - B[(f' (z)) 2 + f ( z ) ' f " ( Z ) ] }
W2 (t)
0 _< z _< z. (15)
SPECIAL I.
Steady
CASES
state p e n e t r a t i o n
In this case the p e n e t r a t o r is d r i v e n at a c o n s t a n t v e l o c i t y w(t) i n t e r a c t i o n p r e s s u r e , in this case, is f o u n d from e q u a t i o n (5): p(Z,t) The c r i t e r i o n
= Pst + {A[f' (z)]2
for s e p a r a t i o n
Pst + {A[f' (z)]2 and the s e p a r a t i o n
2.
+ B f(z).f"(z)}
and the
w2 (t).
at d i s t a n c e z f r o m the n o s e tip is then:
+ B f(z).f" (z)} Q2 (t) = 0
velocity
(16)
is i m m e d i a t e l y
(17)
found.
C_ritical v e l o c i t y
The c r i t i c a l v e l o c i t y is d e f i n e d as the v e l o c i t y w h e r e s e p a r a t i o n o c c u r s at = L (L is the n o s e length). At any l ower v e l o c i t y , full c o n t a c t exists. The c r i t i c a l v e l o c i t y Wc(t) in the s t e a d y state case is: 1/2 We (t) = { - A[f' (L)]~s~ B f(L)f"(L) } In the g e n e r a l
case,
it is f o u n d
B f(L) f' (L)FI(L) Pst [1 m+Am
f r o m the f o l l o w i n g
expression: I B F2(L) + ~ P0F3(L) - {B.f(L)f' (L) m+Am
(18)
172
David
I
2 P0 [f' (L)]2 In a s p e c i a l tangent:
but
f(L)
very
Z. Y a n k e l e v s k y
- B[(f' (L))2
common
case
+ f(L)'f"(L)]}
where
the nose
w 2 (t) c
and
= 0.
the b o d y
(19)
have
a common
= D/2
(20-a)
f' (L) = 0.
(20-b)
Hence: We (t) =
2.Ps t I/2 BD f"(L) ]
[
(21)
In this s p e c i a l c a s e the s t e a d y state c r i t i c a l as the g e n e r a l case c r i t i c a l v e l o c i t y .
3.
Critical
velocity
has
the
same
magnitude
area
The c r i t i c a l a r e a is that area of the p e n e t r a t o r w h i c h r e m a i n s in c o n t a c t w i t h the m e d i u m w h e n the v e l o c i t y a p p r o a c h e s an i n f i n i t e value. F r o m e q u a t i o n (14) it c a n be s h o w n that the c o o r d i n a t e Zcr, w h i c h d e f i n e s t h a t area, is f o u n d from the e q u a t i o n : I B ' F 2 ( Z c r ) + 2 P0F3(Zcr) I B ' f ( Z c r ) ' f ' (Zcr) m+Am = 2 P0 [f' (Zcr)]2
+ B{ [f' (Zcr)] 2 + f ( Z c r ) - f " ( Z c r )
}.
(22)
For a g i v e n p e n e t r a t o r w i t h k n o w n m e d i u m p r o p e r t i e s , s a t i s f i e s e q u a t i o n (22) m a y be f o u n d (see Figs 3-5).
Example
criterion
ogive CRH
where: The
of z
that cr
I
Separation The
a value
R
shape
nose =
shape
for an o g i v e
is d e f i n e d
nose,
shown
in Fig.
by the C a l i b e r - R a d i u s - H e a d
2.
parameter: (23)
--
D
= the
radius
function f(z)
is s t u d i e d
of the n o s e
circular
arc.
is:
= R[cos~
- cos~]
(24)
where: = sin -I
Differentiating
[sin~
equation
- ~].
(24) w i t h
(.25)
respect
to z y i e l d s : (26-a)
f' (z) = tan~ f"(z) The
critical
for ~ = 0
I tan2~ [ cos-----~- cos~
: _ i
velocity
(see Fig.
Wc(t)
is f o u n d
2) we get:
(26-b)
]" from equation
(21).
Using
equation
(26-b)
Cavitation
FIGURE 2.
phenomena
173
The ogive nose.
700
600
500 (.) m
/0=120
400 >I0
B
300 -
;.,,o p [~~-~e-:~]
W
m = 20 Z 0 F-
I11
r = 5 MPG I00 -
hi m
kg" sec2
200 -
e=O.I CRH = 6 J 0.1 DISTANCE
FI~3RE 3.
Effect of m e d i ~
0.2 FROM
I 0.3 NOSE
I 0.4 TIP ( M . )
density on cc~tact area.
0.48
174
David
Wc(t)
= [2
• (I - £)
Z. Y a n k e l e v s k y
• CRH] I/2
That expression has already been derived is o b t a i n e d h e r e as a s p e c i a l case.
directly
in an e a r l i e r
work
[8] a n d
T o f i n d the s e p a r a t i o n v e l o c i t y for a n y v a l u e z = z, we use e q u a t i o n s (24), (26-a) a n d (26-b) t o e x p r e s s t h e c o e f f i c i e n t s F (z) in e q u a t i o n s (10-a) to 1 (10-d). F o r e a c h v a l u e of z, n u m e r i c a l v a l u e s are o b t a i n e d f o r the c o e f f i c i e n t s a n d the c o r r e s p o n d i n g s e p a r a t i o n v e l o c i t y is f o u n d f r o m e q u a t i o n (14). Figures 3-5 s h o w the d e p e n d e n c e of t h e c o n t a c t a r e a o n the i n s t a n t a n e o u s v e l o c i t y a n d the e f f e c t of m e d i u m p a r a m e t e r s s u c h as d e n s i t y (Fig. 3), v o l u m e t r i c s t r a i n (Fig. 4) a n d p r i n c i p a l s t r e s s d i f f e r e n c e at f a i l u r e (Fig. 5).
Kxample
2
To study how some shape parameters s h a p e f u n c t i o n is c h o s e n (Fig. 6): f(z) These
= az
parameters
affect
the
separation
velocity,
- b z n.
a, b, n a r e
found
from the
following
conditions:
700 6=0.4 __
500
--'--
E -- 0. I
....
E = 0.01
-
0.001
I I"
400 )F-
S W
300
>
z
0
200 _
CRH = 6
P= I1:
h,
^^_
uu
/
/
ko, sec 2
~ ' ~ -
.-"-J
m = 20 kg.sec z
,00 --
rrt
r=5MPa I 0.1 DISTANCE
FIGURE 4.
I 0.2
I 0.3
FROM
NOSE
I 0.4 TIP
(M.)
Effect of volumetric strain on contact area.
0.48
a general
Cavitation
f (L)
These
175
D
:
f' (0)
= tans
f' (L)
= tanS.
conditions f(z)
phenomena
lead
= z.tan~
to +
the
shape
D [ ~ - L
n = tan8 - tans D -- - tans 2L
function:
• tans
](
z
)n
B ~ s.
Figures 7 and 8 show the velocity versus contact area
influence of relationship
the for
angles s and 8 on the a certain problem.
700
600
500
:E 400 v
>I-
5 300
q w
zQ
200
F-
UJ o0
r = 5 Mo_p_g____./ P
,00 _ T= I MPo i
I
I
0.1
0.2
0.3
DISTANCE FIE~RE 5.
/
FROM
E
0.4
0.48
NOSE TIP (M.)
Effect of failure criterion on contact area.
separation
176
David
Z. Y a n k e l e v s k y
CONCLUSIONS The c a v i t a t i o n concept in s o i l - p r o j e c t i l e i n t e r a c t i o n is important and has p r a c t i c a l a p p l i c a t i o n in the design of m i l i t a r y p r o j e c t i l e s for both p e n e t r a bility and t r a j e c t o r y s t a b i l i t y points of view. T h e o r e t i c a l models for predicting such p h e n o m e n a are useful for an u n d e r s t a n d i n g of the r e l e v a n t factors that affect cavitation. In this paper an a n a l y t i c a l c r i t e r i o n for the contact area of a p r o j e c t i l e nose with the s u r r o u n d i n g soil m e d i u m is developed. For a g i v e n p e n e t r a t o r and medium, the c r i t e r i o n for full contact is formulated from w h i c h the c o r r e s p o n d i n g c r i t i c a l v e l o c i t y may be found. For higher v e l o c i t i e s than the critical, s e p a r a t i o n occurs and only partial contact exists. It is shown that there is always a finite area in c o n t a c t with the m e d i u m and the c r i t e r i o n for the c r i t i c a l area at a very high v e l o c i t y is formulated. From studying a few problems, it may be c o n c l u d e d that s e p a r a t i o n c r i t e r i o n is st r o n g l y d e p e n d e n t on penetrators' shape and m e d i u m failure criterion, and less s e n s i t i v e to m e d i u m d e n s i t y and v o l u m e t r i c strain.
•!
D/2
~,z
Z
FIGURE 6.
Nose of a general shape.
_
Cavitation
phenomena
177
C o r r e l a t i o n of a t h e o r e t i c a l p r e d i c t e d d e c e l e r a t i o n - t i m e h i s t o r y with the c o r r e s p o n d i n g e x p e r i m e n t a l data, w h i c h has been shown in e a r l i e r works [7,8], alth o u g h e s s e n t i a l in itself, does not n e c e s s a r i l y v a l i d a t e the p r e s e n t model. V e r i f i c a t i o n of a c a v i t a t i o n m o d e l should include e x p e r i m e n t a l o b s e r v a t i o n s . Hill's p a p e r [9] c o n t a i n s a report on e x p e r i m e n t a l work of c a v i t a t i o n in metals. However, m o d e r n e x p e r i m e n t a t i o n should use flash x-ray p h o t o g r a p h y to m e a s u r e the c o n t a c t area of the nose w i t h the target.
700
i
600
i
. . . .
i
-
a
= 60
°
a
=
a
= 30 °
45 °
500
400
0
500
>
i\ "\
o
\ \
z_ o 200
I00
I
I
0.2 0.5 DISTANCE FROM NOSE TIP (M.)
FIC~RE 7.
Effect of shape parameters c~ separation criterion.
0.4
78
David
Z. Y a n k e l e v s k y
700 I
I
k
r
= 5MPo
600 500 • 400
B -7°
"J' 300 > 200 I00
I
I
0.2 0.3 DISTANCE FROM NOSE TIP (M.) FIGURE 8.
0.4
Effect of shape parameters on separation criterion.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9.
L. Thigpen, Projectile penetration of elastic plastic earth media. J. Geotech. Engng. Div., ASCE 100, GT3 (1974). M. H. Wagner et al., Numerical analysis of projectile impact and deep penetration into earth media. WES Contract Report S-75-4, California Research and Technology, Inc. (1975). R. K. Byers and A. J. Chabai, Penetration calculations and measurements for a layered soil target. Int. J. Numer. Analyt. Methods Geomech. I, 107-138 (1977). R. S. Bernard and S. V. Hanagud, Dev~lopment of a projectile penetration theory. U.S. Army Waterways Experiment Station, Technical Report S-75-9 (1975). F. R. Norwood, Cylindrical cavity expansion in a locking soil. Sandia Labs, SLA-74-0201, Albuquerque, New Mexico (1974). P. F. ~a__dala, Evaluation of e~pirical and analytical procedures used for predicting the rigid body motion of an earth penetrator. Waterways Experiment Station, Report S-75-15, Vicksburg, Mississippi (1975). D. Z. Yankelevsky and M. A. Adin, A simplified analytical method for soil penetration analysis. Int. J. Numer. Analyt. Methods Geomech. 4, July-Sept (1980). D. Z. Yankelevsky and J. Gluck, Nose shape effect on high velocity soil penetration. Int. J. Mech. Sci. 22, (1980). R. Hill, Cavitation and the influence of headshape in attack of thick targets by nondeforming projectiles. J. Mech. Phys. Solids 28, 249-263 (1980).
O734-743X/85 $3.00+0.00 © 1985 Pergamon Press Ltd.
C A V I T A T I O N P H E N O M E N A IN SOIL-PROJECTILE INTERACTION David Z. Yankelevsky Department of Civil Engineering, Technion - - Israel Institute of Technology, Haifa, Israel 32000
(Received 12 September 1984; in revised form 12 July 1985) Stlmm~--~ - An analytical criterion is developed for the contact area of a projectile nose penetrating through soil. The cavitation criterion is based on t~he disc penetration model developed earlier and defines the critical velocity at which separation starts as well as the relationship between any higher velocity and the area of contact. At very high velocities the contact area is found to approach an asymptotic value. Some examples are studied from which both nose shape and soil failure criterion are found to have a pronounced effect on separation.
NOTATION A a
B
b CRH
D flZ)_ Fi(Z) m n
Pst p(z,t) R (z ,t) c
(t)
w(t) z z z cr Z
material parameter geometry constant material parameter geometry constant caliber radius head projectile diameter nose shape function functions of nose geometry (i = 1,2,3,4) projectile mass constant material parameter interaction pressure on internal boundary radial displacement of internal boundary radius of curvature critical velocity penetration depth separation distance local coordinate critical separation distance distance from top surface
angle 8 Am
Po T qJ
angle added mass volumetric locking strain initial mass density principal stress difference at failure angle angle
INTRODUCTION D u r i n g the last few y e a r s m u c h t h e o r e t i c a l w o r k has b e e n d i r e c t e d t o w a r d s the d e v e l o p m e n t of c o m p u t e r c o d e s for p r o j e c t i l e p e n e t r a t i o n a n a l y s i s [I-3]. These c o d e s are b a s e d on the f i e l d e q u a t i o n s and on m e d i u m and p r o j e c t i l e p r o p e r t i e s 167
L68
David
Z. Y a n k e l e v s k y
and enable the d e t a i l e d analysis of the responses of p e n e t r a t o r s and targets. It was found that a few c o m p u t e r hours were r e q u i r e d to c o m p l e t e the calculations of a single p e n e t r a t i o n event; thus c u r r e n t use and p a r a m e t e r analysis are p r a c t i c a l l y prevented. In those cases that were studied, very good agreement was found with test data, and further u n d e r s t a n d i n g of the p e n e t r a t i o n p h e n o m e n o n has been achieved. In most of these cases, it is shown that full c o n t a c t does not always exist b e t w e e n the p e n e t r a t o r and the s u r r o u n d i n g soil, s e p a r a t i o n may occur and partial c o n t a c t then exists. A few s i m p l i f i e d models have been p r o p o s e d for p e n e t r a t i o n c a l c u l a t i o n s , based on the s p h e r i c a l or the c y l i n d r i c a l cavity e x p a n s i o n m o d e l s [4,5]. The models do not p r o p e r l y r e p r e s e n t the s o i l - p e n e t r a t o r interaction, since it is assumed that full c o n t a c t exists b e t w e e n the p e n e t r a t o r nose and the soil, and that the i n t e r a c t i o n p r e s s u r e d i s t r i b u t i o n is known in a d v a n c e [4]. These a s s u m p t i o n s are not s u p p o r t e d by c o m p u t e r code results. When the s i m p l i f i e d m o d e l s were c h e c k e d with respect to test data, d i c t i o n s were found to c o r r e s p o n d very p o o r l y with test data [6].
pre-
R e c e n t l y a new a n a l y t i c a l m o d e l has been developed, in w h i c h the m e d i u m is r e p r e s e n t e d by discs of small thickness [7]. The internal b o u n d a r y radius of each disc equals to the local radius of the p r o j e c t i l e nose as long as comp r e s s i v e i n t e r a c t i o n p r e s s u r e exists. With the aid of the discs model, the c r i t e r i a for s e p a r a t i o n and the p r e s s u r e d i s t r i b u t i o n may n a t u r a l l y be found. Earlier c o m p a r i s o n s of the m o d e l ' s p r e d i c t i o n s with test data and with twod i m e n s i o n a l c o m p u t e r code c a l c u l a t i o n s show good a g r e e m e n t [7,8] and encourage further studies of p e n e t r a t i o n p h e n o m e n a with the aid of the m o d e l ~ This paper is d e v o t e d to studying the c a v i t a t i o n phenomena. A separation c r i t e r i o n is d e v e l o p e d for a p e n e t r a t o r having an a r b i t r a r y shape, and the effect of p e n e t r a t o r and m e d i u m p r o p e r t i e s on the size of the contact zone are studied.
MATHEMATICAL FORMULATION C o n s i d e r a p r o j e c t i l e p e n e t r a t i n g into a s e m i - i n f i n i t e m e d i u m (Fig. I). It is assumed that the p r o j e c t i l e impacts normal to the top surface and its d i r e c t i o n is not c h a n g e d d u r i n g the p e n e t r a t i o n event. A s i m p l i f i e d r e p r e s e n t a t i o n of the m e d i u m by discs has been p r o p o s e d [7], where the b a s i c a s s u m p t i o n s of this model are: I. 2. 3. 4. 5. 6. 7.
Heat t r a n s f e r and d i s s i p a t i o n b e t w e e n p r o j e c t i l e and soil is neglected; The p r o j e c t i l e is a rigid body and only soil d e f o r m a t i o n s are considered; The p r i n c i p a l m e c h a n i s m g o v e r n i n g p e n e t r a t i o n is assumed to be that of high local volume changes and only plastic strains are considered; F r i c t i o n b e t w e e n the soil and the nose surface is a s s u m e d to be negligible; The p r o j e c t i l e nose is slender; Only the p r o j e c t i l e nose, or part of it, is in c o n t a c t with the soil; Shear stresses at the i n t e r f a c e s b e t w e e n discs are n e g l e c t e d c o m p a r e d to the radial stresses~
D e r i v a t i o n of the g o v e r n i n g e q u a t i o n s s u b j e c t e d to the above the e x p r e s s i o n for the i n t e r a c t i o n p r e s su r e [7]: p(z,t) Pst
= Pst
+ A'R2 (z,t)
= 1 . (~) 2 T • £n I
I
B = ~ p-Zn(
)
yields
(I) (l-a)
1
A = ~ P0 + 2 p. Zn(
+ B'R(z,t)-R(z,t)
assumptions
)
(l-b) ( 1 -c)
Cavitation
where
p(z,t)
= the radial c o m p o n e n t (considered p o s i t i v e
phenomena
169
of the i n t e r a c t i o n in compression);
pressure,
R(z,t),R(z,t),R(z,t) = the i n t e r n a l b o u n d a r y radial values ment, v e l o c i t y and a c c e l e r a t i o n c o r r e s p o n d i n g l y ; T
= principal
g
= the v o l u m e t r i c
P0 = the p A local
coordinate =
difference
locking
free field mass
= the mass
z
stress
w(t)
density system
-
at depth
z
of d i s p l a c e -
at failure;
strain;
density;
behind
the plastic
z is a t t a c h e d
shock
to the nose
front. tip,
where:
z.
(2)
w(t) is the p e n e t r a t i o n depth of the nose tip, at time t. The nose shape funct i o n f(z) is s u p p o s e d c o n v e x with c o n t i n u o u s first and second d e r i v a t i v e s , and is equal to the p r o j e c t i l e local radius R(z,t) : R(Z,t) Differentiating
= f(z).
(3)
equation
(3) with
~(z,t)
=
~(t)
~f(z) ~z
~(z,t)
=
~'
•
(t)
•
to time and using
÷
and
w(t) (4-b)
~f(z) ~z
(2) yields: (4-a)
-
w2 (t)-f"(z)
into e q u a t i o n
+ w(t)
(I) y i e l d s
~v(t)
dZ
f(z)
FIGURE I.
equation
_ w(t).f' (z)
~z---~r--. ~2f(z)
S u b s t i t u t i n g e x p r e s s i o n s (4-a) for the i n t e r a c t i o n pressure:
respect
Projectile penetrating into a semli-infinite medi~.
• f'
(z).(4-b)
the e x p r e s s i o n
LTO
David
p(Z,t)
= Pst
Z. Y a n k e l e v s k y
+ {A[f' (z)] 2 + B . f ( z ) f " (z) }w 2 (t)
+ B ' f ( z ) . f ' (z).w(t) . (5)
Thus the i n t e r a c t i o n p r e s s u r e on a disc at d e p t h Z d e p e n d s on the i n s t a n t a n e o u s p r o j e c t i l e m o t i o n , on the n o s e s h a p e f u n c t i o n and on the m e d i u m p r o p e r t i e s . C o n t a c t b e t w e e n the p r o j e c t i l e n o s e and the m e d i u m e x i s t s as long as p(Z,t) > 0. The c r i t e r i o n for s e p a r a t i o n at d i s t a n c e z f r o m the n o s e tip therefore: Pst
+ {A[f' (~)]2
+ Bf(~).f.,(~)}~2
The c o n t r i b u t i o n s of all d i s c s y i e l d to t o t a l r e s i s t i v e force: P(t)
(t)
in c o n t a c t
+ Bf(z) f' (z)w(t) with
the nose
is
= 0.
are
(6)
integrated
to
= f 2~.f(z)-f' (z).p(Z,t)-dz. 0
(7)
S u b s t i t u t i n g e q u a t i o n (5) into e q u a t i o n (7), a n d a s s u m i n g that the m e d i u m p a r a m e t e r s have the same v a l u e s for all d i s c s in c o n t a c t w i t h the p r o j e c t i l e at time t, yields: P(t)
= 2~{Ps t / f(z).f' (z)dz 0 • f " ( z ) d z ] . w 2 (t)
By u s i n g
equations
(l-b)
and
+
+
[A f f(z).(f' (z))3dz 0
[B / f2 (z) . (f, (z) ) 2dz]~(t) }. 0
(I-c)
(8)
we obtain: z
P(t)
+ B I f2 (z)'f' (z) 0
= 2Z{Ps t / f(z)f' (z)dz 0
z
+ B[/ f(z)-(f' (z))3dz 0
+ [ f2 (z).f' (z) 0
z
I • f " ( z ) d z ] w 2 (t) + ~ p0[/ f ( z ) - ( f ' ( z ) ) 3 d z ] w 2 0
z
(t) + B[/ f2 (z) 0
(f' (z))2dz]w(t) }. The
expression P(t)
for the r e s i s t i v e = Pst
(9) ~
force
m a y be w r i t t e n
I
" FI (z)
+ B . F 2 ( z ) w 2 (t) + ~
P0 F3(z)
as follows:
.~2
(t)
+ B.F4(z)'Q(t) (10)
where
Fi(z)
are
of s e p a r a t i o n
functions
of the n o s e
geometry,
depending
on the
coordinate
z: z
F I (z)
= 2z / f(z)'f' (z)dz 0
= 2~'f 2 (z)
(10-a)
F2(Z)
= 27 /
F3(z)
= 2~ I f(z)" (f' (z))3dz 0
(I0-c)
F4(z)
= 27 / f2 (z)" (f' (z))2dz. 0
(10-d)
z
[f(z)" (f' (z)) 3 + f2 (z)f' ( z ) f " ( z ) ] d z
0
= 2 ~ [ f ( z ) ' f ' (z)] 2 (10-b)
Z
It is a s s u m e d f (o)
that = 0
the nose
is pointed: (11)
Cavitation
Using Newton's
second
m~(t)
law,
= - PstF1lz)
w h e r e m = the p r o j e c t i l e The
the e q u a t i o n
instantaneous -
The t e r m B'F4(z) (13)
Pst [I -
is t h e r e f o r e : I (~) + ~ P0F3
B F2(z)
F4(~)
is: (12)
w 2 (t).
(13)
m+B F4(~)
m a y be c o n s i d e r e d
into e q u a t i o n
of m o t i o n
mass.
deceleration
m+B
171
I [B F 2 ( z ) - + ~ 0 0 F3lz)]w2 (t) - B F4(z)Q(t)
-
Pst.F1(z) W(t)
phenomena
as the a d d e d m a s s
Am.
Substituting
equation
(6) y i e l d s :
B-f(z) f' (z).F I (~) B F2(z) + I P0F3 (z) m + Am ] - {B.f(z) f' (z) m + Am
I
2 P0 [f' (z)]2
- B[(f' (z)) 2 + f(z) f"(z°)]} ~2 (t) = 0.
In e q u a t i o n (14) the s e p a r a t i o n d e p e n d i n g on the m e d i u m p r o p e r t i e s , o r d i n a t e ~.
(14)
v e l o c i t y may be found for e v e r y nose shape, the shape f u n c t i o n a n d the s e p a r a t i o n cob
W h e n the s e p a r a t i o n v e l o c i t y is d e t e r m i n e d i n t e r a c t i o n p r e s s u r e d i s t r i b u t i o n m a y be found p(z,t)
= Pst[1
for a g i v e n c o o r d i n a t e from e q u a t i o n s (5) and
z, the (13):
I B'F 2(z)+ ~ P0F3(z) - {B.f(z)f' (z) re+Am
F I (z) - B.f(z)f' (z) m - ~ ]
I - ~p0[f' (z) ]2 - B[(f' (z)) 2 + f ( z ) ' f " ( Z ) ] }
W2 (t)
0 _< z _< z. (15)
SPECIAL I.
Steady
CASES
state p e n e t r a t i o n
In this case the p e n e t r a t o r is d r i v e n at a c o n s t a n t v e l o c i t y w(t) i n t e r a c t i o n p r e s s u r e , in this case, is f o u n d from e q u a t i o n (5): p(Z,t) The c r i t e r i o n
= Pst + {A[f' (z)]2
for s e p a r a t i o n
Pst + {A[f' (z)]2 and the s e p a r a t i o n
2.
+ B f(z).f"(z)}
and the
w2 (t).
at d i s t a n c e z f r o m the n o s e tip is then:
+ B f(z).f" (z)} Q2 (t) = 0
velocity
(16)
is i m m e d i a t e l y
(17)
found.
C_ritical v e l o c i t y
The c r i t i c a l v e l o c i t y is d e f i n e d as the v e l o c i t y w h e r e s e p a r a t i o n o c c u r s at = L (L is the n o s e length). At any l ower v e l o c i t y , full c o n t a c t exists. The c r i t i c a l v e l o c i t y Wc(t) in the s t e a d y state case is: 1/2 We (t) = { - A[f' (L)]~s~ B f(L)f"(L) } In the g e n e r a l
case,
it is f o u n d
B f(L) f' (L)FI(L) Pst [1 m+Am
f r o m the f o l l o w i n g
expression: I B F2(L) + ~ P0F3(L) - {B.f(L)f' (L) m+Am
(18)
172
David
I
2 P0 [f' (L)]2 In a s p e c i a l tangent:
but
f(L)
very
Z. Y a n k e l e v s k y
- B[(f' (L))2
common
case
+ f(L)'f"(L)]}
where
the nose
w 2 (t) c
and
= 0.
the b o d y
(19)
have
a common
= D/2
(20-a)
f' (L) = 0.
(20-b)
Hence: We (t) =
2.Ps t I/2 BD f"(L) ]
[
(21)
In this s p e c i a l c a s e the s t e a d y state c r i t i c a l as the g e n e r a l case c r i t i c a l v e l o c i t y .
3.
Critical
velocity
has
the
same
magnitude
area
The c r i t i c a l a r e a is that area of the p e n e t r a t o r w h i c h r e m a i n s in c o n t a c t w i t h the m e d i u m w h e n the v e l o c i t y a p p r o a c h e s an i n f i n i t e value. F r o m e q u a t i o n (14) it c a n be s h o w n that the c o o r d i n a t e Zcr, w h i c h d e f i n e s t h a t area, is f o u n d from the e q u a t i o n : I B ' F 2 ( Z c r ) + 2 P0F3(Zcr) I B ' f ( Z c r ) ' f ' (Zcr) m+Am = 2 P0 [f' (Zcr)]2
+ B{ [f' (Zcr)] 2 + f ( Z c r ) - f " ( Z c r )
}.
(22)
For a g i v e n p e n e t r a t o r w i t h k n o w n m e d i u m p r o p e r t i e s , s a t i s f i e s e q u a t i o n (22) m a y be f o u n d (see Figs 3-5).
Example
criterion
ogive CRH
where: The
of z
that cr
I
Separation The
a value
R
shape
nose =
shape
for an o g i v e
is d e f i n e d
nose,
shown
in Fig.
by the C a l i b e r - R a d i u s - H e a d
2.
parameter: (23)
--
D
= the
radius
function f(z)
is s t u d i e d
of the n o s e
circular
arc.
is:
= R[cos~
- cos~]
(24)
where: = sin -I
Differentiating
[sin~
equation
- ~].
(24) w i t h
(.25)
respect
to z y i e l d s : (26-a)
f' (z) = tan~ f"(z) The
critical
for ~ = 0
I tan2~ [ cos-----~- cos~
: _ i
velocity
(see Fig.
Wc(t)
is f o u n d
2) we get:
(26-b)
]" from equation
(21).
Using
equation
(26-b)
Cavitation
FIGURE 2.
phenomena
173
The ogive nose.
700
600
500 (.) m
/0=120
400 >I0
B
300 -
;.,,o p [~~-~e-:~]
W
m = 20 Z 0 F-
I11
r = 5 MPG I00 -
hi m
kg" sec2
200 -
e=O.I CRH = 6 J 0.1 DISTANCE
FI~3RE 3.
Effect of m e d i ~
0.2 FROM
I 0.3 NOSE
I 0.4 TIP ( M . )
density on cc~tact area.
0.48
174
David
Wc(t)
= [2
• (I - £)
Z. Y a n k e l e v s k y
• CRH] I/2
That expression has already been derived is o b t a i n e d h e r e as a s p e c i a l case.
directly
in an e a r l i e r
work
[8] a n d
T o f i n d the s e p a r a t i o n v e l o c i t y for a n y v a l u e z = z, we use e q u a t i o n s (24), (26-a) a n d (26-b) t o e x p r e s s t h e c o e f f i c i e n t s F (z) in e q u a t i o n s (10-a) to 1 (10-d). F o r e a c h v a l u e of z, n u m e r i c a l v a l u e s are o b t a i n e d f o r the c o e f f i c i e n t s a n d the c o r r e s p o n d i n g s e p a r a t i o n v e l o c i t y is f o u n d f r o m e q u a t i o n (14). Figures 3-5 s h o w the d e p e n d e n c e of t h e c o n t a c t a r e a o n the i n s t a n t a n e o u s v e l o c i t y a n d the e f f e c t of m e d i u m p a r a m e t e r s s u c h as d e n s i t y (Fig. 3), v o l u m e t r i c s t r a i n (Fig. 4) a n d p r i n c i p a l s t r e s s d i f f e r e n c e at f a i l u r e (Fig. 5).
Kxample
2
To study how some shape parameters s h a p e f u n c t i o n is c h o s e n (Fig. 6): f(z) These
= az
parameters
affect
the
separation
velocity,
- b z n.
a, b, n a r e
found
from the
following
conditions:
700 6=0.4 __
500
--'--
E -- 0. I
....
E = 0.01
-
0.001
I I"
400 )F-
S W
300
>
z
0
200 _
CRH = 6
P= I1:
h,
^^_
uu
/
/
ko, sec 2
~ ' ~ -
.-"-J
m = 20 kg.sec z
,00 --
rrt
r=5MPa I 0.1 DISTANCE
FIGURE 4.
I 0.2
I 0.3
FROM
NOSE
I 0.4 TIP
(M.)
Effect of volumetric strain on contact area.
0.48
a general
Cavitation
f (L)
These
175
D
:
f' (0)
= tans
f' (L)
= tanS.
conditions f(z)
phenomena
lead
= z.tan~
to +
the
shape
D [ ~ - L
n = tan8 - tans D -- - tans 2L
function:
• tans
](
z
)n
B ~ s.
Figures 7 and 8 show the velocity versus contact area
influence of relationship
the for
angles s and 8 on the a certain problem.
700
600
500
:E 400 v
>I-
5 300
q w
zQ
200
F-
UJ o0
r = 5 Mo_p_g____./ P
,00 _ T= I MPo i
I
I
0.1
0.2
0.3
DISTANCE FIE~RE 5.
/
FROM
E
0.4
0.48
NOSE TIP (M.)
Effect of failure criterion on contact area.
separation
176
David
Z. Y a n k e l e v s k y
CONCLUSIONS The c a v i t a t i o n concept in s o i l - p r o j e c t i l e i n t e r a c t i o n is important and has p r a c t i c a l a p p l i c a t i o n in the design of m i l i t a r y p r o j e c t i l e s for both p e n e t r a bility and t r a j e c t o r y s t a b i l i t y points of view. T h e o r e t i c a l models for predicting such p h e n o m e n a are useful for an u n d e r s t a n d i n g of the r e l e v a n t factors that affect cavitation. In this paper an a n a l y t i c a l c r i t e r i o n for the contact area of a p r o j e c t i l e nose with the s u r r o u n d i n g soil m e d i u m is developed. For a g i v e n p e n e t r a t o r and medium, the c r i t e r i o n for full contact is formulated from w h i c h the c o r r e s p o n d i n g c r i t i c a l v e l o c i t y may be found. For higher v e l o c i t i e s than the critical, s e p a r a t i o n occurs and only partial contact exists. It is shown that there is always a finite area in c o n t a c t with the m e d i u m and the c r i t e r i o n for the c r i t i c a l area at a very high v e l o c i t y is formulated. From studying a few problems, it may be c o n c l u d e d that s e p a r a t i o n c r i t e r i o n is st r o n g l y d e p e n d e n t on penetrators' shape and m e d i u m failure criterion, and less s e n s i t i v e to m e d i u m d e n s i t y and v o l u m e t r i c strain.
•!
D/2
~,z
Z
FIGURE 6.
Nose of a general shape.
_
Cavitation
phenomena
177
C o r r e l a t i o n of a t h e o r e t i c a l p r e d i c t e d d e c e l e r a t i o n - t i m e h i s t o r y with the c o r r e s p o n d i n g e x p e r i m e n t a l data, w h i c h has been shown in e a r l i e r works [7,8], alth o u g h e s s e n t i a l in itself, does not n e c e s s a r i l y v a l i d a t e the p r e s e n t model. V e r i f i c a t i o n of a c a v i t a t i o n m o d e l should include e x p e r i m e n t a l o b s e r v a t i o n s . Hill's p a p e r [9] c o n t a i n s a report on e x p e r i m e n t a l work of c a v i t a t i o n in metals. However, m o d e r n e x p e r i m e n t a t i o n should use flash x-ray p h o t o g r a p h y to m e a s u r e the c o n t a c t area of the nose w i t h the target.
700
i
600
i
. . . .
i
-
a
= 60
°
a
=
a
= 30 °
45 °
500
400
0
500
>
i\ "\
o
\ \
z_ o 200
I00
I
I
0.2 0.5 DISTANCE FROM NOSE TIP (M.)
FIC~RE 7.
Effect of shape parameters c~ separation criterion.
0.4
78
David
Z. Y a n k e l e v s k y
700 I
I
k
r
= 5MPo
600 500 • 400
B -7°
"J' 300 > 200 I00
I
I
0.2 0.3 DISTANCE FROM NOSE TIP (M.) FIGURE 8.
0.4
Effect of shape parameters on separation criterion.
REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9.
L. Thigpen, Projectile penetration of elastic plastic earth media. J. Geotech. Engng. Div., ASCE 100, GT3 (1974). M. H. Wagner et al., Numerical analysis of projectile impact and deep penetration into earth media. WES Contract Report S-75-4, California Research and Technology, Inc. (1975). R. K. Byers and A. J. Chabai, Penetration calculations and measurements for a layered soil target. Int. J. Numer. Analyt. Methods Geomech. I, 107-138 (1977). R. S. Bernard and S. V. Hanagud, Dev~lopment of a projectile penetration theory. U.S. Army Waterways Experiment Station, Technical Report S-75-9 (1975). F. R. Norwood, Cylindrical cavity expansion in a locking soil. Sandia Labs, SLA-74-0201, Albuquerque, New Mexico (1974). P. F. ~a__dala, Evaluation of e~pirical and analytical procedures used for predicting the rigid body motion of an earth penetrator. Waterways Experiment Station, Report S-75-15, Vicksburg, Mississippi (1975). D. Z. Yankelevsky and M. A. Adin, A simplified analytical method for soil penetration analysis. Int. J. Numer. Analyt. Methods Geomech. 4, July-Sept (1980). D. Z. Yankelevsky and J. Gluck, Nose shape effect on high velocity soil penetration. Int. J. Mech. Sci. 22, (1980). R. Hill, Cavitation and the influence of headshape in attack of thick targets by nondeforming projectiles. J. Mech. Phys. Solids 28, 249-263 (1980).