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An analytical model for tumbling projectile perforation of thin aluminum plates
Pergamon
Int. J. Impact Engng Vol. 18, No. 1, pp. 45-63, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0734-743X(95)00024-0 0734 743X/96 $9.50+ 0.00
AN ANALYTICAL MODEL FOR TUMBLING PROJECTILE P E R F O R A T I O N O F T H I N A L U M I N U M PLATES KEZHUN LI and WERNER GOLDSMITH Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A.
(Received 14 October 1994; in revised form 23 February 1995) Summary---An analytical model for the perforation of thin aluminum targets by tumbling cylindrical projectiles was developed. The target material was considered to be rigid-perfectly plastic without strain hardening, while the projectile was treated as undeformable. The perforation process was experimentally found to consist of three stages: plugging, hole enlargement, and front petaling. Both conservation of energy and conservation of momentum laws were used for modeling the plugging stage, while a lower bound method was employed during the hole enlargement stage. The energy dissipated during the petaling stage consists of shearing fracture of the petal, localized plastic shear in a zone con tiguous with the edges, the momentum of the petal and the bending energy of the petal. The analytical results provided generally good agreement with the corresponding experimental data in terms of the final velocity and final oblique angle of the projectile as well as the crater length of the target.
NOTATION A D E, Era Eq
Es es
F. F~ h H Ip k~ 1 L mp mg
M Mb M~ r re r1
R Ro S
S At U V Vbl Vg ~)m On X
projectile face area projectile diameter E t = E s - Eq + Era, total energy consumed during the plugging stage kinetic energy of plug energy consumed during free inelastic impact between projectile and plug shear and deformation work due to presence of shear area during the plugging stage E s per unit shear area force acting on projectile frontal surface force acting on projectile lateral surface plate thickness variable during axisymmetric hole enlargement plate thickness variable during non-axisymmetric hole enlargement target thickness total height of ridge in ductile hole enlargement projectile moment of inertia thickening coefficient in thin plate hole enlargement arm length projectile length projectile mass plug mass total moment moment acting on projectile frontal surface moment acting on projectile lateral surface radial coordinate radius of hole during hole enlargement radius o:f plastic boundary during hole enlargement projectile radius radius of hole before hole enlargement begins shear zone width contiguous to the fracture edge in petaling stage shear area in plug formation time increment, time interval shear displacement velocity of the projectile minimum perforation velocity in normal impact final velocity of plug normal velocity component on lateral surface of the projectile normal velocity component on frontal surface of the projectile coordinate during hole enlargement 45
46 Yc, zc zp AW fl 6 ~u 70 0 01 ,9 pp Pt ~rr ar ~r0 ~r~ zs co
Kezhun Li and W. Goldsmith coordinates of projectile's center distance between upper point of projectile face and the target energy dissipated during petaling stage yaw angle oblique angle crater length tensile strain ultimate tensile strain maximum shear strain impact angle impact angle at time step i angle of contact between target and projectile during plugging radius of curvature in bending of petal mass density of projectile mass density of target dynamic yield stress of target radial stress circumferential stress stress along the thickness of target plate ultimate shear stress rotational speed of projectile
Superscript e a
experimental analytical
Subscript 0 f
initial final
INTRODUCTION In a companion paper [1], an experimental study was presented describing the response of thin and moderately thick aluminum plates due to impact of tumbling cylindrical projectiles. This rotation was produced by overlapping contact of the plane-faced striker with the edge of a generator placed between the gun and the target plate. The projectiles consisted of 12.7mm diameter cylinders with an aspect ratio of 3, fired from a power gun at speeds ranging from 400 to 800 m/s; the impact angles varied from 0 ° to 50 °. The nomenclature adopted in the present analysis is shown in Fig. 1. The YAW angle, ~, is defined as the angle between the longitudinal axis of the projectile and the velocity vector of the projectile's center of mass. The OBLIQUE angle, 8, is the angle between the velocity vector and the target normal. The IMPACT angle, 0, is the angle between the projectile axis and the normal to the target. Obviously the impact angle is equal to the yaw angle when the oblique angle is zero. Particular emphasis in the experimental study for thin plates was placed on observations of (a) the final velocity and obliquity of the projectile after perforation, and (b) the crater length of the target. These results are compared with the corresponding model predictions developed in the present paper. Among the various topics studied in the field of penetration mechanics, normal impact on a stationary target has been the most thoroughly investigated. An authoritative and comprehensive survey of this subject was given by Backman and Goldsmith [2]. Most of the phenomenological models were presented using the law of conservation of energy, e.g. [3] or conservation of momentum, or both [4-8]. The models given by Recht and Ipson [8], Awerbuch and Bodner [4] which contains three-stage processes, Liss and Goldsmith [5] which contains five-stage processes, and Yuan et al. 1-6] are typical. Compared with normal penetration, tumbling penetration with combined yaw and obliquity has not been studied extensively, though a certain amount of effort has been devoted to oblique impact [8-17] and yawed impact [18-22]. There are several reasons for this, one being the difficulty encountered in controlled impact by a projectile with a precisely predetermined angle of
An analytical model for tumbling projectile perforation of thin aluminum plates
47
s
/
I
Target
iI o it I
vf~
.... ,
r
After Impact
Before Impact
Yl
iI
Z<
/
Fig. 1. Nomenclature relating to impact by a tumbling cylindrical projectile on plates.
yaw. Secondly, the mechanism of the penetration process is very complicated due to its non-axisymraetric geometry. In this paper, a phenomenological model was created on the basis of experimental observation of the major deformation features [1]. The modeling requires reasonable assumptions during perforation such as consecutive stages of plugging, hole enlargement and petaling, etc. The perforation process was modeled using a combined momentum and energy approach. A computer program was generated to evaluate the model. The results obtained provide all :final kinematic parameters such as final velocity and oblique angle of the projectile as well as the crater length of the target. Comparisons of the experimental data and the analytic~tl results were performed. MODEL DEVELOPMENT
H/D<~
Thin targets, which are defined here as those with thickness ranging from 3.2 mm to 6.4 mm (for a 12.7rnm diameter of the projectile, 0.5), are modeled as solids where stress, strain and deformation gradients throughout the thickness do not exist and where, hence, wave propagation in the thickness direction of the plate can be neglected. Experimental results used for the construction of the model consist of 6061-T6 aluminum plates with thicknesses of 3.2 mm, 4.8 mm and 6.4 mm. The analysis presumes perforation of the plates corresponding to the condition of the present associated experimental range. Observations indicate that the penetration process in these targets can be characterized by three deformation stages. The perforation process in a thin target by a tumbling projectile is schematically shown in Figs 2(a)-(d). Two or more deformation stages may happen at the same time. The three stages represent (i) plugging; (ii) hole enlargement; and (iii) front petaling. Usually stages (i) and (ii) happen concurrently except under conditions of normal perforation. It is assumed that stage (iii) is found only when a transition from hole enlargement to petaling occurs. The target material is considered to be rigid-perfectly plastic without strain hardening, while the projectile is regarded as undeformable due to the high hardness of the projectile material. The equations of motion of the projectile were numerically integrated using a step-by-step finite difference method: the new position and orientation of the projectile after a further time increment were computed and the process repeated.
48
Kezhun Li and W. Goldsmith
target
targct
V
.
•
s
4 Co) Initial Perforation
(a) Initial Contact
target
target
Y
% plug
(c) Hole F_.nlargcmcnt
(d) From Petaling
Fig. 2. Schematic of the perforation process by a tumbling blunt-faced cylindrical projectile on thin
plates: (a) initial contact; (b) initial perforation; (c) hole enlargement; (d) front petaling.
Plugging stage Plug formation starts once the projectile touches the target. Unlike normal perforation, where the plug is immediately initiated over the entire circular contact region, this process commences here by initial failure of the target along the upper edge of the projectile/target contact area and spreads gradually by further intersection of the projectile and the target, as shown in Fig. 2(b). This event is similar to the formation of a petal. The part of the target in contact with the face of the projectile is assumed to instantly attain its velocity. Due to the oblique orientation and rotation of the projectile, the shape of the plug is neither circular nor exactly elliptical but approximates an ellipse if the rotation during perforation is not large. When tumbling is present, the oblique position of the projectile leads to indentation between the lateral surface of the projectile and the entry side of the target during the petaling stage. This process results in an elliptical crater. Once the center of the face of the projectile is intercepted by the target, the contact area becomes a semi-ellipse and the indentation ends. At the same time, initiation ensues of either hole enlargement or petaling, depending on a transition criterion. Since this process lasts a very short time, thickening of the plate near the periphery of the crater is very small and is therefore neglected. After hole enlargement or
An analytical model for tumbling projectile perforation of thin aluminum plates
49
petaling occurs, the plugging process continues further until the whole face of the projectile is no longer in touch with the target. Calculation of the motion of the plug and projectile is based on the energy method developed by Recht and Ipson [8]. The energy dissipation is partitioned into three parts: E~, Eq and Em. Term E~ is the energy expended at the periphery of the crater in the deformation process that separates the plug from the target element; Eq is the energy expended throughout the plug in plastic deformation that accounts for the projectile and plug reaching a common velocity; and E m is the kinetic energy of the plug. (a) Calculation of E~. The energy loss, Es, due solely to the presence of the peripheral shear area, might be expected to be relatively insensitive over rather wide ranges of velocity [8] and, for normal perforation, can be written as:
mp mpV21
Es = 2 rnp + m-----~
(1)
or
1 PvL -mpv~, Es = 2 ppL + pt H
(2)
where mp and mg are the masses of the projectile and the plug, Vb~is the minimum perforation velocity, pp and Pt are the densities of the projectile and the target, L is the length of the projectile and H is the target thickness. Here it is assumed that the plug is cylindrical and of the same diameter as the projectile. Since E s is due solely to the presence of the peripheral shear area, it is reasonable to assume that this energy per unit shear area, e~, is constant for a given type of projectile and target material. Thus:
1 E~
(3)
es = 2r~ R H or 1 ppmpLv21 es - 4re (ppL + p t H ) R H
(4)
where R is the radius of the projectile. This energy value for each type of target material was obtained from the normal shots on targets of the same type of material. In tumbling perforation, the energy dissipation in time step i is: AEs(i) = e~AS(i)
(5)
where AS(i) is the area swept out in time step i. From the geometric relation shown in Figs 3(a) and (b): L zv(i) = zc(i) + -~ cos 0~ + R sin 01
(6)
where z c is the coordinate of the projectile center and 0 i is the impact angle at time step i. The equation of the ellipse in Fig. 3(b) is: X2
y2
R 2 Jr (R/cos 01)2 - 1.
(7)
AS(i) ~ 2 c ~ s 0 / {[-xK(i + 1) -- XK(i)]2 + [yr( i + 1) -- yt(i)] 2} 1/2
(8)
From Fig. 3(b),
where the effective thickness, due to the oblique position of the projectile is H / c o s 0 I.
50
Kezhun Li and W. Goldsmith
vo\l
7;: ..............\\, -
~,~
\
o .-°"
Y M
:M,P
N
K
.....
K..
(b) A-A
(c) B-B
Fig. 3. Projectile-target geometry during perforation by a tumbling blunt-faced cylindrical projectile on thin plates.
F r o m E q n (7), we h a v e
(9)
xK(i) = [ R 2 -- y~(i) COS2 Oil 1/2
xz(i + 1) = [R 2 -- y~(i + 1) cos 20i+ 1] 1/2
(I0)
YK(i) = yN(i)
(11)
yK(i+ 1) = YN(i+ 1).
(12)
where
A l s o f r o m Fig. 3(b),
R
yN(i) = T N = T M -- M N = - cos 0~
R ys(i + 1) = cos Oi-~1
Zp(i) sin Oicos 0~
zp(i + 1) sin Oi+ 1 cos Oi+ 1"
(13)
(14)
An analytical model for tumbling projectile perforation of thin aluminum plates
51
(b) Calculation of Eq. The energy expended throughout the plug in plastic deformation Eq, that accounts for the projectile and plug reaching a common velocity, based on Eqn (1) with a modification to account for discretization of the process, can be expressed as follows [8]:
1 Amg(i)mp v~(i)
AEq(i)
-
(15)
2 Amg(i) + mp
where VN(i) is 1:hevelocity at point N normal to the face of the projectile, and Ares(i) is the mass of the plug formed at time i. From Fig. 3(a), VN(i) = ~¢(i)cos 0 i -- ~c(i) sin 0 i - [ -
(~ 1~. - -
Zp(i)"~A
s~nOi)tli
(16)
where y¢ is the coordinate of the projectile's center. Also we have
Amg(i)=2H[Xk(i)+2k(i+l)]lyN(i)-- yN(i+l)lpt.
(17)
Substituting Eqns (16) and (17) into Eqn (15) provides AEq(i). (c) Calculat, ion of Em. In the initial perforation stage, the part of the target material in contact with the face of the projectile is assumed to instantly attain the same velocity as that part of the projectile, or VN(i).Thus, we have AEm(i) = ½Amg(i)v2(i).
(18)
Substituting Eqns (16) and (17) into Eqn (18) determines AEm(i). (d) Force and moment acting on the face of the projectile. The energy rate dissipated during the initial perforation stage is equal to the product of the projectile/target interaction force and the velocity component in the force direction. Since friction forces appear to be small [23,24], they are neglected in the calculation. The direction of the interacting force is normal to the projectile surface as is the velocity component. The relation between the energy rates, and normal fi~rce, and the velocity can be expressed as: AEt(i)
At
-- Fb(i)vN(i )
(19)
or
Fb(i) = AE s + AEq + AE m
(20)
Vs(i) At where At is the time step and Fb(i ) is the force acting on the face of the projectile. Also, from Fig. 3(a), the moment from the face of the projectile is: Mb(i) = Fb(i)(R -
-
-
sinZP(i)01,f ~
(21)
(e) Force and moment actin9 on the lateral surface of the projectile. In the initial perforation, the resisting pressure from the lateral indentation [along arc MK, as shown in Fig. 3(c)] is postulated to be equal to the yield stress of the material try. As shown in Figs 3(a) and (c), this process corresponds to Zp(i)/sin 0 i < R, and the force and the moment from the lateral surface of the projectile can be derived as follows: F~(i) = 2
% sin ~ Rd'9
(22)
,)3o
Ms(i ) = 2
try sin ,9 Rd,9 J #o
l
(23)
52
Kezhun Li and W. Goldsmith
where ,9o, as shown in Fig. 3(c), is UN ~9o = tan-
(24)
1NK
or
[ R--zp(i)/sinO, ~9o = tan-1 I-x/R2 _ (R - zp(i)/sin 0i) 2 J
(25)
1 f_ zp(i) "~ ,~ l=-~L-- Rsin~gtanOi + ~ g--s~nOi)tanvi.
(26)
and the arm length
After integration, we have
H Fs(i ) = 2 ay-----=c 2
(27)
cos 0 i
and
Ms(i)=aYco~iI(L+2cltanOi)c2-tanOi(~R --,9oR2[-ClC211 2
(28)
where
c1= R c2 = ~
Zp(i)
(29)
sin 0~ .
(30)
Hole enlargement stage The experimental observations indicated that, for a relatively low impact angle, an appreciable pile-up of material near the periphery of the hole occurred after the initial perforation. This is similar to the hole enlargement in thin plate perforation studied by Taylor [25]. However there are differences between the Taylor problem and the present one. First, the hole enlargement here is not axisymmetric as in Taylor's case. In axisymmetric hole enlargement, velocities at each point of the crater are directed radially as shown in Fig. 4(a) and the thickening around the edge of the hole is evenly distributed. Here, however, the velocities at each point have the same direction as shown in Fig. 4(b). This results in uneven thickening around the arc, which decreases from the upper point B in Fig. 5 to the intercept of
Fig. 4(a). Velocity distribution in axisymmetric hole enlargement.
An analyticalmodel for tumbling projectileperforationof thin aluminum plates
•
i I
• •
~'l--'-''~'~l :
-
53
•x Ii
............................................... . . . . . . . velocl~"
Fig. 4(b). Velocitydistribution in asymmetrichole enlargement.
B
.............. i........ C O
O'
D
D'
iD
(a) SideView
(b) FrontView A-A
Fig. 5. Schematicofthickeningduringasymmetricholeenlargement:(a) side view;(b)front viewA-A. the target and the axis of the projectile from where the hole enlargement initiates, O in Fig. 5, and where there is no thickening at all. Secondly, the initial radius of the hole in the plate is not zero as it was in Taylor's problem. The current non-axisymmetric hole enlargement model :is still based on the axisymmetric case; it is assumed that the thickening of the upper points B and B' is the same as that of axisymmetric hole-enlargement and the distribution of the thickening around the arc BC is quadratic, or hyperbolic due to symmetry of the arc. Since it is very difficult to analyze the hole enlargement when the initial radius of the hole is not zero by using Taylor's theory [25,1 Bethe's method [26-] is adopted here with modified boundary conditions to incorporate a non-zero initial radius. As mentioned in [27-], Bethe's method leads to a lower bound for the expansion of a hole and is just as useful for the practical calculation of work done, even though it results in a vastly different stress distribution and lip height at the edge of the hole than Taylor's theory predicts. In Bethe's model, shown in Fig. 6, it is assumed that: tr0 = 0 and a z = 0
(31)
where tr0 is the circumferential stress and trz is the stress along the thickness of the plate. The equation of equilibrium in a thin plate of variable thickness h becomes d(htrr) + - ~ = 0
(32)
54
K e z h u n Li and W. G o l d s m i t h t
-I I I I
HI
h
11 I I I I ,
I I
', I
_t_
~ I I
[ N
V"
, I
r
I
I
,, Fig. 6. D i a g r a m of the strain distribution associated with theductile hole formation model of Bethe.
where trr is the radial stress and r the radial coordinate. Equation (32) has the solution c her r = r
(33)
where c is a constant. According to the Von-Mises or Tresca yield criteria, we have O"r = O'y.
(34)
/~r o = H r 1 = constant
(35)
Substitution of ar into Eqn (33) gives
where H is the total height of ridge, r o is the radius of bole and r 1 is the radius of plastic boundary, as shown in Fig. 6. Assuming constancy of volume and referring to Fig. 6, 1
2
1
2
j.o
I*'~
//~ - H'~
36)
where R o is the initial radius. Solving the above equations, we obtain ./~ . k e . 1 +. H
/1.
(R°~ 2 \r o ) "
(37)
Figure 7 shows the thickening coefficient k e as a function of r o / R o. It is found that for a given initial radius R o, the thickness /~ increases with an increase of the radius r o. The equation also shows that for a zero initial radius (R o = 0), or equivalently r o --, ~ , ke is equal to 2, which agrees with the solution by Bethe and is close to Taylor's theory. The distribution of the thickening, shown in Fig. 5, is assumed to be quadratic around the upper half of the hole, so that -h(x) = a 1 + a2x + a3 X2
(38)
56
Kezhun Li and W. Goldsmith
(a) Petaling
(b) Ductile Tearing of Metals
,[
H
(c) Final Tearing
(d) Shearing around the Edge
Fig. 8. Diagram of petaling stage: (a) petaling; (b) ductile tearing of metal; (c) final tearing; (d) shearing around the edge.
Front petaling stage
When the impact angle is relatively large, the failure mode after initial perforation is front petaling as defined by Wu and Goldsmith [28] rather than hole enlargement. The experiment and analysis [1] showed that the petaling will not materialize until the impact angle reaches 15° for 4.8 mm thick 6061-T6 aluminum plates. It is 9.3 ° for 3.2 mm and 18.8° for 6.4 mm thick plates. The target material in front of the contact region of the lateral surface of the projectile was bent after tearing and thus formed a petal, as shown in Fig. 8(a). Observation indicated that the width of the petal is approximately the same as the projectile diameter. This is similar to the tearing of thin ductile metals studied by Mai and Cotterell [291 as shown in Fig. 8(b). As for the first stage, the force and moment are derived based on an energy approach. The energy dissipated in this process is mainly due to shearing fracture of the petal, localized plastic shear work in a zone contiguous with the torn edges, the kinetic energy and the bending energy of the petal. The following derivation is based on the model developed by Mai and CottereU [29]. (i) The energy rate due to fracture. The major fractures in the target producing the petal are two nearly parallel cracks like those generated in the tearing of ductile metals, as shown in Fig. 8(c). The work required to fracture a length Ax of the sheet is given by [29]: AW 1 = 2
zsAx(H - u)du = % H 2 A x
(48)
where % is the shear stress during tearing and is assumed to remain constant. (ii) Localized work in a shear zone contiguous to the torn edge. The plastic work necessary to produce; the thin lip of width s on the edges of the petal shown in Fig. 8(d), is obtained from the relation [29]: A W 2 = 4 H s ~ yT° A x
,/3
(49)
An analytical model for tumbling projectile perforation of thin aluminum plates 2
i
|
i
55
|
1.9 1.8 1.7
~
1.6
~1.5 T:
~1.4 J¢ I--
1.3 1.2 1.1
1
2=
1.5
2 i5 3' 3.5 ' 4' Ratio ot Final Radius to Initial Radius
4 i5
5
Fig. 7. Thickening coefficient as a function of initial radius and final radius.
with the boundary conditions
h(O)= ken
(39)
h(_+ R) = n .
(40)
It is assumed that the point C has the same thickness as the point O, i.e. no thickening at C. This is reasonable since the length of OC was found to be small from the experiments. Solving Eqn (38), we obtain
-h(x)= ken + (1 -ke)nx2 R2 .
(41)
In the hole enlargement, the pressure is still equal to the yield stress try. From Figs 3(a) and (c), we have
Fs(i,=2f]/2aysing(Rd`9c~sOi)
(42,
Ms(i,=2f]/Zaysin`9(Rd`9~)l
(43,
where I is given by Eqn (26). From Fig. 3(c). x = R cos ,9
(44)
so
-h(~ = keH sin 2 `9+
H cos 2 ,9.
(45)
After integration of Eqns (42) and (43), there results
Fs(i)= Ms(i)=
2 HR
3 cos 0 i ay(2ke + 1)
[1
(
zP(i)~
(46) 1
-~L+ R-s~nOijtanOi F~(i)
nHR2tanOiay(l+3ke,. 8cos0i
(47)
An analytical model for tumbling projectile perforation of thin aluminum plates
57
where Yois the maximum shear strain. Yo= x/~e~, and where e, is the ultimate tensile strain. Here, the strain hardening effect is neglected. In the present work, the width s is measured experimentally from the perforated samples. Roughly, s = 3H. (iii) Kinetic energy of the petal. It is assumed that the petal, which is in contact with the lateral surface of the projectile instantly attains the normal component of the velocity of the projectile. Here the velocity is assumed to be that of point M as shown in Fig. 3(a). The velocity is:
Vm(i)=~sinO'+ Y¢C°S0~--(2
COS0,,]ZP(0 ~0 i
(50)
so the kinetic energy of the petal is AWa = ½Amv2(i) = PtnRv2(i)Ax.
(51)
(iv) Bending energy of the petal. The energy dissipated in the bending of the petal is:
r n/2 a 1 tryRH 2 Ax AW,=4RJo ~uduAx=~ x
(52)
where x is the radius of curvature which is given by 1-29] H 2e
x
(53)
where e is the tensile strain. Clearly, the real radius of curvature is dependent on the impact angle. The bJ,gger the impact angle, the larger the radius of curvature, and the less the energy dissipated. In the present model e is taken to be the maximum, or the ultimate tensile strain eu. (v) Force and moment acting on the lateral surface of the projectile. An energy approach is used here similar to that of the initial perforation stage,
AWl + AW~ + AW3 + AW, = Fs(i)vm(i ). At
(54)
The force is assumed to act at point M. 1
F~(i) = Atv~(i)(AW~ + AW2 + AW3 + AW4).
(55)
mx At = vm(i) cos 0 i.
(56)
Also
Substituting Eqns (48), (49), (51), (52) and (56) into Eqn (55),
Fs(i) = "c~H2 +
4Hstryy o
v~ ~
t- ptHRv2(i) +
tryRH 2
2~c
(57)
and / 1
MsO) = l~s(i)~~ L - - Equation of motion of the projectile Once the forces and the moments from both the face and lateral surface of the projectile are known, the equation of motion of the projectile can be derived from Newton's Second Law (Fig. 9).
58
Kezhun Li and W. Goldsmith
Mb
z
Fig. 9. Rigid-body motion of the projectile.
The equations of motion of the projectile are: Fz(i) = mpZ'¢(/)
(59)
Fy(i) = mpy¢(i)
(60)
M(i) = Ip'O(i)
(61)
where (62)
mp = n R 2 L p p
Ip --
+ ]5
(63)
with initial conditions: 0(0) = 0o
(64)
~c(0) = vo cos flo
(65)
3~o(0)= vo sin flo
(66)
0(0) = 09o.
(67)
From the expressions for the forces and moments Fb(i), Fs(i), Mb(i), Ms(i), we have: Fz(i) = - Fb(i) COS0 i -- Fs(i ) sin 0 i
(68)
Fy(i) = Fb(i) sin 0 i -- Fs(i ) cos 01
(69)
M(i) = Ms(i ) - Mb(i).
(70)
A forward Euler method is used in the finite difference calculation. The basic equation is: :,( i)
dy
y(i + 1)-y(i) At
(71)
RESULTS AND DISCUSSION The phenomenological model developed above has been programmed in F O R T R A N language and calculated numerically. The program input data include material constants, problem geometry, and kinematic parameters of the projectile. The computational output provides predictions of the final velocity, oblique angle, and rotational speed of the projectile as well as the crater length of the target. Simulations of 32 runs (23 for 4.8 mm, 5 for 6.4 mm, and 4 for 3.2 mm thick targets) were performed. Four normal perforations on 4.8 mm and 3.2 mm of 6061-T6 aluminum were used to obtain an average value of the energy loss es. The experimental ballistic limit %~ (normal impact) for 6061-T6 aluminum was found to be 89m/s for 6.4mm thick plates, 66m/s for 4.8 mm plates and 41 m/s for 3.2 mm plates. Material properties obtained from [30] are listed in Table 1. The computational results and corresponding experimental data are presented in Table 2. The calculated final velocities, final oblique angles and crater lengths are in good correspondence with the experimental data.
An
analytical model for tumbling projectile perforation of thin aluminum plates
59
Table 1. Properties of projectile and target materials Density
Dynamic yield
Ultimate shear
Ultimate tensile
Material
(kg/m 3)
stress (MPa)
strength (MPa)
strain (%)
Projectile
7977 2780
1393 295
804 190
-18
AL 6061-T6
Table 2. Experimental and analytical results of perforation of plates by tumbling projectiles Initial condition
Final condition
Projectile
Projectile
t
~o
Oo
/~o
O~o
~
~
/~
Crater
/~
~°o
~'o
Run
Mtls
(mm)
(m/s)
(deg)
(deg)
(rad/s)
(m/s)
(m/s)
(deg)
(deg)
(mm)
(mm)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 All A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23
A1 AI A1 A1 AI A1 AI AI AI AI AI A1 A1 AI A1 A1 A1 A1 AI A1 AI AI AI
4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8
477 336 638 318 480 511 445 398 335 488 407 395 453 500 480 315 568 586 324 451 565 409 525
0 5 6 5 20 21 21 28 42 9 23 14 26 23 32 29 2 10 50 4 0 0 0
6 8 3 7 - 7 8 -6 10 8 -3 7 7 7 10 7 12 5 -4 13 3 0 0 0
254 276 1655 170 752 1604 937 1402 979 498 1063 350 1407 2050 1682 1023 554 190 1791 32 0 0 0
451 285 591 281 440 404 393 298 230 465 344 343 396 413 382 238 542 564 214 422 535 392 498
446 297 600 280 443 446 404 317 247 460 346 348 400 430 407 233 536 551 230 423 538 387 499
1 - 7 -2 -9 - 13 -1 - 15 - 1 3 -7 -2 -2 1 - 1 0 -2 - 1 -9 11 -2 0 0 0
0 - 10 - 1 - 11 - 14 1 - 15 1 2 -9 -3 -4 - 1 2 3 -3 1 -9 8 -4 0 0 0
17.0 22.0 21.0 20.0 21.0 33.0 22.0 36.0 40.0 17.0 33.0 28.0 34.0 32.0 36.0 37.0 19.0 16.0 40.0 18.0 13.5 13.5 13.5
17.1 20.3 20.2 19.7 21.5 31.4 22.7 35.1 37.5 16.5 31.2 26.0 32.6 33.0 35.3 35.5 17.6 16.6 40.2 16.8 12.7 12.7 12.7
B1 B2 B3 B4 B5
A1 A1 A1 A1 A1
6.4 6.4 6.4 6.4 6.4
551 516 423 576 446
18 13 46 11 53
- 6 - 7 10 6 10
1089 239 2176 701 2997
479 470 267 495 284
493 472 283 500 310
- 14 - 14 5 -6 11
- 16 - 18 1 -5 8
23.0 18.0 40.0 25.0 41.0
22.0 16.2 39.3 24.4 40.5
C1 C2 C3 C4
A1 A1 A1 A1
3.2 3.2 3.2 3.2
685 395 298 557
4 32 2 0
4 10 10 0
259 1268 94 0
653 356 261 545
660 358 275 540
1 6 -2 0
3 6 - 1 0
20.0 38.0 21.0 14.0
18.4 36.3 20.6 12.7
Figure 10 depicts the experimental and analytical velocity drop as a function of impact angle for a given oblique angle and initial velocities of 350 m/s and 500 m/s for 4.8 mm thick 6061-T6 aluminum targets. It is seen that the velocity drop increases with impact angle and decreases wit]h initial translational velocity. Figure 11 shows the results of the final oblique angle as a function of impact angle for the given oblique angle and initial velocities. The absolute value of the final oblique angle increases with increasing impact angle when the impact is below a certain value (10 ° for 500 m/s and 12 ° for 350m/s), but decreases to zero when the impact angle is above it. A higher initial velocity reduces the magnitude of the final oblique angle, as shown in the figure. The variation of crater length of the target as
60
Kezhun Li and W. Goldsmith
x : v - 350 m/s, experimental data
45
411
~
o : v - 500 m/s, experimental data solid line: analytical results
x
30
~: 25
10
IMPACT ANGLE degree
Fig. 10. Experimental and analytical results of percentage velocity drop as a function of impact angle for 4.8 mm thick 6061-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5 °. 20 x : v = 350 m/s, experimental data
15
tO
o : v = 500 m/s, experimental data solid line: analytical results
10
ut
,
(.o z,<
,,, _o
o'
3-, ,< _z " -lO
-15
-20 IMPACT ANGLE degree
Fig. 11. Experimental and analytical results of final oblique angle as a function of impact angle for 4.8 mm thick 6061-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5 ° .
a function of the impact angle is depicted in Fig. 12, showing an increase for large impact angles. Observations from Figs 10-12 indicate that the velocity drop, final oblique angle, and crater length reach stable conditions when the impact angle is above 50°, similar to the situation encountered in side-on impact (at an angle of 90°). The comparisons illustrated in the tables and the figures validate the model for the range of parameters examined here. Figure 10 shows that the model underestimates the velocity drop as the impact angle increases. It is found from the experiments that bending of the target becomes large as the impact angle increases or the initial velocity decreases. In the model development, however, the bending effect is neglected. This leads to less energy dissipated in the perforation and subsequently, a low velocity drop.
An
analytical model for tumbling projectile perforation of thin aluminum plates
61
50 ~5
valocity 300 ~ 600 m/s
o: experimental data 40
solid line: analytical results Oo
~5 E E :~3 N LU
LU
O
15 11)
IMPACT ANGLE degree
Fig. 12. Experimental and analytical results of crater length as a funcion of impact angle for 4.8 mm thick 606 l-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5°.
In the petaling stage, the energy dissipation consists of four parts: the energy due to fracture AW~, the localized work in a shear zone AW2, the kinetic energy of the petal AW3, and the bending energy of the petal A W4. The following discussion will focus on the effect of these parts. From Eqns (48), (49), (51) and (52), the ratios of the four parts are: AWx:AW2:AW~:AW4=Ts:
12ayTo ptRv~ ayReu i-.-.~ : H : - n "
(72)
For 3.2 mm-6.4 mm thick 6061-T6 aluminum targets, the ratio is: AWx :AW2 :AW3 :AW4 = 1.0:4.6:(0.1 ~ 0.5):(0.3 ~ 0.6).
(73)
It can be seen from Eqn (73) that, in the petaling stage, most of the energy goes to the localized work in a shear zone contiguous to the fracture edge instead of the energy due to fracture. The kinetic and bending energies are not very important in an aluminum plate. This means that the shear zone and the maximum shear strain are the main factors in the petaling stage. Currently, the shear zone is determined from the experiments and assumed to be independent of the impact velocity. In fact, the shear zone size changes with the impact velocity (3.3H to 2.8H). It transpires that the higher the velocity, the smaller the shear zone. Considerable discrepancies between the model calculations and the experiments arise in several cases, such as runs A6, A14, and A15 where relative errors for the velocity drop are 40%, 20%, and 25% and runs A2, A15, and A19 where errors for the final oblique angle are 3 °. This may be due to inaccuracy in the measurement and a certain amount of scatter in the experiments. Further dew.qopment of the analytical model will depend on the observed target response for regimes different from those studied here. For example, for low velocities, high impactangles or thin plates, bending effects and membrane forces tend to dominate the penetration process. This has been neglected in the current model. Also, as the velocity of the projectile or the thickness of the target increases, the size of the shear zone contiguous to the torn edge decreases, similar to what occurs when a beam is loaded impulsively (see Jones [31] regarding this subject).
62
Kezhun Li and W. Goldsmith CONCLUSION
A n a n a l y t i c a l m o d e l was c o n s t r u c t e d to predict the p e r f o r a t i o n of thin a l u m i n u m plates by tumbling, blunt-faced hard-steel cylindrical projectiles. This m o d e l was b a s e d on e x p e r i m e n tal o b s e r v a t i o n s t h a t i n d i c a t e d the presence of three consecutive stages: plugging, hole e n l a r g e m e n t a n d petaling. C a l c u l a t i o n s for the system response were p e r f o r m e d for 32 i m p a c t c o n f i g u r a t i o n s i n c l u d i n g those c o r r e s p o n d i n g to test results d e s c r i b e d elsewhere. Such d a t a , o b t a i n e d for the final velocity a n d final oblique angle of the projectile as well as the c r a t e r length p r o d u c e d in the targets, were c o m p a r e d with c o r r e s p o n d i n g m o d e l predictions. It was f o u n d t h a t increasing i m p a c t angle results in a n o t i c e a b l e increase of the velocity d r o p a n d t r a j e c t o r y c h a n g e of the projectile. The c r a t e r length of the target also increases. H o w e v e r , when the i m p a c t angle is a b o v e 50 °, the velocity d r o p a n d the c r a t e r length seem to be stabilized a n d the final o b l i q u e angle tends to be zero like those in side-on impact. H i g h e r projectile velocity tends to reduce the velocity d r o p a n d c h a n g e of the final o b l i q u e angle. In the p e t a l i n g stage, m o s t of the initial energy is c o n v e r t e d into localized w o r k in a shear zone c o n t i g u o u s to the fracture edge, r a t h e r t h a n into the energy of fracture a n d bending. Acknowledgement--This work constitutes a portion of a doctoral dissertation by the first author at the University of California, Berkeley. The work was sponsored by the Air Force Office of ScientificResearch, Bolling Air Force Base, Washington, D.C., under contract AFOSR F49620-89-8127.
REFERENCES 1. K. Li and W. Goldsmith, Impact on aluminum plates by tumbling projectiles: experimental study. Int. J. Impact Engng 18, 23-43 (1996). 2. M.E. Backman and W. Goldsmith, The mechanics of penetration of projectiles. Int. J. Engng Sci. 16, 1-99 (1978). 3. B. Landkof and W. Goldsmith, Petaling of thin, metallic plates during penetration by cylindro-conical projectiles. Int. J. Solids Struct. 21,245-266 (1985). 4. J. Awerbuch and S. R. Bodner, Analysis of the mechanics of perforation of projectiles in metallic plates. Int. J. Solids and Struct. 10, 671-684 (1974). 5. J. Liss, W. Goldsmith and J. M. Kelly, A phenomenological penetration model of plates. Int. J. Impact Engng l, 321 341 (1983). 6. W. Yuan, L. Zhou, X. Ma and W. J. Stronge, Plate perforation by deformable projectiles--aplastic wave theory. Int. J. Impact Engng 1, 393-412 (1983). 7. M. Zaid and B. Paul, Mechanics of high speed projectile perforation. J. Franklin Inst. 264, 117-126 (1957). 8. R. F. Recht and T. W. Ipson, Ballistic perforation dynamics J. Appl. Mech. 30, 384-390 (1963). 9. D. M. Cunningham and W. Goldsmith, An experimental investigation of beam stresses produced by oblique impact of a steel sphere. J. Appl. Mech. 23, 606-611 (1956). 10. W. Goldsmith and D. M. Cunningham, Kinematic phenomena observed during the oblique impact of a sphere on a beam. J. Appl. Mech. 23, 612-616 (1956). 11. M. Zaid and B. Paul, Oblique perforation of a thin plate by a truncated conical projectile. J. Franklin Inst. 268, 25-45 (1959). 12. M.E. Backman, S.A. Finnegan and K.G. Whitman, Dynamics of the oblique impact and ricochet of nondeforming projectiles against thin plates. In Recent Advances in Engineering Science, pp. 9-20 (Edited by G. C. Sih). Lehigh University Publication (1977). 13. J. Awerbuch and S. R. Bodner, An investigation of oblique perforation of metallic plates by projectiles. Exp. Mech. 17, 147-153 (1977). 14. W. Goldsmith and S. A. Finnegan, Normal and oblique impact of cylindro-conical and cylindrical projectiles on metallic plates. Int. J. Impact Engng 4, 83-105 (1986). 15. J. Falcovitz, M. Mayseless, Z. Tauber, D. Keck, R. Kennedy, K. Ofstedahl and P. Singh, A computer model for oblique impact of a rigid projectile at ductile layered targets. Proc. l l t h Int. Syrup. on Ballistics, pp. 1-11, Brussels, Belgium (1989). 16. R. L. Woodward and N. J. Baldwin, Oblique perforation of targets by small armor-piercing projectiles J. Mech. Engng Sci. 21, 85-91 (1979). 17. S. R. Reid, A. J. Edmunds and W. Johnson, Bending of long steel and aluminum rods during end impact with a rigid target. J. Mech. Engng Sci. 23, 85-92 (1981). 18. S.J. Bless,J. P. Barber, R. S. Bertke and H. F. Swift, Penetration mechanics of yawed rods. Int J. Engng Sci. 16, 829-834 (1978). 19. S.J. Bless and J. P. Barber, Bending waves in yawed rod impacts. J. Ballistics 2, 281-298 (1978). 20. V. Hohler and A. J. Stilp, Influence of the primary yaw angle on the secondary yaw angle during armor plate perforation. Proc. 6th Int. Syrup. on Ballistics, pp. 302 308, Orlando, Florida (1981).
An analytical model for tumbling projectile perforation of thin aluminum plates
63
21. M. Mayseless, Y. Kivity, G. Rosenberg and A. A. Betser, Bending waves in yawed rod impacts. Proc. 6th Int. Syrup. on BaJlistics, pp. 309-314, Orlando, Florida (1981). 22. W. Goldsmith, E. Tam and D. Tomer, Yawing impact on thin plates by blunt projectiles. Int. d. Impact Engng 16, 479-498 (1995). 23. J. M. Krafft, Surface friction in ballistic penetration. J. Appl. Phys. 26, 1248-1253 (1955). 24. W.T. Thomson, An approximate theory of armor penetration. J. Appl. Phys. 26, 80-82 (1955). 25. G.I.Tay~r~Thef~rmati~nandenlargement~facircu~arh~einathinp~asticsheet.Quart.J.Mech.App~.Math. 1, 103-124 (Zt948). 26. H. Bethe, Attempt of a theory of armor penetration. Frankford Arsenal Ordnance Laboratory Technical Report UN-41-5-23 (1941). 27. R. L. Woodward, The penetration of metal targets by conical projectiles. Int. J. Mech. Sci. 20, 349-359 (1978). 28. E. Wu and W. Goldsmith, Normal impact of blunt projectiles on moving targets: experimental studies. Int. J. Impact Engng 9, 389-404 (1990). 29. Y.W. Mai and B. Cotterell, The essential work of fracture for tearing of ductile metals. Int. J. Fracture 24, 229-236 (1984). 30. J. Awerbuch and S. R. Bodner, Experimental investigation of normal impact perforation of projectiles in metallic plates. Int. J. Solids Struct. 10, 685-699 (1974). 31. N. Jones, Structural Impact. Cambridge University Press (1989).
Int. J. Impact Engng Vol. 18, No. 1, pp. 45-63, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0734-743X(95)00024-0 0734 743X/96 $9.50+ 0.00
AN ANALYTICAL MODEL FOR TUMBLING PROJECTILE P E R F O R A T I O N O F T H I N A L U M I N U M PLATES KEZHUN LI and WERNER GOLDSMITH Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A.
(Received 14 October 1994; in revised form 23 February 1995) Summary---An analytical model for the perforation of thin aluminum targets by tumbling cylindrical projectiles was developed. The target material was considered to be rigid-perfectly plastic without strain hardening, while the projectile was treated as undeformable. The perforation process was experimentally found to consist of three stages: plugging, hole enlargement, and front petaling. Both conservation of energy and conservation of momentum laws were used for modeling the plugging stage, while a lower bound method was employed during the hole enlargement stage. The energy dissipated during the petaling stage consists of shearing fracture of the petal, localized plastic shear in a zone con tiguous with the edges, the momentum of the petal and the bending energy of the petal. The analytical results provided generally good agreement with the corresponding experimental data in terms of the final velocity and final oblique angle of the projectile as well as the crater length of the target.
NOTATION A D E, Era Eq
Es es
F. F~ h H Ip k~ 1 L mp mg
M Mb M~ r re r1
R Ro S
S At U V Vbl Vg ~)m On X
projectile face area projectile diameter E t = E s - Eq + Era, total energy consumed during the plugging stage kinetic energy of plug energy consumed during free inelastic impact between projectile and plug shear and deformation work due to presence of shear area during the plugging stage E s per unit shear area force acting on projectile frontal surface force acting on projectile lateral surface plate thickness variable during axisymmetric hole enlargement plate thickness variable during non-axisymmetric hole enlargement target thickness total height of ridge in ductile hole enlargement projectile moment of inertia thickening coefficient in thin plate hole enlargement arm length projectile length projectile mass plug mass total moment moment acting on projectile frontal surface moment acting on projectile lateral surface radial coordinate radius of hole during hole enlargement radius o:f plastic boundary during hole enlargement projectile radius radius of hole before hole enlargement begins shear zone width contiguous to the fracture edge in petaling stage shear area in plug formation time increment, time interval shear displacement velocity of the projectile minimum perforation velocity in normal impact final velocity of plug normal velocity component on lateral surface of the projectile normal velocity component on frontal surface of the projectile coordinate during hole enlargement 45
46 Yc, zc zp AW fl 6 ~u 70 0 01 ,9 pp Pt ~rr ar ~r0 ~r~ zs co
Kezhun Li and W. Goldsmith coordinates of projectile's center distance between upper point of projectile face and the target energy dissipated during petaling stage yaw angle oblique angle crater length tensile strain ultimate tensile strain maximum shear strain impact angle impact angle at time step i angle of contact between target and projectile during plugging radius of curvature in bending of petal mass density of projectile mass density of target dynamic yield stress of target radial stress circumferential stress stress along the thickness of target plate ultimate shear stress rotational speed of projectile
Superscript e a
experimental analytical
Subscript 0 f
initial final
INTRODUCTION In a companion paper [1], an experimental study was presented describing the response of thin and moderately thick aluminum plates due to impact of tumbling cylindrical projectiles. This rotation was produced by overlapping contact of the plane-faced striker with the edge of a generator placed between the gun and the target plate. The projectiles consisted of 12.7mm diameter cylinders with an aspect ratio of 3, fired from a power gun at speeds ranging from 400 to 800 m/s; the impact angles varied from 0 ° to 50 °. The nomenclature adopted in the present analysis is shown in Fig. 1. The YAW angle, ~, is defined as the angle between the longitudinal axis of the projectile and the velocity vector of the projectile's center of mass. The OBLIQUE angle, 8, is the angle between the velocity vector and the target normal. The IMPACT angle, 0, is the angle between the projectile axis and the normal to the target. Obviously the impact angle is equal to the yaw angle when the oblique angle is zero. Particular emphasis in the experimental study for thin plates was placed on observations of (a) the final velocity and obliquity of the projectile after perforation, and (b) the crater length of the target. These results are compared with the corresponding model predictions developed in the present paper. Among the various topics studied in the field of penetration mechanics, normal impact on a stationary target has been the most thoroughly investigated. An authoritative and comprehensive survey of this subject was given by Backman and Goldsmith [2]. Most of the phenomenological models were presented using the law of conservation of energy, e.g. [3] or conservation of momentum, or both [4-8]. The models given by Recht and Ipson [8], Awerbuch and Bodner [4] which contains three-stage processes, Liss and Goldsmith [5] which contains five-stage processes, and Yuan et al. 1-6] are typical. Compared with normal penetration, tumbling penetration with combined yaw and obliquity has not been studied extensively, though a certain amount of effort has been devoted to oblique impact [8-17] and yawed impact [18-22]. There are several reasons for this, one being the difficulty encountered in controlled impact by a projectile with a precisely predetermined angle of
An analytical model for tumbling projectile perforation of thin aluminum plates
47
s
/
I
Target
iI o it I
vf~
.... ,
r
After Impact
Before Impact
Yl
iI
Z<
/
Fig. 1. Nomenclature relating to impact by a tumbling cylindrical projectile on plates.
yaw. Secondly, the mechanism of the penetration process is very complicated due to its non-axisymraetric geometry. In this paper, a phenomenological model was created on the basis of experimental observation of the major deformation features [1]. The modeling requires reasonable assumptions during perforation such as consecutive stages of plugging, hole enlargement and petaling, etc. The perforation process was modeled using a combined momentum and energy approach. A computer program was generated to evaluate the model. The results obtained provide all :final kinematic parameters such as final velocity and oblique angle of the projectile as well as the crater length of the target. Comparisons of the experimental data and the analytic~tl results were performed. MODEL DEVELOPMENT
H/D<~
Thin targets, which are defined here as those with thickness ranging from 3.2 mm to 6.4 mm (for a 12.7rnm diameter of the projectile, 0.5), are modeled as solids where stress, strain and deformation gradients throughout the thickness do not exist and where, hence, wave propagation in the thickness direction of the plate can be neglected. Experimental results used for the construction of the model consist of 6061-T6 aluminum plates with thicknesses of 3.2 mm, 4.8 mm and 6.4 mm. The analysis presumes perforation of the plates corresponding to the condition of the present associated experimental range. Observations indicate that the penetration process in these targets can be characterized by three deformation stages. The perforation process in a thin target by a tumbling projectile is schematically shown in Figs 2(a)-(d). Two or more deformation stages may happen at the same time. The three stages represent (i) plugging; (ii) hole enlargement; and (iii) front petaling. Usually stages (i) and (ii) happen concurrently except under conditions of normal perforation. It is assumed that stage (iii) is found only when a transition from hole enlargement to petaling occurs. The target material is considered to be rigid-perfectly plastic without strain hardening, while the projectile is regarded as undeformable due to the high hardness of the projectile material. The equations of motion of the projectile were numerically integrated using a step-by-step finite difference method: the new position and orientation of the projectile after a further time increment were computed and the process repeated.
48
Kezhun Li and W. Goldsmith
target
targct
V
.
•
s
4 Co) Initial Perforation
(a) Initial Contact
target
target
Y
% plug
(c) Hole F_.nlargcmcnt
(d) From Petaling
Fig. 2. Schematic of the perforation process by a tumbling blunt-faced cylindrical projectile on thin
plates: (a) initial contact; (b) initial perforation; (c) hole enlargement; (d) front petaling.
Plugging stage Plug formation starts once the projectile touches the target. Unlike normal perforation, where the plug is immediately initiated over the entire circular contact region, this process commences here by initial failure of the target along the upper edge of the projectile/target contact area and spreads gradually by further intersection of the projectile and the target, as shown in Fig. 2(b). This event is similar to the formation of a petal. The part of the target in contact with the face of the projectile is assumed to instantly attain its velocity. Due to the oblique orientation and rotation of the projectile, the shape of the plug is neither circular nor exactly elliptical but approximates an ellipse if the rotation during perforation is not large. When tumbling is present, the oblique position of the projectile leads to indentation between the lateral surface of the projectile and the entry side of the target during the petaling stage. This process results in an elliptical crater. Once the center of the face of the projectile is intercepted by the target, the contact area becomes a semi-ellipse and the indentation ends. At the same time, initiation ensues of either hole enlargement or petaling, depending on a transition criterion. Since this process lasts a very short time, thickening of the plate near the periphery of the crater is very small and is therefore neglected. After hole enlargement or
An analytical model for tumbling projectile perforation of thin aluminum plates
49
petaling occurs, the plugging process continues further until the whole face of the projectile is no longer in touch with the target. Calculation of the motion of the plug and projectile is based on the energy method developed by Recht and Ipson [8]. The energy dissipation is partitioned into three parts: E~, Eq and Em. Term E~ is the energy expended at the periphery of the crater in the deformation process that separates the plug from the target element; Eq is the energy expended throughout the plug in plastic deformation that accounts for the projectile and plug reaching a common velocity; and E m is the kinetic energy of the plug. (a) Calculation of E~. The energy loss, Es, due solely to the presence of the peripheral shear area, might be expected to be relatively insensitive over rather wide ranges of velocity [8] and, for normal perforation, can be written as:
mp mpV21
Es = 2 rnp + m-----~
(1)
or
1 PvL -mpv~, Es = 2 ppL + pt H
(2)
where mp and mg are the masses of the projectile and the plug, Vb~is the minimum perforation velocity, pp and Pt are the densities of the projectile and the target, L is the length of the projectile and H is the target thickness. Here it is assumed that the plug is cylindrical and of the same diameter as the projectile. Since E s is due solely to the presence of the peripheral shear area, it is reasonable to assume that this energy per unit shear area, e~, is constant for a given type of projectile and target material. Thus:
1 E~
(3)
es = 2r~ R H or 1 ppmpLv21 es - 4re (ppL + p t H ) R H
(4)
where R is the radius of the projectile. This energy value for each type of target material was obtained from the normal shots on targets of the same type of material. In tumbling perforation, the energy dissipation in time step i is: AEs(i) = e~AS(i)
(5)
where AS(i) is the area swept out in time step i. From the geometric relation shown in Figs 3(a) and (b): L zv(i) = zc(i) + -~ cos 0~ + R sin 01
(6)
where z c is the coordinate of the projectile center and 0 i is the impact angle at time step i. The equation of the ellipse in Fig. 3(b) is: X2
y2
R 2 Jr (R/cos 01)2 - 1.
(7)
AS(i) ~ 2 c ~ s 0 / {[-xK(i + 1) -- XK(i)]2 + [yr( i + 1) -- yt(i)] 2} 1/2
(8)
From Fig. 3(b),
where the effective thickness, due to the oblique position of the projectile is H / c o s 0 I.
50
Kezhun Li and W. Goldsmith
vo\l
7;: ..............\\, -
~,~
\
o .-°"
Y M
:M,P
N
K
.....
K..
(b) A-A
(c) B-B
Fig. 3. Projectile-target geometry during perforation by a tumbling blunt-faced cylindrical projectile on thin plates.
F r o m E q n (7), we h a v e
(9)
xK(i) = [ R 2 -- y~(i) COS2 Oil 1/2
xz(i + 1) = [R 2 -- y~(i + 1) cos 20i+ 1] 1/2
(I0)
YK(i) = yN(i)
(11)
yK(i+ 1) = YN(i+ 1).
(12)
where
A l s o f r o m Fig. 3(b),
R
yN(i) = T N = T M -- M N = - cos 0~
R ys(i + 1) = cos Oi-~1
Zp(i) sin Oicos 0~
zp(i + 1) sin Oi+ 1 cos Oi+ 1"
(13)
(14)
An analytical model for tumbling projectile perforation of thin aluminum plates
51
(b) Calculation of Eq. The energy expended throughout the plug in plastic deformation Eq, that accounts for the projectile and plug reaching a common velocity, based on Eqn (1) with a modification to account for discretization of the process, can be expressed as follows [8]:
1 Amg(i)mp v~(i)
AEq(i)
-
(15)
2 Amg(i) + mp
where VN(i) is 1:hevelocity at point N normal to the face of the projectile, and Ares(i) is the mass of the plug formed at time i. From Fig. 3(a), VN(i) = ~¢(i)cos 0 i -- ~c(i) sin 0 i - [ -
(~ 1~. - -
Zp(i)"~A
s~nOi)tli
(16)
where y¢ is the coordinate of the projectile's center. Also we have
Amg(i)=2H[Xk(i)+2k(i+l)]lyN(i)-- yN(i+l)lpt.
(17)
Substituting Eqns (16) and (17) into Eqn (15) provides AEq(i). (c) Calculat, ion of Em. In the initial perforation stage, the part of the target material in contact with the face of the projectile is assumed to instantly attain the same velocity as that part of the projectile, or VN(i).Thus, we have AEm(i) = ½Amg(i)v2(i).
(18)
Substituting Eqns (16) and (17) into Eqn (18) determines AEm(i). (d) Force and moment acting on the face of the projectile. The energy rate dissipated during the initial perforation stage is equal to the product of the projectile/target interaction force and the velocity component in the force direction. Since friction forces appear to be small [23,24], they are neglected in the calculation. The direction of the interacting force is normal to the projectile surface as is the velocity component. The relation between the energy rates, and normal fi~rce, and the velocity can be expressed as: AEt(i)
At
-- Fb(i)vN(i )
(19)
or
Fb(i) = AE s + AEq + AE m
(20)
Vs(i) At where At is the time step and Fb(i ) is the force acting on the face of the projectile. Also, from Fig. 3(a), the moment from the face of the projectile is: Mb(i) = Fb(i)(R -
-
-
sinZP(i)01,f ~
(21)
(e) Force and moment actin9 on the lateral surface of the projectile. In the initial perforation, the resisting pressure from the lateral indentation [along arc MK, as shown in Fig. 3(c)] is postulated to be equal to the yield stress of the material try. As shown in Figs 3(a) and (c), this process corresponds to Zp(i)/sin 0 i < R, and the force and the moment from the lateral surface of the projectile can be derived as follows: F~(i) = 2
% sin ~ Rd'9
(22)
,)3o
Ms(i ) = 2
try sin ,9 Rd,9 J #o
l
(23)
52
Kezhun Li and W. Goldsmith
where ,9o, as shown in Fig. 3(c), is UN ~9o = tan-
(24)
1NK
or
[ R--zp(i)/sinO, ~9o = tan-1 I-x/R2 _ (R - zp(i)/sin 0i) 2 J
(25)
1 f_ zp(i) "~ ,~ l=-~L-- Rsin~gtanOi + ~ g--s~nOi)tanvi.
(26)
and the arm length
After integration, we have
H Fs(i ) = 2 ay-----=c 2
(27)
cos 0 i
and
Ms(i)=aYco~iI(L+2cltanOi)c2-tanOi(~R --,9oR2[-ClC211 2
(28)
where
c1= R c2 = ~
Zp(i)
(29)
sin 0~ .
(30)
Hole enlargement stage The experimental observations indicated that, for a relatively low impact angle, an appreciable pile-up of material near the periphery of the hole occurred after the initial perforation. This is similar to the hole enlargement in thin plate perforation studied by Taylor [25]. However there are differences between the Taylor problem and the present one. First, the hole enlargement here is not axisymmetric as in Taylor's case. In axisymmetric hole enlargement, velocities at each point of the crater are directed radially as shown in Fig. 4(a) and the thickening around the edge of the hole is evenly distributed. Here, however, the velocities at each point have the same direction as shown in Fig. 4(b). This results in uneven thickening around the arc, which decreases from the upper point B in Fig. 5 to the intercept of
Fig. 4(a). Velocity distribution in axisymmetric hole enlargement.
An analyticalmodel for tumbling projectileperforationof thin aluminum plates
•
i I
• •
~'l--'-''~'~l :
-
53
•x Ii
............................................... . . . . . . . velocl~"
Fig. 4(b). Velocitydistribution in asymmetrichole enlargement.
B
.............. i........ C O
O'
D
D'
iD
(a) SideView
(b) FrontView A-A
Fig. 5. Schematicofthickeningduringasymmetricholeenlargement:(a) side view;(b)front viewA-A. the target and the axis of the projectile from where the hole enlargement initiates, O in Fig. 5, and where there is no thickening at all. Secondly, the initial radius of the hole in the plate is not zero as it was in Taylor's problem. The current non-axisymmetric hole enlargement model :is still based on the axisymmetric case; it is assumed that the thickening of the upper points B and B' is the same as that of axisymmetric hole-enlargement and the distribution of the thickening around the arc BC is quadratic, or hyperbolic due to symmetry of the arc. Since it is very difficult to analyze the hole enlargement when the initial radius of the hole is not zero by using Taylor's theory [25,1 Bethe's method [26-] is adopted here with modified boundary conditions to incorporate a non-zero initial radius. As mentioned in [27-], Bethe's method leads to a lower bound for the expansion of a hole and is just as useful for the practical calculation of work done, even though it results in a vastly different stress distribution and lip height at the edge of the hole than Taylor's theory predicts. In Bethe's model, shown in Fig. 6, it is assumed that: tr0 = 0 and a z = 0
(31)
where tr0 is the circumferential stress and trz is the stress along the thickness of the plate. The equation of equilibrium in a thin plate of variable thickness h becomes d(htrr) + - ~ = 0
(32)
54
K e z h u n Li and W. G o l d s m i t h t
-I I I I
HI
h
11 I I I I ,
I I
', I
_t_
~ I I
[ N
V"
, I
r
I
I
,, Fig. 6. D i a g r a m of the strain distribution associated with theductile hole formation model of Bethe.
where trr is the radial stress and r the radial coordinate. Equation (32) has the solution c her r = r
(33)
where c is a constant. According to the Von-Mises or Tresca yield criteria, we have O"r = O'y.
(34)
/~r o = H r 1 = constant
(35)
Substitution of ar into Eqn (33) gives
where H is the total height of ridge, r o is the radius of bole and r 1 is the radius of plastic boundary, as shown in Fig. 6. Assuming constancy of volume and referring to Fig. 6, 1
2
1
2
j.o
I*'~
//~ - H'~
36)
where R o is the initial radius. Solving the above equations, we obtain ./~ . k e . 1 +. H
/1.
(R°~ 2 \r o ) "
(37)
Figure 7 shows the thickening coefficient k e as a function of r o / R o. It is found that for a given initial radius R o, the thickness /~ increases with an increase of the radius r o. The equation also shows that for a zero initial radius (R o = 0), or equivalently r o --, ~ , ke is equal to 2, which agrees with the solution by Bethe and is close to Taylor's theory. The distribution of the thickening, shown in Fig. 5, is assumed to be quadratic around the upper half of the hole, so that -h(x) = a 1 + a2x + a3 X2
(38)
56
Kezhun Li and W. Goldsmith
(a) Petaling
(b) Ductile Tearing of Metals
,[
H
(c) Final Tearing
(d) Shearing around the Edge
Fig. 8. Diagram of petaling stage: (a) petaling; (b) ductile tearing of metal; (c) final tearing; (d) shearing around the edge.
Front petaling stage
When the impact angle is relatively large, the failure mode after initial perforation is front petaling as defined by Wu and Goldsmith [28] rather than hole enlargement. The experiment and analysis [1] showed that the petaling will not materialize until the impact angle reaches 15° for 4.8 mm thick 6061-T6 aluminum plates. It is 9.3 ° for 3.2 mm and 18.8° for 6.4 mm thick plates. The target material in front of the contact region of the lateral surface of the projectile was bent after tearing and thus formed a petal, as shown in Fig. 8(a). Observation indicated that the width of the petal is approximately the same as the projectile diameter. This is similar to the tearing of thin ductile metals studied by Mai and Cotterell [291 as shown in Fig. 8(b). As for the first stage, the force and moment are derived based on an energy approach. The energy dissipated in this process is mainly due to shearing fracture of the petal, localized plastic shear work in a zone contiguous with the torn edges, the kinetic energy and the bending energy of the petal. The following derivation is based on the model developed by Mai and CottereU [29]. (i) The energy rate due to fracture. The major fractures in the target producing the petal are two nearly parallel cracks like those generated in the tearing of ductile metals, as shown in Fig. 8(c). The work required to fracture a length Ax of the sheet is given by [29]: AW 1 = 2
zsAx(H - u)du = % H 2 A x
(48)
where % is the shear stress during tearing and is assumed to remain constant. (ii) Localized work in a shear zone contiguous to the torn edge. The plastic work necessary to produce; the thin lip of width s on the edges of the petal shown in Fig. 8(d), is obtained from the relation [29]: A W 2 = 4 H s ~ yT° A x
,/3
(49)
An analytical model for tumbling projectile perforation of thin aluminum plates 2
i
|
i
55
|
1.9 1.8 1.7
~
1.6
~1.5 T:
~1.4 J¢ I--
1.3 1.2 1.1
1
2=
1.5
2 i5 3' 3.5 ' 4' Ratio ot Final Radius to Initial Radius
4 i5
5
Fig. 7. Thickening coefficient as a function of initial radius and final radius.
with the boundary conditions
h(O)= ken
(39)
h(_+ R) = n .
(40)
It is assumed that the point C has the same thickness as the point O, i.e. no thickening at C. This is reasonable since the length of OC was found to be small from the experiments. Solving Eqn (38), we obtain
-h(x)= ken + (1 -ke)nx2 R2 .
(41)
In the hole enlargement, the pressure is still equal to the yield stress try. From Figs 3(a) and (c), we have
Fs(i,=2f]/2aysing(Rd`9c~sOi)
(42,
Ms(i,=2f]/Zaysin`9(Rd`9~)l
(43,
where I is given by Eqn (26). From Fig. 3(c). x = R cos ,9
(44)
so
-h(~ = keH sin 2 `9+
H cos 2 ,9.
(45)
After integration of Eqns (42) and (43), there results
Fs(i)= Ms(i)=
2 HR
3 cos 0 i ay(2ke + 1)
[1
(
zP(i)~
(46) 1
-~L+ R-s~nOijtanOi F~(i)
nHR2tanOiay(l+3ke,. 8cos0i
(47)
An analytical model for tumbling projectile perforation of thin aluminum plates
57
where Yois the maximum shear strain. Yo= x/~e~, and where e, is the ultimate tensile strain. Here, the strain hardening effect is neglected. In the present work, the width s is measured experimentally from the perforated samples. Roughly, s = 3H. (iii) Kinetic energy of the petal. It is assumed that the petal, which is in contact with the lateral surface of the projectile instantly attains the normal component of the velocity of the projectile. Here the velocity is assumed to be that of point M as shown in Fig. 3(a). The velocity is:
Vm(i)=~sinO'+ Y¢C°S0~--(2
COS0,,]ZP(0 ~0 i
(50)
so the kinetic energy of the petal is AWa = ½Amv2(i) = PtnRv2(i)Ax.
(51)
(iv) Bending energy of the petal. The energy dissipated in the bending of the petal is:
r n/2 a 1 tryRH 2 Ax AW,=4RJo ~uduAx=~ x
(52)
where x is the radius of curvature which is given by 1-29] H 2e
x
(53)
where e is the tensile strain. Clearly, the real radius of curvature is dependent on the impact angle. The bJ,gger the impact angle, the larger the radius of curvature, and the less the energy dissipated. In the present model e is taken to be the maximum, or the ultimate tensile strain eu. (v) Force and moment acting on the lateral surface of the projectile. An energy approach is used here similar to that of the initial perforation stage,
AWl + AW~ + AW3 + AW, = Fs(i)vm(i ). At
(54)
The force is assumed to act at point M. 1
F~(i) = Atv~(i)(AW~ + AW2 + AW3 + AW4).
(55)
mx At = vm(i) cos 0 i.
(56)
Also
Substituting Eqns (48), (49), (51), (52) and (56) into Eqn (55),
Fs(i) = "c~H2 +
4Hstryy o
v~ ~
t- ptHRv2(i) +
tryRH 2
2~c
(57)
and / 1
MsO) = l~s(i)~~ L - - Equation of motion of the projectile Once the forces and the moments from both the face and lateral surface of the projectile are known, the equation of motion of the projectile can be derived from Newton's Second Law (Fig. 9).
58
Kezhun Li and W. Goldsmith
Mb
z
Fig. 9. Rigid-body motion of the projectile.
The equations of motion of the projectile are: Fz(i) = mpZ'¢(/)
(59)
Fy(i) = mpy¢(i)
(60)
M(i) = Ip'O(i)
(61)
where (62)
mp = n R 2 L p p
Ip --
+ ]5
(63)
with initial conditions: 0(0) = 0o
(64)
~c(0) = vo cos flo
(65)
3~o(0)= vo sin flo
(66)
0(0) = 09o.
(67)
From the expressions for the forces and moments Fb(i), Fs(i), Mb(i), Ms(i), we have: Fz(i) = - Fb(i) COS0 i -- Fs(i ) sin 0 i
(68)
Fy(i) = Fb(i) sin 0 i -- Fs(i ) cos 01
(69)
M(i) = Ms(i ) - Mb(i).
(70)
A forward Euler method is used in the finite difference calculation. The basic equation is: :,( i)
dy
y(i + 1)-y(i) At
(71)
RESULTS AND DISCUSSION The phenomenological model developed above has been programmed in F O R T R A N language and calculated numerically. The program input data include material constants, problem geometry, and kinematic parameters of the projectile. The computational output provides predictions of the final velocity, oblique angle, and rotational speed of the projectile as well as the crater length of the target. Simulations of 32 runs (23 for 4.8 mm, 5 for 6.4 mm, and 4 for 3.2 mm thick targets) were performed. Four normal perforations on 4.8 mm and 3.2 mm of 6061-T6 aluminum were used to obtain an average value of the energy loss es. The experimental ballistic limit %~ (normal impact) for 6061-T6 aluminum was found to be 89m/s for 6.4mm thick plates, 66m/s for 4.8 mm plates and 41 m/s for 3.2 mm plates. Material properties obtained from [30] are listed in Table 1. The computational results and corresponding experimental data are presented in Table 2. The calculated final velocities, final oblique angles and crater lengths are in good correspondence with the experimental data.
An
analytical model for tumbling projectile perforation of thin aluminum plates
59
Table 1. Properties of projectile and target materials Density
Dynamic yield
Ultimate shear
Ultimate tensile
Material
(kg/m 3)
stress (MPa)
strength (MPa)
strain (%)
Projectile
7977 2780
1393 295
804 190
-18
AL 6061-T6
Table 2. Experimental and analytical results of perforation of plates by tumbling projectiles Initial condition
Final condition
Projectile
Projectile
t
~o
Oo
/~o
O~o
~
~
/~
Crater
/~
~°o
~'o
Run
Mtls
(mm)
(m/s)
(deg)
(deg)
(rad/s)
(m/s)
(m/s)
(deg)
(deg)
(mm)
(mm)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 All A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23
A1 AI A1 A1 AI A1 AI AI AI AI AI A1 A1 AI A1 A1 A1 A1 AI A1 AI AI AI
4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8 4.8
477 336 638 318 480 511 445 398 335 488 407 395 453 500 480 315 568 586 324 451 565 409 525
0 5 6 5 20 21 21 28 42 9 23 14 26 23 32 29 2 10 50 4 0 0 0
6 8 3 7 - 7 8 -6 10 8 -3 7 7 7 10 7 12 5 -4 13 3 0 0 0
254 276 1655 170 752 1604 937 1402 979 498 1063 350 1407 2050 1682 1023 554 190 1791 32 0 0 0
451 285 591 281 440 404 393 298 230 465 344 343 396 413 382 238 542 564 214 422 535 392 498
446 297 600 280 443 446 404 317 247 460 346 348 400 430 407 233 536 551 230 423 538 387 499
1 - 7 -2 -9 - 13 -1 - 15 - 1 3 -7 -2 -2 1 - 1 0 -2 - 1 -9 11 -2 0 0 0
0 - 10 - 1 - 11 - 14 1 - 15 1 2 -9 -3 -4 - 1 2 3 -3 1 -9 8 -4 0 0 0
17.0 22.0 21.0 20.0 21.0 33.0 22.0 36.0 40.0 17.0 33.0 28.0 34.0 32.0 36.0 37.0 19.0 16.0 40.0 18.0 13.5 13.5 13.5
17.1 20.3 20.2 19.7 21.5 31.4 22.7 35.1 37.5 16.5 31.2 26.0 32.6 33.0 35.3 35.5 17.6 16.6 40.2 16.8 12.7 12.7 12.7
B1 B2 B3 B4 B5
A1 A1 A1 A1 A1
6.4 6.4 6.4 6.4 6.4
551 516 423 576 446
18 13 46 11 53
- 6 - 7 10 6 10
1089 239 2176 701 2997
479 470 267 495 284
493 472 283 500 310
- 14 - 14 5 -6 11
- 16 - 18 1 -5 8
23.0 18.0 40.0 25.0 41.0
22.0 16.2 39.3 24.4 40.5
C1 C2 C3 C4
A1 A1 A1 A1
3.2 3.2 3.2 3.2
685 395 298 557
4 32 2 0
4 10 10 0
259 1268 94 0
653 356 261 545
660 358 275 540
1 6 -2 0
3 6 - 1 0
20.0 38.0 21.0 14.0
18.4 36.3 20.6 12.7
Figure 10 depicts the experimental and analytical velocity drop as a function of impact angle for a given oblique angle and initial velocities of 350 m/s and 500 m/s for 4.8 mm thick 6061-T6 aluminum targets. It is seen that the velocity drop increases with impact angle and decreases wit]h initial translational velocity. Figure 11 shows the results of the final oblique angle as a function of impact angle for the given oblique angle and initial velocities. The absolute value of the final oblique angle increases with increasing impact angle when the impact is below a certain value (10 ° for 500 m/s and 12 ° for 350m/s), but decreases to zero when the impact angle is above it. A higher initial velocity reduces the magnitude of the final oblique angle, as shown in the figure. The variation of crater length of the target as
60
Kezhun Li and W. Goldsmith
x : v - 350 m/s, experimental data
45
411
~
o : v - 500 m/s, experimental data solid line: analytical results
x
30
~: 25
10
IMPACT ANGLE degree
Fig. 10. Experimental and analytical results of percentage velocity drop as a function of impact angle for 4.8 mm thick 6061-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5 °. 20 x : v = 350 m/s, experimental data
15
tO
o : v = 500 m/s, experimental data solid line: analytical results
10
ut
,
(.o z,<
,,, _o
o'
3-, ,< _z " -lO
-15
-20 IMPACT ANGLE degree
Fig. 11. Experimental and analytical results of final oblique angle as a function of impact angle for 4.8 mm thick 6061-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5 ° .
a function of the impact angle is depicted in Fig. 12, showing an increase for large impact angles. Observations from Figs 10-12 indicate that the velocity drop, final oblique angle, and crater length reach stable conditions when the impact angle is above 50°, similar to the situation encountered in side-on impact (at an angle of 90°). The comparisons illustrated in the tables and the figures validate the model for the range of parameters examined here. Figure 10 shows that the model underestimates the velocity drop as the impact angle increases. It is found from the experiments that bending of the target becomes large as the impact angle increases or the initial velocity decreases. In the model development, however, the bending effect is neglected. This leads to less energy dissipated in the perforation and subsequently, a low velocity drop.
An
analytical model for tumbling projectile perforation of thin aluminum plates
61
50 ~5
valocity 300 ~ 600 m/s
o: experimental data 40
solid line: analytical results Oo
~5 E E :~3 N LU
LU
O
15 11)
IMPACT ANGLE degree
Fig. 12. Experimental and analytical results of crater length as a funcion of impact angle for 4.8 mm thick 606 l-T6 aluminum plates struck by hard-steel, blunt-faced cylindrical projectiles. Initial oblique angle is 5°.
In the petaling stage, the energy dissipation consists of four parts: the energy due to fracture AW~, the localized work in a shear zone AW2, the kinetic energy of the petal AW3, and the bending energy of the petal A W4. The following discussion will focus on the effect of these parts. From Eqns (48), (49), (51) and (52), the ratios of the four parts are: AWx:AW2:AW~:AW4=Ts:
12ayTo ptRv~ ayReu i-.-.~ : H : - n "
(72)
For 3.2 mm-6.4 mm thick 6061-T6 aluminum targets, the ratio is: AWx :AW2 :AW3 :AW4 = 1.0:4.6:(0.1 ~ 0.5):(0.3 ~ 0.6).
(73)
It can be seen from Eqn (73) that, in the petaling stage, most of the energy goes to the localized work in a shear zone contiguous to the fracture edge instead of the energy due to fracture. The kinetic and bending energies are not very important in an aluminum plate. This means that the shear zone and the maximum shear strain are the main factors in the petaling stage. Currently, the shear zone is determined from the experiments and assumed to be independent of the impact velocity. In fact, the shear zone size changes with the impact velocity (3.3H to 2.8H). It transpires that the higher the velocity, the smaller the shear zone. Considerable discrepancies between the model calculations and the experiments arise in several cases, such as runs A6, A14, and A15 where relative errors for the velocity drop are 40%, 20%, and 25% and runs A2, A15, and A19 where errors for the final oblique angle are 3 °. This may be due to inaccuracy in the measurement and a certain amount of scatter in the experiments. Further dew.qopment of the analytical model will depend on the observed target response for regimes different from those studied here. For example, for low velocities, high impactangles or thin plates, bending effects and membrane forces tend to dominate the penetration process. This has been neglected in the current model. Also, as the velocity of the projectile or the thickness of the target increases, the size of the shear zone contiguous to the torn edge decreases, similar to what occurs when a beam is loaded impulsively (see Jones [31] regarding this subject).
62
Kezhun Li and W. Goldsmith CONCLUSION
A n a n a l y t i c a l m o d e l was c o n s t r u c t e d to predict the p e r f o r a t i o n of thin a l u m i n u m plates by tumbling, blunt-faced hard-steel cylindrical projectiles. This m o d e l was b a s e d on e x p e r i m e n tal o b s e r v a t i o n s t h a t i n d i c a t e d the presence of three consecutive stages: plugging, hole e n l a r g e m e n t a n d petaling. C a l c u l a t i o n s for the system response were p e r f o r m e d for 32 i m p a c t c o n f i g u r a t i o n s i n c l u d i n g those c o r r e s p o n d i n g to test results d e s c r i b e d elsewhere. Such d a t a , o b t a i n e d for the final velocity a n d final oblique angle of the projectile as well as the c r a t e r length p r o d u c e d in the targets, were c o m p a r e d with c o r r e s p o n d i n g m o d e l predictions. It was f o u n d t h a t increasing i m p a c t angle results in a n o t i c e a b l e increase of the velocity d r o p a n d t r a j e c t o r y c h a n g e of the projectile. The c r a t e r length of the target also increases. H o w e v e r , when the i m p a c t angle is a b o v e 50 °, the velocity d r o p a n d the c r a t e r length seem to be stabilized a n d the final o b l i q u e angle tends to be zero like those in side-on impact. H i g h e r projectile velocity tends to reduce the velocity d r o p a n d c h a n g e of the final o b l i q u e angle. In the p e t a l i n g stage, m o s t of the initial energy is c o n v e r t e d into localized w o r k in a shear zone c o n t i g u o u s to the fracture edge, r a t h e r t h a n into the energy of fracture a n d bending. Acknowledgement--This work constitutes a portion of a doctoral dissertation by the first author at the University of California, Berkeley. The work was sponsored by the Air Force Office of ScientificResearch, Bolling Air Force Base, Washington, D.C., under contract AFOSR F49620-89-8127.
REFERENCES 1. K. Li and W. Goldsmith, Impact on aluminum plates by tumbling projectiles: experimental study. Int. J. Impact Engng 18, 23-43 (1996). 2. M.E. Backman and W. Goldsmith, The mechanics of penetration of projectiles. Int. J. Engng Sci. 16, 1-99 (1978). 3. B. Landkof and W. Goldsmith, Petaling of thin, metallic plates during penetration by cylindro-conical projectiles. Int. J. Solids Struct. 21,245-266 (1985). 4. J. Awerbuch and S. R. Bodner, Analysis of the mechanics of perforation of projectiles in metallic plates. Int. J. Solids and Struct. 10, 671-684 (1974). 5. J. Liss, W. Goldsmith and J. M. Kelly, A phenomenological penetration model of plates. Int. J. Impact Engng l, 321 341 (1983). 6. W. Yuan, L. Zhou, X. Ma and W. J. Stronge, Plate perforation by deformable projectiles--aplastic wave theory. Int. J. Impact Engng 1, 393-412 (1983). 7. M. Zaid and B. Paul, Mechanics of high speed projectile perforation. J. Franklin Inst. 264, 117-126 (1957). 8. R. F. Recht and T. W. Ipson, Ballistic perforation dynamics J. Appl. Mech. 30, 384-390 (1963). 9. D. M. Cunningham and W. Goldsmith, An experimental investigation of beam stresses produced by oblique impact of a steel sphere. J. Appl. Mech. 23, 606-611 (1956). 10. W. Goldsmith and D. M. Cunningham, Kinematic phenomena observed during the oblique impact of a sphere on a beam. J. Appl. Mech. 23, 612-616 (1956). 11. M. Zaid and B. Paul, Oblique perforation of a thin plate by a truncated conical projectile. J. Franklin Inst. 268, 25-45 (1959). 12. M.E. Backman, S.A. Finnegan and K.G. Whitman, Dynamics of the oblique impact and ricochet of nondeforming projectiles against thin plates. In Recent Advances in Engineering Science, pp. 9-20 (Edited by G. C. Sih). Lehigh University Publication (1977). 13. J. Awerbuch and S. R. Bodner, An investigation of oblique perforation of metallic plates by projectiles. Exp. Mech. 17, 147-153 (1977). 14. W. Goldsmith and S. A. Finnegan, Normal and oblique impact of cylindro-conical and cylindrical projectiles on metallic plates. Int. J. Impact Engng 4, 83-105 (1986). 15. J. Falcovitz, M. Mayseless, Z. Tauber, D. Keck, R. Kennedy, K. Ofstedahl and P. Singh, A computer model for oblique impact of a rigid projectile at ductile layered targets. Proc. l l t h Int. Syrup. on Ballistics, pp. 1-11, Brussels, Belgium (1989). 16. R. L. Woodward and N. J. Baldwin, Oblique perforation of targets by small armor-piercing projectiles J. Mech. Engng Sci. 21, 85-91 (1979). 17. S. R. Reid, A. J. Edmunds and W. Johnson, Bending of long steel and aluminum rods during end impact with a rigid target. J. Mech. Engng Sci. 23, 85-92 (1981). 18. S.J. Bless,J. P. Barber, R. S. Bertke and H. F. Swift, Penetration mechanics of yawed rods. Int J. Engng Sci. 16, 829-834 (1978). 19. S.J. Bless and J. P. Barber, Bending waves in yawed rod impacts. J. Ballistics 2, 281-298 (1978). 20. V. Hohler and A. J. Stilp, Influence of the primary yaw angle on the secondary yaw angle during armor plate perforation. Proc. 6th Int. Syrup. on Ballistics, pp. 302 308, Orlando, Florida (1981).
An analytical model for tumbling projectile perforation of thin aluminum plates
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