Lagrangian hydrocode simulations of rolled-homogeneous-armor plate perforation by a shaped charge jet

Pergamon

Int. J. Impact Engn.q Vol. 15. No. 5, pp. 619-643, 1994 Elsevier Science Ltd Printed in Great Britain 0734-743X(93)E0006-4 0734-743X/94 $7.00+ 0.00

LAGRANGIAN HYDROCODE SIMULATIONS OF ROLLED-HOMOGENEOUS-ARMOR PLATE PERFORATION BY A SHAPED CHARGE JET MARTIN N.

RAFTENBERG

Weapons Technology Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, U.S.A.

(Received 1 March 1993; in revised form 25 July 1993) Summary--Three experiments are performed in which a 13-mm-thick rolled-homogeneous-armor plate is perforated by the leading particle of a copper jet produced by firing a shaped charge warhead at long standoff. These experiments are modelled using the 1986 version of EPIC-2, to which two failure models have been added to simulate effects of tensile voids and shear bands. The tensile void model employs a negative pressure cutoff onset criterion, followed by instantaneous reduction to zero of all deviatoric stresses and hydrostatic tensile stress within the finite element. The shear banding model uses the Zener-Hollomon onset criterion, followed by reduction of deviatoric stresses in proportion to the difference between the element's current level of equivalent plastic strain and that corresponding to onset. The two failure models together with the code's slideline erosion algorithm allow for good agreement with experiment in terms of the final hole radius averaged over the target plate's thickness, the time required for complete hole formation, and the net mass lost by the plate. Mesh sensitivity studies are conducted to validate partially the procedure.

NOTATION

A,B,C.M,N

D~ G Kl, K2, K3 Lc, Ls, LB, Rs, RB T

Tm L T* e f(~Pl P Pfail r /~hole =exp. thole ro Srr, Szz, SO0

Srz

s*,, s*, s~o

s?, t

t',;o'~, 10.95 exp.

/0.95

parameters in the Johnson-Cook strength model component of the plastic part of the rate of deformation tensor elastic shear modulus parameters in the Mie-GriJneison equation of state parameters in the geometric model for the leading jet particle temperature melting temperature room temperature homologous temperature internal energy per underformed volume stress reduction function used in the shear band failure model pressure pressure at which the onset of tensile failure occurs radial coordinate computational hole radius at 500/~s after initial impact, averaged over the target plate's thickness experimental final hole radius averaged over the target plate's thickness radial coordinate of a material point referred to the undeformed geometry at t = 0 radial, axial, circumferential normal components of the deviatoric Cauchy stress tensor following reduction shear component of the (deviatoric) Cauchy stress tensor following reduction radial, axial, circumferential normal components of the deviatoric Cauchy stress tensor prior to reduction shear component of the (deviatoric) Cauchy stress tensor prior to reduction time, measured from the instant of initial impact time at the time step immediately prior to that at which a node erodes from the original exit surface of the target plate time at which the target plate's cumulative mass loss equals 0.95 AM, according to computation time at which the target plate's cumulative mass loss equals 0.95 AMcxp', according to experiment axial coordinate 619

620

M . N . RAFTENBERG _.(-)

"~crod©

AM

AMCXr,.

AeF~i, F £P EProde EPnset /.t P Po

O'y

axial velocity of a node on the target plate's original exit surface at the time step immediately prior to that at which it erodes computational accumulated mass lost by the target plate at 500/as after initial impact experimental total mass lost by the target plate the minimum increment above eoP,~c,of equivalent plastic strain corresponding to which all components of deviatoric Cauchy stress within an element are set to zero Griineison coefficient equivalent plastic strain equivalent plastic strain at which a slideline's master surface element is eroded equivalent plastic strain within an element at the time of the onset of its failure by shear banding time rate of change of equivalent plastic strain fractional change in density from the undeformed state density undeformed density flow stress

1. I N T R O D U C T I O N

Specific problems involving target plate perforation by a shaped charge jet are routinely studied experimentally at the U.S. Army Research Laboratory and at other organizations devoted to terminal ballistics. The cumulative cost is great, and the motivation is therefore strong to develop reliable computational methodologies as less expensive alternatives. The development of slideline erosion algorithms over the past fifteen years has made possible the application of Lagrangian hydrocodes to these problems. Reference [1] describes the slideline erosion algorithm installed by Johnson and co-workers into the 1986 version of EPIC-2 [2]. In Ref. [1] this algorithm was shown capable of obtaining reasonable agreement with an Eulerian hydrocode in terms of depth of penetration into semi-infinite steel targets by aluminium, copper, and tungsten rods impacting at 5 km/s. No failure modeling other than slideline erosion was applied in either the Lagrangian or the Eulerian calculations. All metals were represented as elastic/perfectly plastic. In Ref. [3] EPIC-2, with a slightly revised slideline erosion algorithm, was applied to the perforation of a 25-mm-thick mild steel plate by a 100-mm-long, 10-mm diameter tungsten rod impacting at 1.52 km/s. Reasonable agreement with experiment was obtained in terms of characteristics of the tungsten residual, but no experimental check on the target's hole geometry was applied. Apparently, no previous application of EPIC-2 to penetration problems specifically involving shaped charge jets has appeared in the open literature. In the present study, the 1986 version of EPIC-2 [2] is modified to include models for two material failure mechanisms, tensile voids and shear bands. The resulting code is applied to the case of a 13-mm-thick rolled-homogeneous-armor plate (RHA) plate perforated by the leading copper jet particle from a shaped charge warhead fired at normal incidence and long standoff. Experimental data on final hole geometry, total mass lost by the plate, and the time history of the mass loss for this case provide benchmarks by which to assess accuracy of the computational results. Section 2 presents the experimental results. Features of the 1986 version of EPIC-2 are reviewed in Section 3. Two failure models intended to represent effects of tensile voids and shear bands have been inserted into the code and are described in Section 4. Computational results obtained with and without these models are presented in Section 5. Section 6 follows with conclusions.

2. T H E

EXPERIMENTS

The shaped charge warhead used has a conical, oxygen-free, high-conductivity (OFHC) copper liner with a 42 ° apex angle, an 81.28-mm outer-diameter base, and a 1.91-mm wall thickness. The explosive fill used is Composition-B. RHA is a medium-carbon, quenched and tempered, martensitic steel [4]. Each target

Lagrangian hydrocode simulations

621

plate has a 13-mm thickness and a square entrance face with an edge length of at least 197 mm. Brinell hardness numbers (BHN) for entrance and exit faces are measured to be either 340 or 364. In one experiment a standoff of 15.23 charge diameters (C.D.) was used, and in the other two the standoff was 12.00 C.D. At these long standoffs, the jet has broken up into particles before impacting the plate (Fig. 1). The leading particle has a nominal speed of 7.73 km/s at the time of impact I-5-1. The leading particle completes perforation of the target, with most subsequent particles passing through the hole it has created. Following one experiment, the plate was sectioned along diameters through the perforation hole to reveal the hole geometry. Figure 2 shows a cross section at a region of the hole boundary that seems to have been made entirely by the leading jet particle. The hole radius averaged through the thickness, rhote, -exp. was measured to be 18.2 mm. In another experiment the target plate was weighed before and after impact to give a net mass ,exp. the time, measured from initial impact, at which the target lost loss, AM exp', of 136 g. ,0.95, 95% of mass AM ~xp, is roughly bounded on the basis of post-perforation radiographs in

V'

!

!

!

.a

"

I'

..

4

8

i

D

..,

.~

.

,,

FIG.1. Flash radiographs of thejet and RHA plate at 12.2/isbeforeand 33.1 ps after initial impact.

622

M . N . RAFTENBERG 10 m

-2

-6 ¸

-10 '

-14;

-1B

Ittdius - 15.0 mm

-22

-26

RD 1 0 7 7 1 , SECTION

ROLE P R O F I L E

t

-30 r

MN

FIG 2. Digitized boundary of a slice rrom a perforated plate and calculation o f t h e throat radius and ~ .

Rds. 4190 & 4189.

t = 1.7 ps

t = 7 3 . 9 p.s

Times Relative to Time of Initial Impact

t = 18.2 I~s

t = 143.2 ps FIG. 3. Flash radiographs at various times after initial impact.

t = 3 3 . 2 ,us

t = 2 1 3 . 0 ,us

Lagrangian hydrocode simulations TABLE 1. EXPERIMENTAL RESULTS

623

TABLE 2. PARAMETER VALUES FOR THE LEADING JET PARTICLE'S GEOMETRIC MODEL

Parameter

Value or bound

rhol e~xp. (mm) AM ~p' (g) t~.~ (#s)

18.2 136 > 73.9

Value (mm)

Parameter

Lc Ls L. Rs RB

4.77 12.75 3.08 2.40 3.30

ENTRANCE FACE

Y"~""~

VOID

SHEAR BAND

EXIT FACE FIG. 4. Sketch showing locations of prominent voids and shear bands in a radial slice from the RHA plate of one experiment.

CONE

STALK

Lc

i

BULB

FIG. 5. Five-parameter geometric model for the leading jet particle.

624

M.N. RAFTENBERG

Fig. 3 to be greater than 73.9/~s. These experimental results are collected into Table 1 and will serve as benchmarks for the computations. The slice of target plate that yielded Fig. 2 was polished and examined with an optical microscope at magnifications up to 1000X. Figure 4 sketches the observed pattern of prominent voids and shear bands. Both concentrate in the hole boundary's region near the plate's midsurface. Figure 5 presents a geometrical model for the leading jet particle at the time of initial impact. Its five parameters are assigned the values in Table 2 on the basis of a pre-impact radiograph. 3. E Q U A T I O N - O F - S T A T E ,

STRENGTH,

AND SLIDELINE

REPRESENTATIONS

The two metals are handled in the same fashion in terms of the equation of state and strength effects. Their dilatational deformation is governed by the Mie-Gr/ineisen equation of state

(1)

p = (K~/~ + K2]2 2 q- K3]./3) 1 -- ~ F p

+ F(1 + p)e

(1)

where P =-Po

1

(2)

p is pressure, or negative hydrostatic stress, p is current density, Po is undeformed density, e is the internal energy per undeformed volume, and F is the Griineisen coefficient. Values for K 1, K2, K3 and F are presented for O F H C copper and for 304 stainless steel in Kohn's handbook [6]. These parameter values are displayed in Table 3, in which Kohn's values for 304 stainless have been applied to RHA. The distortional deformation of copper and steel is treated by means of a plasticity algorithm which applies the von Mises yield condition. Flow stress ay is computed as a function of equivalent plastic strain eP, equivalent plastic strain rate kp, and homologous temperature T* by means of the Johnson-Cook strength model [7]. The functional form is

= [A + B( P) N] 1 + Cln

I1 - O'*) M]

TABLE 3. MATERIAL PARAMETER VALUES FOR THE METALS

Material parameter

OFHC copper

RHA

Po (kg/m3) G (GPa) K l (GPa) K 2 (GPa) K 3 (GPa) F A (GPa) B (GPa) N C M T~ (K) Tm (K)

8945 46.31 137.2 175.1 564.2 1.96 0.08963 0.2916 0.31 0.025 1.09 294.3 1355.9

7823 77.50 163.9 294.4 500.0 1.16 0.7922 0,5095 0.26 0.014 1.03 294.3 1793.2

(3)

Lagrangian hydrocode simulations

625

Z4

Z0

IB--

1Z-B--

_~ ~. ××

X X ~. ;'~ >,,I

,Q 5," :" ~', >- >1 >:, 3 m D D D D m D Q m i B m D D D D D D m D W m i X x :x( x , X :, >:~ 8 D ~ D D D D m g m ~ D m D D g ~ m Q D l l [~DQ~QmDm~mD~DDWmOW~[ K× ::,,{ ×, X :
• ".J

.;',,

)q .x. 'z .y..>~ :K >:, >~ ,'( -1Z. X X

iUD:BBB[DBUU[BBDBBBBBU61

-16 0

4

8

17'

16

Z0 r

Z4

Z8

3Z

36

40

(~1

FIG. 6. The initial mesh used in most calculations.

where ~P

/'2 \½ = t ~ DPD~)

(4)

and ~P =

:o

kP dt,

(5)

t is time and D~ is the plastic component of the rate of deformation tensor. T* is related to room temperature T~ and melting temperature Tm by T-T, T* - - T m - T,'

(6)

where T is the current temperature, computed using a thermodynamic relation. Equation (3) expresses strain hardening, strain-rate hardening, and thermal softening as separate factors. Hardening is assumed to be isotropic. Constants G (the elastic shear modulus), A, B, N, C, M, T, and Tm are assigned their values presented for O F H C copper and for 4340 steel by Ref. [7]. These values are listed in Table 3. The axisymmetric mesh used in all calculations unless otherwise stated is shown in Fig. 6. The projectile's mesh closely represents the model of Fig. 5. The target's mesh approximates the square steel plate by a circular disc with a thickness of 13.0mm and a radius of 104.0mm. The mesh contains two slidelines, both coincident with the projectile/target interface. One slideline has the projectile's surface nodes serving as master

626

M.N. RAFTENBERG

nodes and the target's surface nodes as slaves. In the other slideline this assignment is reversed. The erosion feature is operative for both slidelines. This allows an element having one or more corner node on the master surface of the slideline to be discarded from the problem in the sense that all stresses in the element are thereafter set to zero, Since both copper projectile nodes and steel target nodes alternately serve as master nodes for one of the two slidelines, the procedure provides a means for modeling both projectile erosion and target hole formation. EPIC-2 uses three criteria to trigger erosion of a given surface element. One criterion is a user-supplied cutoff value for equivalent plastic strain, ~erode" p Another is an angle cutoff, whereby none of the element's vertex angles is allowed to become less than a certain acute value. These two criteria limit the degree of distortion in the element. The third criterion discards a group of one or more elements that is connected to the master surface by a single vertex node. This criterion is motivated by the need to prevent such elements from crossing the slideline surface. Following erosion of a slideline element, its three associated node points, to which its entire mass has been lumped, remain in the problem. One or two may become effectively disembodied from the remaining projectile or target. When this occurs, those disembodied nodes are converted from master into slave nodes. They subsequently interact with the remaining master surface and participate in m o m e n t u m transfer across the slideline. Furthermore, previously interior nodes that now find themselves located on the master surface become identified as the slideline's master nodes. EPIC-2's eroding slideline algorithm is described further in Ref. [1]. 4. THE TENSILE VOID AND SHEAR BANDING FAILURE MODELS The term failure model is here employed to denote an algorithm used to introduce effects of material damage within a finite element by reducing that element's load bearing capability. That element need not be located along a current slideline. The algorithm consists of two parts: (i) a failure onset criterion and (ii) a prescription for post-onset stress reduction. In contrast, the slideline erosion algorithm, which also introduces effects of material damage, applies only to elements located on the current slidelines. The 1986 version of EPIC-2 contains an algorithm that satisfies the above definition of failure model, namely the J o h n s o n - C o o k fracture model [8]. However, this model will not be applied here, since it was derived to fit data from Hopkinson bar tests and quasistatic tensile tests, both of which involved strain rates below 1000 s - 1 . . Instead, two new failure models are installed into EPIC-2 to represent effects of the two micromechanical damage mechanisms observed in RHA specimens from the experiments of Section 2 [5,10]. These mechanisms are spheroidal void nucleation/growth/coalescence and shear banding.

4.1. Tensile void failure model The tensile void failure model is considered first. The onset of failure is assumed to occur when the pressure, which at a given time step is uniform within a finite element, becomes less than or equal to a negative-valued material parameter pf~,, i.e. P ~ Pfail < 0.

(7)

For each metal, Prail is taken to be a constant, independent of all field variables. A more accurate model would allow for rate effects on pfa, (e.g. [11,12,13]), but this case of constant Pr~, is a useful first approximation. The value of - 3 . 0 0 G P a is assigned for both copper and RHA on the basis of data reported by Rinehart [14] (see Ref. [5] for details). At the time step in which Eqn (7) is first satisfied and at all successive time steps, the element is

* This same criticism can be leveled at the Johnson-Cook strength model as used in this paper. That model's originators have recentlymodifiedits form ofstrain rate dependence in order to extend its range of applicability[9].

Lagrangian hydrocode simulations

627

not allowed to support hydrostatic tensile stress or any component of deviatoric stress. Only hydrostatic compression is allowed. 4.2. Shear banding failure model The shear banding failure model is applied to the RHA only. It employs an onset criterion called Zener-Hollomon, after its apparent originators [15]. In a given time step, the product [A + B(eP)N][1 -- (T*) M] is evaluated for each RHA element that has not yet failed or eroded and that has undergone further plastic flow during that time step. If the element also underwent plastic flow during the immediately preceding time step and if the above product has decreased from its level during that preceding time step, then the element is deemed to have begun to fail by shear banding. In effect, the criterion compares the strength increase due to work hardening to its decrease due to thermal softening. Once this onset criterion has been fulfilled in an element, at that and all subsequent time steps, deviatoric stresses are reduced from their computed values. However, hydrostatic tension is not altered, in contrast to the post-onset modeling for the case of tensile failure. Another difference is that the deviatoric stresses are not instantaneously set to zero, as with the previous model. Instead, their gradual reduction is imposed, qualitatively consistent with results from shear band modeling reported in Ref. [16]. If Sr*r, Sz*z, S~o and s*z are the four components of deviatoric stress computed by EPIC-2 in such an element for a given time step, then each of these is altered to the values given by Sij = S~j f(/~P),

(8)

where for Aeg~i~= O, f(ep) =

; eprEO,e .... t]

(9a)

; e~C(eoPn,o,,o0) while for Ae['ai, > O,

!

; ~PcEO,e.... ,] /~P__

f(eP) =

P

e°nset ,• A~'a. ,"

P P 13P ~ (eonset,/~onset -[-

A/~p il)

(9b)

P E p E ['~onset "[- A/~ ~ail , 0 0 ] .

Here ~onset p is the value of equivalent plastic strain ep that existed in the element at the time step in which the Zener-Hollomon criterion was first met. eonse p t is therefore computed by P the code. On the other hand, Ae['~iz is a user-prescribed material property. (eo.s~, + AsPail) is the level of ep beyond which the element no longer supports deviatoric stresses. The post-onset conditions imposed on an element that has failed according to the tensile void failure model are more stringent than those imposed on elements that have failed by the shear band model. For this reason, elements that have failed by shear banding continue to be checked for tensile failure. 5. COMPUTATIONAL RESULTS 5.1. Results with neither failure model active First a series of five problems is run with neither failure model active. Erosion strain P ~erode is assigned the same value for both slidelines and is varied in the range 0.25, 0.50, 0.75, 1.25 and 2.00. All problems are run until 500 #s after initial impact. Computational results for final hole size averaged over the plate thickness, ?ho~c,net mass lost by the target plate, AM, and the time at which 95% of this mass is lost, measured from the instant of initial impact, t o . 9 5 , a r e presented in Table 4. Comparison with the experimental results

628

M . N . RAFTENBERG

TABLE 4. RESULTS FROM PROBLEM SERIES I (neither failure model activated in RHA or Cu)

rho,~

TABLE 5. CPU TIMES ON A CRAY X-MP/48 FOR SIX PROBLEMS

Problem

e~rode

(mm)

AM (g)

to.95 (#S)

Problem

CPU Time (hrs)

I-A I-B I-C I-D I-E I-C(R)

0.25 0.50 0.75 1.25 2.00 0.75

9.6 10.3 11.0 12.1 13.9 9.7

27.1 27.4 27.7 23.4 20.3 21.3

15.9 16.4 16.1 15.6 19.4 49.9

I-C I-C(R) II-C II-C(R) III-B III-B(R)

0.93 7.19 0.93 7.27 0.53 4.78

in Table 1 shows that throughout the series, the computed holes are too small in terms of both fhole and AM, and their formations are completed prematurely. Problem I-C(R) in Table 4 reruns problem I-C with a finer target mesh. The edge length of each triangular element is halved, so that the number of target elements is quadrupled. This refinement decreases fhoJe by 12% and AM by 23%, but increases to.95 by 210%. Thus, mesh refinement has increased the deviation from experiment in computed results for fholo and AM, which suggests that physical mechanisms have not yet been modeled. Hole formation still completes substantially earlier than in the experiments. CPU time on a Cray X-MP/48 increases from 0.93 hours for problem I-C to 7.19 hours for problem I-C(R). These CPU times are displayed in Table 5. 5.2. Results with only the tensile void failure model active In a second problem series, only the tensile void failure model is activated, with Pfail for both copper and RHA set to -3.00 GPa. Again, e~,od~ p ranges over the values 0.25, 0.50, p 0.75, 1.25 and 2.00. Figure 7 is a mesh plot from problem II-C (e~rod e = 0.75) at 3.02 #s after impact. The phenomenon of spallation, or tensile failure brought on by reflection of a compressive shock at a free surface, is exhibited by the five darkened elements in front of the jet particle. The final target mesh from this problem is shown in Fig. 8. A single failed element remains near the hole boundary and approximately at midsurface elevation. The experimental result in Fig. 4 also shows most voids to be roughly at midsurface elevation. Table 6 displays results for fhole, AM, and to.95 from this second series. AM is seen to P while rhole increases with decrease with increasing ecrod p c for most of the range of /~erode, increasing e~rode p throughout the range. These same trends are seen in Table 4 to apply to the first problem series as well. In both cases, the explanation for the increase in fho,~ lies in the distorted lips at the entrance and exit surfaces that escape erosion for large e~rod~ (see Fig. 9 from problem II-E). The spatial averaging involved in computing fhol~includes these lips. Comparison between Tables 4 and 6 shows that with the mesh in Fig. 6, little change in rho,e, AM, and to.95 has accompanied the addition of the tensile failure model with the value of - 3.00 GPa assigned to Pfail, with one exception: to.95 from problem II-E is larger than from the other problems and is consistent with the experimental bound in Table 1. Problem II-C(R) reruns problem II-C with the finer target mesh used in problem I-C(R). Results for ?hole and AM are 6% and 15%, respectively, smaller in problem II-C(R) than in problem II-C. Once again, mesh refinement has caused Yhol, and AM results to deviate further from experiment. The computed to.95 value is 24% larger in problem II-C(R) than in problem II-C, but still smaller than the experimental bound in Table 1. CPU times from these two problems are added to Table 5. Inclusion of the tensile void failure model has caused CPU times to change insignificantly from their levels for problems I-C and I-C(R). 5.3. Results with both the tensile void and the shear band failure models active Finally, six problem series are run in which both the tensile void and the shear band failure models are activated. Each problem series is characterized by a fixed value of Ae~'~iI,

Lagrangian hydrocode simulations hblcm

II-C! (Twilc

629

Failure in RHA B Cu. /~,~g - -3.00 Gk

No Shear Band Failure; f$d cycle - 186

= 0.79

timc=3.02p

-2

-6

-10

-14

-18

-22,

-26,

-301 0

B

4

12

16

20 I-

24

28

32

36

(Ml91

Dark elements have failed by tensile voids. FIG. 7. Mesh plot at 3.02 PCSafter impact for problem II-C. 1CIhblem

WC (Tensile Failure in RHA & CU. &I = -3.00 GPx No Sbcar Bend Failure; E&,, - 0.75)

E,’ lime = 500.00 ps

cycle = 12983 2

-2

-6 E -10 N -14

-18

-22

-26

-30 0

4

B

12

16

20 r

Dark

element

has

failed

by tensile

24

28

32

(Il.1

voids.

FIG. 8. Mesh plot at 500.00~s after impact for problem II-C.

36

s

I

630

M . N . RAFTENBERG TABLE 6. RESULTS FROM PROBLEM SERIES I1 (tensile failure modeling in R H A & Cu, Pt,il = - 3 . 0 0 GPa; no shear band failure modeling)

e.oa="

r-hole (ram)

AM (g)

to.95

Problem

II-A II-B II-C II-D II-E II-C(R)

0,25 0,50 0,75 1.25 2,00 0.75

9.7 10.6 10.9 12.2 15.0 10.2

27.9 28.6 27.2 24.9 22.8 23.1

10.5 15.3 22.7 23.1 89.3 28.1

(/Is)

10

Proble~mII-E (Tensile Failure in R.HA& Co. Plall = -3.00 GPa: No Shear Band Fnilure; ~,0~ = 2.00) clJc le

time = 500.00 ps

~ "

~<[XXXXXiX~Xxxxx ~×XX]XXX× X x x x x XXNxX × ~XIXXXXXXXXXX:x :,<~x x XXX XXx ~(xx XXx x,~x >
I

-2

-6

== -10 ¸

-1,:1

-18

X

-22

-26

-30 O

4

8

12

16

20

r"

24

28

32

36

40

(~N)

D a r k e l e m e n t s h a v e failed by tensile voids. FIG. 9. Mesh plot at 500.00 l~s after impact for problem II-E.

either 0.00 (instantaneous deviatoric stress reduction to zero), 0.10, 0.25, 0.50, 0.75 or 1.00. Within each series, ee,od = p is varied over the same five values as before. Pr,iJ is again set to - 3 . 0 0 G P a for both copper and RHA. Results for ~hole, AM, and to.95 are collected in Table 7. Figures 10 and 11 plot results for r-,ol= from the six problem series. For a fixed Aerial, r-ho,e is seen to increase with increasing e~rod p ~ for ee~ode p in the range of 0.25 to 2.00. Also, r-hole decreases with increasing Ae~'a~ for eo~ode p in the range of 0.25 to 2.00, with a single slight exception: at an e~rod P c of 2.00, rhol= increases by 0.1 mm when Ae~'aiI is increased from 0.75 to 1.00. With this one insignificant exception ignored, the finding means that the more gradually deviatoric stresses are reduced following the onset of shear band failure, the smaller the final hole size. Activation of the shear band failure model allows for a large increase in computed hole size over those obtained in problem series I and II. For the series with Ae~',iI equal to 0.50,

Lagrangian hydrocode simulations Problem

Seric.s 111l?wugh

VIII

(Tensile

631

Failure in RHA & Cu, P,,,~ = -3.00 GPa;

Shear Band FaiIure in MA.

A&

Varied)

27

21

21 . lB

2 IL

. . . . . . . . . ~. . . . . . . . .

15.

12 9.

6.

3. . ... ... ... .. ... ;gg

L 8.25 ee.ee

8.58

8.75

1.80

1.25

1.58

1.75

I

2.88

P =erodc

FIG. 10. Computed r,,ho,evs E:,,,~~with A&, a parameter for problem series III through VIII. Pmblcm

S&u

III Tbmugh

VIII (Tensile FaiIur~ in RHA B CU. P+I = -3.00 GPa; Shear Band Failure in RHA, A& Vtied)

27 I

1B’

el e.e

8.1

e.2

8.3

e.t

8.5

8.6

8.7

8.8

8.9

1

Agil

FIG. 11. Computed f,,ho,cvs A&

with &Erodea parameter for problem series III through VIII.

M. N. RAFTENBERG

632 TABLE

7. RESULTS

FROM PROBLEM

THROUGH

SERIES III

VIII

(wnsile la~lure modeling in RHA & Cu. P,.,, = - 3.00 GPa: shear band lailure modeling in RHA. AC;.,, varied1

to.95

erode

A%,

rhoI. (mm)

AM

Problem

(g)

(/A

III-A III-B III-C III-D III-E

0.25 0.50 0.75 1.25 2.00

0.00 0.00 0.00 0.00 0.00

15.5 18.1 20.2 23.2 26.3

76.1 105.5 130.3 171.4 220.7

224.7 183.4 192.3 323.3 267.7

IV-A IV-B IV-C IV-D IV-E

0.25 0.50 0.75 1.25 2.00

0.10 0.10 0.10 0.10 0.10

14.1 16.2 18.7 21.9 24.8

65.2 83.3 110.9 152.5 193.9

228.9 228.4 293.3 332.7 381.4

V-A V-B v-c V-D V-E

0.25 0.50 0.75 1.25 2.00

0.25 0.25 0.25 0.25 0.25

12.1 15.3 17.4 18.9 22.3

45.8 72.9 95.3 112.2 158.3

289.5 395.1 329.4 358.7 446.3

VI-A

VI-E

0.25 0.50 0.75 1.25 2.00

0.50 0.50 0.50 0.50 0.50

11.5 12.3 13.1 16.0 17.7

40.5 45.6 49.5 76.7 94.8

305.9 310.2 212.7 389.5 81.6

VII-A VII-B VII-C VII-D VII-E

0.25 0.50 0.75 1.25 2.00

0.75 0.75 0.75 0.75 0.75

1 I.0 11.9 12.8 14.9 17.0

36.7 40.8 47.7 63.8 85.2

382.8 267.6 217.4 192.2 249.7

VIII-A VIII-B VIII-C VIII-D VIII-E

0.25 0.50 0.75 1.25 2.00

1.00 1.oo 1 .oo 1.00 1.00

10.7 11.4 12.3 14.2 17.1

34.7 38.1 41.8 55.6 80.4

323.8 180.0 235.4 180.6 232.7

III-B(R)

0.50

0.00

17.3

97.2

158.3

VI-B

VI-C VI-D

0.75,and 1.0, ?,,ho,eis still consistently at least slightly smaller than the experimental value of 18.2 mm. However, the curves in Fig. 10 corresponding to problem series III, IV and V and respective A&, values of 0.00, 0.10, and 0.25 intersect the experimental result at an P E,,,~~ of about 0.5,0.7, and 1.0, respectively. Four problems that produce j;holevalues within 4% of the experimental result are problem III-B (E!&,~~= 0.50, A&pail= O.OO),problem IV-C = O.lO), problem V-D (E:,,,~~ = 1.25, AE.;=~, = 0.25), and problem VI-E (Glde = 0.75, AE~'~~, (&de

= 2.00,AE~P~, = 0.50).

Mesh plots from problem III-B at 3.01, 100.06, and 500.02 /JSafter impact are presented in Figs 12, 13, and 14, respectively. The blue elements in Fig. 12 and the dark elements in Figs 13 and 14 have failed by shear banding. The magenta element in Fig. 12 has failed successively by shear banding followed by tension. In Fig. 12 the target’s hole is surrounded by RHA material that has failed by shear banding. The single magenta element on the exit surface indicates that spallation has preceded perforation. Perhaps formation of the relatively small, early-time debris fragments seen in Fig. 3 can be attributed to the failure mechanism of tensile voids.

Lagrangian

hydrocode

simulations

Blue elements have failed by shear banding. Magenta element has failed by shear banding FIG. 12. Mesh

plot at 3.01 ps after

impact

followed

for problem

by tensile III-B.

voids.

634

-2

-6 s -I@ N -1C

-‘I i

Blue elements have failed by shear banding. Magenta elements have failed by shear banding FIG.

followed

by tensile

19. Mesh plot at 3.01 ps after impact for problem III-B(R).

voids.

Lagrangian

hydrocode

simulations

635

11 Fmblun

Ill-8

(Tensile Failure in RHA & Shear Band Failure in RHA.

t cycle = 1825

time = 100.06 p.5

i

-2

-6 ii E -18 N -14

-18

-22

-26

-39 4

I

0

12

16

ZB r

Dark

elements

FIG

have

13. Mesh

plot

failed

by shear

at 100.06~ts

24

28

32

36

’

(mm) banding.

after

impact

for problem

III-B

-Roblen

III-B (Tensile Failure in RHA B Cu. p,Sz I -3.03 GPa; Shear Band Failure in RHA. A& s 0.00. e’$& - 0.50)

cycle = 7555

lime-soo.02p

-18

-22.

-26

-30,

e

4

8

12

16

20 I-

Dark

elements FIG. 14. Mesh

have plot

failed

by shear

at 500.02 ~1s after

24

ZB

32

(m)

banding. impact

for

problem

III-B.

36

d

636

M . N . RAFTENBERG

Once an element of RHA material adjacent to the current hole boundary has failed by shear banding, the stress reduction scheme of Eqns (8) and (9) renders it prone to large deformation and hence susceptible to slideline erosion. Figures 13 and 14 show that in problem Ill-B, every RHA element that fails by shear banding and that does not have at least two of its edges bounded by elements that have not failed is discarded via slideline erosion by 500ps. A comparison of Figs 13 and 14 reveals that a substantial amount of RHA material is still eroding and contributing to the final hole size between 100 and 500/~s after impact. It is suggested that the relatively large, late-time fragments seen in the radiographs of Fig. 3 are primarily attributable to the failure mechanism of shear banding rather than tensile voids. The last element failure by shear banding occurs between 250 and 300/~s, and the last instance of slideline erosion at 489.8 ~ls. Figure 15 gives further support to the separation of RHA fragments into two categories: (i) early-time, high-speed fragments that originate from the region nearest the centerline, in which tensile void failure is in evidence in the calculations, and (ii) late-time, slow-moving fragments that originate at larger radial coordinates, in a region that separates to form the spall ring and in which shear banding failure predominates. Let I~-o~ol denote the absolute value of the axial velocity of a node point on the original exit surface of the target plate at the time step immediately prior to that at which the node erodes from the target plate. Let tle~-o~cdenote the time of that time step, and let r 0 denote the node's original radial coordinate. Then Fig. 15 plots values computed from problem III-B for Iz~o~¢l "~-~ and t~rod ~,1-~ both as functions of r o. The innermost eight nodes, with r o values between 0 and 9.1 mm, are seen to erode during the first 7.8/~s and have IZerod~l "~-~ values between 0.69 and 3.13 km/s. The next six nodes, with r o values between 10.4 and 16.9 mm, erode during the interval of 24.0 to 125.9/~s and have I~;-o~ol values between 0.01 and 0.10km/s. The mesh at 500/as shown in Fig. 14 is taken to be the final state for problem III-B. Comparison with Fig. 2 shows that this predicted hole geometry agrees quite closely with the experimental result, both in terms of the average hole radius ?ho~cand the general shape of the hole profile. In particular Fig. 14 contains a prominent protuberance that juts into 10000

- 1000 Problem III-B (Tensile Failure in R H A & Cu, Pl"ait = - 3 . 0 0 GPa; Shear Band Failure in RHA. Ae~ait = 0.00: ~,ode = 0.50)

1000

100

B

100

10 10

.(-) I Z erode I

......... 1

0

t(ero)de

~;

10 ro

FIG. 15. C o m p u t e d

.'~(-)erodea n d

1~5

(mm) tcroa c~-~ vs r o f r o m p r o b l e m III-B.

20

Lagrangian hydrocode simulations Problem Series HI (Tensile Failure in RHA & Cu.

637

pl,,~t = -3.00 GPa:

Shear Band Failure in R.HA. A~, u = 0.00)

.............i..........................................................i...........................................i............................. Z~

~17S .< ISO w Q

136 g

~ 100 [75

0

e

.so

100

150

zoo

z50

300

3(30

400

4se

500

"riME (~s)

FIG. 16. Mass lost from target plate vs time for problem series III. the hole to form a distinguishable throat at about the midsurface elevation. Figure 14 contains no elements that have failed by tensile voids, a result that is clearly inconsistent with the experimental presence of voids in Fig. 4. Figure 14 does display some shear banding within the midsurface-region protuberance that forms the throat. This is the region in which most shear bands are observed in Fig. 4. Moreover, all of the most recently eroded elements that were adjacent to those along the final hole boundary had failed by shear banding prior to erosion. This constitutes some degree of agreement with the presence of shear bands along the boundary of the protuberance in Fig. 4. Figure 14 also displays shear banding near the intersection of the hole boundary and the entrance surface, consistent with Fig. 4. Two elements of shear banding are also contained in Fig. 14 near the exit surface. This last result is inconsistent with Fig. 4. Note, however, that the shear bands denoted in Fig. 4, are those that were observable with an optical microscope at 1000X magnification. Perhaps additional shear bands could have been observed in these and other regions of the cross section by more discriminating means. Figure 16 plots cumulative mass lost by the target plate as a function of time after impact for problem series III. The curve for which eerode p = 0.50 corresponds to problem III-B. This curve can be divided into four regions. For the first 25 ps after impact, mass loss occurs at the rapid average rate of 1.9 g/.ts-x. The average mass loss rate between 25 and lOOps is 0.52 gps -x. From 100 to 200/~s, the average rate further reduces to the still appreciable 0.17 g/~s-i. This is consistent with the continuing mass loss detected in the mesh plots of Figs 13 and 14. By 200/.ts, the curve has reached a plateau; no additional erosion occurs until 489.9 ps. The value attained at 500 ps by each curve in Fig. 16 is identified with AM, the cumulative mass that the plate has lost during the 500ps following initial impact. Results for AM P with m~ail a parameter in Fig. 17. This from the six problem series are plotted vs eerode figure shows that in each of the problem series III through VIII, AM increases with p increasing eerod~ p for e~rod, p in the range of 0.25 to 2.00. The figure also shows that for eerode

M. N. RAFTENBERG

638

Problem Series Hi Through Vlll (Tensile Failure in RHA & C'u. P/aa = -3.00 GPa; Shear Band Failun~ in RHA, ~ . u Varied)

Z~e

175

15o

1Z5

75

/ zs

~.u

= ( 0.25

................ ~ o e.ee

e.zs

e.50

e.Ts

l.eo

1.zs

l o.so 10.75 t i.oo 1.50

1.7s

z.ee

P c with Ae~ij a parameter for problem series III through VIII. FIG 17. Computed AM vs ecroa

fixed at 0.25, 0.50, 0.75, 1.25 or 2.00, AM decreases as AeF.~l is increased in the range of 0.00 to 1.00. AM has been dramatically increased over the values obtained in problem series I and II by the activation of the shear band model. Thus, whereas AM results from problem series I and II never exceeded 28.6 g, in Fig. 17 the curves pertaining to problem series III, IV and V intersect the experimental result of 136 g at a n eeProde of about 0.8, 1.1 and 1.6, respectively. The curves for problem series VI, VII and VIII remain below this experimental result throughout the e~rod~ range of 0.00 to 2.00. Recall that similar observations were made for fhole on the basis of Figs 10 and 11. In Fig. 10 the curves of fhole VS eProde from problem series III, IV and V also intersected the experimental result of 18.2 mm, but at the lower eorod~ p values of about 0.5, 0.7 and 1.0, respectively. As a consequence of this difference, problems Ill-B, IV-C, V-D and VI-E, which all produced fhol~ values within 4% of the experimental result, are now seen in Table 7 to produce AM values that are smaller than the experimental result by 22%, 18%, 18% and 30%, respectively. These discrepancies between computed AM values from problems Ill-B, IV-C, V-D and VI-E and the experimental result of 136g are attributed to the removal of insufficient RHA material in these four problems, despite the fact that the final average hole radius displays close agreement 'with experiment in each case. Apparently too much RHA material has been compressed or bent and retained in the calculations instead of being discarded by slideline erosion. The problem that produces a p AM result in closest agreement with experiment is problem III-C (e,rod e = 0.75, Ae~'aiI = 0.10). AM for this problem is 130.3 g (Table 7), which is only 4% less than the experimental result of 136g. The fho~e value obtained from problem III-C is 20.2mm, which is 11% larger than the experimental result of 18.2 mm. One problem that produces a AM value somewhat closer to AM exp than has problem III-B and an fhol~ value still within 3% ,,..,. r~l¢"hole~Texpis" problem IV-C (/~Prode ~--- 0.75, Ae~'aiI = 0.10).

Lagrangian hydrocode simulations

639

1B Problem IV-C (Tensile Failtu~ in RHA & Cu, Pfail " .3.00 GPa:, Shent Band Failure in RHA, ~fff~ - 0.10; ~,..a. = 0.75)

cycle = 8097

time = 500.00 ps

..4

-7'

-6

-1B

-14

-1B _;r;,

-Z6

-30

e

4

fl

17'

16

~ r

Z'I"

ZB

(m)

D a r k e l e m e n t s h a v e failed by s h e a r b a n d i n g . FIG. 18. Mesh plot at 500.00/is after impact for problem IV-C.

Figure 18 provides a mesh plot from this problem at 500/~s, to be compared with Fig. 14 from problem III-B. The shape of the final hole profile still agrees reasonably well with the experimental profile in Fig. 2. The number of shear banded elements has increased markedly over the result from problem III-B. This illustrates a trend that is found by examining other problems as well (see Ref. [53): For Fholefixed near/'hole, -exp. as m~fail p IS - increased in the range of 0.00 to 0.50 (corresponding to a decrease in the rate at which deviatoric stresses are reduced following the onset of shear banding), the number of shear banded elements that remain attached to the plate at 500 ps increases. Now consider the third benchmark quantity, to.95. It is apparent from the mass loss vs time plots in Fig. 16 that activation of the shear band model has greatly extended the time at which target erosion comes to completion. Results for to.95, the time at which the plate has lost 95% of mass AM, are listed for problem series III through VIII in Table 7. All ,exp. in Table 1. to.95 results in to.95 listings in this table satisfy the experimental bound on -0.95 Table 7 do not exhibit a simple dependence on either ec,o~ p e or Ae['ail. To study mesh sensitivity, problem III-B(R) reruns problem III-B with the finer target mesh employed in problems I-C(R) and II-C(R). Mesh plots for problem III-B(R) are presented at 3.01 and 500.01 ps after impact in Figs 19 and 20, respectively. The radial extent of the spall fracture surface is about 4 mm in Fig. 19 and only about 1 mm in Fig. 12 from problem III-B. Also, the spall fracture surface is recessed about 0.8 mm from the exit surface in problem III-B(R), while no such recession can be resolved with the coarser mesh of problem III-B. Reference [5] presents mesh plots from both problems III-B and III-B(R) at various times after perforation. Following perforation, mesh plots from the two problems continue to display good qualitative agreement through 10/~s, both in terms of the hole boundary's geometry and the general location of zones of elements that have failed. By 20 #s after

640

M. N. RAFTENBERG

I

i

-2

-E E -10 N -11

-18

-22

-26

Dark FIG

elements

20. Mesh

have

failed

by shear

plot at 500.01 jts after

banding.

impact

for problem

Ill-B(R)

-1. ! I

.A. 4y-

..i..

FIG. 21. Mass

lost from

target

plate

YS time for problems

III-B

and III-B(R).

Lagrangian hydrocode simulations

641

impact, differences in hole profiles from problems III-B and III-B(R) become apparent and seem to be driven by differences in the spatial distributions of zones of RHA that have failed by shear banding. At 500/~s, Fig. 20 shows that the final average hole radius, ~ho~, attains the value of 17.3 mm in problem III-B(R). This represents a 4% reduction from its value from problem III-B. Comparison of Figs 14 and 20 shows that the shape of the final hole profile and the location of shear banded elements in the final mesh are highly mesh dependent. Mass lost by the target plate is plotted vs time in Fig. 21 for problems III-B and III-B(R). At all times after 5 tts, the mass loss in problem III-B(R) is less than that in problem III-B. At 500/~s, the net mass loss, AM, is 97.2 g in problem III-B(R) and 105.5 g in problem III-B (Table 7), corresponding to an 8.5% difference. This difference of 8.3 g bounds the difference at all earlier times. Parameter to.95 is found from Fig. 21 to be 158.3#s for problem III-B(R). This represents a 13.7% decrease from the result of 183.4#s from problem III-B (Table 7), but still satisfies the experimental bound in Table 1. CPU times on a Cray X-MP/48 for problems III-B and III-B(R) are 0.53 and 4.78 hours, respectively, and are included in Table 5. For both target meshes, activation of the shear band failure model has been accompanied by substantial reduction in required CPU time. The main reason for this reduction is related to the time step, determined in EPIC-2 by means of the Courant stability condition [17-1. According to this condition, at any given time in the calculation, the time step is limited by the current dimensions of the smallest non-eroded element. In problems with the shear band failure model active, more target elements become eroded and fewer highly distorted elements remain attached to the plate. 6. C O N C L U D I N G

REMARKS

The computational results can be summarized as follows: The 1986 version of EPIC-2, with its slideline erosion algorithm, but without failure models activated, was unable to produce agreement in terms of fho~e, AM, and to.95 with results from one experimental condition. That condition involved an RHA target plate of 13-mm thickness impacted by a particular copper shaped charge jet at long (12.00 or 15.23 C.D.) standoff. Specifically, for slideline erosion parameter eerode p varied in the range of 0 to 2, results for ?hole, AM and to.95 were consistently smaller than according to experiment. The tensile void failure model was then activated with its parameter Prall fixed at - 3.00 G P a for both copper and RHA. This produced little change in fho~eand AM results; both remained substantially smaller than the experimental values, to.95 increased significantly in only one problem, corresponding to an e~rod~ of 2.00, and was placed in agreement with the experimental bound. Effects of variations in PralJ were not examined. The shear banding failure model was then activated in addition to the tensile void failure model with Pfail still fixed at - 3 . 0 0 G P a . The shear banding model introduced input parameter Ae~'~,. The addition of this model allowed for substantial increases in ~ho~e,AM, and to.95. Initially, particular meshes were used for the leading jet particle and for the target, with the latter mesh consisting of triangular elements with a maximum initial edge length of 1.3 mm. Reasonable agreement with experiment in terms of fhole, AM, and to,95 was obtained for a range of/~erode p and Ae~ail combinations. Problems III-B (/~erodeP = 0.50, At;~ail = 0.00), P P P III-C (eerod e ~ 0.75, AePail = 0.00), III-D (e~od ~ = 1.25, Ae~'~, = 0.00), IV-C (Eerod e = 0.75, P ~ P Ae~'~ij = 0.10), IV-D (e~rode 1.25, AeP~il= 0.10), V-C (e~rod~= 0.75, Ae~'ail= 0.25), V-D P e ~ 1.25, Ae~'~iI ~ 0.25), and VI-E (e~rod~ P (e~od = 2.00, Ae~'aiI = 0.50) all produced rhol~ and AM results within 30% of ~hole ,~xp. and AM ~Xp', respectively, and to.95 results within the experimental bound. In particular, problem III-B produced an fho~e value within 1% of ~xp. and a AM value 22% less than AM ~x°'. This problem also yielded a hole profile shape ole in close agreement with experiment and a degree of agreement with experiment in terms of shear band locations within the final cross section. Problem III-B(R) reran problem III-B with a finer target mesh; element dimensions were

642

M.N. RAFTENBERG

halved and the number of target elements was quadrupled. The resulting fho~ was 5% less than rho~e,-exP"and the resulting AM was 29% less than AM exp'. Thus, with the finer mesh, the case of eerode p = 0.50 and Ae~'all = 0.00 still produced r-hole and AM values within 30% of the experimental results. However, the shape of the final hole profile from problem III-B(R) differed substantially from that of problem III-B and no longer agreed well with experiment; the hole throat in problem III-B(R) was located near the entrance surface instead of near the midsurface. Presumably, with this finer mesh another combination of eerode p and Ae~'a, would have produced better overall agreement with experiment. It can be concluded that the computational scheme involving EPIC-2 with both failure models active has not converged to great accuracy with the mesh of Fig. 6. The degree to which convergence has been achieved with the finer mesh of problem III-B(R) was not determined. Recently, a set of experiments was conducted at the U.S. Army Research Laboratory in which the same shaped charge warhead as employed in this study was fired from two different standoffs into RHA target plates of three different thicknesses ['18,19]. It remains to be seen whether the computational scheme presented in this paper can be used to achieve acceptable agreement over the entire range of experimental conditions for a single set of values for ~erode, p Pfail and Ae~',i~. Acknowledgements--The experiments described in Section 2 were performed at the U.S. Army Research Laboratory (ARL) by Messrs Grat E. Blackburn, Sterling C. Shelley, Jr, David R. Schall, Carl V. Paxton and Joseph W. Gardiner. Doctors Jonas A. Zukas and Eitan Hirsch, formerly of ARL and Israel Military Industries, respectively, provided helpful comments regarding mesh sensitivity. Discussions with Doctors Thomas W. Wright and Michael J. Scheidler of ARL influenced the shear band failure model. Doctors Gordon R. Johnson and Robert A. Stryk of Alliant Techsystems, Inc. clarified some issues regarding use of and modification to EPIC-2. Ms Claire D. Krause, formerly of ARL, performed microscopy on RHA specimens and produced Fig. 4. This paper benefitted greatly from comments by the two reviewers. Both raised many helpful points that were incorporated into the revision.

REFERENCES 1. F. P. STECHER and G. R. JOHNSON, Lagrangian computations for projectile penetration into thick plates. Computers in Engineering 1984, Vol. 2 (edited by W. A. GRUVER),pp. 292-299, A.S.M.E., New York (1984). 2. G. R. JOHNSON and R. A, STRVK, User instructions for the EPIC-2 code. AFATL-TR-86-51, Air Force Armament Laboratory, Eglin AFB, FL (1986). 3. G. R. JOHNSON, R. A. STRYK, T. J. HOLMQUISTand O. A. SOUKA, Recent code developments for high velocity impact: 3D element arrangements and 2D fragment distributions. Int. J. Impact Engng 10, 281-294 (1990). 4. U.S. Department of Defense, Military specification: Armor plate, steel, wrought, homogeneous (for use in combat-vehicles and for ammunition testing). MIL-A-12560G(MR), U.S. Army Materials Technology Laboratory, Watertown, MA 0984). 5. M. N. RAFTENBERG, Modeling RHA plate perforation by a shaped charge jet. BRL-TR-3363, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD (1992). 6. B.J. KOHN, Compilation of Hugoniot equations of state. AFWL-TR-69-38, Air Force Weapons Laboratory, Kirtland AFB, NM (1969). 7. G. R. JOHNSON and W. H. COOK, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Seventh International Ballistics Symposium Proceedings, pp. 541-547. The Hague (1983). 8. G. R. JOHNSON and W. H. COOK, Fracture characteristics of three metals subjected to various strains, high strain rates, temperatures and pressures. Enong Fracr Mech. 21, 31-48 (1985). 9. T. J. HOLMQUIST and G. R. JOHNSON, Determination of constants and comparison of results for various constitutive models. Journal de Physique I V l, C3.853-860 (1991). 10. C.D. KRAUSEand M. N. RAFTENBERG,Metallographic observations of r611ed-homogeneous-armor specimens from plates perforated by shaped charge jets. ARL-MR-68, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD (1993). II. B. M. BUTCHER, L. M. BARKER, D. E. MUNSON and C. D. LUNDERGAN, Influence of stress history on time-dependent spall in metals. AIAA Journal 2(6), 977-990 (1964). 12. B. M. BUTCHER, Spallation in 4340 steel. J. appl. Mech. 34(1), 209-210 (1967). 13. F. R. TULER and B. M. BUTCHER, A criterion for the time dependence of dynamic fracture. Int. J. Fracture Mech. 4(4), 431-437 (1968). 14. J. S. RINEHART, Some quantitative data bearing on the scabbing of metals under explosive attack. J. appl. Physics 22, 555-560 (1951). 15. C. ZENER and J. H. J. HOLLOMON, Effect ofstrain rate upon plastic flow of steel. J. appl. Physics 15, 22-32 (1944).

Lagrangian hydrocode simulations

643

16. T. W. WRIGHT and J. W. WALTER.Jr, On stress collapse in adiabatic shear bands. J. Mech. Physics Solids 35, 701-720 0987). 17. G. R. JOHNSON,Analysis of elastic-plastic impact involving severe distortions. J. appl. Mechanics 98, 439-444 0976). 18. M. N. RAFTENBERG,Experimental investigation of RHA plate perforation by a shaped-charge jet. In Proceedings of Twelfth Army Symposium on Solid Mechanics (edited by S.-C. CHOU), pp. 395-410. U.S. Army Materials Technology Laboratory, Watertown, MA (1992). 19. M.N. RAFTENBERG,Experimental investigation of rolled-homogeneous-armor plate perforation by a shaped charge jet. ARL-TR-328, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD (1994).