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Stability and fragmentation of ejecta in hypervelocity impact
Int. J. Impact Engng Vol. 10, pp. 197-212, 1990
0734-743X/90$3.00 + 0.00 PergamonPressplc
Printedin Great Britain
STABILITY AND FRAGMENTATION OF EJECTA IN HYPERVELOCITY IMPACT l
D. E. Grady and S. L. Passman Sandia National Laboratories, Thermomechanical & Physical and Process Research Divisions P.O. Box 5800, Div. 1534 and 6212, Albuquerque, New Mexico 87185-5800
INTRODUCTION
The near-normal hypervelocity impact of a metal projectile on a metal target plate, where the dimensions of the projectile and the thickness of the plate are comparable, results in the ejection of a high-velocity plume of debris. The ejected plume emerges as a contiguous expanding bubble of the projectile and target material. At a critical time during expansion, the bubble of metal undergoes fragmentation with the fragment debris expanding as a thin shell of material along trajectories imparted at the time of impact. Relatively little matter resides within this shell. Although systematic data are scant, there is evidence that as the impact velocity increases, the extent of stable bubble expansion before fragmentation increases and the characteristic particle size in the debris plume decreases (Grady et aL, 1985). Understanding the stability and fragmentation features of the impact ejecta is critical in assessing details of the impulse and damage experienced by a secondary structure. A phenomenological model of hypervelocity impact (Swift et aL, 1983) has been reasonably successful in describing certain features of the ejection and expansion process. In the Swift model, the ejected debris is assumed to be an expanding spherical shell of material. Application of the model leads to predictions of the center-of-mass and expansion velocities and, consequently, details of the momenta imparted to the debris shell. In the present study, we accept the premises of the Swift model of hypervelocity impact and address the issues of stability and fragmentation of a rapidly expanding spherical shell of metal. How well the Swift model describes the nature of the debris is not critical to this investigation. The physical phenomenology which we reveal is expected to be generally applicable to the present, as well as to other divergent flow problems. In this analysis we solve for the symmetric motion of a spherical shell of an incompressible ideal fluid subject to arbitrary pressure conditions on the inner and outer surfaces. We ignore strength initially because, in the regime of interest, inertial forces overwhelm the strength properties of the medium. Combining the calculated motion with the momentum equation, we determine the spatial pressure profile within the body. For the freely expanding shell (zero pressure on the inner and outer surface) representative of the hypervelocity impact ejecta, we find that this pressure is everywhere positive and has positive gradients directed into the body at the inner and outer boundaries. The latter condition is shown to guarantee Rayleigh-Taylor stability of the respective surfaces. This behavior explains the contiguous stable expansion phase of an ejecta bubble prior to fragmentation. We then continue the analysis and introduce material strength (a constant flow stress) as a material property. Through an approximate analysis of stability we show that strength provides the destabilizing force which counters the inertial stability of the motion. A criterion for the onset of fragmentation is determined through a balancing of these opposing forces. Data obtained from impact experiments of lead and uranium projectiles on lead and uranium plates (Grady et aL, 1985) are compared with the analysis. In this comparison, Swift's model of hypervelocity ejecta is XThis work performed at Sandia National Laboratories supported by the U.S. DOE under contract DE-AC04-76DP00789.
1.97
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D. E. Grady and S. L. Passman
combined with the present analysis to evaluate the dependence of the imparted motion on kinematic and material properties, the onset of instability, and fragmentation of the debris ejecta. Reasonable agreement with the impact data is achieved. Finally we reexamine computational simulations of the lead and uranium impact experiments which also show stable expansion and fragmentation. Based on comparison with the present analysis we suggest that stable flow and fragmentation observed in computer simulations is a consequence of natural physical instabilities.
SWIFT'S MODEL OF HYPERVELOCITY IMPACT In the analysis of Swift et al. (1983), an equidimensional projectile traveling at hypervelocity speeds undergoes near-normal impact on a target plate with thickness comparable to that of the projectile. On impact, strong shock waves propagate into both projectile and plate material, converting much of the projectile kinetic energy into potential energy of compression. The proximity of free surfaces leads to rapid decompression of the shocked material and most of it expands violently in a bubble-like structure which emerges from the back of the plate. A smaller portion of the material is ejected as a cone of fine debris on the impact side of the target plate. Such an event is illustrated in the series of photographs in Fig. 1.
40 Fs
• LEAD PROJECTILE IMPACT ON LEAD PLATE • VELOCITY = 1 2 0 0 m / s
6 mm
Fig. 1: A set of high-speed front-lit photographs of the back surface debris ejected by the impact of a lead projectile on a lead target at approximately 1200 m/s.
Stability and fragmentation of ejecta in hypervelocity impact
199
The state of the material in the rearward moving bubble was of principal interest in the analysis of Swift. The distributed impulse and damage imparted to a secondary structure is determined through interaction with this bubble. For sufficiently high impact velocities, much of the debris material may be vaporous. We are interested here, however, with the range of velocities where the debris is in the condensed liquid or solid state. In modelling the hypervelocity impact event, Swift e t a / . (1983), introduced a geometry-approximating observation and some simplifying assumptions which allowed them to evaluate the phenomena with global momentum and energy balance principles. Although some of the assumptions can be argued with in detail, Swift's model provides a reasonable description of some of the essential features of the debris momenta and trajectories. The general geometry and parameters in Swift's model are illustrated in Fig. 2. It is assumed in the development that the momentum of the projectile is totally transferred to the spherical ejecta bubble. Thus, impulse transferred to the plate is ignored and backsplash mass and momentum is assumed sufficiently small as to be discounted. Inelastic collision results in an excess energy which Swift e t a / . (1983), assume goes into spherical bubble expansion about the center-of-mass motion, although an efficiency coefficient Q ~_ 1 is introduced to account for other forms of dissipation, such as heat or radiation.
J J
J J J J J
J
J
v,/
J
Vm
J m
\
J
J J J J J
M
J
J J J
M+rn
(b)
(o)
Fig. 2: Geometry assumed in the Swift model of hypervelocity impact. (a) Before impact, a projectile of mass m is traveling at a velocity v,~ toward the plate. The mass M of the plate which will be ejected is identified. (h) After impact a spherical shell of mass M + m is expanding with velocity ve about a center of mass, traveling at a velocity vc. It is then straightforward, based on momentum and energy conservation, to derive the center-of-mass velocity vc and the expansion velocity ve of the spherical debris bubble, in terms of the incident projectile velocity and participating masses, 1 v< -- 1 + ' (1)
MI---------~"'
,,~
=
OV~/m -
-
1 + M/m
.
(2)
v'~
Swift e t aL (1983), provide estimates for the ratio of ejected plate mass M to projectile mass m, based on the relative dimensions of projectile and plate, and argue that Q is frequently near unity.
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and S. L. P a s s m a n
High speed photographs or flash radiographs of hypervelocity impact events show that the bubble of debris initially expands as an unbroken shell of matter not unlike the spherical model illustrated in Fig. 2. Experiment further shows that as expansion and thinning of the bubble proceeds, a point is reached at which instability and disintegration into a cloud of discrete particles occurs. In subsequent expansion, although trajectories and momenta are maintained, the constitution of the bubble has changed from that of a continuous shell of matter to a distribution of discrete particles. The study of Swift et al. (1983), discusses this aspect of the phenomenology in an observational sense. Physical causes for the behavior and predictions of the fragment process are not put forward, however. It is this aspect of the hypervelocity impact event that we analyze in subsequent sections. What are the physical and material properties in the rapid expansion of a bubble of condensed matter which lead to the initial stability of that expansion, a critical point in the expansion process at which breakup initiates, and ultimately disintegration into a distribution of particles of characteristic size and shape. We proceed in the spirit of Swift's model and focus on details of the expanding motion of a spherical shell of matter. The assumption of spherical symmetry leads to a tractable analysis, but it is certainly not restrictive. The observations and conclusions provided by the analysis are recognized to be generally applicable to motions involving more complex divergent flow. The relations of Swift in Equations 1 and 2 will be useful in estimating the initial conditions for the analysis in the present study.
MOTION OF AN EXPANDING SPHERICAL SHELL In pursuing the investigation of the stability and fragmentation of an expanding shell of ejecta we will solve, in the present section, for the symmetric radial motion of a spherical shell (Passman and Grady, 1989). The material in the shell is assumed to be an ideal incompressible fluid. Initially we will consider the shell to be subjected to arbitrary pressure boundary conditions on the inner and outer surfaces. An arbitrary configuration of the shell is illustrated in Fig. 3.
Pa(a)
Pb(b) Fig. 3: Arbitrary configuration of a spherical shell of inner radius a and outer radius b. The coordinate r references a spatial point a < r < b. P~(a) a n d P~(b) represent uniform pressures over the inner and outer surfaces, respectively.
Stability and fragmentation of ejecta in hypervelocity impact
201
We look first for the velocity field v(r,t) over the interval a < r < b. It will be convenient to assume that the solution will lead to a time dependence for the outer radius b(t) which is invertible to t(b) and, instead, solve for v(r, b). Incompressible flow implies that the motion is divergence free, thus;
cgr=v(r, b) _ 0 ,
(3)
ar which has the solution, r2u(r,b) = g(b) ,
(4)
where g(b) is an arbitrary function. The boundary condition v(b, b) = b(b) provides the solution, b2 .
~(r, b) = -~b(b) .
(5)
It is useful at this point to note that,
av
. a~
b (
~dbl
(8)
We will define y(b) = b2(b) and write Equation 6 as,
Ov
b~ / dy
4
(7)
0-7-- 2~-~~~ + ~Y) Momentum balance in spherical coordinates provides, Oo~
+ ¥
-
at
(s)
Defining 7r = P/p, where P is the pressure and p is the density, and using Equations 5 and 7, we obtain,
0-7 ~ + ~-J~,Y = - ~ - J
+ ~y
(9)
It is convenient to introduce dimensionless independent variables, • = b/a and ~ = r/a which are defined over the domain 1 < ~ < co; 1 _< ~ _< 8. The resulting partial differential equation and boundary conditions are, ,~ +
=
(#~ - l / y ' - ~ y = ,~°(#)
,
,~(#,#) = ,~(#)
,~(1,#)
.
,
(10)
In Equation 10, y' = dy/d~. The momentum balance relation (Equation 10) is readily integrated,
+
=
(,0s-1)y w - # y
+f(fl) ,
(11)
where f(#) is an arbitrary integration function. Applying the first boundary condition of Equation 10 provides an expression for the pressure, ~r(~,~)=~rb(~)
'('2--1)
(~--l)
y' - 1
202
D.E.
Grady
and S.
L. P a s s m a n
Applying the second boundary condition leads to the following ordinary differential equation for y, 2 ]~(~2 + / 3 + 1)yt + (fl~ + 2~ + 3)y -- (/3 _ 1) 5 (~ra(~) - ~ra(~))
.
(13)
T w o useful relations in the algebra leading to E q u a t i o n 13 are ~a _ 1 -- (~ - 1)(~2 + ~ + 1) and fir _ 4/3 + 3 -(~ - 1)2(Z 2 + ~/~ + 3).
Finally a change in dependent variable is in order which considerably simplifies the form of Equation 13. Introducing 3 ~s (14)
2~2 + ~ + i y~'j
replaces Equation 13 with T' -
3/~2 (Z~-=-~)2 ( ~ ( Z ) - ~ ° ( ~ ) )
(15)
An interpretation of r is useful: The kinetic energy of the spherical shell is provided by, T =
4rr2dr
.
(16)
Using the solution, v = (b2/r2)v/-~ (Equation 5) and performing the algebra leads to, T
3
/~3
~rPbos - 2 ~ 2 + ~ + 1
y ,
(17)
where bo is the outer radius at a reference configuration corresponding to a -- 0. Comparison of Equations 14 and 17 reveals that r is the mean specific kinetic energy for a shell with configuration characterized by the parameter/3. Finally, using Equation 14, we obtain the following expression from Equation 12 for the pressure distribution through the shell,
Thus, Equations 15 and 18 provide the solution for the symmetric motion and pressure distribution of a spherical shell. For an arbitrary pressure on the boundaries, Equation 15 is integrated for the motion. The subsequent solution is substituted in Equation 18 for the pressure.
PRESSURE STATES IN A FREELY EXPANDING SPHERICAL SHELL For the problem of a freely expanding spherical shell modelling the ejecta from a hypervelocity impact event, the solution of the previous section can be used to calculate pressure states in the body. Assuming zero pressure boundary conditions on the inner and outer surface, Equation 15 yields r t = 0. Equivalently, the kinetic energy of the body is constant. From Equation 18, the pressure distribution reduces to
where ro is the constant mean specific kinetic energy. Note also that as the shell becomes thin (/3 --+ 1), and 2 = ~b from Equation 14, that ro --+ !y 1 "2 ~- ~Vo 1 where vo is the limiting expansion velocity of the spherical shell.
Stability and fragmentation of ejecta in hypervelocity impact
203
It will be useful to calculate the magnitude of the pressure gradients at the inner and outer surface of the shell from Equation 19. They are, respectively,
~s
1
3/~2 ,/~S _
(3/~ 2 +
2~ +
1) ~'o
(20)
,
1
3/~ 4 (/~ + 2/~ + 3) ro .
(21)
It is readily seen that the pressure through the body, 1 _~ ~ _~ /~, is everywhere compressive and has a m a x i m u m at 4~s (22) This m a x i m u m is centered through the thickness of the shell when the shell is thin (fl --+ 1), b u t occurs near the inner wall, ~ma®--~ 4 l/s, as ~ ~ oo. This is clearly seen in the pressure profiles shown in Fig. 4.
I
I
I
I
I
I
I
I
I
I
Fig. 4: Inertial pressure profiles through the thickness of a spherical shell with inner radius a and outer radius b. Dimensionless variables are ~ = r / a , = b/a. Pressures are normalized with respect to the m a x i m u m . The m a x i m u m inertial pressure normalized by the m e a n specific kinetic energy, as a function of the shell configuration, is plotted in Fig. 5, illustrating the extremely rapid decrease in pressure as the shell thins. An interesting reference occurs at 1 / ~ --~ 0.7(a --~ 0.Tb), at which time the m a x i m u m inertial pressure equals the 1 2 specific kinetic energy, ~r,~ffi = To, or P,~® m ~pvo, where, as noted previously, vo is the limiting expansion velocity of the shell. For metals with a density near 104 k g / m s, such as lead or copper, a n d for an expansion velocity of 100 m / s , P, naz -~ 50 MPa, which illustrates that the inertial pressures can be substantial in hypervelocity, impact events.
204
D.
100
E.
Grady and S. L .
I
Passman
!
I
!
|
I
!
!
10 0
b-
X 0
E
.1
.0 1 .1
1
1/fl Fig. 5: M a x i m u m inertial pressure in the spherical shell normalized by the mean specific kinetic energy. As noted above and illustrated in Fig. 4, as the shell becomes increasingly thinner, (8 --* 1), the pressure distribution through the thickness becomes increasingly more symmetric with the m a x i m u m pressure approaching the center. This is seen by expanding the pressure in Equation 19 in terms of small quantities appropriate to the thinning shell. We write h = 1+ a
and =1+-
z a
,
, (0
(23)
(24)
where h = b - a is the shell thickness and z = r - a is a coordinate through the thickness. Expansion of Equation 19, and retention of the lowest order terms, results in ~r = 6~o(~ - 1)(8 - ~) = 3e2z(h - z)
(25)
where we define a stretching rate ~ = Vo/r. Equation 25 provides a parabolic pressure profile through the thickness and is identical to the solution for an isotropically stretching plate of thickness h (Romero, 1989b). The m a x i m u m pressure in the thin shell limit is, 7r,,~az = ~Vo(~ - 1) 2 = -3e2h~4 "
(26)
Equation 25 is a valid representation if the shell is sufficiently thin (8 sufficiently close to one). An indication of the degree of thinning required is given by a plot of the pressure gradients at the inner and outer walls of the shell provided in Equations 20 and 21 and shown in Fig. 6. This plot suggests that Equation 25 is a reasonable representation of the pressure through the thickness of the shell, provided 1 / 8 is greater t h a n about 0.9.
Stability and fragmentation of ejecta in hypervelocity impact
10.0
!
I
I
!
!
!
205
i
o '/'I"61
0.0 0.0
I
I
I
I
1.0
Fig. 6: Plot of the pressure gradient at the inner and outer wall of the shell. Note that gradients approach equality as/~ ~ 1.
STABILITY AND
FRAGMENTATION
With the solutions from the preceeding section we now address the central issue of this work; namely the onset of instability and breakup of the expanding spherical shell. The solutions for the motion and the pressure field assume spherical symmetry. The stabilityof this spherical flow to asymmetric perturbations is pertinent to the fragmentation processes. W e wish to show that an expanding sphericalshellis initiallystablewith respect to asymmetric perturbations in the motion. As expansion proceeds, however, a criticalconfiguration is achieved for which subsequent motion is unstable, leading to breakup and fragmentation of the body. This criticalconfiguration will be shown to depend on both kinetic properties of the motion and on material properties of the expanding shell. A rigorous stabilityanalysis is not performed here. W e instead use an approximate analysis based on earlier work which illustratesthe underlying physics governing stability,and arrive at a criterionfor predicting the onset of breakup and fragmentation. Further details concerning instabilityand fragmentation of stretching sheets have been provided by Grady (1987), Fressengeas and Molinari (1988), and Romero (1989b). Similar concerns relating to the breakup of stretching jets are addressed by Frankel and Weihs (1985), Curtis (1987), and Romero (1989a). Although the analysis provided in preceeding sections assumed ideal fluid response, it is now important to include the plastic dissipation appropriate to flow in metals. Without such dissipation the flow is stable at all configurations and uninteresting for the present application. For simplicity, we will assume that plastic dissipation is characterized by a constant flow stress Y. In the previous solution for a freely expanding shell the mean specific kinetic energy ro provides a measure of the kinematic state of the body. We can now introduce a dimensionless parameter ,!.o = p r o / Y and note that our concern is with problems for which .!.o > > 1. Note that Jo is a measure of the ratio of the kinetic energy of the body to the volumetric dissipation rate. Before addressing instability it is important to recognize that for a large portion of the motion the inertial pressure field through the thickness of the shell stabilizes the flow and resists the growth of perturbations leading to breakup. This is seen qualitatively by considering an element of the shell near either the inner
E=. E. Crady and S. L .
206
Passman
or the outer surface and imagining a small perturbation such as a dimple in this surface. Consider the dimple to have a depth of order ~ and to perturb the flow over a distance of ~. Since the inertial pressure gradient is positive (OP/Or > 0 directed in from the surface), the perturbing dimple will lead to a relative low pressure immediately under the dimple and a high pressure near the wings of the perturbation, leading to a pressure gradient directed away from the center of the dimple. Consequently, a restoring body force of order fb ~ (,5/~)ap/Or will seek to stabilize the flow. This behavior is akin to a dimple in a lake where gravitational pressure provides the stabilizing restoring force as predicted by classic Rayleigh-Taylor theory. Thus, we emphasize one of the key observations of this study: Namely, in high velocity divergent flow, inertial compressive pressure fields are brought about which lead to stability of the corresponding motion. This has been discussed previously by Grady (1987) and addressed in related work by Frankel and Weihs (1985), Fressengeas and Molinari (1989), and Romero (1989a). Now consider an ideal incompressible plastic material characterized by a constant flow stress Y. It is found that for a limiting thin expanding shell an expression for the mean stress equivalent to Equation 25 is given by 2 Y (27) The corresponding radial and circumferential stresses are, respectively, 2 Y 7frr =
- - T f - - X/r~ P
,
(28)
1 Y
';tee = --/r -'1- . ~
(29)
P
Here the ~r's refers to stresses divided by the density and ~r = ( r . + 2tee)/3. The circumferential stress is plotted in Fig. 7. Note that the stretching shell is not necessarily in tension. Rather, a large portion of the thickness is in compression with only thin regions near the inner and outer surface in tension. This assumes, of course, that a state of motion exists such that the inertial pressure is still sufficiently high. As expansion ensues, the circumferential stress profile will evolve and eventually achieve a configuration in which the entire thickness is in tension as is illustrated with the stress profiles in Fig. 7.
2
t
fonsion
o4
0 ~
n
-1I -2
4 4~ p -" 1.1D
1
1 Fig. 7: Plot of the circumferential stress through the thickness of the shell for different degrees of expansion. A value of Jo = P~'o/Y = 100 was used to calculate the profiles. The tensile skin depth is also shown.
Stability and fragmentation of ejecta in hypervelocity impact
207
The tensile skin depth Ao is calculated by identifying the depth in from the surface at which ~r00 -- 0. We obtain, A°
,I
4Y
2 ~ -- 1 - ~ 1 V'3pro(flor, if
A,/h
(30) 1)' '
< < 1, A, _ Y h v~p~o(Z - 1)2
(31)
The ductility of materials is known to increase with confining pressure. Consequently, the interior of the shell thickness, where the inertial pressure is high, should provide a favorable environment for ductile flow. Conditions for material within the tensile skin depth, on the other hand, are not so favorable and we suspect that this environment will provide the precursors to breakup, which will occur when the global conditions for instability are satisfied. Those concerned with the development of materials which perform satisfactorily in shaped-charge jet applications would do well to consider these details of the flow field in which the material is deforming. Finally, we wish to establish a criterion at which the expanding shell becomes unstable and undergoes fragmentation. Early on, we have shown that the large inertial pressures developed in the rapidly expanding shell stabilize the motion. These pressures die out as the shell thins and eventually a configuration is reached where the dissipative plastic processes destabilize the motion and fragmentation follows. We will use as our criterion one which has been shown by Romero (1989a) to predict the onset of instability in rapidly stretching jets. We determine the circumferential stretching force through the thickness of the shell as a function of the motion and identify the maximum in the force curve where subsequent expansion reduces, rather than increases, this force. This can be viewed as a loose application of Le Chatelier's principle which simply states that for a system to be stable any deviation from equilibrium should bring about processes which would tend to restore equilibrium (e.g. Reif, 1965). A negative force-displacement behavior would fail to do this. A similar criterion has been stated by Considiere (1885) concerning the onset of instability in the quasistatic stretching of plastic bodies. To obtain the required force we integrate the circumferential stress through the thickness of the shell,
F : foh~resdz=afl~r, ed~
,
(32)
where F is also normalized by the density. Using Equations 27 and 29, we obtain,
which integrates to, F
v~Y(~-
1)
r o ( f l - 1) s
(34)
a
and is plotted in Fig. 8. As the shell expands, the corresponding stretching force moves from right to left in Fig. 8 (toward fl - 1 =
h/a
: 0). Initially the force is compressive but becomes tensile at a value of fl - 1 : ~y/-~-Y/pro. Continued thinning of the shell leads to a maximum, 'c-l:h-a
=~/~ro
'
(35)
which we identify as the critical configuration for onset of instability and fragmentation.
APPLICATION TO IMPACT FRAGMENTATION The expanding impact debris for the impact of lead on lead experiment shown in Fig. 1 provides a good example of the phenomena that the present analysis attempts to explain. This experiment is one of a number of such tests which were performed on lead and uranium (Grad},, eta/. 1985). In most of these tests, however, a single flash radiograph obtained well after fragmentation was the sole diagnostic. Thus experimental data which can be compared with the present study, focused on the onset of instability, is limited. Nonetheless,
208
D.E.
Grady and S. L. Passman
comparisons where possible are worthwhile. Attempts at such comparisons will identify a methodology for relating analysis and experiment. Also comparisons will bring to light the properties to measure in future experimental studies.
critical configuration
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unstable
~ com~cession
Fig. 8: This figure shows the stretching force as a function of configuration and identifies the point of instability. Equation 35 is the principal relation in the present analysis which predicts the onset of instability and fragmentation in an expanding impact debris cloud. We assume that the debris can be modelled as an expanding spherical shell of matter after Swift et al. (1983). Equation 35 relates two dimensionless parameters. The first, Jo = p r o / Y is a measure of the intensity of expansion relative to plastic dissipation in the body. The second, ~ = b / a is a measure of the distention of the spherical shell. A graph of these two parameters identifies regions of stable or unstable expansion and a boundary demarcating onset of instability through Equation 35. Such graphs are shown in Figs. 10 and 12, and will be discussed shortly. First, it is necessary to discuss the experiments which are compared with the analytic predictions. The experiments were selected from the collection of some forty impact tests performed on lead and uranium by Grady et al. (1985). We will not use the experiment shown in Fig. 1. It was unique in being the only experiment in which high speed photography was used to diagnose the impact debris. It was also unique in being the only experiment which used an ogive-nosed projectile which lent a pointed character to the debris cloud not observed in the other tests. On most of the experiments in this study a single flash radiograph of the debris cloud late in time was obtained. However, on two experiments (one on lead and one on uranium), which were comparable in geometry and impact velocity, multiple flash radiographs were obtained. The two experiments are shown in Fig. 9. The velocities in these tests were near the upper end of the range investigated in that work. It is interesting to note differences in the character of debris breakup in the lead and uranium experiments. In the lead experiment debris emerges as a continuous expanding ductile bubble. The bubble is still apparently intact at 50 ~s but has fragmented by 100 ~s. In contrast uranium appears to fragment almost immediately upon emerging from the back of the target. We will conclude from experimental observation that in 5 of the 6 images shown in Fig. 9 fragmentation has occurred. That is, in each case the expanding bubble has achieved instability onset and fragmentation. In one image (50 ~8 for lead) fragmentation and hence instability has not been achieved. We recognize that instability is probably not synonymous with fragmentation. The lead bubble at 50 #s could be unstable but not yet fragmented. The X-ray as a shadowgraph would not resolve the evolution of instability growth so nicely shown with the front lit photography in Fig. 1. Nevertheless as a criterion based on flash-radiography diagnostics the 50 Aus lead image does not show fragmentation and will be identified as stable expansion.
Stability and fragmentation of ejecta in hypervelocity impact
Projectile: uranium cylinder (6 m m x 6 mm) Target: uranium plate (6 mm thk.) Impact Velocity: 1245 m/s
Projectile: lead cylinder (6 mm x 6 mm) Target: lead plate (6 mm thk.) Impact Velocity: 1330 m/s
T
25ram
/k lid
iF
50gs
100gs
209
T
25mm
50gs
~
(r~ .41
100p,s
." 175/Zs
HI l
~
..
Fig. 9: Multiple flash radiographs of behind target debris caused by projectiles undergoing normal impact on target plates. Using this data to place points on the stability plot in Fig. 10 requires that ,7o = pro/Y and ~ = h/a be determined for each of the six images. This can be done but the uncertainties are fairly significant. The flow stress Y used for lead and uranium was 10 and 700 MPa, respectively. These are consistent with data in the literature, although the unknown temperature and strain rate dependence probably lend a factor of two or more uncertainty. We calculated Tofrom the expansion velocity predicted from Swift's Equation 2. The mass in the debris cloud is reasonably well known from the measured mass missing from the plate. The calculated expansion velocity is about 1/3 of the impact velocity. It was difficult to calculate an expansion velocity directly from the radiographs due to the changing shape of the cloud and ambiguity as to the location of the center of mass. Finally, h/a was estimated from the average cloud diameter in each radiograph and the mass in the debris cloud. Experimental data points for each of the six images of Fig. 9 are plotted in Fig. 10. The size of the symbols provides a measure of the uncertainty alluded to previously. The proximity of the one stable data point for lead to the boundary, which determines the transition from stable to unstable expansion lends a measure of support for the physical ideas and analysis put forth in the present study. Now we will make some different comparisons with the present analysis and offer some observations concerning computational simulations of impact debris clouds. Bergstresser, in the study of Grady eta/. (1984), performed computer simulations of the lead and uranium impact experiments from that work. The twodimensional Eulerian hydrocode CSQII (Thompson, 1988) was used for the calculations. The code uses a realistic three-phase equation of state and an elastic-perfectly plastic model for material strength with a yon Mises yield criterion. Plots for several of these calculations are shown in Fig. 11. Note that in two of the plots a stable debris cloud in observed whereas, in the other two, either incipient or complete fragmentation has occurred. The plots for the impact of lead on lead are at different times for the same impact velocity of 1.3 km/s. The plots for the impact of uranium on uranium are at approximately the same configuration but for impact velocities of 1.3 and 3.0 km/s, respectively.
210
D. E. Grady and S. L. Passman
We have treated these computer images much like the experimental impact data described earlier. The computational data points are shown on the stability plot in Fig. 12. Again, the results are reasonably consistent with proposed mechanisms of inertial stability governing the stable growth and onset of fragmentation in expanding debris clouds. The tentative agreement between the results of computational simulations and the present theory of inertially controlled onset of instability lends support for concluding that the breakup of debris clouds in calculations is a consequence of quite reasonable physical instabilities rather than numerical instabilities. This observation has some precedence. Work of Swegle and Robinson (1989) has shown that hydrocodes using general elasticplastic equations of state are quite capable of simulating transient loading of bodies leading to unstable flow and fragmentation.
1000
. . . . . . . .
I
. . . . . . . .
I
. . . . . . . .
:
stable ,
lead ~
>-
'
100
.~--
t
1
~
0
l.c~ II
sfabinty /boundary
unstable
10 uranium ~
,
,
,
I
I
I
I
,
0
I I
1
oi .00 1
i
,
,
,
, , , , I
*
*
*
*
' I I I I
.0 1
#-1
l
I
I
I
I
I I
.1
1
- h/o
Fig. I0: Stability plot for the spherical expansion of plastic shells. Experimental points are from Fig. 9.
CONCLUSIONS
The present study offers an explanation for the behavior of debris ejected from the back surface of a target impacted by a high velocity projectile. Such debris is observed to expand as a ductile bubble and then fragment through surface instabilities when a critical expansion is achieved. An approximate model of hypervelocity impact developed by Swift et al. (1983), which assumes an expanding spherical shell of matter for the debris cloud, is introduced to motivate the analysis. We then solve for the spherically symmetric motion of, and pressures within, a spherical shell of ideal incompressible fluid subjected to inner and outer pressure boundary conditions. The solution is specialized to the freely expanding spherical shell (no pressure on the inner and outer boundaries). It is shown that the pressure within the shell, which is brought about by inertia of the motion, is everywhere compressive. The character of this compressive pressure profile guarantees stability of the spherical motion. That is, nonspherical perturbations in the motions will die out rather than grow unstably. We then specialize further to the thin freely expanding spherical shell. Material response is extended to ideal incompressible plasticity and solutions for the radial and circumferential stress distributions are determined. It is shown that as the stabilizing inertial forces decrease with increasing expansion of the shell, plasticity provides the destabilizing force which leads to instability and fragmentation. From the solution, a criterion is established for the transition from stable flow to instability and fragmentation.
Stability
and fragmentation
of ejecta
in hypervelocity
impact
211
~, 3 v
,<
_<
'1_
,..j
""~{~~ "e:~''~;:"~:~"~ ' ~ I " ~~:': " "'"~£'.~"
"< o
"<
0
-1
-I o
1
2
3
1
4
(a)
2
3
4
RADIALDISTANCE (cm)
(o)
RADIALDISTANCE (cm)
4
3
3
o
~2
~2
Z ,<
Z
<
.... ~.~-: ~ ", .~..~.' .~,.~.~, ~ ,~. ~ ~ ~; , n~..-~ .....
<
0 q, #
-I
-1 1
2
3
1
4
2
3
4
RADIALDISTANCE (cm)
RADIALDISTANCE (cm)
(b)
(d)
Fig. 11: Computational simulations of lead and uranium impact experiments of Bergstresser from Grady eta/. (1985). (a) Lead on lead impact. Velocity of 1.3 km/s. Time of 50/~s. (b) Lead on lead impact. Velocity of 1.3 km/s. Time of 35/~s. (c) Uranium on uranium impact. Velocity of 1.3 km/s. Time of 35/~s. (d) Uranium on uranium impact. Velocity of 3.0 km/s. Time of 14/~s.
1000
,
i
J
i
i
i
i
i[
i
i
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i
i
i
stab~ty _._._.~ boundary
>--
i
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q.
stable
10
II -'3
o
1
• 1
.0
,
,
,
,
,
,,I
.1
,8-1 = h / a Fig.
12:
Stability plot for the spherical expansion of plastic shells. Data points are from computer simulations in Fig. 11.
212
D.E.
Grady and S. L. Passman
Experimental results from high velocity impact tests of lead and uranium projectiles on lead and uranium targets using multiple flash radiography diagnostics (Grady et a/., 1985) are compared with the theory. Although data are limited and the material and kinematic parameters somewhat uncertain, the comparison supports the physics and analysis put forth in this study. Similar comparisons are made between the analysis and computational simulations of the lead and uranium impact experiments. Again the results support the proposed mechanisms of initial inertial stability followed by plasticity-induced onset of instability and fragmentation.
REFERENCES Consid~re, A. (1885). M~moire sur l'emploi du fer et de l'acier dans les constructions. Ann. Ponts Chaussges 9, 574-775. Curtis, J. P. (1987). Axisymmetric instability model for shaped charge jets. J. Appl. Phys. 61, 4978-4985. Frankel, I. and D. Weihs (1985). Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 155, 289-307. Fressengeas, C. and A. Molinari (1989). Fragmentation of rapidly stretching jets. J. Fluid Mech., submitted. Grady, D. E., T. K. Bergstresser and J. M. Tayler (1985). Impact fragmentation of lead and uranium plates. Sandia National Laboratories Report, SAND84-1545, unpublished. Grady, D. E. (1987). Fragmentation of rapidly expanding jets and sheets. Int. J. Impact Eng. 5, 285-292. Passman, S. L., and D. E. Grady (1989). Exact solutions for symmetric deformations of hollow bodies of ideal fluids with application to inertial stability. Sandia National Laboratories Report, SAND88-3383, unpublished. Reif, F. (1965). "Fundamentals of Statistical and Thermal Physics," McGraw Hill. Romero, L. A. (1989a). The instability of rapidly stretching plastic jets. J. Appl. Phys., 65, 3006-3016. Romero, L. A. (1989b). The stability of stretching and accelerating plastic sheets. J. Appl. Phys., submitted. Swegle, J. W., and A. C. Robinson (1989). Acceleration instability in elastic-plastic solids. J. Appl. Phys., 66, 2838-2858. Swift, H. F., R. Bamford and R. Chem (1983). Designing space vehicle shields for meteoroid protection: A new analysis. Adv. Space Res. 2, 219-234. Thompson, S. L. (1988). CSQIII an eulerian finite difference code for two-dimensional material response. Sandia National Laboratories Report, SAND87-2763, unpublished.
0734-743X/90$3.00 + 0.00 PergamonPressplc
Printedin Great Britain
STABILITY AND FRAGMENTATION OF EJECTA IN HYPERVELOCITY IMPACT l
D. E. Grady and S. L. Passman Sandia National Laboratories, Thermomechanical & Physical and Process Research Divisions P.O. Box 5800, Div. 1534 and 6212, Albuquerque, New Mexico 87185-5800
INTRODUCTION
The near-normal hypervelocity impact of a metal projectile on a metal target plate, where the dimensions of the projectile and the thickness of the plate are comparable, results in the ejection of a high-velocity plume of debris. The ejected plume emerges as a contiguous expanding bubble of the projectile and target material. At a critical time during expansion, the bubble of metal undergoes fragmentation with the fragment debris expanding as a thin shell of material along trajectories imparted at the time of impact. Relatively little matter resides within this shell. Although systematic data are scant, there is evidence that as the impact velocity increases, the extent of stable bubble expansion before fragmentation increases and the characteristic particle size in the debris plume decreases (Grady et aL, 1985). Understanding the stability and fragmentation features of the impact ejecta is critical in assessing details of the impulse and damage experienced by a secondary structure. A phenomenological model of hypervelocity impact (Swift et aL, 1983) has been reasonably successful in describing certain features of the ejection and expansion process. In the Swift model, the ejected debris is assumed to be an expanding spherical shell of material. Application of the model leads to predictions of the center-of-mass and expansion velocities and, consequently, details of the momenta imparted to the debris shell. In the present study, we accept the premises of the Swift model of hypervelocity impact and address the issues of stability and fragmentation of a rapidly expanding spherical shell of metal. How well the Swift model describes the nature of the debris is not critical to this investigation. The physical phenomenology which we reveal is expected to be generally applicable to the present, as well as to other divergent flow problems. In this analysis we solve for the symmetric motion of a spherical shell of an incompressible ideal fluid subject to arbitrary pressure conditions on the inner and outer surfaces. We ignore strength initially because, in the regime of interest, inertial forces overwhelm the strength properties of the medium. Combining the calculated motion with the momentum equation, we determine the spatial pressure profile within the body. For the freely expanding shell (zero pressure on the inner and outer surface) representative of the hypervelocity impact ejecta, we find that this pressure is everywhere positive and has positive gradients directed into the body at the inner and outer boundaries. The latter condition is shown to guarantee Rayleigh-Taylor stability of the respective surfaces. This behavior explains the contiguous stable expansion phase of an ejecta bubble prior to fragmentation. We then continue the analysis and introduce material strength (a constant flow stress) as a material property. Through an approximate analysis of stability we show that strength provides the destabilizing force which counters the inertial stability of the motion. A criterion for the onset of fragmentation is determined through a balancing of these opposing forces. Data obtained from impact experiments of lead and uranium projectiles on lead and uranium plates (Grady et aL, 1985) are compared with the analysis. In this comparison, Swift's model of hypervelocity ejecta is XThis work performed at Sandia National Laboratories supported by the U.S. DOE under contract DE-AC04-76DP00789.
1.97
198
D. E. Grady and S. L. Passman
combined with the present analysis to evaluate the dependence of the imparted motion on kinematic and material properties, the onset of instability, and fragmentation of the debris ejecta. Reasonable agreement with the impact data is achieved. Finally we reexamine computational simulations of the lead and uranium impact experiments which also show stable expansion and fragmentation. Based on comparison with the present analysis we suggest that stable flow and fragmentation observed in computer simulations is a consequence of natural physical instabilities.
SWIFT'S MODEL OF HYPERVELOCITY IMPACT In the analysis of Swift et al. (1983), an equidimensional projectile traveling at hypervelocity speeds undergoes near-normal impact on a target plate with thickness comparable to that of the projectile. On impact, strong shock waves propagate into both projectile and plate material, converting much of the projectile kinetic energy into potential energy of compression. The proximity of free surfaces leads to rapid decompression of the shocked material and most of it expands violently in a bubble-like structure which emerges from the back of the plate. A smaller portion of the material is ejected as a cone of fine debris on the impact side of the target plate. Such an event is illustrated in the series of photographs in Fig. 1.
40 Fs
• LEAD PROJECTILE IMPACT ON LEAD PLATE • VELOCITY = 1 2 0 0 m / s
6 mm
Fig. 1: A set of high-speed front-lit photographs of the back surface debris ejected by the impact of a lead projectile on a lead target at approximately 1200 m/s.
Stability and fragmentation of ejecta in hypervelocity impact
199
The state of the material in the rearward moving bubble was of principal interest in the analysis of Swift. The distributed impulse and damage imparted to a secondary structure is determined through interaction with this bubble. For sufficiently high impact velocities, much of the debris material may be vaporous. We are interested here, however, with the range of velocities where the debris is in the condensed liquid or solid state. In modelling the hypervelocity impact event, Swift e t a / . (1983), introduced a geometry-approximating observation and some simplifying assumptions which allowed them to evaluate the phenomena with global momentum and energy balance principles. Although some of the assumptions can be argued with in detail, Swift's model provides a reasonable description of some of the essential features of the debris momenta and trajectories. The general geometry and parameters in Swift's model are illustrated in Fig. 2. It is assumed in the development that the momentum of the projectile is totally transferred to the spherical ejecta bubble. Thus, impulse transferred to the plate is ignored and backsplash mass and momentum is assumed sufficiently small as to be discounted. Inelastic collision results in an excess energy which Swift e t a / . (1983), assume goes into spherical bubble expansion about the center-of-mass motion, although an efficiency coefficient Q ~_ 1 is introduced to account for other forms of dissipation, such as heat or radiation.
J J
J J J J J
J
J
v,/
J
Vm
J m
\
J
J J J J J
M
J
J J J
M+rn
(b)
(o)
Fig. 2: Geometry assumed in the Swift model of hypervelocity impact. (a) Before impact, a projectile of mass m is traveling at a velocity v,~ toward the plate. The mass M of the plate which will be ejected is identified. (h) After impact a spherical shell of mass M + m is expanding with velocity ve about a center of mass, traveling at a velocity vc. It is then straightforward, based on momentum and energy conservation, to derive the center-of-mass velocity vc and the expansion velocity ve of the spherical debris bubble, in terms of the incident projectile velocity and participating masses, 1 v< -- 1 + ' (1)
MI---------~"'
,,~
=
OV~/m -
-
1 + M/m
.
(2)
v'~
Swift e t aL (1983), provide estimates for the ratio of ejected plate mass M to projectile mass m, based on the relative dimensions of projectile and plate, and argue that Q is frequently near unity.
200
D. E. Grady
and S. L. P a s s m a n
High speed photographs or flash radiographs of hypervelocity impact events show that the bubble of debris initially expands as an unbroken shell of matter not unlike the spherical model illustrated in Fig. 2. Experiment further shows that as expansion and thinning of the bubble proceeds, a point is reached at which instability and disintegration into a cloud of discrete particles occurs. In subsequent expansion, although trajectories and momenta are maintained, the constitution of the bubble has changed from that of a continuous shell of matter to a distribution of discrete particles. The study of Swift et al. (1983), discusses this aspect of the phenomenology in an observational sense. Physical causes for the behavior and predictions of the fragment process are not put forward, however. It is this aspect of the hypervelocity impact event that we analyze in subsequent sections. What are the physical and material properties in the rapid expansion of a bubble of condensed matter which lead to the initial stability of that expansion, a critical point in the expansion process at which breakup initiates, and ultimately disintegration into a distribution of particles of characteristic size and shape. We proceed in the spirit of Swift's model and focus on details of the expanding motion of a spherical shell of matter. The assumption of spherical symmetry leads to a tractable analysis, but it is certainly not restrictive. The observations and conclusions provided by the analysis are recognized to be generally applicable to motions involving more complex divergent flow. The relations of Swift in Equations 1 and 2 will be useful in estimating the initial conditions for the analysis in the present study.
MOTION OF AN EXPANDING SPHERICAL SHELL In pursuing the investigation of the stability and fragmentation of an expanding shell of ejecta we will solve, in the present section, for the symmetric radial motion of a spherical shell (Passman and Grady, 1989). The material in the shell is assumed to be an ideal incompressible fluid. Initially we will consider the shell to be subjected to arbitrary pressure boundary conditions on the inner and outer surfaces. An arbitrary configuration of the shell is illustrated in Fig. 3.
Pa(a)
Pb(b) Fig. 3: Arbitrary configuration of a spherical shell of inner radius a and outer radius b. The coordinate r references a spatial point a < r < b. P~(a) a n d P~(b) represent uniform pressures over the inner and outer surfaces, respectively.
Stability and fragmentation of ejecta in hypervelocity impact
201
We look first for the velocity field v(r,t) over the interval a < r < b. It will be convenient to assume that the solution will lead to a time dependence for the outer radius b(t) which is invertible to t(b) and, instead, solve for v(r, b). Incompressible flow implies that the motion is divergence free, thus;
cgr=v(r, b) _ 0 ,
(3)
ar which has the solution, r2u(r,b) = g(b) ,
(4)
where g(b) is an arbitrary function. The boundary condition v(b, b) = b(b) provides the solution, b2 .
~(r, b) = -~b(b) .
(5)
It is useful at this point to note that,
av
. a~
b (
~dbl
(8)
We will define y(b) = b2(b) and write Equation 6 as,
Ov
b~ / dy
4
(7)
0-7-- 2~-~~~ + ~Y) Momentum balance in spherical coordinates provides, Oo~
+ ¥
-
at
(s)
Defining 7r = P/p, where P is the pressure and p is the density, and using Equations 5 and 7, we obtain,
0-7 ~ + ~-J~,Y = - ~ - J
+ ~y
(9)
It is convenient to introduce dimensionless independent variables, • = b/a and ~ = r/a which are defined over the domain 1 < ~ < co; 1 _< ~ _< 8. The resulting partial differential equation and boundary conditions are, ,~ +
=
(#~ - l / y ' - ~ y = ,~°(#)
,
,~(#,#) = ,~(#)
,~(1,#)
.
,
(10)
In Equation 10, y' = dy/d~. The momentum balance relation (Equation 10) is readily integrated,
+
=
(,0s-1)y w - # y
+f(fl) ,
(11)
where f(#) is an arbitrary integration function. Applying the first boundary condition of Equation 10 provides an expression for the pressure, ~r(~,~)=~rb(~)
'('2--1)
(~--l)
y' - 1
202
D.E.
Grady
and S.
L. P a s s m a n
Applying the second boundary condition leads to the following ordinary differential equation for y, 2 ]~(~2 + / 3 + 1)yt + (fl~ + 2~ + 3)y -- (/3 _ 1) 5 (~ra(~) - ~ra(~))
.
(13)
T w o useful relations in the algebra leading to E q u a t i o n 13 are ~a _ 1 -- (~ - 1)(~2 + ~ + 1) and fir _ 4/3 + 3 -(~ - 1)2(Z 2 + ~/~ + 3).
Finally a change in dependent variable is in order which considerably simplifies the form of Equation 13. Introducing 3 ~s (14)
2~2 + ~ + i y~'j
replaces Equation 13 with T' -
3/~2 (Z~-=-~)2 ( ~ ( Z ) - ~ ° ( ~ ) )
(15)
An interpretation of r is useful: The kinetic energy of the spherical shell is provided by, T =
4rr2dr
.
(16)
Using the solution, v = (b2/r2)v/-~ (Equation 5) and performing the algebra leads to, T
3
/~3
~rPbos - 2 ~ 2 + ~ + 1
y ,
(17)
where bo is the outer radius at a reference configuration corresponding to a -- 0. Comparison of Equations 14 and 17 reveals that r is the mean specific kinetic energy for a shell with configuration characterized by the parameter/3. Finally, using Equation 14, we obtain the following expression from Equation 12 for the pressure distribution through the shell,
Thus, Equations 15 and 18 provide the solution for the symmetric motion and pressure distribution of a spherical shell. For an arbitrary pressure on the boundaries, Equation 15 is integrated for the motion. The subsequent solution is substituted in Equation 18 for the pressure.
PRESSURE STATES IN A FREELY EXPANDING SPHERICAL SHELL For the problem of a freely expanding spherical shell modelling the ejecta from a hypervelocity impact event, the solution of the previous section can be used to calculate pressure states in the body. Assuming zero pressure boundary conditions on the inner and outer surface, Equation 15 yields r t = 0. Equivalently, the kinetic energy of the body is constant. From Equation 18, the pressure distribution reduces to
where ro is the constant mean specific kinetic energy. Note also that as the shell becomes thin (/3 --+ 1), and 2 = ~b from Equation 14, that ro --+ !y 1 "2 ~- ~Vo 1 where vo is the limiting expansion velocity of the spherical shell.
Stability and fragmentation of ejecta in hypervelocity impact
203
It will be useful to calculate the magnitude of the pressure gradients at the inner and outer surface of the shell from Equation 19. They are, respectively,
~s
1
3/~2 ,/~S _
(3/~ 2 +
2~ +
1) ~'o
(20)
,
1
3/~ 4 (/~ + 2/~ + 3) ro .
(21)
It is readily seen that the pressure through the body, 1 _~ ~ _~ /~, is everywhere compressive and has a m a x i m u m at 4~s (22) This m a x i m u m is centered through the thickness of the shell when the shell is thin (fl --+ 1), b u t occurs near the inner wall, ~ma®--~ 4 l/s, as ~ ~ oo. This is clearly seen in the pressure profiles shown in Fig. 4.
I
I
I
I
I
I
I
I
I
I
Fig. 4: Inertial pressure profiles through the thickness of a spherical shell with inner radius a and outer radius b. Dimensionless variables are ~ = r / a , = b/a. Pressures are normalized with respect to the m a x i m u m . The m a x i m u m inertial pressure normalized by the m e a n specific kinetic energy, as a function of the shell configuration, is plotted in Fig. 5, illustrating the extremely rapid decrease in pressure as the shell thins. An interesting reference occurs at 1 / ~ --~ 0.7(a --~ 0.Tb), at which time the m a x i m u m inertial pressure equals the 1 2 specific kinetic energy, ~r,~ffi = To, or P,~® m ~pvo, where, as noted previously, vo is the limiting expansion velocity of the shell. For metals with a density near 104 k g / m s, such as lead or copper, a n d for an expansion velocity of 100 m / s , P, naz -~ 50 MPa, which illustrates that the inertial pressures can be substantial in hypervelocity, impact events.
204
D.
100
E.
Grady and S. L .
I
Passman
!
I
!
|
I
!
!
10 0
b-
X 0
E
.1
.0 1 .1
1
1/fl Fig. 5: M a x i m u m inertial pressure in the spherical shell normalized by the mean specific kinetic energy. As noted above and illustrated in Fig. 4, as the shell becomes increasingly thinner, (8 --* 1), the pressure distribution through the thickness becomes increasingly more symmetric with the m a x i m u m pressure approaching the center. This is seen by expanding the pressure in Equation 19 in terms of small quantities appropriate to the thinning shell. We write h = 1+ a
and =1+-
z a
,
, (0
(23)
(24)
where h = b - a is the shell thickness and z = r - a is a coordinate through the thickness. Expansion of Equation 19, and retention of the lowest order terms, results in ~r = 6~o(~ - 1)(8 - ~) = 3e2z(h - z)
(25)
where we define a stretching rate ~ = Vo/r. Equation 25 provides a parabolic pressure profile through the thickness and is identical to the solution for an isotropically stretching plate of thickness h (Romero, 1989b). The m a x i m u m pressure in the thin shell limit is, 7r,,~az = ~Vo(~ - 1) 2 = -3e2h~4 "
(26)
Equation 25 is a valid representation if the shell is sufficiently thin (8 sufficiently close to one). An indication of the degree of thinning required is given by a plot of the pressure gradients at the inner and outer walls of the shell provided in Equations 20 and 21 and shown in Fig. 6. This plot suggests that Equation 25 is a reasonable representation of the pressure through the thickness of the shell, provided 1 / 8 is greater t h a n about 0.9.
Stability and fragmentation of ejecta in hypervelocity impact
10.0
!
I
I
!
!
!
205
i
o '/'I"61
0.0 0.0
I
I
I
I
1.0
Fig. 6: Plot of the pressure gradient at the inner and outer wall of the shell. Note that gradients approach equality as/~ ~ 1.
STABILITY AND
FRAGMENTATION
With the solutions from the preceeding section we now address the central issue of this work; namely the onset of instability and breakup of the expanding spherical shell. The solutions for the motion and the pressure field assume spherical symmetry. The stabilityof this spherical flow to asymmetric perturbations is pertinent to the fragmentation processes. W e wish to show that an expanding sphericalshellis initiallystablewith respect to asymmetric perturbations in the motion. As expansion proceeds, however, a criticalconfiguration is achieved for which subsequent motion is unstable, leading to breakup and fragmentation of the body. This criticalconfiguration will be shown to depend on both kinetic properties of the motion and on material properties of the expanding shell. A rigorous stabilityanalysis is not performed here. W e instead use an approximate analysis based on earlier work which illustratesthe underlying physics governing stability,and arrive at a criterionfor predicting the onset of breakup and fragmentation. Further details concerning instabilityand fragmentation of stretching sheets have been provided by Grady (1987), Fressengeas and Molinari (1988), and Romero (1989b). Similar concerns relating to the breakup of stretching jets are addressed by Frankel and Weihs (1985), Curtis (1987), and Romero (1989a). Although the analysis provided in preceeding sections assumed ideal fluid response, it is now important to include the plastic dissipation appropriate to flow in metals. Without such dissipation the flow is stable at all configurations and uninteresting for the present application. For simplicity, we will assume that plastic dissipation is characterized by a constant flow stress Y. In the previous solution for a freely expanding shell the mean specific kinetic energy ro provides a measure of the kinematic state of the body. We can now introduce a dimensionless parameter ,!.o = p r o / Y and note that our concern is with problems for which .!.o > > 1. Note that Jo is a measure of the ratio of the kinetic energy of the body to the volumetric dissipation rate. Before addressing instability it is important to recognize that for a large portion of the motion the inertial pressure field through the thickness of the shell stabilizes the flow and resists the growth of perturbations leading to breakup. This is seen qualitatively by considering an element of the shell near either the inner
E=. E. Crady and S. L .
206
Passman
or the outer surface and imagining a small perturbation such as a dimple in this surface. Consider the dimple to have a depth of order ~ and to perturb the flow over a distance of ~. Since the inertial pressure gradient is positive (OP/Or > 0 directed in from the surface), the perturbing dimple will lead to a relative low pressure immediately under the dimple and a high pressure near the wings of the perturbation, leading to a pressure gradient directed away from the center of the dimple. Consequently, a restoring body force of order fb ~ (,5/~)ap/Or will seek to stabilize the flow. This behavior is akin to a dimple in a lake where gravitational pressure provides the stabilizing restoring force as predicted by classic Rayleigh-Taylor theory. Thus, we emphasize one of the key observations of this study: Namely, in high velocity divergent flow, inertial compressive pressure fields are brought about which lead to stability of the corresponding motion. This has been discussed previously by Grady (1987) and addressed in related work by Frankel and Weihs (1985), Fressengeas and Molinari (1989), and Romero (1989a). Now consider an ideal incompressible plastic material characterized by a constant flow stress Y. It is found that for a limiting thin expanding shell an expression for the mean stress equivalent to Equation 25 is given by 2 Y (27) The corresponding radial and circumferential stresses are, respectively, 2 Y 7frr =
- - T f - - X/r~ P
,
(28)
1 Y
';tee = --/r -'1- . ~
(29)
P
Here the ~r's refers to stresses divided by the density and ~r = ( r . + 2tee)/3. The circumferential stress is plotted in Fig. 7. Note that the stretching shell is not necessarily in tension. Rather, a large portion of the thickness is in compression with only thin regions near the inner and outer surface in tension. This assumes, of course, that a state of motion exists such that the inertial pressure is still sufficiently high. As expansion ensues, the circumferential stress profile will evolve and eventually achieve a configuration in which the entire thickness is in tension as is illustrated with the stress profiles in Fig. 7.
2
t
fonsion
o4
0 ~
n
-1I -2
4 4~ p -" 1.1D
1
1 Fig. 7: Plot of the circumferential stress through the thickness of the shell for different degrees of expansion. A value of Jo = P~'o/Y = 100 was used to calculate the profiles. The tensile skin depth is also shown.
Stability and fragmentation of ejecta in hypervelocity impact
207
The tensile skin depth Ao is calculated by identifying the depth in from the surface at which ~r00 -- 0. We obtain, A°
,I
4Y
2 ~ -- 1 - ~ 1 V'3pro(flor, if
A,/h
(30) 1)' '
< < 1, A, _ Y h v~p~o(Z - 1)2
(31)
The ductility of materials is known to increase with confining pressure. Consequently, the interior of the shell thickness, where the inertial pressure is high, should provide a favorable environment for ductile flow. Conditions for material within the tensile skin depth, on the other hand, are not so favorable and we suspect that this environment will provide the precursors to breakup, which will occur when the global conditions for instability are satisfied. Those concerned with the development of materials which perform satisfactorily in shaped-charge jet applications would do well to consider these details of the flow field in which the material is deforming. Finally, we wish to establish a criterion at which the expanding shell becomes unstable and undergoes fragmentation. Early on, we have shown that the large inertial pressures developed in the rapidly expanding shell stabilize the motion. These pressures die out as the shell thins and eventually a configuration is reached where the dissipative plastic processes destabilize the motion and fragmentation follows. We will use as our criterion one which has been shown by Romero (1989a) to predict the onset of instability in rapidly stretching jets. We determine the circumferential stretching force through the thickness of the shell as a function of the motion and identify the maximum in the force curve where subsequent expansion reduces, rather than increases, this force. This can be viewed as a loose application of Le Chatelier's principle which simply states that for a system to be stable any deviation from equilibrium should bring about processes which would tend to restore equilibrium (e.g. Reif, 1965). A negative force-displacement behavior would fail to do this. A similar criterion has been stated by Considiere (1885) concerning the onset of instability in the quasistatic stretching of plastic bodies. To obtain the required force we integrate the circumferential stress through the thickness of the shell,
F : foh~resdz=afl~r, ed~
,
(32)
where F is also normalized by the density. Using Equations 27 and 29, we obtain,
which integrates to, F
v~Y(~-
1)
r o ( f l - 1) s
(34)
a
and is plotted in Fig. 8. As the shell expands, the corresponding stretching force moves from right to left in Fig. 8 (toward fl - 1 =
h/a
: 0). Initially the force is compressive but becomes tensile at a value of fl - 1 : ~y/-~-Y/pro. Continued thinning of the shell leads to a maximum, 'c-l:h-a
=~/~ro
'
(35)
which we identify as the critical configuration for onset of instability and fragmentation.
APPLICATION TO IMPACT FRAGMENTATION The expanding impact debris for the impact of lead on lead experiment shown in Fig. 1 provides a good example of the phenomena that the present analysis attempts to explain. This experiment is one of a number of such tests which were performed on lead and uranium (Grad},, eta/. 1985). In most of these tests, however, a single flash radiograph obtained well after fragmentation was the sole diagnostic. Thus experimental data which can be compared with the present study, focused on the onset of instability, is limited. Nonetheless,
208
D.E.
Grady and S. L. Passman
comparisons where possible are worthwhile. Attempts at such comparisons will identify a methodology for relating analysis and experiment. Also comparisons will bring to light the properties to measure in future experimental studies.
critical configuration
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Fig. 8: This figure shows the stretching force as a function of configuration and identifies the point of instability. Equation 35 is the principal relation in the present analysis which predicts the onset of instability and fragmentation in an expanding impact debris cloud. We assume that the debris can be modelled as an expanding spherical shell of matter after Swift et al. (1983). Equation 35 relates two dimensionless parameters. The first, Jo = p r o / Y is a measure of the intensity of expansion relative to plastic dissipation in the body. The second, ~ = b / a is a measure of the distention of the spherical shell. A graph of these two parameters identifies regions of stable or unstable expansion and a boundary demarcating onset of instability through Equation 35. Such graphs are shown in Figs. 10 and 12, and will be discussed shortly. First, it is necessary to discuss the experiments which are compared with the analytic predictions. The experiments were selected from the collection of some forty impact tests performed on lead and uranium by Grady et al. (1985). We will not use the experiment shown in Fig. 1. It was unique in being the only experiment in which high speed photography was used to diagnose the impact debris. It was also unique in being the only experiment which used an ogive-nosed projectile which lent a pointed character to the debris cloud not observed in the other tests. On most of the experiments in this study a single flash radiograph of the debris cloud late in time was obtained. However, on two experiments (one on lead and one on uranium), which were comparable in geometry and impact velocity, multiple flash radiographs were obtained. The two experiments are shown in Fig. 9. The velocities in these tests were near the upper end of the range investigated in that work. It is interesting to note differences in the character of debris breakup in the lead and uranium experiments. In the lead experiment debris emerges as a continuous expanding ductile bubble. The bubble is still apparently intact at 50 ~s but has fragmented by 100 ~s. In contrast uranium appears to fragment almost immediately upon emerging from the back of the target. We will conclude from experimental observation that in 5 of the 6 images shown in Fig. 9 fragmentation has occurred. That is, in each case the expanding bubble has achieved instability onset and fragmentation. In one image (50 ~8 for lead) fragmentation and hence instability has not been achieved. We recognize that instability is probably not synonymous with fragmentation. The lead bubble at 50 #s could be unstable but not yet fragmented. The X-ray as a shadowgraph would not resolve the evolution of instability growth so nicely shown with the front lit photography in Fig. 1. Nevertheless as a criterion based on flash-radiography diagnostics the 50 Aus lead image does not show fragmentation and will be identified as stable expansion.
Stability and fragmentation of ejecta in hypervelocity impact
Projectile: uranium cylinder (6 m m x 6 mm) Target: uranium plate (6 mm thk.) Impact Velocity: 1245 m/s
Projectile: lead cylinder (6 mm x 6 mm) Target: lead plate (6 mm thk.) Impact Velocity: 1330 m/s
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210
D. E. Grady and S. L. Passman
We have treated these computer images much like the experimental impact data described earlier. The computational data points are shown on the stability plot in Fig. 12. Again, the results are reasonably consistent with proposed mechanisms of inertial stability governing the stable growth and onset of fragmentation in expanding debris clouds. The tentative agreement between the results of computational simulations and the present theory of inertially controlled onset of instability lends support for concluding that the breakup of debris clouds in calculations is a consequence of quite reasonable physical instabilities rather than numerical instabilities. This observation has some precedence. Work of Swegle and Robinson (1989) has shown that hydrocodes using general elasticplastic equations of state are quite capable of simulating transient loading of bodies leading to unstable flow and fragmentation.
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CONCLUSIONS
The present study offers an explanation for the behavior of debris ejected from the back surface of a target impacted by a high velocity projectile. Such debris is observed to expand as a ductile bubble and then fragment through surface instabilities when a critical expansion is achieved. An approximate model of hypervelocity impact developed by Swift et al. (1983), which assumes an expanding spherical shell of matter for the debris cloud, is introduced to motivate the analysis. We then solve for the spherically symmetric motion of, and pressures within, a spherical shell of ideal incompressible fluid subjected to inner and outer pressure boundary conditions. The solution is specialized to the freely expanding spherical shell (no pressure on the inner and outer boundaries). It is shown that the pressure within the shell, which is brought about by inertia of the motion, is everywhere compressive. The character of this compressive pressure profile guarantees stability of the spherical motion. That is, nonspherical perturbations in the motions will die out rather than grow unstably. We then specialize further to the thin freely expanding spherical shell. Material response is extended to ideal incompressible plasticity and solutions for the radial and circumferential stress distributions are determined. It is shown that as the stabilizing inertial forces decrease with increasing expansion of the shell, plasticity provides the destabilizing force which leads to instability and fragmentation. From the solution, a criterion is established for the transition from stable flow to instability and fragmentation.
Stability
and fragmentation
of ejecta
in hypervelocity
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Fig. 11: Computational simulations of lead and uranium impact experiments of Bergstresser from Grady eta/. (1985). (a) Lead on lead impact. Velocity of 1.3 km/s. Time of 50/~s. (b) Lead on lead impact. Velocity of 1.3 km/s. Time of 35/~s. (c) Uranium on uranium impact. Velocity of 1.3 km/s. Time of 35/~s. (d) Uranium on uranium impact. Velocity of 3.0 km/s. Time of 14/~s.
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12:
Stability plot for the spherical expansion of plastic shells. Data points are from computer simulations in Fig. 11.
212
D.E.
Grady and S. L. Passman
Experimental results from high velocity impact tests of lead and uranium projectiles on lead and uranium targets using multiple flash radiography diagnostics (Grady et a/., 1985) are compared with the theory. Although data are limited and the material and kinematic parameters somewhat uncertain, the comparison supports the physics and analysis put forth in this study. Similar comparisons are made between the analysis and computational simulations of the lead and uranium impact experiments. Again the results support the proposed mechanisms of initial inertial stability followed by plasticity-induced onset of instability and fragmentation.
REFERENCES Consid~re, A. (1885). M~moire sur l'emploi du fer et de l'acier dans les constructions. Ann. Ponts Chaussges 9, 574-775. Curtis, J. P. (1987). Axisymmetric instability model for shaped charge jets. J. Appl. Phys. 61, 4978-4985. Frankel, I. and D. Weihs (1985). Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 155, 289-307. Fressengeas, C. and A. Molinari (1989). Fragmentation of rapidly stretching jets. J. Fluid Mech., submitted. Grady, D. E., T. K. Bergstresser and J. M. Tayler (1985). Impact fragmentation of lead and uranium plates. Sandia National Laboratories Report, SAND84-1545, unpublished. Grady, D. E. (1987). Fragmentation of rapidly expanding jets and sheets. Int. J. Impact Eng. 5, 285-292. Passman, S. L., and D. E. Grady (1989). Exact solutions for symmetric deformations of hollow bodies of ideal fluids with application to inertial stability. Sandia National Laboratories Report, SAND88-3383, unpublished. Reif, F. (1965). "Fundamentals of Statistical and Thermal Physics," McGraw Hill. Romero, L. A. (1989a). The instability of rapidly stretching plastic jets. J. Appl. Phys., 65, 3006-3016. Romero, L. A. (1989b). The stability of stretching and accelerating plastic sheets. J. Appl. Phys., submitted. Swegle, J. W., and A. C. Robinson (1989). Acceleration instability in elastic-plastic solids. J. Appl. Phys., 66, 2838-2858. Swift, H. F., R. Bamford and R. Chem (1983). Designing space vehicle shields for meteoroid protection: A new analysis. Adv. Space Res. 2, 219-234. Thompson, S. L. (1988). CSQIII an eulerian finite difference code for two-dimensional material response. Sandia National Laboratories Report, SAND87-2763, unpublished.