Int. J. Impact I~non# Vol. 9, No. 1, pp. 89-113, 1990 Printed in Great Britain
0734-743X/890 $3.00+ 0.00 © 1990 Pergamon Press plc
DEBRIS CLOUD DYNAMICS CHARLES E. ANDERSON, J R . , t TIMOTHY G . TRUCANO~ a n d SCOTT A. M U L L I N t 1"Southwest Research Institute, San Antonio, TX 78228, U.S.A. and J;Sandia National Laboratories, Albuquerque, NM 87185, U.S.A. (Received 28 January 1988; and in final form 11 August 1989)
Summary--The hypervelocity impact of a projectile upon a thin metal plate and subsequent formation of back-surface debris is reviewed. At sufficiently high impact velocities, roughly greater than 3.0 km/s (depending upon the shock impedances of the materials involved), shock formation and interaction dominate and control the overall response of both the projectile and the target plate. We focus upon the importance of shock heating, melting, and vaporization in this application. Because of the complexity of the physical interactions, numerical simulation of such problems is necessary to draw quantitative conclusions. Thus, we also assess the current status of computational modeling of this kind of impact event, specifically addressing recent work bearing on the sensitivity of such modeling to the equations of state and certain numerical issues.
NOMENCLATURE a A b B co Cp k E Eo Es E', n us up P S T Tm Tv Vo V Vt fl y p F #
Tillotson EOS constant Tillotson EOS constant (A = pock) Tillotson EOS constant (F o = a + b) Tillotson EOS constant bulk sound speed specific heat at constant pressure slope of u,-up curve internal energy Tillotson EOS constant Energy at sublimation point equivalent to the total heat at the boiling point E, + sE, where e is the fraction of vaporization energy added to E. to guarantee a gas-like behavior as p ~ 0 specific energy required for a phase transformation shock velocity particle velocity pressure entropy temperature melt temperature vaporization temperature impact velocity specific volume ( = I/p) final (release) volume Tillotson EOS constant coefficient of volumetric expansion Tillotson EOS constant density Gruneisen coefficient reduced density (P/Po - 1) compression (1 - Po/P)
projectile material target material
Hugoniot initial condition portion of this paper was presented at SMIRT-9-Post Conference Seminar on Impact, Lausanne, Switzerland, 26-27 August, 1987. 89
(?HARLES E. ANI)t!RSON,JR. et al. INTRODUCTION
The impact of a particle or projectile onto a target initially produces a shock wave that propagates into both the projectile and the target. Depending upon the amplitude of the shock and the attendant release waves, a complex process of material failure in the form of dynamic fracture and fragmentation of the projectile and target may occur. If the impact velocity is sufficiently high, this process is accompanied by melting and vaporization of the target and projectile material. Ultimately, at extremely high velocities, as may be found in the impact of meteorites, for example, complete vaporization of the projectile and part of the target can occur. In this paper, the evolution of the impact interaction of a projectile upon a thin plate is reviewed. For a given class of impact velocities and projectile properties, the impact of the projectile upon a thin plate results in complete perforation of the target, accompanied by the disintegration of the projectile and a portion of the target. This results in the formation and subsequent propagation of a debris cloud consisting of both projectile and target material. The dispersal of the forward momentum of the projectile into the relatively diffuse structure of the debris cloud suggested a possible means of protecting spacecraft from micrometeoroid impacts [1-5], first suggested by Whipple [6, 7]. The difficulty of analyzing this type of impact phenomenology can be formidable, yet it still has great practical interest. This paper provides a review of current work of this phenomenology, with special attention focused upon the particular problems that arise from impact velocities sufficiently high to cause vaporization of projectile and target materials. We review the details of hypervelocity impact, shock interaction, and debris formation which occur in thin plate impacts at high velocity. Specifically, the discussions focus on recent experiments performed in this area. While general analytic considerations can provide insight in interpreting these experiments, quantitative analysis requires the use of complex numerical simulation techniques. We assess current computational capabilities for modeling this interaction. Two major areas of interest in any such simulations are (1) the sensitivity of the simulation to the material constitutive models, specifically, within the context of this paper, the equation of state, and (2) the overall accuracy of the numerical algorithms. We will show that a reasonable discussion of these issues is closely linked to the interpretation of experimental data, and that contemporary experimental techniques are capable of providing data which can serve as points from which numerical extrapolation can reliably take place.
Hypervelocity impact produces extreme pressures in the projectile and target materials. When the target is thin relative to the projectile diameter, the propagating shock accelerates the target material. Subsequent rarefaction waves shatter the projectile and target, forming a debris cloud behind the target plate. The debris cloud consists of both projectile and target materials. The physical process of the cloud formation can be illustrated qualitatively through several figures and flash radiographs. A typical debris cloud is shown in Fig. 1. The cloud resulted from the impact of a spherical projectile at 6.6 km/s, where the diameter of the projectile is 2.4 times the plate thickness. A qualitative description of the early stages of debris cloud formation is obtained from the diagram of Maiden [1, 2], shown in Fig. 2. A right circular cylinder, with a length-to-diameter ratio of one, impacts a thin target plate at a velocity Vo. Shortly after impact, the two shocks $1 and S2 are shown propagating away from the impact surface (Fig. 2a). Because the projectile is finite, rarefaction waves R1 and R 2 are unloading the shocked material. A little later after impact, shock $2 strikes the free surface of the target and reflects as the rarefaction wave R 3 shown in Fig. 2b. The portion of the rarefaction wave R 3 transmitted across the target-projectile interface (I) is rarefaction wave R4. From a physical viewpoint, rarefaction waves R~ to R 4 are generated to satisfy boundary conditions; in particular, R 1, R 2, and R 3 a r e generated to satisfy the condition of zero stress at a free surface. Regions of tension form where the rarefaction waves interact.
Debris cloud dynamics
FIG. 1. Typical debris cloud from hypervelocity impact of a spherical projectile into a thin target.
" " I~''~'
Bubble of Debris FIG. 2. Schematic of early stages of impact on a thin target (a) shortly after impact; (b) after reflection of the shock in the target (from Maiden  and Maiden and McMillan ).
Fracture will occur in the projectile or target where the net tensile stress exceeds the fracture stress of the material. When the material has fractured, rarefaction waves will be generated to satisfy the conditions of zero stress at the new free surfaces. Thus, the whole process of fracture of the projectile and target can be interpreted as a multiple-spalling phenomenon. Particle size of the debris is a function of the impact velocity, i.e. shock
CHARLES E. ANDERSON,JR. et al.
strength, which affects the magnitude of the rarefaction waves. Also, if the shock strength is sufficiently high, then heating of the material can result in degradation of the strengths of the projectile and target materials. The tensile stress which can be supported by the material becomes less, decreasing the particle size in the debris. In the limit of total melt, only surface tension forces need be exceeded to create droplets. Finally, if the impact velocity is sufficiently high, release from the impact shock can result in a vaporized debris cloud. The dispersion of the debris cloud depends, then, upon the thermodynamic state of the materials. The formation of the debris bubble is just beginning in Fig. 2b. Small particles of debris are being ejected backwards because of the tensile stresses and excessive material strains. Projectile geometry is known to have an effect upon the shape of the debris cloud. Various investigators have examined projectiles with different shapes, including spheres, right circular cylinders, cones, and disks . It is reasonable to state that the mechanism of formation and general shape of the debris cloud is similar for all these geometries, though the actual 'fine structure' of the debris cloud is dependent on the projectile shape. The differences in cloud shape result from the differences in the shape of the shock reaching the free surface of the target. For example, the divergence of the cloud is dependent on the geometry of the projectile: for a cylindrical projectile, the shock near the axis is planar. For a spherical projectile, the shock is more nearly hemispherical, resulting in a larger dispersion of the debris cloud . Overall, though, a qualitative discussion of the formation of the debris cloud is the same for differently shaped projectiles. Figure 3 depicts some of the same information as Fig. 2, except for a spherical projectile. Figure 3a depicts the initial impact conditions, and Fig. 3b shows the shocked regions shortly after impact, similar to Fig. 2a but without the detail. When the shock front reaches the free surface of the target, the pressure pulse is relieved, causing a rapid expansion of the compressed material accompanied by a decrease in the internal energy, and a corresponding increase in the kinetic energy (Fig. 3c; similar to Fig. 2b). Finally, after some time has elapsed, the debris cloud has a shape depicted in Fig. 3d. Note that a cross section of the cloud will show that the debris cloud is largely hollow. When looking at flash X-ray photographs, the cloud looks 'solid' because the picture is taken through the sides of the debris cloud. Qualitatively, then, the projectile and target material are distributed mainly in a thin shell on the surface of a growing elongated bubble attached to the lip of the hole forming in the target plate . Analytical models of the dispersion of the debris cloud have been formulated with some success [10-12]. These approximate theories are based on momentum and energy arguments to estimate the characteristics of the debris cloud. An excellent review of this treatment is given by Herrmann and Wilbeck . Herrmann and Wilbeck point out that
~ in Projectile Shield L-~
\ ~- .Shoclt.Front ~ S h n e l d
(c) (d) FIG. 3. Schematic of formation of debrisbubble(after Rincy and HaJda ).
Debris cloud dynamics
Proj~'tiltffTarget - - ~ Interface
(b) Fic. 4. Numerical simulation of debris cloud formation: material distribution (frominformation in Rosenblatt et al. ). the most suspect assumption of the theory is that the material is distributed unifomaly over the surface of a spherically expanding debris cloud, as opposed to the majority of the mass being concentrated near the axis at the front of the cloud (e.g. Figs 1 and 5). The recent analytic work by Passman  has modified the work of Swift e t al. , to reduce cloud dispersion, though Passman's work specifically addresses impacts on the order of 100 km/s. Rosenblatt e t al.  examined the early stages of the formation of the debris bubble through numerical simulations. A number of computations were made, but the 7.5 km/s impact of an aluminum sphere into an aluminum target with plate thickness being one-fourth the diameter of the sphere is typical of their results. The hypervelocity impact produces three general regions of debris dynamics (Fig. 4). The dotted line represents the projectile-target interface. Region 1 contains the main body of the cloud, composed of projectile material and target material. The target material in Region 1 comes from an area slightly less than the original projectile radius. This region contains approximately four times the kinetic energy and twice the axial momentum of the other two regions. The neck of the debris cloud, Region 2, is composed primarily of an expanding annular region which accounts for most of the angular divergence. The periphery of the neck region defines Region 3, which contains target mass originating in regions out to the final target hole. Very little kinetic energy and axial momentum are contained in this portion of the cloud. Transition from one region to the next is not sharply defined, but the regions serve to illustrate where different portions of the debris cloud originate. The inferences drawn from the computational results depicted in Fig. 4 are confirmed by tests conducted by Piekutowski . Two experiments were performed to determine
FIG. 5. Experimental investigation of material distribution in the debris cloud. (a) 1 g, cylindrical ( L / D = 1) aluminum projectile into 0.61 mm thick copper plate at 6.39 km/s; (b) 1 g, 7.72 mm diameter copper disc into 2.03 mm thick aluminum plate at 6.23 km/s (from Piekutowski ; flash radiographs courtesy of A. Piekutowski).
CHARLES E. ANDERSON, JR. et al.
the origin of material making up the debris cloud. In the first experiment, copper was used as the target, impacted by a right circular cylinder aluminum projectile. In the second experiment, a copper disc, with the same areal density as the aluminum projectile, impacted an aluminum target. The areal densities of the two targets were kept the same. Copper was used since it is considerably more opaque to X-rays than aluminum, thereby allowing a distinction to be made between target and projectile material. Flash radiographs are given in Fig. 5; the initial conditions of the tests are given in the figure. Double-exposed images of the debris cloud were recorded on a single piece of film. Slight differences in the front end of the debris cloud are evident from the differences in the geometry of the projectile; the reasons for this have already been discussed. The radiographs permit the following observations : the front cone and the front edge of the debris cloud are composed of target material (Fig. 5a); and the projectile is contained in a well-defined region inside the outer cloud of target material (Fig. 5b). Thus, experiment confirms the computational analysis. The advantage of the computations is that they provide substantially more detail than it is possible to measure with existing experimental techniques. The combination of experiment and computational analysis promises to provide considerable information on the physics of debris cloud formation and subsequent expansion. Accurate computations require realistic material models. At the velocities of hypervelocity impacts, thermodynamic considerations become very important. It is therefore instructive to examine the specifics of the impact and the subsequent release from the shocked state, and examine some consequences of this shock heating. HYPERVELOCITY IMPACT, MELTING, AND VAPORIZATION The discussion so far has been qualitative in that we have used experimental observations and computer calculations to infer information concerning the appearance of the debris cloud. We will now be interested in depicting differences in the debris cloud as a result of melting, and in particular, vaporization. A one-dimensional analysis will not permit an estimate of the total amount of material shocked to pressures sufficiently high to lead to melting and vaporization since edge relief waves will attenuate the shock as it moves into the projectile and target. However, knowing the velocities for the onset of melting and vaporization is extremely illuminating, and these velocities can be inferred from relatively simple one-dimensional considerations. Initial shock conditions
Consider the one-dimensional impact of a projectile, velocity Vo, with a target, assumed to be of different materials. Superscripts 'p' and 't' represent the projectile and target materials, respectively. A shock will propagate both into the target and projectile. Conservation of mass, momentum, and energy across the shock yields the famous Rankine-Hugoniot jump conditions: poUs = p(us - up)
E . = PH(V0 -- ) I . ) / 2
(1) (2) (3)
where the initial conditions ahead of the shock have been taken as zero (uo = Po = Eo = 0). It is assumed that a linear shock velocity-particle velocity relationship holds for the materials: Us =- C O + k u p .
(A quadratic term in Up can be accommodated easily; thus, there is no loss of generality in the analysis.) The shock equations can be written for both materials (Table 1). (The shock velocity in the projectile is taken as the velocity relative to a 'stationary' projectile; see Ravid et al.  for corresponding expressions assuming a laboratory coordinate
Debris cloud dynamics TABLE l. IMPACT EQUATIONS
u~ = cg + kPu~ p_
P H - - poUs Ur,
u~. .-. . c. o + k ut, t
Pit - -
Vf FIG. 6. Schematic of Hugoniot, isentropic release and internal energies.
system.) At the target-projectile interface, there is pressure equilibrium: P. = P~ = Ph.
The other condition at the interface is continuity of the velocity, i.e. the sum of the particle velocities is equal to the impact velocity: v0 = u~ + ut.
These equations are solved simultaneously for the particle velocities of the projectile and target material, which then gives the impact pressure. Release f r o m the shocked state
A requirement in the calculation of hypervelocity impact is an accurate thermodynamic description of the interacting materials through a wide range of pressure and density, generally referred to as the equation of state (EOS). The passage of a shock is an irreversible process resulting in an increase of the internal energy of the material. This is depicted in Fig. 6 which shows the Hugoniot of a material. The material is shocked to the thermodynamic state of (P., V.) along the Rayleigh line, The total shaded triangular area in Fig. 6 represents the internal energy of the shocked state I-equation (3)]. However, the release occurs isentropically. The recovered energy is the area under the isentrope, denoted by the lightly speckled region. Thus, a residual amount of energy remains in the material, represented by the darker shaded area. Therefore, the equation of state must describe not only the conditions after shock impact, but also its release, and changes of phase if the shock energy is sufficient to melt or vaporize the material. Asay and Kerley 1-18] give an excellent overview of the state of the art in modeling material response to impact; approximately half of the article describes EOS modeling, while the other half is concerned with constitutive modeling. But for the purposes here, to keep the analysis relatively simple, we have adopted two different analytical equations
CHARLES E. ANDERSON, JR. el al.
of state: the Mie-Gruneisen [19, 20] and Tillotson  equations of state. Difficulties with these formulations will be alluded to in subsequent discussion. The Mie-Gruneisen EOS has the form: P = PH + p F ( E -- EH)
where it is usually assumed for metals that the dynamic Gruneisen coefficient is given by: (8)
F = FoPo/p.
In one sense, the Mie-Gruneisen equation of state provides a relation between thermodynamic states relative to a reference state--the reference state being the Hugoniot. To compute release from the shocked state, the second law of thermodynamics: dE = T dS-
is used with the isentropic constraint dS = 0: (10)
dE = -PdV.
Following the treatment of McQueen et al. [22-], a specific position in the P, V, E plane is denoted by a subscript i. This point is found from the Hugoniot at the corresponding volume: (11)
A finite difference scheme is used to march down the release isentrope, starting at the Hugoniot impact state. The change in volume, AV, is used as the marching step size. The following approximations are used: d E ~ A E = E i -- E i_ 1
P ~ (P, + P,_ , ) / 2 dV'~AV
= V I - VI_I
Substituting equations (12) into (10) gives: El = E l - 1
(Pi + Pi - 1 ) 2
The substitution of equation (13) into equation (11), and rearranging, gives:
Pi=E(PH),~( ,_-~',_,AV (EH)i)]/[I + F
[AVIv 2 _]"
Equation (14) is started at the Hugoniot impact state, and a volume step size AV (usually 1/100 of the interval Vo - Va) is used, along with equation (13) to determine P, V, E values along the isentropic release path. The energy under the isentrope is computed by summing the increments: A E - P' - P ' - ' A V
2 along the entire path until Pi becomes zero. The energy computed from underneath the release path is subtracted from the original energy [equation (3)] to obtain the residual internal energy (waste heat). This energy is then compared to the energy required to melt or vaporize the material. It is convenient to use the temperature, though Herrmann and Wilbeck report : The calculation of temperature is uncertain since most equations cf state are based on mechanical measurements along Hugoniots, and the temperature is estimated from theoretical arguments . It is more direct to calculate the internal energies, and to compare residual energies with melting and sublimation energies, but the same uncertainties arise in integrating along the release isentrope.
Debris cloud dynamics
The methodology reported here is an approximation in that we have ignored the entropy change in the presence of a phase change. However, though material phase changes are nonisentropic processes, the work at Sandia National Laboratories  indicates that release from the shock state is nearly isentropic even when crossing phase lines. Thus, we will compute the temperature of the released state within these assumptions, realizing that the procedure is only an approximation. It can be shown using classical thermodynamics that the change in internal energy can be related to the change in temperature by : dE= CpdT---Tfl dPP
P d V + n.
We will compute the temperature when the material has returned to the ambient pressure. As the final state is at the same pressure as the initial state, the second term on the right-hand side of equation (t6) is zero. Though it is usual to take Po as zero relative to the shock pressure, the third term on the right-hand side of equation (16) requires that Po be one atmosphere. In theory, the energy required to bring the material to incipient melting, total melt, incipient vaporization, and total vaporization can be computed knowing Tin, Tv, and the heats of fusion and vaporization. The Mie-Gruneisen EOS is an accurate thermodynamic description of most metals in the solid regime. The Mie-Gruneisen EOS has its origins in the theory of condensed matter; specifically, the internal energy is computed from summing the energies of the simple harmonic oscillators of a crystal lattice. Thus, the Mie-Gruneisen equation cannot be expected to give good results in the expanded liquid, and in particular, the vapor region. As the energy increases, the assumption that the Gruneisen coefficient is a function only of volume is no longer justified; the dependence of F on temperature becomes important when electron excitation or chemical dissociation occurs [ 18]. To remedy these objections, Tillotson developed an analytic EOS applicable to hypervelocity impact 1-21]. In compression, the Tillotson EOS has the form: Pc = a + E (po~ 2
G-/+ 1 The constants a, b and A are derived from Hugoniot experiments. E o and B are adjusted to give the best fit for the EOS surface. The above formula is used for compression and a small region of expansion where the energy is less than the sublimation energy. This models the situation in which a material is shocked to a sufficiently low pressure that it remains a solid when it returns isentropically to zero pressure. At higher energies in expansion, the gas phase of the material is represented by:
[- E / ~bEp Pe = a E p + lt_ S-2
+ A # e r'~] e -~'~2 E > E's.
The constants ~ and 7 control the rate of convergence to an ideal gas. In the mixed phase
region, the pressure is given by: P = P,(E - E~) + Pc(E; - E) E's- E s
E, < E < E~.
Across the change of phase line at Vo, the pressure and its first derivatives are continuous. While the Tillotson EOS is easy to use and asymptotically correct (it goes to the Thomas-Fermi result for very high compression, and an ideal gas for large expansion), it does not accurately describe the gas-liquid and liquid-solid phase transformation regions. It also does n6t accurately reproduce isobaric expansion experiments which measure enthalpy and density at constant pressure .
CHARLES E. ANDERSON, JR. et el.
TABLE 2. MIE-GRUNE1SEN EOS COEFFICIENTS 
2.712 10.208 11.346 8.639
5.38 5.14 2.03 2.48
1.34 1.22 1.47 1.64
2.13 1.52 2.77 2.20
AI Mo Pb Cd
TABLE 3. TILLOTSON EOS COEFFICIENTS 
A1 Cd Mo Pb
0.5 0.5 0.5 0.4
5.0 4.5 4.5 1.5
78.50 53.13 269.69 46.76
65.0 45.0 165.0 15.0
2.76 0.23 1.62 0.20
14.10 1.43 6.74 1.14
1.63 1.70 1.02 2.37
5.0 10.0 5.0 13.0
5.0 10.0 5.0 15.0
== 200 --
Mie-Gruneisen & Tillotson EOS
~ ~X / I ~ ~ /
F -Mie'GruneisenEOS /Impact at 25 km/s
~x~ I ~ ' ~ ' ~
Impact at 25 km/s
Specific Volume (cm3/g)
FIG. 7. A1 into AI impact: Hugoniot and isentropic release paths.
Isentropic release from the impact state using the Tillotson EOS follows the procedures already outlined for the Mie-Gruneisen EOS. Closed form expressions for Pi can be obtained using equations (17) and (18), although substantially more algebra is required. The parameters for the Mie-Gruneisen and Tillotson equations of state for four materials, aluminum, cadmium, molybdenum, and lead, are given in Tables 2 and 3, respectively. Some slight adjustments were made of the constants in the Tillotson EOS to be consistent with the Hugoniot properties obtained from Ref. 1-26]. The release paths for aluminum using the Mie-Gruneisen and Tillotson EOS are shown for several different impact velocities in Fig. 7. The release paths of the two equations of state are essentially the same until impact pressures become greater than 174 GPa. As will be seen, aluminum begins to vaporize upon release from this pressure. As already discussed, the Mie-Gruneisen EOS does not account for the vapor phase and the associated expansion.
Debris cloud dynamics 6000
/ i /
/ / I
E Q I,--
Particle Velocity (km/s)
FIG. 8. Estimates of residual temperature versus particle velocity using Tillotson EOS. TABLE 4.
T~ (°c) AI Cd Mo Pb
660 321 2610 327
Incipient melt up PH (km/s) (GPa) 2.85 0.90 2.75 0.80
71 31 238 29
FOR P H A S E C H A N G E S
Complete melt up Pri (km/s) (GPa) 3.45 1.15 3.15 0.95
94 43 289 37
Incipient vaporization T, up PH (°C) (km/s) (GPa) 2480 765 5560 1740
5.20 1.50 4.00 1.85
174 64 409 100
The Tillotson EOS, on the other hand, through equation (18), provides for the expansion of the vaporized material. In Fig. 7, it can be seen that considerable expansion, i.e. a large change in volume, occurs as the shocked material returns to ambient pressure. End state calculations and discussion
Residual temperatures versus particle velocity are shown in Fig. 8 for four metals: aluminum, cadmium, lead, and molybdenum. For impacts where the projectile and target are the same materials, the impact velocity is twice the particle velocity. Table 4 lists the particle velocity and impact pressures for the transition points of incipient melt, total melt, and incipient vaporization for the four metals of interest. There has been no attempt to estimate the particle velocity for total vaporization. With
CHARLES E. ANDERSON,JR. et al.
the exception of cadmium, impact velocities of tens of kilometers per second are required for total vaporization. The reason for this can be understood qualitatively. The isentropic release curve becomes more and more asymptotic to ambient pressure as more and more energy is delivered in the initial shock state; this effect is depicted in Fig. 7. Physically, even though a large amount of energy is available from the shock process, the 'adiabatic' expansion of the vaporized material absorbs a significant amount of the available energy, evident from the very slow increase in temperature of cadmium (Fig. 8) after complete vaporization. For example, cadmium begins to vaporize upon release from a shock pressure of 64 GPa. Using the complete, thermodynamically consistent equation of state ANEOS 1-28], it is estimated that only 80% of the cadmium is vaporized even at an impact pressure of 500 GPa. This compares reasonably well with the predictions of the simple model discussed in this paper using the Tillotson EOS, considering that the ANEOS cadmium EOS was adjusted to obtain better agreement with experiment . Hopkins et al.  conducted a series of experiments which examined the impact of aluminum spheres into aluminum plates, and cadmium spheres into cadmium plates (Fig. 9a). Information concerning the debris clouds resulting from the impacts can be inferred from the ballistic limit thickness of the backup plate as a function of the impact velocity of the sphere. The results are shown in Fig. 9b. It is instructive to compare the entries in Table 4 with the abrupt changes in the curves in Fig. 9b. Lead (Pb) and molybdenum (Mo) are interesting because the momentum and kinetic energies of the Pb or Mo projectiles striking Pb or Mo plates (respectively) are almost identical because of their similar densities (Pb = 11.346 g/cm3; Mo -- 10.208 g/cm3). Thus, any differences in the results of the debris clouds can be attributed almost entirely to differences in their equations of state 1-30]. Figure 10 shows two flash radiographs of Mo into Mo (Mo --* Mo) at 6.6 km/s; Fig. 11 shows two flash radiographs of Pb --* Pb at the same impact velocity. The projectiles were spheres; the diameter of the spheres was approximately 2.4 times the plate thickness. From Fig. 8, it is expected no Mo would vaporize, and little would be melted (only that portion of the Mo shocked to the full
AI or Cd
~ ~ -
0.787 mm - ~
AI or Cd
~ 50.Smrrl~-- L ---~
3 4 5 Projectile Velocity(km/s)
(b) Fie. 9. Ballistic limit thickness of a backup plate for AI into A1 and Cd into Cd impacts. (a) Schematic of experimental setup; (b) ballistic limit thickness versus impact velocity (from Hopkins et al. I-5]).
Debris cloud dynamics
FIG. 10. Debris cloud for Mo into Mo impact at 6.58 km/s (courtesy of D. Liquornik).
CHARLES E. ANDERSON, JR. et ~d.
FIG. 11. Debris cloud for Pb into Pb impact 6.58 km/s (courtesy of D. Liquornik).
Debris cloug clynamlcs
impact pressure would melt--relief waves would attenuate the impact stress to levels below those required to melt most of the projectile and target). However, Fig. 8 indicates that Pb would melt and vaporize for these impact conditions. There are several distinct differences in the flash radiographs. The Mo debris cloud has the appearance of a 'conventional' debris cloud, consisting of many small particles. The dispersion of the cloud is evident by the 'growth' of the debris cloud from comparisons of two radiographs at times 31 and 40 ps after impact. The Pb ~ Pb impact, on the other hand, shows a diffuse debris cloud. No discrete solid particles are distinguishable in the forward part of the cloud, though streamers reminiscent of liquid connect the main part of the cloud to the plate. From the discussion of the results by Rosenblatt et al. , these liquid-like streamers originate from the neck region of the target, and would not be shocked to as high a pressure. Another prominent difference between the Mo and Pb impacts is the rate of expansion of the debris cloud. The lead debris cloud is expanding considerably faster than the Mo debris cloud. Vaporization would tend to superimpose an overall spherical expansion on the translational motion of a solid-liquid debris cloud because of the high gas pressure (the experiments are conducted in a vacuum). Both the diameter and leading edge of the Pb debris cloud are expanding faster than the discrete particle Mo cloud. These experiments will be discussed further in the next section. COMPUTATIONAL
Review of previous work The one-dimensional model is limited in its predictive ability because of rarefaction waves due to the finite size of the projectile. To account for these effects requires a much more sophisticated two-dimensional calculation. Numerical simulations of the hypervelocity impact of a projectile against a thin bumper plate are generally performed with computer programs referred to as hydrocodes [31, 32]. The time-dependent conservation equations are advanced in time on a spatial mesh, subject to material response modeling. An equation of state computes the hydrostatic pressure component of the stress tensor, and some form of constitutive relationship with plastic yielding describes strength effects. Material failure also can be described. An artificial viscosity is used to broaden shocks and dampen numerical oscillations. The earliest work in analyzing the dynamics of the debris cloud is that of Riney and Halda , and Rosenblatt et al. . Riney and Halda examined impacts into a multi-layered target; they showed that the important parameter for the axial momentum and kinetic energy of the debris cloud is the mass per unit area of the target. Swift et al.  used the results of Rosenblatt et al.  to compare computations against experimental data of aluminum into aluminum (AI ~ AI), copper into copper (Cu ~ Cu) and cadmium into cadmium (Cd ~ Cd) impacts at approximately 7 km/s. The computational and experimental results for the A1 ~ A1, and Cd ~ Cd impacts are shown in Fig. 12. The authors found that fair agreement existed for the velocities of the leading edge of the debris cloud for A1 ~ A1 and Cu ~ Cu, where the debris consisted of solid material; however, the code overpredicted the expansion velocity of the vaporized Cd ~ Cd impact. In all cases, the numerical simulations predicted a relatively thick cloud, whereas experimental results showed a very thin cloud . Contemporary analysis A hiatus of approximately 15 years occurred before the next computations were performed . During this period of time, tremendous advances were made in computer technology; yet, the hypervelocity impact of a projectile into a thin target with its resultant debris cloud remains a most challenging computational problem. Two issues, intimately coupled in numerical simulations, exist: what is the correct material description, and are current numerical algorithms sufficiently accurate for the computations? The remaining paragraphs address these issues.
CHARLES E. ANDERSON,JR. et al. 2 cm Optical bding Edge)
Cu--~Cu 18.23 p.s After Impact
I Code Prediction Plate
I I iptical
Cd-~.d 15.52 Its
FIG. 12. Comparison of computations to experiments of debris clouds. (a) Cu --* Cu 18.23/~s after impact; (b) Cd --* Cd 15.52 ps after impact (from Swift et al. ).
Though theoretical EOSs are available, reliable and reproducible experimental techniques do not currently exist for investigating impact at velocities higher than the 7-8 km/s achievable with two-stage light gas guns. It is of considerable interest to understand the response of metals at higher impact velocities, and though numerical simulations can be made, the degree of confidence varies among researchers. An area which has received considerable attention during the last several years is the use of metals which have low energies of sublimation, such as Pb and Cd 1-35], and comparison of numerical simulations with experiment. Parallel experiments were conducted using Mo and Pb. Spherical projectiles of Mo and Pb with a mass of 20 g were impacted into Mo and Pb target plates, approximately 6.6 mm thick at 6.6 km/s i-25, 30]. As already discussed, the Pb would be expected to melt and vaporize from the impact; but, no Mo would vaporize, though some would be melted. The experiments confirmed this analysis. An aluminum witness plate was placed 50 cm behind the target plates. In the Mo shots, distinct craters were observed in the witness plate. In the Pb shots, no surface pitting was visible; rather, the aluminum witness plate was lightly covered with a deposit of lead 1-30]. Flash X-rays (Figs 10 and 11) were made of the resultant debris cloud. Densitometer measurements were made to assess the spatial
Debris cloud dynamics 0.8 ~'~
OLD T6 EOS -----
...: / / ~
NEW T6 EOS
" - - TILLOTSON EOS
......... EXPERIMENTAL DATA
WI•0 . 4 0.2
POSITION Z-DIRECTION (cm)
FIG. 13. Comparison of computations to experimental data for the Pb --, Pb 30 #s after impact (after Holian and Burkett ). mass distribution (g/cm 2) of the debris cloud. Further details of the experiments and densitometer measurements are described by Bjork . Two separate investigations were made of these experiments. Bjork  compares up-to-date numerical simulations to experiment for both Mo and Pb. Holian and Burkett  discuss three different equations of state (the Tillotson EOS, an old tabular EOS for lead, and a new tabular EOS for lead developed specifically for their study), and then use the three EOSs in numerical simulations of the Pb -~ Pb experiments. Bjork states that the computations of Mo ~ Mo, where the debris cloud consists mainly of solid particles, agreed quite well with experiment . Both Bjork and Holian and Burkett showed poor correlation between computations and experiment for the Pb --+ Pb impact. Though the position and velocity of the leading edge of the debris cloud are well reproduced by the computations, a major discrepancy exists in the areal density of the debris cloud, depicted in Fig. 13. (The presence of a curve labeled CSQIII in this figure will be explained in the Discussion section.) The figure shows that the computations predict a ball of gas with greatest density at the center of the cloud. The experiment, on the other hand, shows that the highest density is at the leading edge of the debris cloud. Holian and Burkett point out : Since all three equations of state have quite different two-phase regions, one might expect significant differences in the calculations. However, there are only 10-15 % differences in the mass distributions predicted by the three hydrocode calculations along the center line of the cloud. Qualitatively, they are the same. In fact, the calculations are more similar to each other than they are to the experiments. The principal difference is that the experimental debris cloud appears to be hollow, while the calculated cloud is not. One-dimensional debris cloud experiments and computations Before discussing further these discrepancies between computations and experiments, it is instructive to examine the work of Trucano et al. [29, 36, 37]. This work is distinguished by the experimental procedure whereby multi-dimensional geometric dispersion of the vaporized debris cloud is isolated from thermodynamic effects. Two-dimensional calculations have confirmed the one-dimensional nature of the experiment. In these planar impact experiments, tantalum impactors were launched into thin samples of cadmium, lead, and polymethyl methacrylate (PMMA) at velocities of up to 6.7 km/s. The resulting debris crossed a vacuum gap and stagnated against a laser window, buffered by aluminum, allowing the acquisition of time-resolved particle velocity data. Recent work of Kerley and Wise [-38] suggests that this technique can be extended to porous metals. Except for the
CHARLES E. ANDERSON,JR. et al. 4.0
CHARTD ( 4 0 ZONES IN CD) CHARTD ( 5 0 0 ZONES IN CD) EXPERIMENT
TIME (~s) FIG. 14. Course and fine zoning comparison of one-dimensional Cd ~ Cd.
Pb samples, these essentially one-dimensional experiments exhibit a characteristic double wave structure--the first wave is from the impact of the sample debris, and the second signals debris recompression and impact of the remnants of the tantalum projectile. One-dimensional calculations can be performed to simulate the experiments and determine, for example, the influence of the sample equation of state on the particle velocity data. Here, we will focus on the experiments having Cd and Pb samples. In Fig. 14, experimental data are shown from one of the experiments having a Cd sample 0.25 mm thick. The impact velocity was 5.23 km/s, which we estimate (using ANEOS ) produces a mass fraction of vapor in cadmium of roughly 33%. Also in the figure are results for two calculations using the one-dimensional Lagrangian wavecode CHART-D . Computations were made with 40 zones (solid line) in the sample, and with 500 zones (dashed line) in the sample. The calculation with 500 zones is basically 'converged'additional zones will not change the agreement with the data in the absence of EOS variations. This demonstrates the advantage of dealing with an experiment that is one-dimensional, for there is little likelihood that numerical convergence in this sense could be achieved for multi-dimensional impact experiments. Simulations were also performed in which the Cd EOS alone was varied 1-29]. These variations were: (1) Cd treated as a frozen solid (Mie-Gruneisen) EOS; (2) Cd treated as a solid-vapor (this treatment was different from the Tillotson EOS--the ANEOS solid model was used, with a direct conversion to ideal gas, without performing a Maxwell construction to create a liquid phase); and (3) Cd treated as a thermodynamically consistent three-phase substance, using ANEOS. The agreement between computations and experiments was very poor for the frozen solid EOS, indicating that little or no metastable condensed phase material initially impacts the laser window. Both the second and the third EOS gave substantially better agreement with the experiment with the agreement improving as the EOS tended to better agreement with the known Cd EOS data. For example, the three-phase model gave better results than the two-phase, and the correlation improved by further adjusting the zero Kelvin contribution, by modifying the melt boundary description, and by shifting the critical point closer to estimated values 1-29]. An elastic precursor to the initial stagnation wave was noted in the experimental data indicating that low density vapor arrived first at the laser window. The computations over-drove this precursor until sufficient zoning was used (that is, 500 zones in the Lagrangian calculations). This is one illustration of the difficulty of drawing conclusions about EOS from multi-dimensional experiments in the absence of sufficiently converged numerical hydrodynamics. The above EOS variations demonstrate that comparisons of computations with
Debris cloud dynamics 5.0
E >. (J O
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0
TIME (/~s) FIG. 15. Comparison of computations and experimental data for one-dimensional Pb--* Pb.
these experiments are sensitive to fine details of vapor dome physics, unlike the situation where coarse, multi-dimensional experimental data are used. Thus, the experimental technique appears to have the capability to assess directly the thermophysics associated with phase transition, in particular vapor-state properties. Both Eulerian and Lagrangian computational techniques can be utilized to simulate these plate impact experiments; whereas, standard Lagrangian techniques are impossible to use in simulating multi-dimensional debris cloud formation. The Eulerian wavecode CSQIII 1,40], run in a one-dimensional mode, gave the same kind of correlation with experiment that CHART-D did in Fig. 14, providing enough zones in the region of impact and void are used (on the order of 1000 zones). There are some specific differences between the Lagrangian and Eulerian calculations, but these can be traced to differences in advecting material, numerical mass diffusion, and finite mass effects in the leading Lagrangian zones of the debris. Now consider Fig. 15, where the data for a Pb experiment  are compared with both a CHART-D calculation (500 zones in the sample) and a CSQIII calculation (1000 zones in the void/impact region). Both of these calculations are numerically converged. The impact velocity was 6.12 km/s in this particular experiment. The estimate of the mass fraction of vapor produced by this impact is roughly 33 %, so it is expected this experiment will produce data similar to that of the Cd experiments. However, note the remarkable difference between the data in Fig. 15 and those in Fig. 14. Also, note the excellent overall agreement between the Lagrangian and Eulerian calculations, admittedly at the expense of many zones, and the strong disagreement between the calculations and the experiment. The EOS used in these calculations is as accurate as the best EOS used in the Cd simulations. The experiment is also repeatable, so the data must be viewed as accurate. We are then left with the problem of understanding why calculations that, a priori, should be as accurate as in the case of the Cd experiments are instead in disagreement with the Pb experiments. Various possibilities for this discrepancy, all conjecture at this point, are: (1) the Pb experiments are extraordinarily sensitive to the location of the critical point in the EOS (while there is overall good agreement between the Pb EOS used in these calculations and the Pb EOS developed by Holian 1-25], the ANEOS critical temperature is considerably higher than that obtained by Holian 1,41]); (2) non-equilibrium EOS effects are important to the Pb experiments; or (3) mixed-phase flow effects are dominant in the Pb experiments. Continued experimentation in the future will shed additional light on this. But, in view of this discussion, it is certainly understandable that there could be considerable difficulty in achieving agreement between coarsely zoned Eulerian calculations (because of hardware
CHARLES E. ANDERSON, JR. el al.
limitations) and the complex multi-dimensional Pb -o Pb impact experiments discussed in the previous section. Discussion
What conclusions can be drawn from the above work? Bjork has argued that an error exists in the EOS description of Pb 1-30]. This remark is also supported by the above 1-D vaporization work on Pb, even though recent equations of state for Pb, including the new tabular EOS developed by Holian and the ANEOS EOS used in the above simulations, agree well with a full range of previously acquired thermophysical data. We emphasize that the sensitivity of the comparisons of calculations and experiments for the 1-D experiments to sample EOS is considerably greater than in the case of the comparison of 2-D axisymmetric Eulerian codes with the available X-ray imaging data. Holian and Burkett  made the same point when noting the relative insensitivity of their 2-D calculations to rather large EOS variations. We are led to consider additional aspects of the 2-D calculations that can contribute to the disagreement with experiment. Bjork, and Holian and Burkett, used contemporary Eulerian computer codes for their numerical simulations. Bjork used the Eulerian code called PEACH ; Holian and Burkett used the Eulerian code LASOIL . Holian and Burkett noted the inability of LASOIL (and all current versions of other Eulerian hydrocodes) to calculate the separate transport of liquid and vapor (two-phase flow). This could be approximately treated by creating a new material (the vapor) within a computational cell by using the EOS to determine the fractions of gas and liquid in the cell, similar to the way void is created in CSQIII. The vapor would then be permitted to advect independent of the liquid, and expand faster and in a more spherical manner than the liquid. While such flow mechanisms might be necessary to achieve the best agreement with experiment (especially with the data of the 1-D vaporization experiments), we note that the velocity of expansion and the location of the 2-D debris cloud computed by LASOIL were in good agreement with experiment (although the cloud front velocity is largely determined by the impact velocity, large differences are seen in the expansion of the debris clouds of Figs 10 and 11, where the initial impact velocities are approximately the same).* Another issue concerns the nature of the advection schemes used by the Eulerian codes that simulate multi-dimensional hypervelocity impacts. Referring to Fig. 13, we note that CSQIII predicts a somewhat more accurate areal density than LASOIL, although it is still not excellent. In particular, CSQ predicts that the mass is more concentrated at the front of the debris cloud, though the data do indicate more mass than is being predicted. The CSQIII calculation was performed with the same grid as the LASOIL calculation mentioned above, and we have already argued that EOS variations will not make a significant difference here. CSQ, unlike LASOIL, has a second-order accurate advection transport algorithm, in which quantities such as internal energy, momentum, and kinetic energy have a linear distribution across a cell during Eulerian remaps, rather than the constant distributions assumed by first-order advection schemes. Higher order advection schemes are designed to move material through the grid with significantly decreased mass diffusion, especially near large gradients. These are the conditions prevalent in the expanding front of the debris cloud, and we argue that it is mainly this effect that accounts for the differences we see in the computations of Fig. 13. We note that in the comparisons by Swift e t al. 1-33] a code with a first-order advection scheme also was used. They reported that the computed debris cloud was too thick, once again indicating some difficulty with mass diffusion during the advection of material through the grid. A very nice study by Holian and Burkett 1-42] quantifies the effects on the solution for the Pb ~ Pb impact using different advection schemes, cell size, and using a sophisticated versus a simple EOS. The density contours within the debris cloud were compared at 10 ~ts. CAVEAT , a new ALE code developed at Los Alamos, was used for the numerical study. The baseline case used cells 0.5 mm on a side, and the tabular Pb EOS from Reference *This would argue that the EOS for cadmium used in the comparisonsof Swift et
1-33]was in error.
. Similar to Fig. 13, a second-order advection scheme produced a much more hollow debris cloud than a first-order advection scheme. The second study increased the zone size to 0.15 cm on a side, and used the second-order accurate advection scheme. With the large cell size, the results appeared somewhat similar to the fine grid first-order scheme, with the densest part of the cloud in the center. This again demonstrates the importance of numerical convergence emphasized previously. Finally, Holian and Burkett compared the results of the tabular EOS against an ideal gas EOS, using the second-order advection scheme with the fine grid. The overly simplistic ideal gas EOS resulted in a very diffuse debris cloud expanding much too quickly. Thus, we can draw several conclusions. Something as simple as the grid resolution can create major difficulties, especially since current two-dimensional computations may be almost one order of magnitude too large in resolution to represent numerically converged calculations. Second, the accuracy of the numerical results is sensitive to the advection scheme used in Eulerian codes. The first-order advection scheme gives an answer that is not even qualitatively correct in the mass distribution of the vapor debris cloud; whereas, a second-order scheme does predict that the densest part of the expanding debris cloud is at the leading edge of the expansion. And finally, a major question to resolve is whether or not non-equilibrium effects are present in the Pb experiments. However, one major numerical question still needs to be addressed--two-phase flow. In the present calculations, a particle velocity representing some sort of integrated average of the condensed mattervapor state is computed. It can be argued that some of the discrepancies between computations and experiment could be rectified by incorporating a two-phase flow algorithm since the vapor, with less inertia, would tend to expand more rapidly than the condensed matter. Thus, additional mass flux towards the front of the debris cloud would occur. Modeling of these effects requires relative velocity between the vapor and liquid phases. We note that Eulerian codes, as opposed to traditional Lagrangian, would be required for two-phase flow calculations of this type.
SUMMARY We have discussed some aspects of the hypervelocity impact of a projectile against a thin target. We have chosen to focus on the particular problem of simulating such impacts with hydrocodes. We have noted that over the full range of phenomenology associated with these impacts, from structural response, to fragmentation under shock loading, and ultimately to full vaporization and perhaps ionization, there can be rather severe difficulties in extracting good agreement between computation and experiment. The problem is not simply one of comparing experimental data with computer calculations, and deducing obvious differences that can be attributed to specific problems with the computations. Rather, workers in this area must deal with experimental data that may be insufficiently quantitative to allow the unfolding of the discrepancies into distinct areas such as: (1) unsuitable EOS; (2) insufficient accuracy in the numerical grid and hydrodynamics algorithms; or (3) need for new phenomena, such as non-equilibrium EOS formulations or multi-phase flow. Our opinion is that contemporary experiments, with modern optical and time-resolved diagnostics, can provide data of sufficiently quantitative accuracy that more stringent and precise comparisons of computer simulations and experiments will result. The 1-D shock vaporization experiments described above provide one example of how fruitful accurate, time-resolved data can be when performing computational benchmarking for hypervelocity impact phenomenology. More accurate data in multi-dimensional experiments are also becoming available. It is with this type of information that we may be able to resolve the question of what are the major causes of differences between calculations and experiments. Acknowledgements--The authors would like to express their appreciation to the management at Southwest ResearchInstituteand SandiaNational Laboratoriesfor support and encouragementduringthe progress of this work. We are especiallygrateful to Dr KathleenS. Holian for many helpfulconversations,and for providing
CHARLES E. ANDERSON, JR. eta[,
details of her work. We would also like to thank Drs Marlin E. Kipp and Stephen L. Passman for their patient review of the manuscript. The work performed at Sandia National Laboratories was supported by the U.S. Department of Energy under contract DE-ACO4-76-DP00789.
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