The fragmentation behavior of a metal powder compact which is ductile in compression but brittle in tension is studied via impact experiments and analytical models. Consolidated metal compacts were prepared via cold-isostatic pressing of <10 *μ*m zinc powder at 380 MPa followed by moderate annealing at 365 °C. The resulting zinc material is ductile and strain-hardening in high-rate uniaxial compression like a traditional metal, but is elastic-brittle in tension with a fracture toughness comparable to a ceramic. Cylindrical samples were launched up to 800 m/s in a gas gun into thin aluminum perforation targets, subjecting the projectile to a complex multiaxial and time-dependent stress state that leads to catastrophic fracture. A soft-catch mechanism using low-density artificial snow was developed to recover the impact debris, and collected fragments were analyzed to determine their size distribution down to 30 *μ*m. Though brittle fracture occurs along original particle boundaries, no power-law fragmentation behavior was observed as is seen in other low-toughness materials. An analytical theory is developed to predict the characteristic fragment size accounting for both the sharp onset of fragmentation and the effect of increasing impact velocity.

## I. INTRODUCTION

Fragmentation remains one of the most challenging material behaviors to predict, particularly for the type of complex, time-dependent stress loadings that occur during ballistic impact or other high-rate loading. For many materials, impact results in a fragment distribution with a characteristic length scale. This length scale, often associated with an average fragment size, is commonly estimated using energy-balance methods introduced by Grady,^{1–3} statistical fracture models developed by Mott,^{1,4} the cohesive-zone model proposed by Drugan,^{5} or other methods.^{6–9} For brittle materials, additional complications arise. Under dynamic loading, frequently these materials fragment according to a power-law distribution that has no characteristic length scale.^{10–12} Several authors have studied this power-law behavior^{13–24} and its potential relationship with a separate fragmentation mechanism that arises from microbranching of high-velocity cracks.^{25–27} In our previous study, we noted a transition to power-law type fragmentation with increasing impact velocity for very brittle cold-isostatically pressed aluminum compacts.^{28} Ramesh and coworkers provide a recent review of work on dynamic fragmentation mechanisms in a range of materials.^{29} The general lack of experimental fragment distributions on brittle materials at high rates of loading has made development of analytical and computational models challenging.

In this work, we fabricate a zinc material that is ductile and robust in uniaxial compression, but elastic-brittle in tension. Our goal is to study this material under ballistic impact with thin plates and develop theories that describe its fragmentation. Zinc compacts were produced in-house via cold-isostatic pressing (CIP) of gas-atomized zinc powder followed by an air-free annealing. A combination of quasistatic, dynamic, and gas gun testing was used to characterize the dynamic mechanical response and fragmentation. The zinc is highly ductile in uniaxial dynamic compression in a Hopkinson bar. In uniaxial tension, however, it is elastic-brittle and fails at a stress approximately three times lower than the compressive strength. Cylinders of the zinc material were fired at normal incidence into 1.62 mm Al 7075 plates to study the impact fragmentation at normal obliquity. The debris from this impact is very fragile, and to successfully recover the fragments we describe a new soft-catch procedure using artificial snow. The snow is evaporated away post-shot, leaving the fine debris intact and with no surface residue. Typically greater than 98% of the debris, much of which is micron-scale, can be recovered with this technique.

The debris is then analyzed with a video camsizer that images all particles down to 30 *μ*m. We introduce a single-parameter fragment distribution form, originally suggested by Mott, that describes the recovered debris. Though a separate mode of fine fragments does appear at the highest impact velocities, no power-law behavior is observed in any recovered fragments and there is always an identifiable characteristic fragment size. Scanning electron microscopy (SEM) analysis of debris shows that fracture occurs along original particle boundaries; this is expected, as the annealing treatment improves ductility within the zinc particles but does not result in significant sintering at their interface. An analysis of the largest recovered fragment shows a sharp transition centered at 517 m/s, corresponding to the point at which the mass of the largest recovered fragment is half the original projectile mass. We define this as the shattering velocity, and work to develop an analytical model for the characteristic fragment size that incorporates behavior both below and above this shattering point. A residual velocity model for the projectile perforation is first introduced and is then used to estimate a global strain-rate that can be incorporated into a fragmentation model. The final theoretical form provides a special treatment for the region around the shattering point, smoothly transitioning into an energy-balance fragmentation model at higher impact velocities. This is shown to be in good agreement with experimental results for the zinc compact. This theory requires only standard material parameters as well as a shattering velocity and fracture toughness to describe the characteristic fragment size. A simplified form of the model that applies at typical ballistic conditions is developed and used to show that the characteristic size of the fragments produced by this manner of impact should scale as the projectile size to the 4/3 power and with target thickness to the -2/3 power.

## II. EXPERIMENTAL METHODOLOGY

Sigma Aldrich zinc dust (CAS number 7440‐66-6), of which ≥98% is composed of spherical zinc particles nominally <10 *μ*m in diameter, was cold-isostatically pressed at 380 MPa into a monolith using a wet-bag technique with a polyurethane mold. The pressed zinc rod was then heated at 336 °C in an argon-filled tube furnace for four hours to improve ductility. As shown later via electron microscopy of recovered fragments, this annealing treatment does not result in significant sintering or mass diffusion between the particles. The resulting density of the compacts was 6.80 g/cm^{3}, about 4.75% porosity compared to that of pure bulk zinc.

Zinc samples were tested in a split-pressure Hopkinson bar, composed of C350 maraging steel 0.75 in. in diameter, to measure properties of the material when subject to dynamic loading conditions. Cylindrical samples 10 mm in length and 10 mm in diameter were tested in compression at three different strain-rates. A momentum trap system and a copper pulse shaper of 0.53 mm thickness were used in each shot. A small amount of MoS_{2} grease was used to minimize friction of the expanding sample against the bars. A high-speed video was used to analyze the Hopkinson bar compression tests, though no failure was observed either visually or from a drop in stress in the measured waveforms.

Due to the brittleness of the zinc compacts, manufacturing coherent samples for a traditional tensile test was challenging. Instead, a 10 mm diameter by 5 mm thick sample was used in a Brazilian configuration in the Hopkinson bars to obtain a rough tensile strength. This methodology is outlined by Xia and coworkers in Ref. 31, and has been used to establish dynamic tensile failure criteria for several rocks and geomaterials.^{32–34} When the Brazilian disc is in force equilibrium between the incident and transmission Hopkinson bar, the tensile strength can be estimated from the standard Brazilian equation as

where $Fmax$ is the maximum force prior to failure, *r _{p}* is the radius of the disc, and

*h*is its thickness.

_{p}The quasistatic fracture toughness of the pressed zinc was measured through a three-point bend test using a rectangular zinc sample. The samples, measuring 74 mm × 16.8 mm× 8.3 mm with an average density of 6.83 g/cm^{3}, followed ASTM E399 specifications^{35} with the exception of the notch. Rather than an angled notch of 30°, the notch was 1 mm wide with a flat end made by a single cut from a saw blade.

A zinc disc for sound speed measurements was also isostatically pressed from starting powders to determine elastic moduli. The disc measured 47 mm in diameter and 7.92 mm in height with a density of 6.84 g/cm^{3}, and sound speed was measured using a pulse/echo technique with appropriate ultrasonic couplant.

Impact fragmentation testing was conducted with 10 mm diameter, 10 mm long cylinders. The projectiles were launched in a 4-piece sabot at a thin aluminum plate in the 0.5 in. gas gun at the Naval Postgraduate School. A schematic of the impact is given in Fig. 1. A high speed video on a Phantom v711 was used to ensure that the projectiles impacted at normal incidence, and any shots with detectable tilt were discarded. Impact velocities of the projectiles were 763 m/s, 650 m/s, 610 m/s, 509 m/s, 470 m/s, and 323 m/s. These velocities were measured using Oehler Model 57 infrared light screens between the sabot stripper and the catch chamber. High-speed video analysis was used to confirm the impact velocity from the light screens and measure the residual projectile velocity following perforation. Projectiles impacted 7075 aluminum plates of 240 mm width, 184 mm height, and 1.62 mm thickness bolted to a supporting steel frame. Relevant material properties of the porous zinc material and impact plate are listed in Tables I and II. For all shot velocities, the projectiles fully perforated the aluminum plate and fragmented into a debris cloud inside the catch chamber.

Material property . | Value . |
---|---|

Mean zinc projectile density | 6.80 ± 0.09 g/cm^{3} |

Mean zinc projectile porosity | 4.75 ± 0.1% |

Zinc compacted density | 7.139 g/cm^{3} |

Projectile compressive strength σ _{c} | 275 MPa |

Projectile tensile strength σ _{t} | 97.4 MPa |

Quasistatic fracture toughness K _{c} | 5.58 ± 0.3 MPa $m$ |

c_{l} (longitudinal sound speed) | 4.42 km/s |

c_{s} (shear sound speed) | 2.26 km/s |

c_{b} (bulk sound speed) | 3.57 km/s |

Young's modulus | 93.8 GPa |

Poisson's ratio | 0.323 |

Hugoniot (bulk zinc, Ref. 30) | Value |

A | 3030 m/s |

b | 1.55 |

γ _{o} | 2.01 |

Material property . | Value . |
---|---|

Mean zinc projectile density | 6.80 ± 0.09 g/cm^{3} |

Mean zinc projectile porosity | 4.75 ± 0.1% |

Zinc compacted density | 7.139 g/cm^{3} |

Projectile compressive strength σ _{c} | 275 MPa |

Projectile tensile strength σ _{t} | 97.4 MPa |

Quasistatic fracture toughness K _{c} | 5.58 ± 0.3 MPa $m$ |

c_{l} (longitudinal sound speed) | 4.42 km/s |

c_{s} (shear sound speed) | 2.26 km/s |

c_{b} (bulk sound speed) | 3.57 km/s |

Young's modulus | 93.8 GPa |

Poisson's ratio | 0.323 |

Hugoniot (bulk zinc, Ref. 30) | Value |

A | 3030 m/s |

b | 1.55 |

γ _{o} | 2.01 |

Property . | Value . |
---|---|

Density ρ _{t} | 2.81 g/cm^{3} |

Bulk sound speed c_{b} | 5.24 km/s |

Young's modulus E | 71.7 GPa |

Poisson's ratio ν | 0.33 |

Yield strength Y | 503 MPa |

Width | 240 mm |

Height | 184 mm |

Thickness h _{o} | 1.62 mm |

Hugoniot (Ref. 30) | Value |

A | 5200 m/s |

b | 1.36 |

Property . | Value . |
---|---|

Density ρ _{t} | 2.81 g/cm^{3} |

Bulk sound speed c_{b} | 5.24 km/s |

Young's modulus E | 71.7 GPa |

Poisson's ratio ν | 0.33 |

Yield strength Y | 503 MPa |

Width | 240 mm |

Height | 184 mm |

Thickness h _{o} | 1.62 mm |

Hugoniot (Ref. 30) | Value |

A | 5200 m/s |

b | 1.36 |

The debris produced during these shots is quite brittle, and considerable care must be taken to gently decelerate the fragments without additional damage. For this, we developed a simple catch medium of artificial snow that allowed for gradual deceleration and left no organic residue on any fracture surface. Large blocks of distilled water were frozen and shaved into a fine powder using a Hatsuyuki HF-500E ice shaver. The resulting artificial snow had an average density of 0.23 g/cm^{3}, and was loaded into a catch chamber behind the impact plate (Fig. 2). The projectile is fired through the target and into the snow medium within two minutes of shaving to prevent any sintering or melting of the ice. Wilson, Zhang, and Kim also report the use of an artificial snow medium for recovering warhead fragments from explosive shots.^{36}

Following each shot, the artificial snow with entrained fragments was carefully scooped out and placed into a Büchner funnel with 20 *μ*m filter paper. The snow was melted away with a mild heat lamp, and the remaining fragments were heated briefly at 60 °C in a muffle furnace to remove any residual moisture. Fragments from the ejected plug of Al 7075 from the target plate were large and distinct and easily separated manually at this stage. With this technique, we are typically able to collect 98% or better of the original projectile mass, and the snow does not add any unwanted residue that can skew the total mass, as can occur when using ballistic gelatin or polyethylene glycol.

To determine fragment distributions, the debris was analyzed with a Haver Tyler computerized particle analyzer (Model 2‐1) to measure the mass distribution over a linear particle size. Fragments are fed into a vibrating pan, which separates particles before dropping them in front of a high-resolution CCD camera running at 20 kHz. Fragments are individually imaged and measured and can then be binned into a distribution similar to traditional sieve data. Manual analysis of the fragments is not possible, as even the lower velocity shots presented here result in thousands of fragments in the 30–300 *μ*m range. We have found that fragment statistics below 1 mm are highly reproducible across several analysis runs with this technique, but large asymmetric fragments show higher variance in the 2D presented area as it falls in front of the camera. Several analysis runs are thus combined to account for this variation.

## III. RESULTS AND DISCUSSION

### A. Mechanical properties

The basic mechanical properties of the zinc materials derived from sound speed and quasistatic measurements are shown in Table I. The value of Young's modulus calculated from longitudinal and shear wave measurements is 93.8 GPa and the Poisson's ratio was 0.323, comparable to that of bulk, non-porous zinc.^{37} The fracture toughness was determined to be 5.58 ± 0.3 MPa $m$, closer to that of ceramics such as Al_{2}O_{3} or SiC than a typical metal.^{38}

There is a strong asymmetry between the mechanical response to uniaxial compression and uniaxial tension. Dynamic compression data from our split Hopkinson pressure bar measurements are shown in Fig. 3. Compression measurements are taken at three different strain rates, and all show highly ductile, work-hardening behavior with a yield strength around 275 MPa. Little discernible strain-rate hardening occurs over this range of rates. In all cases, there is no observable mechanical failure in the sample nor drop in stress that would indicate material fracture. In contrast, zinc samples shot in a dynamic Brazilian configuration fracture rapidly in tension. The tensile strength extracted from this test using Eq. (1) is shown in the horizontal line in Fig. 3, and is approximately one-third the value of the compressive strength (97.4 MPa). Minimal ductility is observed in these Brazilian-style tensile tests.

A simple linear fit is given in the plastic regime, to roughly estimate a constant plastic wave speed in a state of uniaxial stress, as would be relevant in a Taylor-test impact analysis. This wave speed *c _{pl}* estimated from the fit shown in Fig. 3 is 544 m/s.

### B. Residual velocity

We next wish to consider how these zinc materials behave under ballistic impact. Though the material fragments heavily at low velocities compared to a traditional metal projectile, we expect that ballistic properties for thin-plate perforation will not be significantly altered. We begin with the most straightforward: the residual velocity of the projectile following perforation of the Al 7075 plates. A snapshot of a typical high-speed video of the impact is shown in Fig. 4. The zinc cylinders strike at normal incidence, producing a small impact flash. They then plug the aluminum plate cleanly, potentially resulting in fracture of the plate material as well as the zinc. The final frame in Fig. 4 shows the radial expansion of the zinc cylinder and a crack oriented along the cylinder axis.

The residual velocity *V _{r}* of the projectile's center of mass following plate perforation was measured from high-speed imaging and is displayed versus impact velocity in Fig. 5. Errors in measured

*V*are comparable to the size of the data points. We next develop a basic analytical theory for the impact and show that it is consistent with the measured data.

_{r}We assume that two energy loss mechanisms are prominent when the zinc cylinders impact and perforate the thin plates. The first is a plastic work term *W _{p}* associated with driving a cylindrical plug from the target plate. The second is a dynamic work term

*W*required to accelerate the plug. We then form a simple energy balance in the spirit of classic work by Thomson

_{d}^{39}

where *V _{o}* is the impact velocity,

*M*is the initial projectile mass, and

_{o}*m*is the plug mass. This approach neglects the small amount of zinc lost to produce the impact flash seen in Fig. 4. For the dynamic work term

*W*, we assume that the plug is accelerated to the initial velocity. This is a suitable assumption for the thin plates considered here.

_{d}*W*can then be approximated as

_{d}where *r _{p}* is the radius of the projectile,

*h*is the target plate thickness, and

_{t}*ρ*is the target plate density. The resisting force from the plate as a plug is being driven forward by the projectile is assumed to arise from resistance to shearing and can be written as

_{t}where 0 ≤ *x *≤* h _{t}* is the distance into the target plate and

*Y*is the yield strength of the plate in uniaxial tension. The total work required to displace the plug against the resisting force is then

Substituting these results for *W _{p}* and

*W*yields the final residual velocity

_{d}This form contains only basic material properties of the plate and projectile, and is in good agreement with the experimental residual velocities shown in Fig. 5. Equation (6) predicts a ballistic limit velocity *V*_{50} (the velocity at which the cylinder begins perforating the plate) of 69 m/s, though further shots would be needed to properly verify this value. The limit velocity of the plate is not our primary concern in this work, and all shots performed here are well above the expected *V*_{50}.

### C. Shattering velocity

The simple residual velocity model given above is very similar to what one might use for a traditional, ductile metal projectile. However, as it passes through the plate the low tensile strength and fracture toughness of the zinc compact lead to considerable fragmentation into a fine particulate. Figure 6 shows all fragments recovered from the artificial snow soft-catch medium, with a clear trend towards increased fine fragments below 300 *μ*m at the highest velocities. Quantifying this fragmentation is one of our primary interests in this work. The character of the crack surfaces is shown in Fig. 7. The starting powder is shown in frame (a), and is consistent with spherical gas-atomized feedstock. Frames (b) and (c) denote typical crack surfaces seen throughout the recovered debris. In nearly all cases, the crack moves through the original boundaries between particles, as would be expected. Occasionally, a larger zinc particle (typically >20 *μ*m in size) will show transgranular cracking through the particle, but this is not the norm.

We begin by examining only the largest fragment mass *M _{c}*. This manner of analysis is common in examining behind-target debris,

^{40}especially for hypervelocity impact.

^{41,42}The normalized value $Mc/Mo$ is plotted in Fig. 8. The solid curve is a fit using the sigmoidal form

where *α* is a parameter controlling the width of the shattering transition, and has the value *α* = 50 m/s for this material. We define the shattering velocity *V _{s}* as the point at which $Mc/Mo=0.5$, consistent with previous treatments of thin-plate ballistics.

^{43}This is also similar to the transition point between regimes of damage and fragmentation introduced by Kun and Herrmann.

^{21}These authors develop a numerical model of impacting brittle spheres, and use the mass of the largest fragment divided by the total mass as the order parameter describing the damage to fragmentation transition. For our zinc cylinders, the largest residual mass shows a rapid drop above 400 m/s, giving an estimated shattering velocity of 517 m/s for this size projectile and particular target plate. This rapid drop near the shattering velocity is similar to what is seen with blunt impact of standard metals, albeit at a much lower velocity in our case.

^{40}

### D. Impact pressure

The induced shock from a blunt impact is often correlated with the onset of shattering during thin-plate perforation.^{40} Here, we write a basic equation of state for the porous zinc and use it to estimate the maximum shock pressure seen during these impact events. The Hugoniot is assumed to have a linear relationship $Us=A+bup$ where *U _{s}* is the shock velocity and

*u*is the particle velocity. Values from the Los Alamos shock Hugoniot handbook (Ref. 30) are used for fully dense zinc and for the aluminum plate, and are given in Tables I and II. The porosity in our zinc samples is accounted for following the procedure given by McQueen and coworkers.

_{p}^{44,45}The non-porous Hugoniot can be written as

where *v* is the specific volume following shock loading and *v _{o}* is the pre-shock specific volume of bulk zinc. The porous Hugoniot, referenced to

*P*using the Mie-Grüneisen approach, is

_{H}where *γ _{o}* is the Grüneisen parameter under ambient conditions and

*v*is the specific volume of the porous zinc. Implicit in this equation is that the ratio

_{oo}*γ*∕

*v*is a constant and thus, $\gamma /v=\gamma o/vo$.

^{44}The porous Hugoniot

*P*can be recast into pressure/particle-velocity space using the relation

_{oo}Figure 9(a) shows an impedance match for the porous zinc projectile striking the Al 7075 plate at the experimental shattering velocity of 517 m/s, corresponding to $Mc/Mo=0.5$. The peak impact shock pressure in the projectile at the shattering velocity is 4.2 GPa. Figure 9(b) shows the peak shock pressure as a function of impact velocity of the porous zinc cylinder.

We expect that the initial shock may generate both rapid radial expansion leading to tensile failure (consistent with the radial expansion seen in Fig. 4) and traditional spall from rarefactions at the edges. The fracture toughness can be used to make an estimate of the spall strength of the material *σ _{s}*, using the relationship suggested by Grady

^{46}

where *c _{b}* is the bulk sound speed,

*K*is the dynamic fracture toughness, and $\u03f5\u0307$ is the strain-rate in the spall plane. Taking a sensible value of $\u03f5\u0307$ =10

_{f}^{5}s

^{−1}and assuming that the quasi-static fracture toughness can be used, the spall strength is approximately 0.6 GPa.

### E. Fragmentation theory

Our next task is to quantify the full spread of fragments produced by the impact. All experimental data shown in this section were taken from the computerized particle size analysis discussed above. We first consider appropriate theoretical forms for the recovered fragment distribution. This includes two main parts; the first is a fragment distribution form, and the second is a characteristic fragment size. The fragment distribution is usually assumed or taken from general physical arguments, and the characteristic fragment size is often estimated using methods such as Grady's energy-balance model. Grady^{1} and Elek and Jaramaz^{47} review many of the common distributions used to treat fragmentation during high strain-rate events.

We have found that a particular fragment distribution form matches the cold-pressed zinc compacts here as well as other similar materials.^{28,48} Unlike thin cases around an expanding cylinder for which the fragmentation is effectively one- or two-dimensional, our fragments are generated by impact and have a definite three-dimensional character to them. We begin with the general form

where *F*(*m*) represents a cumulative distribution function (CDF), normalized to the total number of fragments *N _{o}*. This form describes the percentage of fragments distributed over their mass

*m*. Here,

*μ*is a characteristic mass that describes the distribution, and

_{m}*β*is an arbitrary parameter. Mott argued for a value $\beta =12$ to describe artillery shell fragments,

^{4}and this choice leads to what is often referred to as simply the Mott distribution. A plot of ln $(1\u2212F)$ versus

*m*in this case is often referred to as a “Mott plot” and has been widely used in historical fragmentation data.

^{13}Setting

*β*= 1 corresponds to the Grady-Kipp distribution,

^{2}also widely used for general fragmentation. Brown and Wohletz discuss possible physical interpretations of the

*β*parameter, though frequently it is treated as an empirical choice.

^{49}For artillery shells, Mott and Linfoot argued that if the fragments were smaller than the case thickness, a value $\beta =13$ is warranted, reflecting the three-dimensional nature of the fracture. Here, we will refer to this as the three-dimensional Mott distribution, and we will use this basic form to describe all experimental results.

The probability density function (PDF) of the three-dimensional Mott distribution is

and the distribution has an expected value of 6*μ _{m}*. It is common in fragmentation literature to express distributions in terms as a function of a linear size

*s*rather than mass. This is also a more natural way to present the results from a traditional or computerized sieving process. Additionally, while our analysis does produce a fragment count, presenting the distribution in terms of mass is more natural since an enormous number of <300

*μ*m particles are produced. Thus, we also convert these expressions into a mass-based CDF and PDF rather than a fragment count in the following way. Suppose an impact event generates a collection of

*N*fragments with a total mass

_{o}*M*. We can relate normalized number (

_{o}*N*) and mass (

*M*) distributions via the expression

where the average fragment mass $Mo/No$ is 6*μ _{m}* as discussed above. We then perform a change of variables to express the PDFs as a function of linear size

*s*using the relationships $m=\rho s3,\u2009dm=3\rho s2ds$ where

*ρ*is the density of the material. The mass PDF expressed in this way is

where we have also converted the characteristic mass *μ _{m}* to a characteristic linear size

*μ*. The corresponding CDF is

_{s}The expected value for this distribution is 4*μ _{s}*.

*M*(

*s*) can also be expressed in terms of special functions as

where *γ*(*a*,*x*) and Γ(*a*) are the lower incomplete gamma function and the complete gamma function, respectively. Equation (15) for the PDF and Eq. (17) for the CDF will be used in the rest of this work to directly analyze recovered fragments.

Experimentally, the fragment distributions for the zinc compacts are fit more accurately with bimodal fragment distributions. We have seen similar behavior in related materials such as pressed aluminum compacts.^{28,48} The bimodal 3-D Mott PDF takes the form

Here, *s* denotes the linear size scale of the fragments, *m*(*s*) is the mass PDF, and *μ*_{1} and *μ*_{2} are size scales for the two 3-D Mott distributions. As noted above, the peaks in the fragment distribution correspond to values of 4*μ*_{1} and 4*μ*_{2}. *χ* is a weighting factor for the two distributions, and is expected to be velocity/strain-rate dependent. One possible explanation for the bimodal fragment distribution is the following. During impact, we hypothesize that two major fragmentation mechanisms are at play. The first is an uncorrelated cracking of the material that leads to relatively large fragments (the larger of the two distributions). These fragments are assumed to form due to an energy-balance type criterion proposed by Grady^{1} and Glenn and Chudnovsky.^{3} These authors treat the primary cracking as a completely uncorrelated, Poisson process with a critical energy criterion governed by the dynamic fracture toughness of the material. The smaller distribution is hypothesized to arise from an alternate mechanism that occurs primarily in zones around the main cracks. A combination of microbranching, friction between primary fragments, or other mechanisms gives rise to fine debris that peaks around 650 *μ*m, which we note is still quite large compared to the starting zinc powder size. With increasing impact velocity, we would expect the characteristic size of the large distribution to decrease and the quantity of the material affected by the second mechanism to increase. Ultimately, for catastrophic fragmentation, the entire fragment pattern would be expected to be unimodal and comprised entirely of fine fragments.

### F. Fragmentation results

Experimental data for the mass PDF and CDF distributed over a linear size *s* are shown in Figs. 10 and 11. Data are plotted on a logarithmic scale in linear size to more clearly see the features arising from the fine fragments. Solid lines in these figures denote fits to the data using the theoretical forms given in Eq. (18) and with the corresponding bimodal CDF form. The theoretical curve was fit to the experimental PDF and the same parameters were used to plot the CDF lines in Fig. 11. The characteristic fragment sizes *μ*_{1} and *μ*_{2} as well as the weighting factor *χ* are treated as fit parameters, and will ultimately be compared to a theoretical model for the fragment size. Values for these are listed in Table III. The fitting used a Levenberg-Marquardt method and the parameter 4*μ*_{1} describing the fine fragments was constrained to be less than or equal to 2 mm. In presenting the experimental fragmentation data, we have done this in a way that it can be paired with the largest residual mass data shown in Fig. 8. Inclusion of the largest fragment results in a distribution that is dominated by this one data point, particularly for low impact velocities. Thus, each fragment distribution is scaled such that the total mass percentage only goes up to the value $1\u2212Mc/Mo$. For example, for the shot at 459 m/s in Fig. 11, the CDF asymptotes at the value 0.22; here, 78% of the recovered mass was in a single large cylindrical fragment that can be seen in Fig. 6. A Feret diameter *L* of the largest fragment is also given in Table III.

Impact velocity . | χ
. | 4μ_{1} (mm)
. | 4μ_{2} (mm)
. | L (mm) . |
---|---|---|---|---|

763 m/s | 0.19 | 0.472 | 1.90 | 6.68 |

680 m/s | 0.19 | 0.613 | 3.31 | 8.81 |

600 m/s | 0.18 | 0.867 | 3.67 | 7.45 |

545 m/s | 0.17 | 0.585 | 2.67 | 9.71 |

459 m/s | 0.06 | 0.661 | 2.09 | 10.1 |

386 m/s | 0.00 | … | 1.90 | 10.4 |

Impact velocity . | χ
. | 4μ_{1} (mm)
. | 4μ_{2} (mm)
. | L (mm) . |
---|---|---|---|---|

763 m/s | 0.19 | 0.472 | 1.90 | 6.68 |

680 m/s | 0.19 | 0.613 | 3.31 | 8.81 |

600 m/s | 0.18 | 0.867 | 3.67 | 7.45 |

545 m/s | 0.17 | 0.585 | 2.67 | 9.71 |

459 m/s | 0.06 | 0.661 | 2.09 | 10.1 |

386 m/s | 0.00 | … | 1.90 | 10.4 |

We notice the following trends from the data fit. While the bimodal fit is superior to any unimodal distribution we considered, there are not always obvious distinct peaks in the mass PDF. While the 600 m/s and 680 m/s impact velocities show clear bimodal trends, other shots tend to have a broader distribution that could potentially be fit by other assumed forms. We have retained the simple bimodal 3-D Mott form, as we have found it also describes brittle aluminum compacts and requires a minimal number of physical parameters. Repeat shots at impact velocities above the shattering transition resulted in small changes to the largest fragments (affecting 4*μ*_{2}), but similar behavior for the fines.

The smaller mode becomes prominent only above the shattering velocity. The peak of this smaller mode 4*μ*_{1} is below 1 mm in all cases. While this mode may be related to microbranching or friction between larger primary fragments, we see no evidence of a power-law fragment distribution emerge as is often seen in other brittle materials. This is demonstrated by a Schuhmann plot in Fig. 12; we note that this plot also has the largest fragment removed. Schuhmann suggested a fragment distribution which is widely used for brittle materials of the form

where *n* is the Schuhmann index.^{12} As discussed by Turcotte^{50} and Grady,^{13} the quantity 3 − *n* is often associated with the fractal dimension of a power-law fragmentation. Solid lines denote fractal dimensions of 2 ($M(s)\u221ds$) and 1 ($M(s)\u221ds2$) in Fig. 12. Grady proposed that this behavior can arise when far more kinetic and strain energy is dissipated in the system than is required by the fracture energy. This non-equilibrium case is observed experimentally for several very brittle materials; for example, Hogan and coworkers^{51} observe a fractal dimension of approximately 2 in drop-hammer crush tests of granite, and Grady and Kipp^{52} observe a dimension of 1 in shock loading of fused silica. In the latter case, the power-law region spanned nearly three decades of particle size. Examining our fragment distributions, we see no clear evidence in this range of impact velocities for a power-law behavior in the Schuhmann plot. We note that Kun and coworkers have studied fragmentation of several geomaterials experimentally and with discrete element models, and shown evidence that power-law behavior similar to Eq. (19) can also occur for fragments that are self-affine rather than self-similar.^{22–24} Though it is challenging to examine the thousands of fine particles produced by our impacts and quantify their 3D geometry, we do not appear to see evidence for self-affine scaling of fragment shapes as observed in Refs. 23 and 24.

Additionally, though the starting powders are not sintered together by the heat treatment, we observe no evidence of impact resulting in a return to the original starting powder state. Thus, despite the brittle tensile response of the zinc compacts, we pursue a theory that assumes that the energy dissipated in the sample by impact is balanced by the creation of new fracture surfaces. Our primary goal is to describe the change in the characteristic fragment size with impact velocity, and account for the largest monolithic fragment of mass *M _{c}* which requires special treatment. The largest fragment size

*L*decreases with increasing

*V*, but the larger mode size 4

_{o}*μ*

_{2}in the fragment distribution has a more complex dependence. These sizes are shown graphically in Fig. 13 along with the analytical model for the characteristic fragment size discussed below.

### G. Characteristic fragment size model

We next develop a simple model to predict the characteristic or expected size fragment *λ*. This is the key parameter for predicting the fragmentation from blunt impact. For the bimodal fragmentation pattern we typically observe for these materials, this is assumed to describe the larger mode of size 4*μ*_{2}. Here, we begin with the traditional Grady criterion for equilibrium three-dimensional fragmentation^{1,13}

where *K _{f}* is a dynamic fracture toughness in Pa $\xb7\u2009m$,

*ρ*is the material density of the zinc projectile, and

_{p}*c*is its bulk sound speed. The average density of the zinc compacts

_{b}*ρ*is 6.80 g/cm

_{p}^{3}, and the bulk sound speed

*c*is 3570 m/s. Here, we approximate

_{b}*K*with our quasistatic fracture toughness $Kc$, measured via a three-point bend test to be 5.58 MPa $\xb7m$. In general, one might expect the dynamic fracture toughness to vary with the rate of loading.

_{f}^{1}We attempted one toughness measurement where a notched three-point bend sample was measured, but with the load frame driven at 12.7 cm/s rather than the normal speed of 0.017 cm/s used for the ASTM-compliant quasistatic test. The resulting fracture toughness was identical to the low-rate value. While not a true dynamic test, in what follows we assume that the constant quasistatic value

*K*is sufficient for the general theory developed.

_{c}Unlike cases such as an expanding cylinder or rod, the strain rate appearing in Eq. (20) during cylinder impact is very complex. Compressive and tensile stresses in the cylinder are highly time-dependent and vary considerably with the spatial location. In light of these complexities, we introduce a simple way to estimate an approximate strain-rate that describes the overall impact. We assume that the strain rate at impact is related to the relative interface velocity between the cylinder and the plate plug. Following impact at a velocity *V _{o}*, there will be a transient shock which will quickly evolve into elastic and plastic waves of uniaxial stress propagating down the cylinder. The elastic wave propagation results in a deceleration force that is proportional to the compressive yield strength of the projectile. Here for simplicity, we take the peak interface velocity and estimate the strain rate as

where *r _{p}* is the cylinder radius and

*V*is given in Eq. (6). The resulting strain-rate is shown in Fig. 5; at the shattering velocity of 517 m/s, Eq. (21) predicts an effective strain rate of 7580 s

_{r}^{−1}. Note that below the ballistic limit the projectile will impact the plate but not pass through, resulting in a higher strain-rate on the projectile as would be expected.

The parameter *χ* is related to the mass fraction present in the smaller fragment mode, which is assumed to be independent of velocity. If this mode is related to microcracking, we would expect that it would grow with impact velocity as cracks are driven at higher velocities and the microbranching zone around a primary crack increases in size.^{28} In this range of impact velocities however, this trend is not observed. As shown in Table III, *χ* is roughly zero below the shattering velocity (indicating no distinct fine fragment mode), and jumps to roughly 0.18 after shattering.

We collect the key fragmentation parameters in Fig. 13. Two values are shown for the characteristic size of the larger fragments. The first is the value 4*μ*_{2} with the largest fragment removed and corresponds to the fit values from the mass PDF shown in Fig. 10. The second metric, denoted by a solid red line, is a mass-fraction mixture of this fit value and the largest fragment size *L*

By 600 m/s, these two values begin to approach each other as the largest fragment is no longer a large, monolithic portion of the cylinder that accounts for much of the mass. The solid blue line shows the estimated characteristic size using a Grady-type energy balance model [Eq. (20)] with the global strain-rate estimate given in Eq. (21). This model clearly underestimates the size of the very largest fragment, but is a reasonable estimate for the other larger fragments. Well above the shattering velocity, the largest fragment will no longer be an outlier in terms of its mass, and thus, a traditional fragmentation size estimate such as Eq. (20) provides a reasonable estimate using the approximate global strain-rate given in Eq. (21).

To fully describe the fragmentation of this material following thin-plate impact, it seems necessary to consider the mass of the largest fragment independent of the other brittle debris produced. We thus introduce a simple theory which incorporates this concept and can account for the characteristic size of larger fragments both prior to and following the shattering transition. The sigmoidal form Eq. (7) is used to weigh the large monolithic fragment, whose size merges smoothly into the energy balance theory [Eq. (20)] above *V _{s}*. Combining all these forms yields the characteristic size

*λ*

where

A plot of Eq. (23) is shown in Fig. 13, and shows good agreement with the experimental data. If the fine fragment mode (with characteristic size 4*μ*_{1}) is ignored, the fragment spray consists of two components. The first is a monolithic large fragment with velocity-dependent size $2rp\kappa $. The second is a distribution of smaller debris characterized by unnormalized mass PDF *m _{c}* and CDF

*M*over linear size

_{c}where the terms have identical meanings to Eqs. (15) and (17). We note that this model is explicitly for the case of blunt impact onto a perforation target at velocities above the ballistic limit *V*_{50}. The non-trivial material parameters that go into this theory are the shattering velocity *V _{s}* and the fracture toughness

*K*. A model for variations in

_{f}*V*with impact obliquity, sample geometry, target thickness, target density, and other parameters is highly desirable but still lacking. Additionally, it is currently unknown whether the second mode of fine fragments of characteristic size 4

_{s}*μ*

_{1}would become more prominent at higher impact velocities and warrant inclusion in a model. In the range accessible on the current gas gun, it is roughly constant above the shattering velocity.

### H. Simplification at high velocities

At velocities well above the ballistic limit *V*_{50}, we can simplify the above expressions and note a number of useful trends. We assume for simplicity that the projectile is a cylinder of diameter 2*r _{p}* and length 2

*r*. The ballistic limit from Eq. (6) in this case is

_{p}For impact velocities well above this point, we assume that the residual velocity can be estimated from simple momentum conservation of the projectile and plug

For the common ballistics scenario of high velocities and small projectiles, the strain-rate in Eq. (21) can then be estimated using this residual velocity as

Combining this with Eq. (20) gives an estimate for the characteristic fragment size

Note that *c _{b}* here is still the bulk sound speed of the projectile and not the target. A plot of Eq. (29) is given in Fig. 13 for our experimental impact conditions, showing that meaningful deviations from the full theory in this case are only observed below the shattering velocity.

Equations (28) and (29) indicate that under normal ballistic conditions the characteristic strain rate for the event should vary with impact velocity as $\u03f5\u0307\u221dVo$ and with projectile size as $\u03f5\u0307\u221drp\u22122$. This suggests that well above the ballistic limit velocity, the characteristic size *λ* of the debris from a blunt impact that plugs a thin-plate target should scale with the projectile size as $\lambda \u221drp4/3$ and with plate thickness as $\lambda \u221dht\u22122/3$.

## IV. CONCLUSIONS

Zinc compacts made via cold isostatic pressing and moderate annealing were used to examine the impact fragmentation of a material that is ductile in compression but elastic-brittle in tension. The final zinc materials were very robust under dynamic uniaxial compression, surviving without fracture at strains up to 15% in split Hopkinson bar testing. The material is brittle in tension, with a tensile strength approximately a third that of the compressive strength, no tensile ductility, and a fracture toughness comparable to a ceramic (5.58 MPa $m$). Samples were sabot-launched in a gas gun into thin aluminum 7075 perforation targets, and the debris was recovered from a soft-catch medium composed of low-density artificial snow. Analysis of the fragment distribution shows that no power-law behavior is observed. Rather, all distributions can be described by a distribution proposed by Mott which contains a characteristic fragment size as the sole parameter. At higher velocities, a second fine-particle mode appears, which we hypothesize to come from crack microbranching or friction along primary cracks. An analytical theory is introduced to predict the characteristic fragment size across the shattering transition. At and below the shattering velocity, the fragment distribution is dominated by a single monolithic piece. Above *V _{s}*, the theory then merges into a traditional equilibrium energy-balance fragmentation model. The anisotropic zinc compacts are thus treated reasonably well by traditional fragmentation theories and by accounting for the reduced shattering velocity, lower fracture toughness, and additional production of fine fragments.

## ACKNOWLEDGMENTS

The authors acknowledge support from the Office of Naval Research Grant No. N0001417WX00882 (program managers Clifford Bedford, Chad Stoltz, and Matt Beyard). The authors would also like to thank Jacob Kline for assistance on gas gun experimentation.