Sufficient computational resolution for the basalt spheres is first determined by assessing at what resolutions total damage within the target begins to plateau. We ran seven different scans of resolution, increasing the particles per centimeter for each successive run by 30–100%. The results are presented in Figure 2. At low resolution, damage decreases with increasing resolution, until converging at resolutions above approximately 50 particles across the diameter of the target. This trend is consistent with previous work using Spheral, which demonstrated damage can be overestimated with insufficient resolution (Benz & Asphaug, 1995; Bruck Syal, Owen, et al., 2016; Bruck Syal, Rovny, et al., 2016). For this work, we subsequently adopt a resolution of 150 particles across the diameter of the basalt sphere, corresponding to a total number of ~1.75 million particles within the target volume, requiring 512 processors per simulation.

Different strain models in Spheral produce large differences in damage morphologies. Using the experimental data (a) to guide proper strain model selection in our code, we find that employing the Benz‐Asphaug strain model produces an undamaged inner core, illustrated in (b), where the cross section shows this preserved, undamaged “core” in blue (b.2). This morphology is not observed in the target when using the pseudo‐plastic strain model shown in (c.2). Snapshots taken at 200 μs, and undamaged material is illustrated in blue, and fully damaged material (no strength) is red.

As a part of this study, we compare two models of strain available in Spheral. A strain model is used by a brittle‐damage model to determine when the stress in a piece of material (represented by an ASPH particle) exceeds a local critical strength threshold and begins to accumulate damage. Essentially, the strain model bridges the strength and damage models within the SPH framework. The Benz‐Asphaug strain model (Benz & Asphaug,) is used in many SPH codes, and the tensor generalization of this algorithmis available in Spheral, whereis the Young's modulus,is the deviatoric stress, andis the pressure. The other strain model we consider follows the progression of the deviatoric stress () in the absence of plastic yielding:whereis the shear modulus. This is known as the “pseudo‐plastic strain model,” as it is intended to mimic the progression of plastic strain in the material. We find that the Benz‐Asphaug strain model produces a characteristic spall wall and undamaged inner “core,” as shown in Figure 3 b (the right‐hand panel illustrates the cross section of the target to its left, where the spall wall is green and intact core is blue), which was observed in the experiments (Figure 3 a). Note that the photograph shown on the right in Figure 3 a is not the core fragment from the photograph shown to the left but rather an example core fragment from the replication of the experiment. In contrast, the pseudo‐plastic strain model produces no spall wall, and, in addition, the damage propagates throughout the entire sphere (Figure 3 c), unlike the experiment. We therefore choose to employ the Benz‐Asphaug strain model in Spheral for further modeling of the experiment.

3.2 Strength Models and Parameter Selection

2004 Y 0 (zero pressure) to Y m (the von Mises plastic limit at large confining pressures): (3) Y i is the yield strength of the rock at pressure P and μ i is the coefficient of internal friction (Collins et al., 2004 , 1968 μ i , Y 0 , and Y m . Since these values are not available for the specific Yakuno basalt used in Nakamura and Fujiwara ( 1991 μ i , Y 0 , and Y m for similar materials from previous studies. A summary of the basalt parameters used in our simulations is shown in Table We investigate two specific strength models available for use in Spheral. The first is a pressure‐dependent strength model (Collins et al.,), which is widely used for geologic materials simulated in shock physics and hydrodynamic codes. It describes how the shear strength of a rock increases from(zero pressure) to(the von Mises plastic limit at large confining pressures):whereis the yield strength of the rock at pressureandis the coefficient of internal friction (Collins et al.,; Lundborg). This strength model requires experimentally determined material values for, and. Since these values are not available for the specific Yakuno basalt used in Nakamura and Fujiwara (), we use estimates of, andfor similar materials from previous studies. A summary of the basalt parameters used in our simulations is shown in Table 1

Table 1. Parameter Definitions and the Corresponding Values Used in the Simulations Presented in This Work for Basalt Parameter Definition Value Y 0 Shear strength at zero pressure 66, 130, 600, and 1,000 MPa Y m Shear strength at infinite pressure 3.5 GPa μ i Coefficient of internal friction 0.6 G Shear modulus 22.7 GPa (Takagi et al., 1984

While the von Mises strength of basalt may vary between samples, a value of 3.5 GPa has been widely used in small‐body fragmentation studies (Benz & Asphaug, 1999; Pierazzo et al., 2005; Senft & Stewart, 2007); hence, we utilize Y m = 3.5 GPa in our simulations. There is more variability present in the range of measured Y 0 values (Table 1), so we examine the sensitivity of our simulation results to the selected Y 0 for values of 66 MPa (Schultz, 1993), 130 MPa (Grady & Hollenbach, 1979), 600 MPa (basalt experiments by Stickle A. at the Applied Physics Laboratory, Johns Hopkins University), and 1 GPa by examining the largest‐remaining target‐fragment mass (M L ), also referred to as the “core.” Y 0 equal to 1 GPa is not an experimentally measured parameter but rather is chosen for scanning purposes to approach the von Mises strength of basalt. These values of shear strength at zero pressure are well above the tensile strength of basalt, ranging between 11–30 MPa (Housen, 2009; Schultz, 1995) and measured by Nakamura et al. (2007) to be Y tensile = 19 MPa for the specific Yakuno basalt used in their 1991 experiment. We find that using Y 0 = 66 MPa and Y 0 = 130 MPa (Figures 4b and 4c) does not produce an M L that resembles the “core” recovered from the experiment (Figure 4a) but rather results in complete shattering of the basalt sphere into small fragments and dust. However, values of Y 0 = 600 MPa and Y 0 = 1 GPa do result in a surviving inner “core” (shown in blue in Figures 6d and 6e) within the target.

Figure 4 Open in figure viewer PowerPoint Damage morphology of the target is sensitive to the material parameters selected in the pressure‐dependent strength model. As the shear strength at zero pressure is increased from 66 MPa to 1 GPa, the damage within the target decreases, creating a larger intact core. Using the constant‐strength model with Y m = 3.5 GPa also produces a large intact core, morphologically similar to the largest fragment recovered from the experiments. Snapshots taken at 70 μs (b–f).

The other strength model under investigation in Spheral is the constant‐strength (von Mises) model. It maintains a fixed yield strength, Y m , and shear modulus, G. Using the von Mises value as the yield strength, Y m = 3.5 GPa, in the constant‐strength model, we find that the resultant M L (Figure 4f) morphologically resembles the “core” from the experiment, similar to using the pressure‐dependent strength model with Y 0 = 600 MPa and Y 0 = 1 GPa. The results from this morphology study show that Y 0 must be larger than 130 MPa when utilizing a strength model in Spheral for a basalt material, where Y 0 equal to 600 MPa, 1 GPa, and 3.5 GPa are all acceptable values.

Expanding the strength investigation, we also compare the fragment masses and velocities from the simulation results to the experiment, using the pressure‐dependent and constant‐strength models. We find that using Y 0 = 600 MPa in the pressure‐dependent strength model produces fragments with large spans in velocities 5.5–80 m/s (Figure 5a. Utilizing a value of Y 0 = 1 GPa slightly decreases the fragment velocity dispersion (Figure 5b). When using Y m = 3.5 GPa and constant strength, the fragment velocity dispersion decreases further (Figure 5c), narrowing to a range of 18–50 m/s (not including the largest fragment). The lower values of Y 0 compared to Y m in the pressure‐dependent strength model allow the material to have a wide range of strengths at varying pressures, resulting in a significant variance of failure strengths within the target. This results in the wide span of fragment velocities seen in Figure 5a. By decreasing the range between Y 0 and Y m , the target is described by a more homogeneous strength; thus, the fragment velocities have a narrower distribution.

Figure 5 Open in figure viewer PowerPoint Strength models and the material parameters selected affect the fragment velocities of the target: (a) Y 0 = 600 MPa and Y m = 3.5 GPa; (b) Y 0 = 1 GPa and Y m = 3.5 GPa; and (c) constant‐strength model using Y m = 3.5 GPa. Fragments smaller than 1 × 10−5 were not included due to the mass resolution of the simulation, where individual fragments at this size become the mass of a single particle.