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Determination of an empirical model for the prediction of penetration hole diameter in thin plates from hypervelocity impact
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International Journal of Impact Engineering 30 (2004) 303–321
Determination of an empirical model for the prediction of penetration hole diameter in thin plates from hypervelocity impact Scott A. Hill NASA/Langley Research Center, MS 431, Hampton, VA 23681, USA Received 28 June 2001; received in revised form 8 November 2002; accepted 24 June 2003
Abstract The purpose of this work is to identify an empirical relationship that describes the size of the hole created in a thin plate from a hypervelocity impact in terms of the material properties and geometry of both the projectile and target. A multivariable power series was selected as the form of the mathematical model to develop this empirical relationship. Material properties and geometry of both the projectile and target were selected as the independent variables of this model to predict the hole diameters in targets. Comparison with historical equations reveals that these new models are more accurate predictors of target hole diameters. This statement is based on a one-to-one comparison of the equations using both the data utilized in developing the new models and ‘‘new’’, independent data. Published by Elsevier Ltd.
1. Introduction One of the challenges in spacecraft design is the development of a shielding system to protect the spacecraft from orbital debris and micrometeoroids. In the past, light gas gun technology could launch particles of engineering interest at velocities approaching 8 km/s. In order to determine shielding performance above this velocity, simulations were performed using hydrocodes such as HULL and CTH. Shielding systems for spacecraft were then designed based on the known performance of the system at impact velocities less than 8 km/s and the anticipated performance at impact velocities greater than 8 km/s. This process has led to some doubt about shielding system performance since the anticipated average impact velocity of orbital debris for spacecraft in low earth orbit is approximately 11 km/s and of micrometeoroids is approximately E-mail address: [email protected] (S.A. Hill). 0734-743X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/S0734-743X(03)00079-4
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18 km/s. This velocity discrepancy leaves the shielding system performance unverified for the majority of the anticipated threat. The purpose of this work is to identify an empirical relationship that describes the size of the hole created in a thin plate from a hypervelocity impact in terms of the material properties and geometry of both the projectile and target. This work is an outgrowth of a broader project to develop a relationship to equate the amount of damage from hypervelocity projectiles of different shapes impacting Whipple shields. Whipple shields are named for their inventor, Fred Whipple, who in 1947, proposed placing a thin metal plate outboard of a spacecraft’s wall to break up an incoming meteorite and reduce the subsequent loading on the spacecraft wall [1]. This design increases the spacecraft’s resistance to penetration or failure from meteorite impacts. A relationship of the aforementioned type is needed in order to convert ballistic limit data acquired using flat plates impacting a Whipple shield configuration [2] into equivalent spherical projectiles. This equivalent spherical projectile would cause the same damage against the back wall of the shield structure as the original flat plate projectile. A search of hypervelocity impact test data acquired through testing at the Marshall Space Flight Center (MSFC) Space Debris Impact Facility (SDIF) yielded a significant amount of data that could be used to develop this relationship. A literature search was also conducted to obtain data involving projectiles and targets composed of materials other than aluminum since the vast majority of tests conducted at the SDIF were of aluminum projectiles impacting aluminum targets. Four reports [3–6] were identified that contained data from numerous hypervelocity impact tests involving projectiles and targets of various materials. Finally, additional information for the relationship was gathered from numerous HULL hydrocode hypervelocity impact simulations performed by the US Army Corps of Engineers (USACOE) [7].
2. Discussion of historical models As was stated earlier, this work is directed towards developing a relationship that predicts the diameter of the hole created in thin plates from a hypervelocity impact by spherical and cylindrical projectiles. Therefore, two basic mathematical models had to be developed, one each for spherical and cylindrical projectiles, to relate the hole diameter in a thin sheet to the projectile geometry and material properties. 2.1. Spherical projectile models Previous work by other researchers in predicting hole diameters in thin sheets from a hypervelocity impact has yielded several models. Rolsten et al. [8] developed a model for spherical projectiles based on a mechanics approach where target material surrounding the impacting projectile is displaced by a radial flow of material. From Rolsten’s mechanics-based model, the following equation for spherical projectiles was derived: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi rt ; ð1Þ dh ¼ 0:9dmax ¼ dp 2 þ rp
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where dh is the diameter of the hole, dmax the maximum diameter of the hole during the hole formation process, dp the projectile diameter, rt the density of the target, and rp the density of the projectile. Immediately noticeable in Eq. (1) is the lack of dependence of the target hole diameter to the projectile velocity or the target thickness. Also noticeable in Rolsten’s equation is the apparent rebound of the material surrounding the hole by 10% from the maximum value seen during the penetration event. Maiden et al. [9] proposed an empirical model developed from test data comprised of spherical aluminum projectiles impacting aluminum targets given as dh V tt 2=3 ¼ 2:4 þ0:9; ð2Þ c dp dp where dp is the diameter of the projectile, V the velocity of the projectile, c the speed of sound, and tt is the thickness of the target. Unfortunately, Eq. (2) is limited when compared against or used to predict target hole sizes involving projectiles and targets other than aluminum due to the limited scope of the test materials. From an equation developed by Charters and Summers (referred to as the Ames equation) that yields the penetration depth in a semi-infinite solid, Sawle [10] developed a model to describe the diameter of a hole in a thin sheet caused by a hypervelocity impact. This model is given by the equation 0:22 2=3 rp dh V tt ¼ C2 þ1:0; ð3Þ c dp rt dp where coefficient C2 was theoretically determined to have a value of 2.6, but a value of 3.2 was shown to yield a slightly better fit of the data. Sawle developed an alternative model to that of Eq. (3) by using the same technique outlined in his text, but starting with a penetration model for single sheets that incorporates the target shear strength that was developed by Sorenson [11]. This alternative formulation of the target hole diameter equation is given as 0:055 2 0:1 2=3 rp dh V tt ¼ rp þ1:0; ð4Þ dp rt sUS dp where sUS is the shear strength of the target material. Nysmith and Denardo [12] developed an empirical model to fit data generated from 3.2 mm spherical aluminum and Pyrex projectiles impacting thin aluminum targets. The equation developed has a form similar to that of Eq. (2) and is given as qffiffiffiffiffiffiffiffiffi t 0:45 dh t ¼ 0:88 rp V : ð5Þ dp dp One limitation identified by Nysmith and Denardo for the use of Eq. (5) is that the front and rear free surfaces of the target plate should have minimal effect on the penetration event, thus requiring application of the equation to impacts on thin sheets. Nysmith and Denardo defined the limit for a thin target, as applied to Eq. (5), as the ratio of the target thickness to the projectile diameter ðtt =dp Þ equal to or less than 0.938.
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Schonberg [13] developed two equations through multivariable linear regression of target hole diameter data obtained from tests conducted at the MSFC SDIF. These equations determine the major and minor hole diameters for oblique impact of spherical projectiles into thin aluminum targets. The equations for major and minor target hole diameters, respectively, are given as 0:782 1:043 dhmax V tt 0:283 ¼ 2:825 cos y þ1:01; ð6Þ c dp dp 0:672 0:851 dhmin V 1:064y tt ¼ 1:250 e þ1:40: c dp dp
ð7Þ
where y is the impact angle of obliquity given in radians. Theoretically, Eqs. (6) and (7) are only valid for impact obliquities measured from the target normal, between 0 and 75 . Finally, Piekutowski [14] developed a simple model for predicting hole diameters in thin aluminum sheets when impacted by 2017-T4 aluminum projectiles at impact velocities of approximately 6.7 km/s. This relationship is given as dh tt þ 1:0: ð8Þ ¼ 4:5 dp dp Piekutowski states that this relationship is not valid for ratios of target thickness to projectile diameter less than 0.08 because of the nonlinear nature of the relationship below this level. 2.2. Cylindrical projectile models Previous hypervelocity impact research with other than spherical projectiles is very limited due to the consistency and repeatability with which spherical projectiles can be launched against targets. Schonberg et al. [15] review of tests conducted at MSFC’s SDIF identified a total of 13 tests involving cylindrical projectiles. The hole diameters in the thin bumpers were all of elliptical shape due to a slight yaw of the projectile prior to impact as surmised by Schonberg. Therefore, Schonberg developed models for both major and minor diameters using multiple linear regression analysis with the models given as 0:617 1:639 dhmaj V tt ¼ 8:323 e1:664y þ 1:40; ð9Þ c dp dp 0:302 0:561 dhmin V tt ¼ 2:309 cos0:177 y þ 1:0: c dp dp
ð10Þ
3. Description of proposed models For this application, an empirically based model was selected over deriving one based on mechanics, physics, or elasticity. Several physics-based models had previously been developed by other researchers in an effort to describe the penetration process; however, these models are primarily for projectiles with specific geometries (e.g. spherical-nose long-rod penetrators) into thick targets and for specific velocity or material behavior regimes. A typical Whipple shield has
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to provide protection to a spacecraft over a wide variety of material behavior and velocity regimes. The velocity regimes are generally classified as ballistic (low velocity), shatter (intermediate velocity), and melt/vaporization (high velocity). An empirical model was selected to treat the wide variety of materials, geometries, and velocities with a single relationship. The models selected were generic multivariable power functions with the independent and dependent variables applied to the model in such a manner as to maintain dimensionality. Hypervelocity impact event parameters that were needed, as a minimum, to describe the target hole diameter were the basic impact geometry (projectile diameter, target thickness, and projectile thickness), projectile velocity, and basic material properties (projectile density, projectile sound speed, target density, and target sound speed). Since normal incidence impacts were the only impact obliquities considered, the penetration event is fully quantified from the geometry aspect by the basic impact geometry parameters. Material density was chosen as one of the material properties to incorporate since it is the only material property involved in the basic shock jump relationship. With the addition of the material sound velocity, all the material properties required to describe the shock velocity in the Rankine–Hugoniot relation are available. Therefore, all the basic material properties used to describe the behavior of shock waves in solid materials are utilized in the model proposed herein. The models for spherical and flat plate projectiles are given by Eqs. (11) and (12), respectively, c1 c2 c3 c4 rp V V tt ; ð11Þ dh ¼ c 0 dp cp ct rt dp c1 c2 c3 c4 c5 rp V V tp tt ; dh ¼ c 0 dp cp ct rt tt dp
ð12Þ
where the subscripts p and t refer to the projectile and target, respectively, dh the diameter of the hole, and c the speed of sound in the material. A second pair of models was developed to determine if the addition of a constant to the models proposed in Eqs. (11) and (12) would predict more accurate target hole diameters. The purpose of this additional constant is to serve as an intercept to the target hole diameter axis. The models for spherical and flat plate projectiles are given as Eqs. (13) and (14), respectively c1 c2 c3 c4 rp V V tt þc5 dp ; ð13Þ dh ¼ c 0 dp cp ct rt dp c1 c2 c3 c4 c5 rp V V tp tt þc6 dp : ð14Þ dh ¼ c 0 dp cp ct rt tt dp Eq. (13) can be used for direct comparison of numerical results against Eq. (11) while Eq. (14) can be compared against Eq. (12).
4. Numerical results The analysis and optimization code, MULTIVAR, was written and used to develop coefficients for Eqs. (11)–(14) that generated the best fit of the mathematical models to the test data.
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MULTIVAR utilizes a Levenberg–Marquardt algorithm for non-linear function minimization and uses the sum of residuals squared as a basis for this minimization. The results from MULTIVAR were compared with those generated by the Microsoft Excel solver for a similar model and showed excellent correlation. Also, MULTIVAR was able to accurately reproduce the coefficients for a single independent variable test case. 4.1. Spherical projectile models Using data from Refs. [3–7] (1262 total data points), the coefficients for Eq. (11) were generated using MULTIVAR. This data consists of numerous target and projectile material and geometry combinations. For example, the values for the ratio of target hole diameter to projectile diameter range from 0.998 to 7.564, the ratio of projectile density to target density range from 0.082 to 3.296, and the ratio of target thickness to projectile diameter range from 0.008 to 3.994. Inserting the calculated coefficients in Eq. (11) yields the relationship 0:033 0:298 0:022 0:359 rp V V tt : ð15Þ dh ¼ 3:309dp rt dp cp ct The resultant sum of residuals squared is 1558.8 and the corresponding correlation coefficient is 0.957. The normalized residuals for Eq. (15) and the test data are presented in Fig. 1. Using the same set of data as that for the previous analysis and the second spherical projectile model, Eq. (13), the following coefficients were generated using MULTIVAR: 0:055 0:339 0:028 0:414 rp V V tt þ0:342dp : ð16Þ dh ¼ 2:947dp cp ct rt dp The MULTIVAR generated coefficients result in a sum of the residuals squared of 1544.2 and a correlation coefficient of 0.957, which shows a comparable fit as that from Eq. (15). The normalized residuals are similar to those presented in Fig. 1.
1
Normalized Residual
0.67 0.33
−0.33 −0.67 −1
Fig. 1. Normalized residuals for Eq. (15).
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1
Normalized Residual
0.67 0.33
−0.33 −0.67 −1
Fig. 2. Normalized residuals for Eq. (17).
4.2. Cylindrical projectile models Using data from Refs. [5–7,15] (102 total data points), coefficients for Eq. (12) were generated using MULTIVAR. As with the spherical data, this data consists of numerous target and projectile material and geometry combinations. For example, the values for the ratio of target hole diameter to projectile diameter range from 0.622 to 2.750, the ratio of the projectile length to that of its diameter range from 0.047 to 3.087, the ratio of projectile density to target density range from 0.361 to 5.357, and the ratio of projectile thickness to target thickness range from 0.027 to 13.270. Inserting the calculated coefficients into Eq. (12) yields the relationship 0:016 0:213 0:147 0:145 0:285 rp V V tp tt : ð17Þ dh ¼ 2:627dp cp ct rt tt dp The MULTIVAR generated coefficients result in a sum of the residuals squared of 85.8 and a correlation coefficient of 0.990. The normalized residuals for Eq. (17) are presented in Fig. 2. Using the same set of data as that for Eq. (17) and the second flat plate projectile model, Eq. (14), the following coefficients where generated using MULTIVAR: 0:021 0:165 0:105 0:105 0:207 rp V V tp tt 0:671dp : ð18Þ dh ¼ 3:274dp rt tt dp cp ct The MULTIVAR generated coefficients result in a sum of the residuals squared of 84.9 and a correlation coefficient of 0.990. The normalized residuals are similar to those presented in Fig. 2.
5. Discussion of results 5.1. Spherical projectile models First of all, since the form of the Rolsten equation (Eq. (1)) is not representative of any of the models developed, it is difficult to make any comparisons between this equation and Eqs. (15) or
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(16). Secondly, since the experimental data used contains only impacts occurring normal to the surface of the target, the terms involving angle of incidence in the Schonberg equations (Eqs. (6) and (7)) reduce to a value of one, thus simplifying the equations. Finally, since the Piekutowski equation (Eq. (8)) is limited to a particular aluminum alloy for the projectiles and to a constant velocity, comparison to the more general equations is not warranted. The Nysmith and Denardo equation (Eq. (5)) is the only other model that is not dimensionally correct and that also lacks the addition of a constant term at the end of the equation. The only two terms with any degree of correlation between Eqs. (15) and (16) and this one are the projectile diameter that is raised to a power of approximately six tenths and the target thickness that is raised to a power of approximately four tenths. Other than those two points, there is considerable variation in the remaining parameters. Of the remaining historical models (Eqs. (2)–(4), (6) and (7)), there are several common points existing between the models. The first is that they all use dimensionless quantities in the form of a power function with the addition of a constant term at the end. Secondly, the ratio of the target hole diameter to projectile diameter is proportional to the two-thirds power of the ratio of the target thickness to projectile diameter. Schonberg’s equation for the major hole diameter (Eq. (6)) does provide an exception for this observation. The final common point is that the trailing constant term has a magnitude approximately equal to one in all cases. The remaining terms have significant variation when comparing from model to model. There is not any significant correlation between the coefficients in Eqs. (15) and (16) with those in the historical models with the exception of the constant multiplier used in Eq. (3). The recommended value of 3.2 compares favorably with the constant multiplier used in both Eqs. (15) and (16). Even the approximately two-thirds power for the target thickness to projectile diameter ratio that is established in the historical models is not adhered to in Eqs. (15) and (16). Even Schonberg’s models, developed through regression techniques, bear out a nearly two-thirds power for this ratio. It could be that the coefficient for the ratio of the target thickness to the projectile diameter is a function of the ratio of the projectile density to the target density; thus, generating a nearly two-thirds coefficient for aluminum on aluminum impacts and a nearly one-half coefficient for Pyrex projectiles impacting aluminum targets. This would explain the major difference in this coefficient in Eqs. (15) and (16) as compared to the other equations, since both equations were developed from a wide range of projectile and target material combinations. Table 1 is given as a direct comparison of Eqs. (2)–(7) (excluding Eq. (4)) to Eqs. (15) and (16). Each model was compared using the original data set employed to develop Eqs. (15) and (16), and a corresponding sum of residuals squared and correlation coefficient was calculated. The reason Eq. (4) was not used in the comparison was that the ultimate shear strength of all the target materials in the database used to develop the models was not known. Review of the data in
Table 1 Comparison of spherical projectile models
Sum of residuals squared Correlation coefficient
Eq. (2)
Eq. (3)
Eq. (5)
Eq. (6)
Eq. (7)
Eq. (15)
Eq. (16)
5630.9 0.834
4914.8 0.856
6094.2 0.818
12485.0 0.569
2550.8 0.928
1558.8 0.957
1544.2 0.957
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Table 1 reveals that only Schonberg’s equation for the hole minor diameter (Eq. (7)) is the only historical model that is even close to the same accuracy as that from Eqs. (15) and (16). Using independent hypervelocity impact data obtained from MSFC SDIF tests, Reynolds et al. [16], and Piekutowski [17], a comparison of the hole diameter prediction capabilities for each of the models, both historic and new, are evaluated. A case-by-case comparison of Eqs. (2)–(7) to Eqs. (15) and (16) is shown in Table 2. Independent data (i.e. data not used in the development of Eqs. (15) and (16)) is used for the purpose of this comparison to see if these models could accurately predict target hole diameters for ‘‘new’’ impact situations. The test parameters required for the target hole diameter predictions are as follows: – D2031—1.52 mm aluminum sphere impacting a 0.813 mm 7075-T6 aluminum alloy target at 7.32 km/s (source: MSFC SDIF); – D2033—1.57 mm aluminum sphere impacting a 1.60 mm 7075-T6 aluminum alloy target at 7.44 km/s (source: MSFC SDIF); – A-11—1.60 mm Pyrex sphere impacting a 0.533 mm 2024-T3 aluminum alloy target at 5.85 km/s (source: Reynolds); – H-14—1.07 mm steel sphere impacting a 0.305 mm 2024-T3 aluminum alloy target at 6.86 km/s (source: Reynolds); – 4-1318—6.35 mm 2017 aluminum sphere impacting a 0.305 mm 1100-0 aluminum alloy target at 6.64 km/s (source: Piekutowski); – 4-1285—9.53 mm 2017 aluminum sphere impacting a 0.305 mm 1100-0 aluminum alloy target at 6.67 km/s (source: Piekutowski); – 4-1765—12.70 mm 2017 aluminum sphere impacting a 1.549 mm 6061-T6 aluminum alloy target at 6.30 km/s (source: Piekutowski). The ultimate shear strength of 7075-T6 aluminum alloy used in Eq. (4) for these predictions was 298.7 MPa and that of 2024-T3 aluminum alloy was 255.1 MPa. Examination of the results in Table 2 reveals that the equation by Sawle (Eq. (4)) predicts fairly accurate target hole diameters when compared to the limited test data. Since shear strength data was not in place for the experimental data used to develop Eqs. (15) and (16), the accuracy of the Sawle equation with the original test data is unknown. The Maiden equation (Eq. (2)) also
Table 2 Predicted target hole diameters for spherical projectiles Test
D2031 D2033 A-11 H-14 4-1318 4-1285 4-1765
Hole diam (mm)
Predicted hole diameter (mm) Eq. (2) (% err)
Eq. (3) (% err)
Eq. (4) (% err)
Eq. (5) (% err)
Eq. (6) (% err)
4.50 5.60 3.05 2.03 7.67 10.69 21.13
4.86 7.05 3.56 2.46 8.37 11.63 20.85
4.98 7.08 4.04 3.05 9.22 12.82 23.28
3.36 4.50 2.96 2.08 8.18 11.63 18.44
4.50 6.26 3.16 3.91 6.14 7.70 18.20
5.42 8.34 3.83 2.62 8.64 12.07 21.61
(8.0) (25.8) (16.9) (21.0) (9.1) (8.7) (1.4)
(10.6) (26.4) (32.4) (50.1) (20.2) (19.9) (10.2)
(25.2) (19.6) (3.0) (2.4) (6.6) (8.7) (12.7)
(0.0) (11.8) (3.8) (92.4) (20.0) (28.0) (13.9)
Eq. (7) (% err)
Eq. (15) (% err)
(20.4) 3.84 (14.6) 4.54 (49.0) 4.97 (11.3) 5.95 (25.6) 3.32 (8.8) 3.75 (29.1) 2.24 (10.0) 2.55 (12.6) 10.19 (32.9) 7.74 (12.9) 14.84 (38.8) 10.06 (2.2) 22.47 (6.3) 21.30
(1.0) (6.2) (22.8) (25.4) (0.9) (5.9) (0.8)
Eq. (16) (% err) 4.52 5.97 3.73 2.54 8.11 10.81 21.49
(0.5) (6.6) (22.2) (24.9) (5.7) (1.1) (1.7)
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showed fairly good correlation with the independent data and performed well with the original experimental data based on the value of the sum of residuals squared. Eqs. (15) and (16) consistently over predicted the target hole diameter from the independent test data, but had the most accurate predictions on average. Also, these two equations generated better fits (by 64% when comparing sum of residuals squared data) of the original experimental data than any of the historical models. One final observation is Eqs. (15) and (16) appear to predict target hole diameters equally well. The tolerance on the thickness associated with sheet stock has led to concern over the accuracy of target thicknesses reported in some of the early literature. An evaluation of the performance of the models was conducted by introducing a random error to all target thicknesses. When applying a random error of no more than 10% to the target thickness data, Eq. (15) shows a sum of residuals squared of 1630.2 and a correlation coefficient of 0.955. With the same data, Eq. (16) has a sum of residuals squared of 1613.6 and a correlation coefficient of 0.955.
5.2. Cylindrical projectile models Schonberg’s cylindrical projectile models are very similar in form to those he developed for spherical projectiles with the exception that the form of the equations for major and minor diameters is reversed. There are only a few common points between Schonberg’s models (Eqs. (9) and (10)) and Eqs. (17) and (18). The constant multiplier and the exponential coefficient for the ratio of projectile velocity to target material sound velocity for the minor diameter equation compares fairly closely with those terms from both Eqs. (17) and (18). The only other common point is that the trailing constant term has a magnitude approximately equal to one in the Schonberg equations and Eq. (18). The remaining terms have significant variation between the models. As with Table 1 for the spherical projectiles, Table 3 is given as a direct comparison of Schonberg’s equations (Eqs. (9) and (10)) with Eqs. (17) and (18). As before, each model was compared using the data employed to develop Eqs. (17) and (18), and a corresponding sum of residuals squared and correlation coefficient was calculated. Eqs. (17) and (18) are significantly more accurate at predicting the hole diameters than the other two models, with Eq. (18) slightly outperforming Eq. (17). As with the spherical projectile models, independent hypervelocity impact data obtained from Mortensen et al. [18] and Schonberg [19] is used to determine if these models can accurately predict target hole diameters for ‘‘new’’ impact situations. The data from Schonberg is actually the results from numerical simulations of hypervelocity impacts utilizing the HULL hydrocode.
Table 3 Comparison of cylindrical projectile models
Sum of residuals squared Correlation coefficient
Eq. (9)
Eq. (10)
Eq. (17)
Eq. (18)
7952.0 o0
407.2 0.950
85.8 0.990
84.9 0.990
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Table 4 Predicted target hole diameters for cylindrical projectiles Test
1 2 3 4 NR-38 NR-42 NR-43
Hole diam (mm) 11.43 13.72 16.76 17.02 10.00 15.00 18.00
Predicted hole diameter (mm) Eq. (9) (% err)
Eq. (10) (% err)
Eq. (17) (% err)
Eq. (18) (% err)
12.00 13.62 15.58 19.27 13.92 17.67 23.81
12.17 13.60 14.74 16.25 9.00 17.51 24.63
13.68 14.64 15.29 16.05 9.44 14.14 17.13
13.79 14.75 15.39 16.12 9.33 14.24 16.87
(4.9) (0.8) (7.1) (13.4) (39.2) (17.8) (32.3)
(6.5) (0.9) (12.1) (4.5) (10.0) (16.7) (36.8)
(19.7) (6.7) (8.8) (5.7) (5.6) (5.7) (4.8)
(20.6) (7.5) (8.2) (5.3) (6.7) (5.0) (6.3)
These predictions are given in Table 4. The test parameters used in generating the target hole diameter predictions for Table 4 are as follows: – 1—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 0.81 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 2—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 1.32 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 3—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 1.80 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 4—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 2.54 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – NR-38—3.50 mm diameter (l/d=2.0) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg). – NR-42—9.13 mm diameter (l/d=0.11) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg); – NR-43—14.40 mm diameter (l/d=0.0277) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg). Examination of the results in Table 4 reveals a trend of increased accuracy in the prediction of target hole diameters as the ratio of target thickness to projectile diameter increases for Eqs. (17) and (18). Using the sum of residuals squared value from the fit to the test data and the relative error measurement from the independent data predictions; Eqs. (17) and (18) offer equal value for predicting target hole diameters from hypervelocity impacts from cylindrical projectiles. Again, as with the spherical projectiles, an evaluation of the performance of the models was conducted by introducing a random error to all target thicknesses. When applying a random error of no more than 10% to the target thickness data, Eq. (17) shows a sum of residuals squared of 86.4 and a correlation coefficient of 0.989. With the same data, Eq. (18) has a sum of residuals squared of 85.7 and a correlation coefficient of 0.990.
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6. Conclusions It is apparent from the results presented in Table 1, that Eqs. (15) and (16) are more accurate predictors of the target hole diameter created from spherical projectiles than the historical models based upon the data used to generate these equations. It is also evident from the results in Table 1 that the model with the added constant (Eq. (16)) slightly outperforms the other model (Eq. (15)). The independent data comparison results in Table 2 show the higher accuracy generated by Eqs. (15) and (16) in comparison to the other spherical projectile models. Comparison of these models with additional independent test data consisting of dissimilar material impacts should continue to demonstrate the increased prediction accuracy over the historical models. As with the spherical projectile models, review of the results presented in Table 3, reveals that Eqs. (17) and (18) are significantly more accurate predictors than those developed by Schonberg et al. Also from Table 3, Eqs. (17) and (18) are nearly identical in their accuracy. Review of the data presented in Table 4 reveals that Schonberg’s major diameter equation (Eq. (8)) is more accurate in a greater number of the test cases, but Eq. (17) has the overall lowest average error among the test cases with Eq. (18) having the second lowest average error. Finally, Eqs. (15)–(18) were shown to be fairly insensitive to small, random variations in the target sheet thickness. For Eqs. (15) and (16), the sum of residuals squared had an increase of less than five percent for a ten percent random variation on the target thickness variable. Eqs. (17) and (18) had a corresponding increase of less than one percent. The results of this work are a set of equations that more accurately predict hole diameters in thin targets for a variety of impact geometries, material combinations, and impact velocities than any models identified during the author’s literature search. Now the process of developing a relationship to convert cylindrical projectile ballistic limit data to an equivalent spherical projectile ballistic limit can be continued. Appendix A See Table 5. Table 5 NASA/MSFC SDIF spherical projectile hypervelocity impact test data Index x 43 44 45 46 47 49 50 51 52 53 54
dp (mm) 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
10.41 10.67 12.45 12.95 14.22 14.99 10.41 10.41 13.21 12.70 14.73
2.75 2.99 4.90 4.95 6.90 6.95 2.93 2.96 5.25 5.04 6.63
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Table 5 (continued) Index x 57 58 59 60 61 62 63 64 65 66 67 69 70 71 72 73 74 75 77 79 80 81 84 85 88 89 92 93 97 98 100 101 102 103 104 105 106 107 108 109 110 111 112 114 115 116 117 118 119
dp (mm) 6.35 6.35 6.35 6.35 6.35 6.35 4.75 4.75 4.75 4.75 3.18 3.18 3.18 4.75 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 6.35 6.35 6.35 6.35 4.75 4.75 4.75 4.75 4.75 4.75 6.35 7.62 7.62 7.62 7.62 7.62 4.75 4.75 4.75 4.75 4.75
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
13.97 8.89 13.97 16.00 13.46 13.97 7.62 9.91 9.65 9.65 5.84 5.33 5.84 9.14 13.72 14.48 14.73 15.24 15.75 16.51 15.49 15.49 15.24 14.99 16.51 15.24 13.97 12.19 13.46 13.46 13.21 11.68 9.40 8.89 8.38 6.86 8.64 8.38 12.19 13.72 12.45 12.95 12.95 13.72 10.41 9.65 10.41 9.65 8.64
6.83 4.77 4.90 6.15 5.79 5.98 3.72 4.18 3.71 3.74 2.85 2.11 3.01 3.26 5.14 6.04 6.33 6.63 6.78 7.18 7.13 7.01 6.73 6.73 6.98 6.63 4.96 4.25 5.80 5.98 5.88 4.31 3.71 3.27 2.59 1.62 2.95 2.77 3.94 4.82 3.37 3.76 4.23 4.68 4.53 3.87 4.15 3.68 3.08
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316 Table 5 (continued) Index x 120 121 122 123 124 125 126 127 128 129 130 134 135 136 137 138 139 140 141 152 153 154 155 163 164 165 166 168 169 173 176 179 181 182 183 184 205 206 290 291 293 294 296 297 298 299 300 332 333
dp (mm) 4.75 3.18 4.75 6.35 6.35 4.75 7.62 7.62 7.62 7.62 7.62 4.75 4.75 4.75 7.62 7.62 7.62 7.62 7.62 7.62 4.75 8.89 8.89 4.75 4.75 4.75 4.75 8.89 8.89 8.89 6.35 6.35 6.35 6.35 6.35 6.35 7.62 7.62 6.53 6.53 6.05 6.05 6.05 5.49 5.49 5.49 6.05 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 1.588 1.588 2.032 1.588 1.588 1.016 1.016 1.016 1.016 2.032 2.032 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 3.175 3.175
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 2.71 2.71
8.13 8.64 7.62 11.68 11.43 7.62 15.75 15.75 16.00 15.49 14.99 8.89 9.40 11.18 18.29 17.02 16.51 15.49 14.48 18.03 13.46 17.53 18.67 10.92 9.91 8.89 8.13 16.76 16.26 16.00 14.48 12.70 13.21 11.18 11.18 11.18 16.76 16.00 11.68 12.95 12.95 13.72 12.70 13.97 12.70 13.97 11.94 15.49 14.22
2.83 3.00 2.25 4.33 3.96 2.54 6.63 6.47 6.89 6.60 5.85 3.09 3.70 4.27 7.20 5.35 5.96 4.74 3.83 7.13 7.39 6.85 6.80 4.06 3.61 2.56 2.00 6.69 6.30 5.72 7.06 5.17 7.21 4.85 5.26 5.53 6.82 6.55 6.50 6.00 3.45 4.63 5.35 4.60 3.69 4.82 2.76 5.04 4.43
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Table 5 (continued) Index x 388 389 391 392 393 395 396 424 425 426 447 448 451 475 476 477 478 479 480 481 483 484 485 486 487 488 489 490 491 494 495 496 542 543 606 613 618 619 620 636 660 661 675 676 680 681 703 726 729
dp (mm)
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
3.18 3.18 7.95 7.95 7.95 6.35 6.35 6.35 6.35 6.35 4.75 4.75 4.75 8.89 8.89 8.89 8.89 6.35 6.35 4.75 4.75 8.89 8.89 8.89 9.53 8.89 8.89 8.89 9.53 9.53 9.53 8.89 8.89 8.89 7.95 7.95 7.95 7.95 7.95 7.95 12.70 12.70 7.95 7.95 6.35 6.35 7.95 9.53 9.53
7.66 7.66 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.270 1.270 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 3.175 3.175 1.588 1.588 1.588 1.588 1.588 1.588 1.016
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
9.40 9.14 14.22 13.72 14.22 10.67 11.68 12.70 11.68 10.92 7.37 7.87 8.05 12.19 11.68 12.70 12.45 11.43 10.67 7.87 6.86 15.49 15.24 14.73 17.78 13.97 16.00 13.46 15.49 13.21 13.46 14.48 15.75 15.75 16.10 19.46 15.72 16.13 14.83 15.52 27.18 27.43 15.24 16.76 14.48 12.95 14.48 17.32 14.15
6.95 7.35 4.29 4.35 4.37 3.64 4.26 5.74 4.59 3.63 2.15 2.45 2.52 4.62 3.95 3.39 3.26 4.32 3.00 3.30 2.54 4.64 5.63 4.09 5.05 3.26 4.73 3.00 4.63 5.02 5.41 2.92 7.04 6.92 7.06 6.67 6.64 6.66 6.79 6.64 6.13 6.10 6.13 6.76 6.36 5.88 6.64 6.77 6.67
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318 Table 5 (continued) Index x 731 732 756 798 799 800 801 802 803 804 806 813 814 815 816 817 825 826 834 844 845 852 853 854 855 856 859 869 872 873 874 875 876 877 878 879 880 881 882 883 894 896 897 898 929 930 931 939 940
dp (mm) 9.53 7.95 7.95 4.75 4.75 4.75 4.75 4.75 6.35 6.35 6.35 4.75 4.75 4.75 6.35 6.35 7.95 7.95 7.95 7.95 7.95 6.35 6.35 7.95 7.95 7.95 7.95 6.35 7.95 7.95 6.35 6.35 4.75 4.75 7.95 7.95 6.35 6.35 4.75 4.75 7.95 9.53 4.75 4.75 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.016 1.588 0.813 0.813 0.813 0.813 0.813 0.813 0.813 0.813 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.588 1.588 1.588 2.032 2.032 2.032 2.032 2.032 1.588 2.032 0.813 0.813 0.813 0.813 0.813 0.813 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
16.28 14.02 16.15 7.19 8.81 8.76 13.21 8.76 11.68 10.67 11.68 9.02 8.03 10.67 13.00 11.63 13.21 13.59 16.38 16.00 16.43 11.99 11.91 15.67 14.43 16.74 17.78 14.25 11.68 11.68 10.03 11.18 7.24 7.72 14.73 14.22 11.53 11.81 9.19 9.78 16.51 19.10 8.66 9.65 12.80 14.43 12.90 14.00 13.46
5.85 6.84 6.93 3.99 5.78 6.27 7.24 6.82 7.03 7.39 7.67 5.95 7.12 7.45 7.29 6.81 6.83 7.07 7.10 6.70 7.22 3.84 3.44 4.45 3.65 5.62 6.88 5.59 5.59 4.62 4.94 5.44 4.25 4.78 5.27 4.57 4.27 4.76 3.60 4.21 7.13 6.86 7.24 7.76 6.08 6.50 6.07 6.93 6.74
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Table 5 (continued) Index x 941 949 960 961 962 965 966 967 969 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 1000 1002 1005 1006 1007 1008 1009 1010 1011 1012 1023 1024 1025 1026 1027 1028 1029 1030 1031 1138 1139 1140 1141 1142 1143
dp (mm) 6.35 6.35 4.75 4.75 4.75 4.75 4.75 6.35 6.35 7.95 7.95 6.35 6.35 4.75 6.35 6.35 7.95 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 6.35 6.35 6.35 6.35 9.53 7.95 7.95 7.95 4.75 4.75 4.75 6.35 6.35 6.35 6.35 7.95 6.35 7.95 7.95 7.95 7.95 7.95 7.95
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.016 1.016 0.813 0.813 0.813 0.813 0.813 1.588 1.588 1.588 0.813 0.813 1.588 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 2.032 2.032 0.813 2.032 2.032 2.032 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
14.48 14.00 9.07 9.17 9.04 9.93 10.69 12.62 11.76 11.96 11.89 9.73 10.19 7.95 13.34 12.45 15.65 12.65 12.60 16.36 15.93 16.23 17.75 16.69 17.09 16.38 12.60 12.50 14.05 15.01 13.41 17.70 16.76 18.52 10.11 10.97 9.60 13.26 13.18 12.95 13.06 16.76 13.69 14.76 15.49 17.15 14.40 15.49 17.15
7.23 7.07 3.75 4.09 3.93 6.62 7.19 6.76 6.79 5.96 6.51 6.15 6.38 5.53 5.26 5.16 5.85 5.94 5.88 5.55 5.61 5.96 6.80 6.56 6.89 6.64 3.88 4.26 6.26 5.47 6.24 6.21 6.42 6.66 4.17 3.81 4.63 5.23 7.05 6.98 4.98 6.22 6.98 5.27 6.15 7.23 5.04 6.19 7.21
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320 Table 5 (continued) Index x 1144 1145 1146 1147 1148 1149 1150 1152 1153 1154 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169
dp (mm) 7.95 7.95 7.95 7.95 7.95 7.95 7.95 9.53 9.53 9.53 9.53 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
17.02 16.76 17.12 16.26 16.76 16.26 16.64 16.64 16.71 16.33 17.78 13.34 13.21 11.81 11.18 11.43 12.70 12.19 12.70 10.92 12.19 13.46 13.21 12.70
7.11 7.13 5.07 6.22 7.13 7.17 7.19 6.85 6.85 6.23 5.52 5.83 4.81 3.97 2.92 3.08 4.07 5.18 5.58 3.06 3.86 5.13 6.19 6.13
References [1] Whipple FL. Meteorites and space travel. Astron J 1947;52(1161):131. [2] Chhabildas LC, Hertel Jr, ES, Reinhart WD, Miller JM. Whipple Bumper Shield Results and CTH Simulations at Velocities in Excess of 10 km/s. Sandia Report SAND91-2683, 1992. [3] Maiden CJ, Gehring JW, McMillan AR. Investigation of fundamental mechanism of damage to thin targets by hypervelocity projectiles. TR63-225, GM Defense Research Laboratory, 1963. [4] McMillan AR. Experimental investigations of simulated meteoroid damage to various spacecraft structures. NASA CR-915, Washington, DC, 1968. [5] Turpin WC, Carson JM. Hole growth in thin plates perforated by hypervelocity pellets. AFML-TR-70–83, 1970. [6] Herrmann W, Jones AH. Survey of Hypervelocity Impact Information. Massachusetts Institute of Technology, ASRL Report No. 99-1, 1961. [7] J. Tipton. HULL hydrocode analysis results presented at NASA/MSFC WP01 Meteoroid/Orbital Debris Working Group, NASA Purchase Order, USACOE, Huntsville, Alabama, 1991–1993. [8] Rolsten RF, Wellnitz JN, Hunt HH. An example of hole diameter in thin plates due to hypervelocity impact. J Appl Phys 1964;34(3):556–9. [9] Maiden CJ, McMillan AR, Sennett RE. Thin Sheet Impact. NASA CR-295, Washington, DC, 1965. [10] Sawle DR. Hypervelocity Impact in Thin Sheets, Semi-Infinite Targets at 15 km/s, AIAA Paper No. 69-378, AIAA Hypervelocity Impact Conference, Cincinnati, OH, 1969. [11] Sorenson NR. Systematic investigation of crater formations in metals. Proceedings of the Seventh Hypervelocity Impact Symposium, vol. VI, 1964. p. 281–325.
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[12] Nysmith CR, Denardo BP. Experimental investigation of the momentum transfer associated with impact into thin aluminum targets. NASA TN D-5492, 1969. [13] Schonberg WP. Hypervelocity impact penetration phenomena in Aluminum space structures. J Aerospace Eng 1990;3(3):173–85. [14] Piekutowski AJ. Formation and description of debris clouds produced by hypervelocity impact. NASA CR-4707, 1996. [15] Schonberg WP, Bean AJ, Darzi K. Hypervelocity impact physics. NASA CR-4343, Washington, DC, 1991. [16] Reynolds BW, Emmons RH. A new system of protection from hypervelocity particles. Proceedings of the Sixth Symposium on Hypervelocity Impact, vol. III, Cleveland, Ohio, USA. 1963. p. 249–79. [17] Piekutowski AJ. Holes produced in thin Aluminum sheets by the hypervelocity impact of Aluminum spheres. Int J Impact Eng 1999;23:711–22. [18] Mortensen RB, Ferguson JE, Joyce JP, Kreyenhagen KN. Effects of 3 to 12 km/s Impacts on finite targets. Proceedings of the Sixth Symposium on Hypervelocity Impact, Cleveland, Ohio, USA. 1963. p. 157–205. [19] Schonberg WP. Predicting Multi-Wall Structural Response to Hypervelocity Impact Using the Hull Code. NASA CR-4486, Washington, DC, 1993.
International Journal of Impact Engineering 30 (2004) 303–321
Determination of an empirical model for the prediction of penetration hole diameter in thin plates from hypervelocity impact Scott A. Hill NASA/Langley Research Center, MS 431, Hampton, VA 23681, USA Received 28 June 2001; received in revised form 8 November 2002; accepted 24 June 2003
Abstract The purpose of this work is to identify an empirical relationship that describes the size of the hole created in a thin plate from a hypervelocity impact in terms of the material properties and geometry of both the projectile and target. A multivariable power series was selected as the form of the mathematical model to develop this empirical relationship. Material properties and geometry of both the projectile and target were selected as the independent variables of this model to predict the hole diameters in targets. Comparison with historical equations reveals that these new models are more accurate predictors of target hole diameters. This statement is based on a one-to-one comparison of the equations using both the data utilized in developing the new models and ‘‘new’’, independent data. Published by Elsevier Ltd.
1. Introduction One of the challenges in spacecraft design is the development of a shielding system to protect the spacecraft from orbital debris and micrometeoroids. In the past, light gas gun technology could launch particles of engineering interest at velocities approaching 8 km/s. In order to determine shielding performance above this velocity, simulations were performed using hydrocodes such as HULL and CTH. Shielding systems for spacecraft were then designed based on the known performance of the system at impact velocities less than 8 km/s and the anticipated performance at impact velocities greater than 8 km/s. This process has led to some doubt about shielding system performance since the anticipated average impact velocity of orbital debris for spacecraft in low earth orbit is approximately 11 km/s and of micrometeoroids is approximately E-mail address: [email protected] (S.A. Hill). 0734-743X/$ - see front matter Published by Elsevier Ltd. doi:10.1016/S0734-743X(03)00079-4
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18 km/s. This velocity discrepancy leaves the shielding system performance unverified for the majority of the anticipated threat. The purpose of this work is to identify an empirical relationship that describes the size of the hole created in a thin plate from a hypervelocity impact in terms of the material properties and geometry of both the projectile and target. This work is an outgrowth of a broader project to develop a relationship to equate the amount of damage from hypervelocity projectiles of different shapes impacting Whipple shields. Whipple shields are named for their inventor, Fred Whipple, who in 1947, proposed placing a thin metal plate outboard of a spacecraft’s wall to break up an incoming meteorite and reduce the subsequent loading on the spacecraft wall [1]. This design increases the spacecraft’s resistance to penetration or failure from meteorite impacts. A relationship of the aforementioned type is needed in order to convert ballistic limit data acquired using flat plates impacting a Whipple shield configuration [2] into equivalent spherical projectiles. This equivalent spherical projectile would cause the same damage against the back wall of the shield structure as the original flat plate projectile. A search of hypervelocity impact test data acquired through testing at the Marshall Space Flight Center (MSFC) Space Debris Impact Facility (SDIF) yielded a significant amount of data that could be used to develop this relationship. A literature search was also conducted to obtain data involving projectiles and targets composed of materials other than aluminum since the vast majority of tests conducted at the SDIF were of aluminum projectiles impacting aluminum targets. Four reports [3–6] were identified that contained data from numerous hypervelocity impact tests involving projectiles and targets of various materials. Finally, additional information for the relationship was gathered from numerous HULL hydrocode hypervelocity impact simulations performed by the US Army Corps of Engineers (USACOE) [7].
2. Discussion of historical models As was stated earlier, this work is directed towards developing a relationship that predicts the diameter of the hole created in thin plates from a hypervelocity impact by spherical and cylindrical projectiles. Therefore, two basic mathematical models had to be developed, one each for spherical and cylindrical projectiles, to relate the hole diameter in a thin sheet to the projectile geometry and material properties. 2.1. Spherical projectile models Previous work by other researchers in predicting hole diameters in thin sheets from a hypervelocity impact has yielded several models. Rolsten et al. [8] developed a model for spherical projectiles based on a mechanics approach where target material surrounding the impacting projectile is displaced by a radial flow of material. From Rolsten’s mechanics-based model, the following equation for spherical projectiles was derived: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi rt ; ð1Þ dh ¼ 0:9dmax ¼ dp 2 þ rp
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where dh is the diameter of the hole, dmax the maximum diameter of the hole during the hole formation process, dp the projectile diameter, rt the density of the target, and rp the density of the projectile. Immediately noticeable in Eq. (1) is the lack of dependence of the target hole diameter to the projectile velocity or the target thickness. Also noticeable in Rolsten’s equation is the apparent rebound of the material surrounding the hole by 10% from the maximum value seen during the penetration event. Maiden et al. [9] proposed an empirical model developed from test data comprised of spherical aluminum projectiles impacting aluminum targets given as dh V tt 2=3 ¼ 2:4 þ0:9; ð2Þ c dp dp where dp is the diameter of the projectile, V the velocity of the projectile, c the speed of sound, and tt is the thickness of the target. Unfortunately, Eq. (2) is limited when compared against or used to predict target hole sizes involving projectiles and targets other than aluminum due to the limited scope of the test materials. From an equation developed by Charters and Summers (referred to as the Ames equation) that yields the penetration depth in a semi-infinite solid, Sawle [10] developed a model to describe the diameter of a hole in a thin sheet caused by a hypervelocity impact. This model is given by the equation 0:22 2=3 rp dh V tt ¼ C2 þ1:0; ð3Þ c dp rt dp where coefficient C2 was theoretically determined to have a value of 2.6, but a value of 3.2 was shown to yield a slightly better fit of the data. Sawle developed an alternative model to that of Eq. (3) by using the same technique outlined in his text, but starting with a penetration model for single sheets that incorporates the target shear strength that was developed by Sorenson [11]. This alternative formulation of the target hole diameter equation is given as 0:055 2 0:1 2=3 rp dh V tt ¼ rp þ1:0; ð4Þ dp rt sUS dp where sUS is the shear strength of the target material. Nysmith and Denardo [12] developed an empirical model to fit data generated from 3.2 mm spherical aluminum and Pyrex projectiles impacting thin aluminum targets. The equation developed has a form similar to that of Eq. (2) and is given as qffiffiffiffiffiffiffiffiffi t 0:45 dh t ¼ 0:88 rp V : ð5Þ dp dp One limitation identified by Nysmith and Denardo for the use of Eq. (5) is that the front and rear free surfaces of the target plate should have minimal effect on the penetration event, thus requiring application of the equation to impacts on thin sheets. Nysmith and Denardo defined the limit for a thin target, as applied to Eq. (5), as the ratio of the target thickness to the projectile diameter ðtt =dp Þ equal to or less than 0.938.
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Schonberg [13] developed two equations through multivariable linear regression of target hole diameter data obtained from tests conducted at the MSFC SDIF. These equations determine the major and minor hole diameters for oblique impact of spherical projectiles into thin aluminum targets. The equations for major and minor target hole diameters, respectively, are given as 0:782 1:043 dhmax V tt 0:283 ¼ 2:825 cos y þ1:01; ð6Þ c dp dp 0:672 0:851 dhmin V 1:064y tt ¼ 1:250 e þ1:40: c dp dp
ð7Þ
where y is the impact angle of obliquity given in radians. Theoretically, Eqs. (6) and (7) are only valid for impact obliquities measured from the target normal, between 0 and 75 . Finally, Piekutowski [14] developed a simple model for predicting hole diameters in thin aluminum sheets when impacted by 2017-T4 aluminum projectiles at impact velocities of approximately 6.7 km/s. This relationship is given as dh tt þ 1:0: ð8Þ ¼ 4:5 dp dp Piekutowski states that this relationship is not valid for ratios of target thickness to projectile diameter less than 0.08 because of the nonlinear nature of the relationship below this level. 2.2. Cylindrical projectile models Previous hypervelocity impact research with other than spherical projectiles is very limited due to the consistency and repeatability with which spherical projectiles can be launched against targets. Schonberg et al. [15] review of tests conducted at MSFC’s SDIF identified a total of 13 tests involving cylindrical projectiles. The hole diameters in the thin bumpers were all of elliptical shape due to a slight yaw of the projectile prior to impact as surmised by Schonberg. Therefore, Schonberg developed models for both major and minor diameters using multiple linear regression analysis with the models given as 0:617 1:639 dhmaj V tt ¼ 8:323 e1:664y þ 1:40; ð9Þ c dp dp 0:302 0:561 dhmin V tt ¼ 2:309 cos0:177 y þ 1:0: c dp dp
ð10Þ
3. Description of proposed models For this application, an empirically based model was selected over deriving one based on mechanics, physics, or elasticity. Several physics-based models had previously been developed by other researchers in an effort to describe the penetration process; however, these models are primarily for projectiles with specific geometries (e.g. spherical-nose long-rod penetrators) into thick targets and for specific velocity or material behavior regimes. A typical Whipple shield has
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to provide protection to a spacecraft over a wide variety of material behavior and velocity regimes. The velocity regimes are generally classified as ballistic (low velocity), shatter (intermediate velocity), and melt/vaporization (high velocity). An empirical model was selected to treat the wide variety of materials, geometries, and velocities with a single relationship. The models selected were generic multivariable power functions with the independent and dependent variables applied to the model in such a manner as to maintain dimensionality. Hypervelocity impact event parameters that were needed, as a minimum, to describe the target hole diameter were the basic impact geometry (projectile diameter, target thickness, and projectile thickness), projectile velocity, and basic material properties (projectile density, projectile sound speed, target density, and target sound speed). Since normal incidence impacts were the only impact obliquities considered, the penetration event is fully quantified from the geometry aspect by the basic impact geometry parameters. Material density was chosen as one of the material properties to incorporate since it is the only material property involved in the basic shock jump relationship. With the addition of the material sound velocity, all the material properties required to describe the shock velocity in the Rankine–Hugoniot relation are available. Therefore, all the basic material properties used to describe the behavior of shock waves in solid materials are utilized in the model proposed herein. The models for spherical and flat plate projectiles are given by Eqs. (11) and (12), respectively, c1 c2 c3 c4 rp V V tt ; ð11Þ dh ¼ c 0 dp cp ct rt dp c1 c2 c3 c4 c5 rp V V tp tt ; dh ¼ c 0 dp cp ct rt tt dp
ð12Þ
where the subscripts p and t refer to the projectile and target, respectively, dh the diameter of the hole, and c the speed of sound in the material. A second pair of models was developed to determine if the addition of a constant to the models proposed in Eqs. (11) and (12) would predict more accurate target hole diameters. The purpose of this additional constant is to serve as an intercept to the target hole diameter axis. The models for spherical and flat plate projectiles are given as Eqs. (13) and (14), respectively c1 c2 c3 c4 rp V V tt þc5 dp ; ð13Þ dh ¼ c 0 dp cp ct rt dp c1 c2 c3 c4 c5 rp V V tp tt þc6 dp : ð14Þ dh ¼ c 0 dp cp ct rt tt dp Eq. (13) can be used for direct comparison of numerical results against Eq. (11) while Eq. (14) can be compared against Eq. (12).
4. Numerical results The analysis and optimization code, MULTIVAR, was written and used to develop coefficients for Eqs. (11)–(14) that generated the best fit of the mathematical models to the test data.
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MULTIVAR utilizes a Levenberg–Marquardt algorithm for non-linear function minimization and uses the sum of residuals squared as a basis for this minimization. The results from MULTIVAR were compared with those generated by the Microsoft Excel solver for a similar model and showed excellent correlation. Also, MULTIVAR was able to accurately reproduce the coefficients for a single independent variable test case. 4.1. Spherical projectile models Using data from Refs. [3–7] (1262 total data points), the coefficients for Eq. (11) were generated using MULTIVAR. This data consists of numerous target and projectile material and geometry combinations. For example, the values for the ratio of target hole diameter to projectile diameter range from 0.998 to 7.564, the ratio of projectile density to target density range from 0.082 to 3.296, and the ratio of target thickness to projectile diameter range from 0.008 to 3.994. Inserting the calculated coefficients in Eq. (11) yields the relationship 0:033 0:298 0:022 0:359 rp V V tt : ð15Þ dh ¼ 3:309dp rt dp cp ct The resultant sum of residuals squared is 1558.8 and the corresponding correlation coefficient is 0.957. The normalized residuals for Eq. (15) and the test data are presented in Fig. 1. Using the same set of data as that for the previous analysis and the second spherical projectile model, Eq. (13), the following coefficients were generated using MULTIVAR: 0:055 0:339 0:028 0:414 rp V V tt þ0:342dp : ð16Þ dh ¼ 2:947dp cp ct rt dp The MULTIVAR generated coefficients result in a sum of the residuals squared of 1544.2 and a correlation coefficient of 0.957, which shows a comparable fit as that from Eq. (15). The normalized residuals are similar to those presented in Fig. 1.
1
Normalized Residual
0.67 0.33
−0.33 −0.67 −1
Fig. 1. Normalized residuals for Eq. (15).
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1
Normalized Residual
0.67 0.33
−0.33 −0.67 −1
Fig. 2. Normalized residuals for Eq. (17).
4.2. Cylindrical projectile models Using data from Refs. [5–7,15] (102 total data points), coefficients for Eq. (12) were generated using MULTIVAR. As with the spherical data, this data consists of numerous target and projectile material and geometry combinations. For example, the values for the ratio of target hole diameter to projectile diameter range from 0.622 to 2.750, the ratio of the projectile length to that of its diameter range from 0.047 to 3.087, the ratio of projectile density to target density range from 0.361 to 5.357, and the ratio of projectile thickness to target thickness range from 0.027 to 13.270. Inserting the calculated coefficients into Eq. (12) yields the relationship 0:016 0:213 0:147 0:145 0:285 rp V V tp tt : ð17Þ dh ¼ 2:627dp cp ct rt tt dp The MULTIVAR generated coefficients result in a sum of the residuals squared of 85.8 and a correlation coefficient of 0.990. The normalized residuals for Eq. (17) are presented in Fig. 2. Using the same set of data as that for Eq. (17) and the second flat plate projectile model, Eq. (14), the following coefficients where generated using MULTIVAR: 0:021 0:165 0:105 0:105 0:207 rp V V tp tt 0:671dp : ð18Þ dh ¼ 3:274dp rt tt dp cp ct The MULTIVAR generated coefficients result in a sum of the residuals squared of 84.9 and a correlation coefficient of 0.990. The normalized residuals are similar to those presented in Fig. 2.
5. Discussion of results 5.1. Spherical projectile models First of all, since the form of the Rolsten equation (Eq. (1)) is not representative of any of the models developed, it is difficult to make any comparisons between this equation and Eqs. (15) or
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(16). Secondly, since the experimental data used contains only impacts occurring normal to the surface of the target, the terms involving angle of incidence in the Schonberg equations (Eqs. (6) and (7)) reduce to a value of one, thus simplifying the equations. Finally, since the Piekutowski equation (Eq. (8)) is limited to a particular aluminum alloy for the projectiles and to a constant velocity, comparison to the more general equations is not warranted. The Nysmith and Denardo equation (Eq. (5)) is the only other model that is not dimensionally correct and that also lacks the addition of a constant term at the end of the equation. The only two terms with any degree of correlation between Eqs. (15) and (16) and this one are the projectile diameter that is raised to a power of approximately six tenths and the target thickness that is raised to a power of approximately four tenths. Other than those two points, there is considerable variation in the remaining parameters. Of the remaining historical models (Eqs. (2)–(4), (6) and (7)), there are several common points existing between the models. The first is that they all use dimensionless quantities in the form of a power function with the addition of a constant term at the end. Secondly, the ratio of the target hole diameter to projectile diameter is proportional to the two-thirds power of the ratio of the target thickness to projectile diameter. Schonberg’s equation for the major hole diameter (Eq. (6)) does provide an exception for this observation. The final common point is that the trailing constant term has a magnitude approximately equal to one in all cases. The remaining terms have significant variation when comparing from model to model. There is not any significant correlation between the coefficients in Eqs. (15) and (16) with those in the historical models with the exception of the constant multiplier used in Eq. (3). The recommended value of 3.2 compares favorably with the constant multiplier used in both Eqs. (15) and (16). Even the approximately two-thirds power for the target thickness to projectile diameter ratio that is established in the historical models is not adhered to in Eqs. (15) and (16). Even Schonberg’s models, developed through regression techniques, bear out a nearly two-thirds power for this ratio. It could be that the coefficient for the ratio of the target thickness to the projectile diameter is a function of the ratio of the projectile density to the target density; thus, generating a nearly two-thirds coefficient for aluminum on aluminum impacts and a nearly one-half coefficient for Pyrex projectiles impacting aluminum targets. This would explain the major difference in this coefficient in Eqs. (15) and (16) as compared to the other equations, since both equations were developed from a wide range of projectile and target material combinations. Table 1 is given as a direct comparison of Eqs. (2)–(7) (excluding Eq. (4)) to Eqs. (15) and (16). Each model was compared using the original data set employed to develop Eqs. (15) and (16), and a corresponding sum of residuals squared and correlation coefficient was calculated. The reason Eq. (4) was not used in the comparison was that the ultimate shear strength of all the target materials in the database used to develop the models was not known. Review of the data in
Table 1 Comparison of spherical projectile models
Sum of residuals squared Correlation coefficient
Eq. (2)
Eq. (3)
Eq. (5)
Eq. (6)
Eq. (7)
Eq. (15)
Eq. (16)
5630.9 0.834
4914.8 0.856
6094.2 0.818
12485.0 0.569
2550.8 0.928
1558.8 0.957
1544.2 0.957
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Table 1 reveals that only Schonberg’s equation for the hole minor diameter (Eq. (7)) is the only historical model that is even close to the same accuracy as that from Eqs. (15) and (16). Using independent hypervelocity impact data obtained from MSFC SDIF tests, Reynolds et al. [16], and Piekutowski [17], a comparison of the hole diameter prediction capabilities for each of the models, both historic and new, are evaluated. A case-by-case comparison of Eqs. (2)–(7) to Eqs. (15) and (16) is shown in Table 2. Independent data (i.e. data not used in the development of Eqs. (15) and (16)) is used for the purpose of this comparison to see if these models could accurately predict target hole diameters for ‘‘new’’ impact situations. The test parameters required for the target hole diameter predictions are as follows: – D2031—1.52 mm aluminum sphere impacting a 0.813 mm 7075-T6 aluminum alloy target at 7.32 km/s (source: MSFC SDIF); – D2033—1.57 mm aluminum sphere impacting a 1.60 mm 7075-T6 aluminum alloy target at 7.44 km/s (source: MSFC SDIF); – A-11—1.60 mm Pyrex sphere impacting a 0.533 mm 2024-T3 aluminum alloy target at 5.85 km/s (source: Reynolds); – H-14—1.07 mm steel sphere impacting a 0.305 mm 2024-T3 aluminum alloy target at 6.86 km/s (source: Reynolds); – 4-1318—6.35 mm 2017 aluminum sphere impacting a 0.305 mm 1100-0 aluminum alloy target at 6.64 km/s (source: Piekutowski); – 4-1285—9.53 mm 2017 aluminum sphere impacting a 0.305 mm 1100-0 aluminum alloy target at 6.67 km/s (source: Piekutowski); – 4-1765—12.70 mm 2017 aluminum sphere impacting a 1.549 mm 6061-T6 aluminum alloy target at 6.30 km/s (source: Piekutowski). The ultimate shear strength of 7075-T6 aluminum alloy used in Eq. (4) for these predictions was 298.7 MPa and that of 2024-T3 aluminum alloy was 255.1 MPa. Examination of the results in Table 2 reveals that the equation by Sawle (Eq. (4)) predicts fairly accurate target hole diameters when compared to the limited test data. Since shear strength data was not in place for the experimental data used to develop Eqs. (15) and (16), the accuracy of the Sawle equation with the original test data is unknown. The Maiden equation (Eq. (2)) also
Table 2 Predicted target hole diameters for spherical projectiles Test
D2031 D2033 A-11 H-14 4-1318 4-1285 4-1765
Hole diam (mm)
Predicted hole diameter (mm) Eq. (2) (% err)
Eq. (3) (% err)
Eq. (4) (% err)
Eq. (5) (% err)
Eq. (6) (% err)
4.50 5.60 3.05 2.03 7.67 10.69 21.13
4.86 7.05 3.56 2.46 8.37 11.63 20.85
4.98 7.08 4.04 3.05 9.22 12.82 23.28
3.36 4.50 2.96 2.08 8.18 11.63 18.44
4.50 6.26 3.16 3.91 6.14 7.70 18.20
5.42 8.34 3.83 2.62 8.64 12.07 21.61
(8.0) (25.8) (16.9) (21.0) (9.1) (8.7) (1.4)
(10.6) (26.4) (32.4) (50.1) (20.2) (19.9) (10.2)
(25.2) (19.6) (3.0) (2.4) (6.6) (8.7) (12.7)
(0.0) (11.8) (3.8) (92.4) (20.0) (28.0) (13.9)
Eq. (7) (% err)
Eq. (15) (% err)
(20.4) 3.84 (14.6) 4.54 (49.0) 4.97 (11.3) 5.95 (25.6) 3.32 (8.8) 3.75 (29.1) 2.24 (10.0) 2.55 (12.6) 10.19 (32.9) 7.74 (12.9) 14.84 (38.8) 10.06 (2.2) 22.47 (6.3) 21.30
(1.0) (6.2) (22.8) (25.4) (0.9) (5.9) (0.8)
Eq. (16) (% err) 4.52 5.97 3.73 2.54 8.11 10.81 21.49
(0.5) (6.6) (22.2) (24.9) (5.7) (1.1) (1.7)
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showed fairly good correlation with the independent data and performed well with the original experimental data based on the value of the sum of residuals squared. Eqs. (15) and (16) consistently over predicted the target hole diameter from the independent test data, but had the most accurate predictions on average. Also, these two equations generated better fits (by 64% when comparing sum of residuals squared data) of the original experimental data than any of the historical models. One final observation is Eqs. (15) and (16) appear to predict target hole diameters equally well. The tolerance on the thickness associated with sheet stock has led to concern over the accuracy of target thicknesses reported in some of the early literature. An evaluation of the performance of the models was conducted by introducing a random error to all target thicknesses. When applying a random error of no more than 10% to the target thickness data, Eq. (15) shows a sum of residuals squared of 1630.2 and a correlation coefficient of 0.955. With the same data, Eq. (16) has a sum of residuals squared of 1613.6 and a correlation coefficient of 0.955.
5.2. Cylindrical projectile models Schonberg’s cylindrical projectile models are very similar in form to those he developed for spherical projectiles with the exception that the form of the equations for major and minor diameters is reversed. There are only a few common points between Schonberg’s models (Eqs. (9) and (10)) and Eqs. (17) and (18). The constant multiplier and the exponential coefficient for the ratio of projectile velocity to target material sound velocity for the minor diameter equation compares fairly closely with those terms from both Eqs. (17) and (18). The only other common point is that the trailing constant term has a magnitude approximately equal to one in the Schonberg equations and Eq. (18). The remaining terms have significant variation between the models. As with Table 1 for the spherical projectiles, Table 3 is given as a direct comparison of Schonberg’s equations (Eqs. (9) and (10)) with Eqs. (17) and (18). As before, each model was compared using the data employed to develop Eqs. (17) and (18), and a corresponding sum of residuals squared and correlation coefficient was calculated. Eqs. (17) and (18) are significantly more accurate at predicting the hole diameters than the other two models, with Eq. (18) slightly outperforming Eq. (17). As with the spherical projectile models, independent hypervelocity impact data obtained from Mortensen et al. [18] and Schonberg [19] is used to determine if these models can accurately predict target hole diameters for ‘‘new’’ impact situations. The data from Schonberg is actually the results from numerical simulations of hypervelocity impacts utilizing the HULL hydrocode.
Table 3 Comparison of cylindrical projectile models
Sum of residuals squared Correlation coefficient
Eq. (9)
Eq. (10)
Eq. (17)
Eq. (18)
7952.0 o0
407.2 0.950
85.8 0.990
84.9 0.990
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Table 4 Predicted target hole diameters for cylindrical projectiles Test
1 2 3 4 NR-38 NR-42 NR-43
Hole diam (mm) 11.43 13.72 16.76 17.02 10.00 15.00 18.00
Predicted hole diameter (mm) Eq. (9) (% err)
Eq. (10) (% err)
Eq. (17) (% err)
Eq. (18) (% err)
12.00 13.62 15.58 19.27 13.92 17.67 23.81
12.17 13.60 14.74 16.25 9.00 17.51 24.63
13.68 14.64 15.29 16.05 9.44 14.14 17.13
13.79 14.75 15.39 16.12 9.33 14.24 16.87
(4.9) (0.8) (7.1) (13.4) (39.2) (17.8) (32.3)
(6.5) (0.9) (12.1) (4.5) (10.0) (16.7) (36.8)
(19.7) (6.7) (8.8) (5.7) (5.6) (5.7) (4.8)
(20.6) (7.5) (8.2) (5.3) (6.7) (5.0) (6.3)
These predictions are given in Table 4. The test parameters used in generating the target hole diameter predictions for Table 4 are as follows: – 1—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 0.81 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 2—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 1.32 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 3—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 1.80 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – 4—7.62 mm diameter (l/d=1.0) aluminum cylinder impacting a 2.54 mm 2014-T6 aluminum alloy target at 3.7 km/s (source: Mortensen); – NR-38—3.50 mm diameter (l/d=2.0) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg). – NR-42—9.13 mm diameter (l/d=0.11) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg); – NR-43—14.40 mm diameter (l/d=0.0277) aluminum cylinder impacting a 1.60 mm 6061-T6 aluminum alloy target at 6.0 km/s (source: Schonberg). Examination of the results in Table 4 reveals a trend of increased accuracy in the prediction of target hole diameters as the ratio of target thickness to projectile diameter increases for Eqs. (17) and (18). Using the sum of residuals squared value from the fit to the test data and the relative error measurement from the independent data predictions; Eqs. (17) and (18) offer equal value for predicting target hole diameters from hypervelocity impacts from cylindrical projectiles. Again, as with the spherical projectiles, an evaluation of the performance of the models was conducted by introducing a random error to all target thicknesses. When applying a random error of no more than 10% to the target thickness data, Eq. (17) shows a sum of residuals squared of 86.4 and a correlation coefficient of 0.989. With the same data, Eq. (18) has a sum of residuals squared of 85.7 and a correlation coefficient of 0.990.
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6. Conclusions It is apparent from the results presented in Table 1, that Eqs. (15) and (16) are more accurate predictors of the target hole diameter created from spherical projectiles than the historical models based upon the data used to generate these equations. It is also evident from the results in Table 1 that the model with the added constant (Eq. (16)) slightly outperforms the other model (Eq. (15)). The independent data comparison results in Table 2 show the higher accuracy generated by Eqs. (15) and (16) in comparison to the other spherical projectile models. Comparison of these models with additional independent test data consisting of dissimilar material impacts should continue to demonstrate the increased prediction accuracy over the historical models. As with the spherical projectile models, review of the results presented in Table 3, reveals that Eqs. (17) and (18) are significantly more accurate predictors than those developed by Schonberg et al. Also from Table 3, Eqs. (17) and (18) are nearly identical in their accuracy. Review of the data presented in Table 4 reveals that Schonberg’s major diameter equation (Eq. (8)) is more accurate in a greater number of the test cases, but Eq. (17) has the overall lowest average error among the test cases with Eq. (18) having the second lowest average error. Finally, Eqs. (15)–(18) were shown to be fairly insensitive to small, random variations in the target sheet thickness. For Eqs. (15) and (16), the sum of residuals squared had an increase of less than five percent for a ten percent random variation on the target thickness variable. Eqs. (17) and (18) had a corresponding increase of less than one percent. The results of this work are a set of equations that more accurately predict hole diameters in thin targets for a variety of impact geometries, material combinations, and impact velocities than any models identified during the author’s literature search. Now the process of developing a relationship to convert cylindrical projectile ballistic limit data to an equivalent spherical projectile ballistic limit can be continued. Appendix A See Table 5. Table 5 NASA/MSFC SDIF spherical projectile hypervelocity impact test data Index x 43 44 45 46 47 49 50 51 52 53 54
dp (mm) 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
10.41 10.67 12.45 12.95 14.22 14.99 10.41 10.41 13.21 12.70 14.73
2.75 2.99 4.90 4.95 6.90 6.95 2.93 2.96 5.25 5.04 6.63
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Table 5 (continued) Index x 57 58 59 60 61 62 63 64 65 66 67 69 70 71 72 73 74 75 77 79 80 81 84 85 88 89 92 93 97 98 100 101 102 103 104 105 106 107 108 109 110 111 112 114 115 116 117 118 119
dp (mm) 6.35 6.35 6.35 6.35 6.35 6.35 4.75 4.75 4.75 4.75 3.18 3.18 3.18 4.75 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 7.62 6.35 6.35 6.35 6.35 4.75 4.75 4.75 4.75 4.75 4.75 6.35 7.62 7.62 7.62 7.62 7.62 4.75 4.75 4.75 4.75 4.75
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
13.97 8.89 13.97 16.00 13.46 13.97 7.62 9.91 9.65 9.65 5.84 5.33 5.84 9.14 13.72 14.48 14.73 15.24 15.75 16.51 15.49 15.49 15.24 14.99 16.51 15.24 13.97 12.19 13.46 13.46 13.21 11.68 9.40 8.89 8.38 6.86 8.64 8.38 12.19 13.72 12.45 12.95 12.95 13.72 10.41 9.65 10.41 9.65 8.64
6.83 4.77 4.90 6.15 5.79 5.98 3.72 4.18 3.71 3.74 2.85 2.11 3.01 3.26 5.14 6.04 6.33 6.63 6.78 7.18 7.13 7.01 6.73 6.73 6.98 6.63 4.96 4.25 5.80 5.98 5.88 4.31 3.71 3.27 2.59 1.62 2.95 2.77 3.94 4.82 3.37 3.76 4.23 4.68 4.53 3.87 4.15 3.68 3.08
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316 Table 5 (continued) Index x 120 121 122 123 124 125 126 127 128 129 130 134 135 136 137 138 139 140 141 152 153 154 155 163 164 165 166 168 169 173 176 179 181 182 183 184 205 206 290 291 293 294 296 297 298 299 300 332 333
dp (mm) 4.75 3.18 4.75 6.35 6.35 4.75 7.62 7.62 7.62 7.62 7.62 4.75 4.75 4.75 7.62 7.62 7.62 7.62 7.62 7.62 4.75 8.89 8.89 4.75 4.75 4.75 4.75 8.89 8.89 8.89 6.35 6.35 6.35 6.35 6.35 6.35 7.62 7.62 6.53 6.53 6.05 6.05 6.05 5.49 5.49 5.49 6.05 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 2.032 1.588 1.588 2.032 1.588 1.588 1.016 1.016 1.016 1.016 2.032 2.032 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.016 3.175 3.175
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 8.66 2.71 2.71
8.13 8.64 7.62 11.68 11.43 7.62 15.75 15.75 16.00 15.49 14.99 8.89 9.40 11.18 18.29 17.02 16.51 15.49 14.48 18.03 13.46 17.53 18.67 10.92 9.91 8.89 8.13 16.76 16.26 16.00 14.48 12.70 13.21 11.18 11.18 11.18 16.76 16.00 11.68 12.95 12.95 13.72 12.70 13.97 12.70 13.97 11.94 15.49 14.22
2.83 3.00 2.25 4.33 3.96 2.54 6.63 6.47 6.89 6.60 5.85 3.09 3.70 4.27 7.20 5.35 5.96 4.74 3.83 7.13 7.39 6.85 6.80 4.06 3.61 2.56 2.00 6.69 6.30 5.72 7.06 5.17 7.21 4.85 5.26 5.53 6.82 6.55 6.50 6.00 3.45 4.63 5.35 4.60 3.69 4.82 2.76 5.04 4.43
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Table 5 (continued) Index x 388 389 391 392 393 395 396 424 425 426 447 448 451 475 476 477 478 479 480 481 483 484 485 486 487 488 489 490 491 494 495 496 542 543 606 613 618 619 620 636 660 661 675 676 680 681 703 726 729
dp (mm)
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
3.18 3.18 7.95 7.95 7.95 6.35 6.35 6.35 6.35 6.35 4.75 4.75 4.75 8.89 8.89 8.89 8.89 6.35 6.35 4.75 4.75 8.89 8.89 8.89 9.53 8.89 8.89 8.89 9.53 9.53 9.53 8.89 8.89 8.89 7.95 7.95 7.95 7.95 7.95 7.95 12.70 12.70 7.95 7.95 6.35 6.35 7.95 9.53 9.53
7.66 7.66 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.270 1.270 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 3.175 3.175 1.588 1.588 1.588 1.588 1.588 1.588 1.016
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
9.40 9.14 14.22 13.72 14.22 10.67 11.68 12.70 11.68 10.92 7.37 7.87 8.05 12.19 11.68 12.70 12.45 11.43 10.67 7.87 6.86 15.49 15.24 14.73 17.78 13.97 16.00 13.46 15.49 13.21 13.46 14.48 15.75 15.75 16.10 19.46 15.72 16.13 14.83 15.52 27.18 27.43 15.24 16.76 14.48 12.95 14.48 17.32 14.15
6.95 7.35 4.29 4.35 4.37 3.64 4.26 5.74 4.59 3.63 2.15 2.45 2.52 4.62 3.95 3.39 3.26 4.32 3.00 3.30 2.54 4.64 5.63 4.09 5.05 3.26 4.73 3.00 4.63 5.02 5.41 2.92 7.04 6.92 7.06 6.67 6.64 6.66 6.79 6.64 6.13 6.10 6.13 6.76 6.36 5.88 6.64 6.77 6.67
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318 Table 5 (continued) Index x 731 732 756 798 799 800 801 802 803 804 806 813 814 815 816 817 825 826 834 844 845 852 853 854 855 856 859 869 872 873 874 875 876 877 878 879 880 881 882 883 894 896 897 898 929 930 931 939 940
dp (mm) 9.53 7.95 7.95 4.75 4.75 4.75 4.75 4.75 6.35 6.35 6.35 4.75 4.75 4.75 6.35 6.35 7.95 7.95 7.95 7.95 7.95 6.35 6.35 7.95 7.95 7.95 7.95 6.35 7.95 7.95 6.35 6.35 4.75 4.75 7.95 7.95 6.35 6.35 4.75 4.75 7.95 9.53 4.75 4.75 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.016 1.588 0.813 0.813 0.813 0.813 0.813 0.813 0.813 0.813 1.016 1.016 1.016 1.016 1.016 1.016 1.016 1.588 1.588 1.588 2.032 2.032 2.032 2.032 2.032 1.588 2.032 0.813 0.813 0.813 0.813 0.813 0.813 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
16.28 14.02 16.15 7.19 8.81 8.76 13.21 8.76 11.68 10.67 11.68 9.02 8.03 10.67 13.00 11.63 13.21 13.59 16.38 16.00 16.43 11.99 11.91 15.67 14.43 16.74 17.78 14.25 11.68 11.68 10.03 11.18 7.24 7.72 14.73 14.22 11.53 11.81 9.19 9.78 16.51 19.10 8.66 9.65 12.80 14.43 12.90 14.00 13.46
5.85 6.84 6.93 3.99 5.78 6.27 7.24 6.82 7.03 7.39 7.67 5.95 7.12 7.45 7.29 6.81 6.83 7.07 7.10 6.70 7.22 3.84 3.44 4.45 3.65 5.62 6.88 5.59 5.59 4.62 4.94 5.44 4.25 4.78 5.27 4.57 4.27 4.76 3.60 4.21 7.13 6.86 7.24 7.76 6.08 6.50 6.07 6.93 6.74
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Table 5 (continued) Index x 941 949 960 961 962 965 966 967 969 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 1000 1002 1005 1006 1007 1008 1009 1010 1011 1012 1023 1024 1025 1026 1027 1028 1029 1030 1031 1138 1139 1140 1141 1142 1143
dp (mm) 6.35 6.35 4.75 4.75 4.75 4.75 4.75 6.35 6.35 7.95 7.95 6.35 6.35 4.75 6.35 6.35 7.95 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 9.53 6.35 6.35 6.35 6.35 9.53 7.95 7.95 7.95 4.75 4.75 4.75 6.35 6.35 6.35 6.35 7.95 6.35 7.95 7.95 7.95 7.95 7.95 7.95
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.016 1.016 1.016 1.016 0.813 0.813 0.813 0.813 0.813 1.588 1.588 1.588 0.813 0.813 1.588 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 2.032 2.032 0.813 2.032 2.032 2.032 1.588 1.588 1.588 1.588 1.588 1.588 2.032 2.032 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
14.48 14.00 9.07 9.17 9.04 9.93 10.69 12.62 11.76 11.96 11.89 9.73 10.19 7.95 13.34 12.45 15.65 12.65 12.60 16.36 15.93 16.23 17.75 16.69 17.09 16.38 12.60 12.50 14.05 15.01 13.41 17.70 16.76 18.52 10.11 10.97 9.60 13.26 13.18 12.95 13.06 16.76 13.69 14.76 15.49 17.15 14.40 15.49 17.15
7.23 7.07 3.75 4.09 3.93 6.62 7.19 6.76 6.79 5.96 6.51 6.15 6.38 5.53 5.26 5.16 5.85 5.94 5.88 5.55 5.61 5.96 6.80 6.56 6.89 6.64 3.88 4.26 6.26 5.47 6.24 6.21 6.42 6.66 4.17 3.81 4.63 5.23 7.05 6.98 4.98 6.22 6.98 5.27 6.15 7.23 5.04 6.19 7.21
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320 Table 5 (continued) Index x 1144 1145 1146 1147 1148 1149 1150 1152 1153 1154 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169
dp (mm) 7.95 7.95 7.95 7.95 7.95 7.95 7.95 9.53 9.53 9.53 9.53 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35 6.35
rp (g/cm3)
tt (mm)
rt (g/cm3)
dh (mm)
V (km/s)
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588 1.588
2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71
17.02 16.76 17.12 16.26 16.76 16.26 16.64 16.64 16.71 16.33 17.78 13.34 13.21 11.81 11.18 11.43 12.70 12.19 12.70 10.92 12.19 13.46 13.21 12.70
7.11 7.13 5.07 6.22 7.13 7.17 7.19 6.85 6.85 6.23 5.52 5.83 4.81 3.97 2.92 3.08 4.07 5.18 5.58 3.06 3.86 5.13 6.19 6.13
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