Is3 062 Review 55c2t

3rd International Conference on Protective Structures (IS3) Newcastle, Australia, 3-6 February 2015, M.G. Stewart & M.D. Netherton (Eds.)

AN EVALUATION OF THE FORRESTAL SCALING LAW FOR PREDICTING THE PERFORMANCE OF TARGETS PERFORATED IN DUCTILE HOLE FORMATION Shannon Ryan* Defence Science and Technology Organisation Fishermans Bend, VIC, 3207, Australia. [email protected]

ABSTRACT Algorithms for predicting the outcome of terminal velocity events can be generally categorized as highly simplified analytical methods, empirically-adjusted analytical methods, or fully empirical methods. Typically, they are valid for a range of interactions in which the target fails in a consistent manner. Forrestal and co-workers developed a scaling law for the case of terminal ballistic events resulting in penetration and perforation of the target plate by ductile hole formation based on cavity expansion theory. Requiring a minimum of two ballistic limit data points and large strain compression curves, the Forrestal scaling law was found to be highly accurate when applied to a wide range of aluminium alloy ballistic limit data, including both 7.62 mm APM2 and 12.7 mm APM2 projectiles. Additionally, the accuracy of the scaling law was consistent when applied to a range of steel plates for which the perforation mechanism (i.e. ductile hole formation) was experimentally observed. Typically predicting a target V50 to within 5% of the experimentally-measured value, the Forrestal scaling law was found to be more accurate than an earlier, simplified cavity expansion equation from Woodward and the sharp projectile against thick metallic target equation from the JTCG/ME penetration equation handbook. KEYWORDS Ballistic protection, terminal ballistics, armour, aluminium, predictive modelling. INTRODUCTION Predicting the outcome of ballistic interactions is a highly complex problem due to the wide range of materials, geometries, loading rates, and subsequent failure mechanisms of projectile and armour components. As such, there are a vast range of methods available for application to the problem. Initially, these methods can be categorised as analytical (e.g. Birkhoff et al., 1948), empirical (e.g. Ballistic Analysis Laboratory, 1961), or numerical (e.g. Walker and Anderson, 1995) in nature, although it should be considered that most analytical and numerical techniques also include some degree of empiricism. Backman and Goldsmith (1978) provided an overview of target failure modes, reproduced in Figure 1, which serves to illustrate the complexity of the problem. Typically, the applicability of a penetration algorithm is limited to a single failure mode. The impact of a hard, sharp-nosed penetrator on a thick, ductile target is amongst the simplest conditions to evaluate (ductile hole enlargement in Figure 1). For this condition, the ballistic limit of the target armour can be estimated by the work required to expand a hole from zero to the projectile diameter according to cylindrical cavity expansion theory.

1

   

   

 

 

 

  Figure 1. Classification of failure modes to illustrate the influence of material properties, structure, projectile and plate geometry on the outcome of a ballistic event (Backman and Goldsmith, 1990). FORRESTAL CAVITY EXPANSION LAW The perforation of aluminium alloy plates by steel non-deforming projectiles has been experimentally observed to occur through ductile hole formation (e.g. Woodward, 1978). The energy required for this process can be equated to the work required to expand a cylindrical hole in the plate from a zero initial radius to that of the penetrating projectile. Forrestal et al. (1990) derive a relationship for the ballistic limit of a target perforated in ductile hole formation in of a quasi-static radial stress required to open a cylindrical cavity from zero initial radius, s.. Assuming the effects of target inertia are negligible, the ballistic limit velocity, Vbl, is calculated as: ⁄

2

where  √3

(1)

 

1 ln 1 1

 

√3   , 0   2 1

 

1 

(3)

(4)

 

√3

(2)

 

p is the projectile density, h is the target thickness, L is the projectile shank length, l is the projectile nose length, k1 is related to the projectile nose geometry (see Forrestal and Warren, 2009), Y is the target yield strength, E is the target elastic modulus,  is the target Poisson’s ratio (assumed incompressible, hence = 0.5), and n is the strain hardening exponent used to fit the stress-strain curve, , ,

⁄

 

(5)

  For a given projectile, Eq. (1) simplifies to: ⁄

(6)

  where K is an empirically determined constant related to the projectile material and geometry. Thus, if the compressive stress-strain behaviour of a material is known, and provided that material will be perforated in ductile hole formation, Eq. (6) enables the ballistic performance of a wide range of different materials and thicknesses to be predicted from only a small number of ballistic firings.

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MECHANICAL TESTING To derive the material parameters required for the Forrestal model (Eq. (5)) a large magnitude, quasistatic compression test is required. Skube (2009) prescribes a technique to measure the required properties using an L/D = 1.5 cylindrical sample with grooved and lubricated ends to minimize friction on the plattens and prevent barrelling of the sample. The test specimen geometry is shown in Figure 2.

Figure 2. Sample geometry after Skube (2009) for large strain compression testing of aluminium. The power law fit has been published for a number of aluminium alloys by Forrestal and co-workers (see Table 1). Additional data has been generated for 2024-T351 and 5083-H112 aluminium, the results of which are also provided in Table 1. Compression testing was performed on a minimum of four samples of each alloy, machined from the through-thickness direction of nominally 38-40 mm thick plates, and loaded at a constant engineering strain rate of 10-4/s. Selected photographs from the compression tests are provided in Figure 3 together with an example of the power law fit for 2024T351. Some difficulties were experienced during the compression testing, primarily associated with ensuring consistent axial loading of the specimens and fracturing of the 2024-T351 specimens. A floating pivot was used to ensure the test specimen was centred. 7.E+05

True stress (kPa)

6.E+05 5.E+05 4.E+05 3.E+05 2.E+05 2024‐T351c

1.E+05

Power fit 0.E+00 0

0.1

0.2

0.3 True strain (‐)

0.4

0.5

0.6

Figure 3. Left: Large strain compression testing of aluminium alloys after Skube (2009); Right: fit of the strain hardening power-law to a representative true stress-true strain curve for 2024-T351. Table 1. Material stress-strain power law data. Alloy 2024-T351 2139-T8 5083-H112 5083-H116 5083-H131 6061-T651 6082-T651 7075-T651

E (GPa) 73.1 73.1 70.3 71 70.3 69 69 71.1

Y (MPa) 324 430 200 240 276 262 265 520

n (-) 0.12 0.085* 0.108 0.108 0.084 0.085 0.06 0.06

Source New Casem and Dandekar, 2012 New Børvik et al., 2010 Forrestal et al., 1990 Piekutowski et al., 1996 Forrestal et al., 2014 Forrestal et al., 2010

*Fit to quasi-static (0.001/s) true stress-true strain curve (Casem and Dandekar, 2012)

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BALLISTIC TESTING Ballistic testing has been performed on a range of aluminium alloys and thicknesses to determine the V50, as per MIL-STD-662F (US Department of Defence, 1997). All tests have been performed with 7.62 mm APM2 projectiles at normal incidence (i.e. impact at 90 to the target surface). For details of the projectile see Moynihan et al. (2000). A summary of the test results is provided in Table 2, including additional data from literature. Table 2. Ballistic limit data with 7.62 mm APM2 projectiles. Alloy 2024-T351 2024-T351 2024-T351 2024-T351 2139-T8 2139-T8 2139-T8 2139-T8 5083-H112 5083-H112 5083-H112 5083-H116 5083-H116 5083-H116 5083-H131 5083-H131 5083-H131 5083-H131 5083-H131 5083-H131 6061-T651 6061-T651 6061-T651 6061-T651 6061-T651 6061-T651 6082-T651 7075-T651 7075-T651 7075-T651 7075-T651 7075-T651 7075-T651

Thickness (mm) 26 30 34 38 25.2 32.3 39.0 40.9 20 30 40 20 40* 60** 26 37.8 38 50.9 54.7 57.2 25 25.7 26 38.8 50 51.2 20 20 26 30 32 38 40*

*2 × 20 mm thick plates 

V50 (m/s) 651.7 715.4 773.9 830.6 682.1 782.7 860.1 892.5 474.3 588.6 697.7 492 722 912 588 712 738.5 876 890 927 578.3 596 583 754 896.2 883 501.0 628 718.4 784.3 817.5 909.5 909

Ref New New New New Cheeseman et al. (2008) Cheeseman et al. (2008) Cheeseman et al. (2008) Cheeseman et al. (2008) New New New Børvik et al. (2010) Børvik et al. (2010) Børvik et al. (2010) Forrestal et al. (2014) Forrestal et al. (2014) New Forrestal et al. (2014)  Forrestal et al. (2014)  Forrestal et al. (2014)  New Gooch Jr. et al. (2007) Gooch Jr. et al. (2007)  Gooch Jr. et al. (2007)  New Gooch Jr. et al. (2007)  Forrestal et al. (2014)  Forrestal et al. (2010) New New New New Forrestal et al. (2010)

**3 × 20 mm thick plates

RESULTS The ballistic test data (V50) from Table 2, excluding that on 2139-T8, is plotted in Figure 4 in of (sh)1/2. Using a least squares fit the constant K is determined, as per Eq.(6), equal to 110.25. The ballistic data on 2139-T8 has been excluded from the calculation as it will be used to assess the predictive accuracy of the derived relationship. The coefficient of determination, R2, for the linear fit is 0.966, indicating a very high level of agreement. In order to predict the ballistic performance of other aluminium alloys impacted by 7.62 mm APM2 projectiles, only the power law parameters of the material stress-strain relationship and the scaling parameter, K = 110.25, are required. For 2139-T8, the ballistic performance predicted by the Forrestal law is compared in Table 3 to experimental measurements from Cheeseman et al. (2008).

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1000

1000 900

800

800 V50 (m/s)

V50 (m/s)

K = 110.25 900

700

700

600

600

500

500 400

400 15

20

25

30

35 40 45 50 Thickness (mm)

55

60

4

65

5

6

7

8

9

(sh)1/2

 

 

Figure 4. Ballistic limit test results for impact of 7.62 mm APM2 projectiles on aluminium targets. Left: in of target material and thickness; right: in of Forrestal et al.’s scaling law.

Table 3. Predicting the ballistic limit of 2139-T8 aluminium plates impacted by 7.62 mm APM2 projectiles using the Forrestal scaling law for ductile hole formation. Projectile

Target

7.62 mm APM2 7.62 mm APM2 7.62 mm APM2 7.62 mm APM2

2139-T8 2139-T8 2139-T8 2139-T8

Thickness (mm) 25.2 32.3 39.0 40.9

V50 (m/s) Predicted Measured 713.5 682.1 808.0 782.7 887.7 860.1 909.1 892.5

Variance (m/s) % 31.4 4.6 25.3 3.2 27.6 3.2 16.7 1.9

Application of the Scaling Law to Other Projectiles and Target Materials Additional data from testing on aluminium alloys with 12.7 mm APM2 projectiles was published by Kymer and Farzinger (1957) and Showalter et al. (2008) and can be used to assess the applicability of the Forrestal et al. scaling law for projectiles other than the 7.62 mm APM2. The results are plotted in Figure 5 together with an evaluation of the predictive accuracy. The quasi-static stress parameter, s, is defined for 7075-T6 using 7075-T651 data from Table 1, for 2024-T4 using 2024-T351 data from Table 1, while for 5059-H131 and 5059-H136 yield strength and elastic modulus are sourced from product technical data sheets and a power coefficient of n = 0.115 is determined from a fit to quasistatic compression data from Pérez-Bergquist et al. (2011). With the exception of the 19.05 mm thick 7075-T6 result, the scaling law is fitted to within 33 m/s (5.2%) of the measured V50. Dey et al. (2004) performed ballistic tests on three steel grades with yield strengths ranging from 499 – 992 MPa. The conical nose, hardened tool steel projectile (51-53 RC) was reported as having perforated all target materials by ductile hole enlargement. As such, the Forrestal scaling law should be applicable. To assess the accuracy of the scaling law applied to steel plates perforated by ductile hole formation, the scaling parameter K was derived from ballistic impact data of the 20 mm calibre conical nose projectile on 5083-H116 aluminium (from Børvik et al., 2004). Mechanical properties Y and E were defined for the three steels in Dey et al. (2004), and strain hardening parameters, n, were empirically fit to quasi-static representations of the Johnson-Cook strength model (modified to include elastic strain). The results of the assessment are summarised in Figure 6. As shown, provided the perforation mechanisms remains ductile hole formation, the Forrestal scaling law remains highly accurate, predicting the V50 of the three steel types within 7.7%. In all cases, the predicted performance was below that of the measured V50, indicating that the calculated stress parameter s may have been overly high. IS3 2015

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Alloy

900

2024-T4 2024-T4 2024-T4 7075-T6 7075-T6 7075-T6 5059-H131 5059-H131 5059-H131 5059-H131 5059-H131 5059-H131 5059-H131 5059-H131 5059-H136 5059-H136 5059-H136

800

V50 (m/s)

700

600 2024‐T4 500

7075‐T6 5059‐H131 5059‐H136

400

K = 80.26 300 5

7

9

11

(sh)1/2

s

h (mm) 19.05 25.4 38.1 19.05 25.4 38.1 47.8 51.2 51.4 61.6 64.2 64.5 76.4 76.6 51.0 64.3 77.1

V50 (m/s) Predict. Meas. 420.8 416.1 485.9 509.0 595.1 618.7 470.6 384.0 543.4 516.6 665.5 632.5 648.2 660.5 671.1 679.7 672.1 670.3 736.3 756.2 751.3 753.5 753.1 768.7 819.5 833.9 820.5 819.6 652.1 647.1 731.7 742.5 801.6 830.3

(GPa) 1.452 1.452 1.452 1.816 1.816 1.816 1.374 1.374 1.374 1.374 1.374 1.374 1.374 1.374 1.302 1.302 1.302

 

Figure 5. Forrestal scaling law applied to aluminium targets impacted by 12.7 mm APM2 projectiles.  

450

Material

400

Weldox 460E Weldox 700E Weldox 900E

V50 (m/s)

350

 

 

Y (MPa) 499 859 992

E (GPa) 210 210 210

n* (-) 0.065 0.030 0.028

 

 

 

* Fit to quasi-static Johnson-Cook strength curve (including elastic strain)

300

 

Material

250 5083‐H116

5083-H116 5083-H116 5083-H116 5083-H116 Weldox 460E Weldox 700E Weldox 900E

Weldox 460E

200

Weldox 700E 150

Weldox 900E K = 52.07

100 3

4

5 (sh)1/2

6

7

8

 

h (mm) 15 20 25 30 12 12 12

 

s (GPa) 1.109 1.109 1.109 1.109 2.209 3.155 3.532

 

V50 (m/s) Predict. Meas. 212.4 216.8 245.2 249 274.2 256.6 300.3 309.7 268.1 290.6 320.4 335 339.0 340.1

 

Figure 6. Application of the Forrestal scaling law to steel targets perforated by 20 mm calibre conical nose hardened steel projectiles by ductile hole formation. Projectile schematic from Dey et al. (2004). Comparison with other methods In Figure 7 the accuracy of the Forrestal scaling law is compared to a cavity expansion theory by Woodward (1978) and the semi-empirical approach specified in the t Technical Coordinating Group for Munitions Effectiveness (JTCG/ME), Anti-Air handbook (1990). The results are plotted in of the ratio of predicted ballistic limit to measured ballistic limit, V50c/V50m. As such, a value smaller than unity indicates a conservative prediction while a value greater than unity is nonconservative. Although all three models consistently provide predictions within 20% of the measured result, the Forrestal model is clearly the most consistently accurate. The Woodward model, also based on cavity expansion theory, is shown to behave similarly to the Forrestal scaling law. The primary difference between the two methods is in specification of the stress term. Forrestal et al. define a quasi-static radial stress based on the target material mechanical properties and strain hardening power law, while Woodward simply uses a flow stress value measured at a true strain of 1.0. For the 7.62 mm APM2 projectiles the flow stress value is shown to provide an excellent agreement with the test data, however for the 12.7 mm APM2 the flow stress term provides an over-prediction of target resistance, while for the steel targets the flow stress term provides an under-prediction (more so for the harder

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steels which display less strain hardening). The predictive accuracy of the JTCG/ME equation varies significantly with target material, and is highly sensitive to the definition of the empirical constants. Of the 53 target configurations (i.e. material type and thickness), empirical constants were defined in the JTCG handbook for 18 configurations, while for the remaining 35 the constants were estimated using the recommended method. 1.4 1.3

V50c/V50m (‐)

1.2 1.1 1 0.9 0.8 0.7

7.62 mm APM2

12.7 mm APM2

0.6 Forrestal

Woodward

20 mm  conical

JTCG/ME

Figure 7. Comparing the predictive accuracy of the Forrestal scaling law, JTCG/ME penetration equation for a sharp attack against a thick , and Woodward’s ductile hole formation model in of the ratio of predicted ballistic limit, V50c, to measured ballistic limit, V50m.

CONCLUSIONS The ballistic limit of plates perforated in ductile hole formation can be estimated by the work done in expanding a cavity from zero initial radius to that of the projectile calibre. Forrestal and co-workers defined a method based on cylindrical cavity expansion theory in which the ballistic limit scales linearly in of plate thickness and a stress term, s, calculated from material elastic modulus, yield strength, and a strain hardening power law fit. The proposed technique was evaluated for a wide range of ballistic tests against aluminium targets with 7.62 mm and 12.7 mm calibre armour piercing projectiles, as well as a selection of steel targets against 20 mm conical-nose hardened steel projectiles that were experimentally observed to fail in ductile hole formation. The predictions of the Forrestal scaling law were consistently in very good agreement with the experimentally-measured V50 (typically  5%). The accuracy of the approach was found to be superior to that of an earlier, more simplified ductile hole formation theory and a widely used empirical equation from the t Technical Coordinating Group for Munitions Effectiveness (JTCG/ME) penetration equation handbook.

ACKNOWLEDGMENTS The contributions of Frank Griffo (DSTO Melbourne), Steve Pattie (formerly DSTO Melbourne), and Darren Edwards (DSTO Melbourne) in conducting the reported ballistic tests and providing additional ballistic data are gratefully acknowledged.

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REFERENCES Backman, M.E., Goldsmith, W. (1978) “The Mechanics of Penetration of Projectiles into Targets”, International Journal of Engineering Science, Vol. 16(1), pp. 1-99. Ballistic Analysis Laboratory (1961) The Resistance of Various Metallic Materials to Perforation by Steel Fragments, Empirical Relationships for Fragment Residual Velocity and Residual Weight, Ballistic Research Laboratories, Aberdeen Proving Ground. Birkhoff, G., MacDougall, D.P., Pugh, E.M., Taylor, G. (1948) “Explosives with Lined Cavities”, Journal of Applied Physics, Vol. 19, No. 6, pp. 563-582. Børvik, T., Clausen, A.H., Hopperstad, O.S., Langseth, M. (2004) “Perforation of AA5083-H116 Aluminium Plates with Conical-Nose Steel Projectiles – Experimental Study”, International Journal of Impact Engineering, Vol. 30(4), pp. 367-384. Børvik, T., Forrestal, M.J., Warren, T.L. (2010) “Perforation of 5083-H116 Aluminum Armor Plates with Ogive-Nose Rods and 7.62 mm APM2 Bullets”, Exp. Mechanics, Vol. 50, pp. 969-978. Casem, D.T., Dandekar, D.P. (2012) “Shock and Mechanical Response of 2139-T8 Aluminum”, Journal of Applied Physics, Vol. 142(3-4), pp. 269-301. Cheeseman, B., Gooch Jr, W.A., Burkins, M.S. (2008) “Ballistic Evaluation of Aluminum 2139-T8”, Proceedings of the 24th International Symposium on Ballistics, 22-26 September, New Orleans. Dey, S., Børvik, T., Hopperstad, O.S., Leinum, J.R., Langseth, M. (2004) “The Effect of Target Strength on the Perforation of Steel Plates Using Three Different Projectile Nose Shapes”, International Journal of Impact Engineering, Vol. 30, pp. 1005-1038. Forrestal, M.J., Luk, V.K., Brar, N.S. (1990) “Perforation of Aluminum Armor Plates with ConicalNose Projectiles”, Mechanics of Materials, Vol. 10, pp. 97-105. Forrestal, M.J., Warren, T.L. (2009) “Perforation Equations for Conical and Ogival Nose Rigid Projectiles into Aluminum Target Plates”, Int. J. of Impact Engineering, Vol. 36, pp. 220-225. Forrestal, M.J., Børvik, T., Warren, T.L. (2010) “Perforation of 7075-T651 Aluminum Armor Plates with 7.62 mm APM2 Bullets”, Experimental Mechanics, Vol. 50, pp. 1245-1251. Forrestal, M.J., Børvik, T., Warren, T.L., Chen, W. (2014) “Perforation of 6082-T651 Aluminum Plates with 7.62 mm APM2 Bullets at Normal and Oblique Impacts”, Experimental Mechanics, Vol. 54, pp. 471-481. Gooch Jr, W.A., Burkins, M.S., Squillacioti, R.J. (2007) “Ballistic Testing of Commercial Aluminum Alloys and Alternative Processing Technologies to Increase the Availability of Aluminum Armor”, Proceedings of the 23rd International Symposium on Ballistics, 16-20 April, Tarragona. t Technical Coordinating Group for Munitions Effectiveness, Anti-Air (1985), Penetration Equations Handbook for Kinetic Energy Penetrators, Aberdeen Proving Ground. Kymer, J.R., Farzinger, H.E. (1957) Ballistic Tests of 2024-T4 and 7075-T6 Aluminum Alloys, Frankford Arsenal, Philadelphia, R-1395. Moynihan, T.J., Chou, S.-C., Mihalcin, A.L. (2000) Application of the Depth-of-Penetration Test Methodology to Characterize Ceramics for Personnel Protection, Army Research Laboratory, Aberdeen Proving Ground, ARL-TR-2219. Pérez-Bergquist, S.J., Gray III, G.T., Cerreta, E.K., Trujillo, C.P., Pérez-Bergquist, A. (2011) “The Dynamic and Quasi-Static Mechanical Response of Three Aluminum Armor Alloys: 5059, 5083 and 7039”, Materials Science and Engineering A, Vol. 528, pp. 8733-8741. Piekutowski, A.J., Forrestal, M.J., Poorman, K.L., Warren, T.L. (1996) “Perforation of Aluminum Plates with Ogive-Nose Steel Rods at Normal and Oblique Impacts”, International Journal of Impact Engineering, Vol. 18, pp. 877-887. Showalter, D.D., Placzankis, B.E., Burkins, M.S. (2008) Ballistic Performance Testing of Aluminum 5059-H131 and 5059-H136 for Armour Applications, US Army Research Laboratory, Aberdeen Proving Ground, ARL-TR-4427. Skube, S. (2009) Large Strain Compression Testing of Ductile Materials at Quasi-Static and Dynamic Strain Rates, Masters Dissertation, Purdue University. US Department of Defense. (1997) V50 Ballistic Test for Armor, MIL-STD-662F. Walker, J.D., Anderson Jr., C.E. (1995) “A Time-Dependent Model for Long-Rod Penetration”, International Journal of Impact Engineering, Vol. 16, No. 1, pp. 19-48. Woodward, R. (1978) “The Penetration of Metal Targets by Conical Projectiles”, International Journal of Mechanical Sciences, Vol. 20, No. 9, pp. 599-607.

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