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Scale-independent description of the rigid-body penetration of spherical projectiles into semi-infinite adobe targets
International Journal of Impact Engineering 75 (2015) 27e29
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Short communication
Scale-independent description of the rigid-body penetration of spherical projectiles into semi-infinite adobe targets A. Heine*, M. Wickert Fraunhofer EMI, Eckerstr. 4, 79104 Freiburg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 June 2014 Received in revised form 18 July 2014 Accepted 19 July 2014 Available online 31 July 2014
In impact experiments, the penetration depth of spherical projectiles into semi-infinite adobe targets has been investigated. In total, 20 tests were performed with steel projectiles of diameters 7.5, 10.0 and 13.5 mm in the velocity regime from 240 to 3230 m/s. In addition, 2 single experiments with aluminum and titanium projectiles were conducted at impact velocities of 830 and 920 m/s, respectively. After an appropriate normalization, all penetration depth vs. velocity data up to impact velocities of around 1500 m/s can be described by one linear curve. This is the velocity regime where projectile fragmentation does not yet occur. The observed linear relation is discussed in the context of a recently published analytical model and shows the scalability of penetration tests also for the considered adobe target material of low strength and low density. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Penetration Adobe
1. Introduction The penetration depth of a projectile in a semi-infinite target is the simplest standard test in ballistics. It also depicts probably the kind of tests that mostly provides information on the target rather than on the projectile material, especially in the case of non-fragmenting projectiles and symmetrical impact conditions. Immense data exists for that kind of experiments, especially for traditional metallic armor materials such as steel or aluminum alloys, e.g. Refs. [1e4]. The semi-infinite penetration has also been investigated for a variety of other materials including ceramics, e.g. Refs. [5], soil [6], concrete [7e10], or sand [11e14]. A material of recent interest regarding the experimental and numerical analysis of the involved penetration processes is adobe, which is a building material characterized by a low strength and a low density. Interesting penetration phenomena have been observed for this material [15e18], requiring a deeper scientific investigation. In this context, aspects of the semi-infinite penetration of non-deforming projectiles and the scalability of ballistic tests are the most obvious questions. 2. Experimental data The performed tests are summarized in Table 1. Except for in two tests, all projectiles were steel spheres with diameters D of * Corresponding author. Tel.: þ49 761 2714 435; fax: þ49 761 2714 1435. E-mail address: [email protected] (A. Heine). http://dx.doi.org/10.1016/j.ijimpeng.2014.07.009 0734-743X/© 2014 Elsevier Ltd. All rights reserved.
7.5, 10.0 or 13.5 mm, i.e. there is a factor of 1.8 between the dimensions of the smallest and the largest projectiles used. The investigated impact velocities were 690e2040 m/s for the D ¼ 7.5 mm projectiles (4 tests), 260e3230 m/s for the D ¼ 10.0 mm projectiles (6 tests), and 240e2040 m/s for the D ¼ 13.5 mm projectiles (10 tests). In two additional tests with D ¼ 10.0 mm aluminum and titanium spheres, the impact velocities were 830 and 920 m/s. With these parameter ranges for projectile diameter and velocity, the full regime of rigid-body penetration could be covered, as for the highest velocity investigated for each diameter, projectile fragmentation already occurred. Furthermore, the two tests with non-steel projectiles allow for an analysis of the influence of the projectile material (density) within the rigid-body penetration regime. A subset of the test data has already been presented in Ref. [17]. The laboratory targets were composed of adobe bricks in front of a steel backing, laterally confined by a steel frame as shown schematically in Fig. 1. The target thickness was in the range of 140 to 280 mm and at least about two times the measured depth of penetration in each test. The typical lateral dimensions of the adobe package inside the steel frame were 120 mm by 240 mm for the adobe material. This means that the penetration crater was far away from any target edges, thus the target was effectively semi-infinite. The adobe material is characterized by a density of 1.8 g/cm3 and a quasi-static compressive strength of 4 N/mm2.
28
A. Heine, M. Wickert / International Journal of Impact Engineering 75 (2015) 27e29
Table 1 Summary of all experiments (semi-infinite penetration data). Test No.
Projectile material
Projectile D [mm]
Projectile v [m/s]
Penetration P [mm]
Projectile fragmented?
12756 12754 12758 12759 12760 12750 12753 12751 12752 12138 12757 12749 11938 11937 12027 12029 11939 12035 12038 11949 12136 12137
Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Aluminum Titanium
7.5 7.5 7.5 7.5 10.0 10.0 10.0 10.0 10.0 10.0 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 10.0 10.0
690 1290 1530 2040 260 670 680 1190 1460 3230 240 480 610 900 1150 1200 1460 1510 1950 2040 830 920
41 75 79 77 16 63 61 90 104 67 23 58 61 90 126 129 125 157 131 125 24 42
No No No Yes No No No No No Yes No No No No No No Yes No Yes Yes No No
3. Analysis and interpretation The experimental penetration depths are plotted in Fig. 2 versus the impact velocity v. In order to guide the analysis, all penetration depths P were divided by the diameter D and the density rP of the used projectiles in Fig. 3. Several observations are made from Fig. 3. There is a maximum for the normalized penetration depth of 1.5 cm3/g that occurs for an impact velocity of around 1500 m/s. If the velocity is increased above 1500 m/s, the depth of penetration does not further increase, but drops below the already achieved level. Such phenomena have been observed for different target materials, e.g. for aluminum alloys [4], sand [12], or graphite [19]. The onset of such a change in behavior typically characterizes a change in the penetration mechanics, such as from rigid-body penetration to projectile failure. Also in the present case, the transition observed around 1500 m/s is correlated with a fracturing of the projectiles, see Table 1. Furthermore, in the velocity regime below 1500 m/s impact velocity, i.e. in the velocity regime below the transition from rigidbody penetration to projectile failure, as a result of the normalization all data coalesce on one straight line. Qualitatively, this indicates that the involved processes are scalable. In addition, this result may be interpreted through the application of a model description given in Ref. [19] for a graphite target. According to this, the penetration depth can be calculated as:
Fig. 1. Schematic target configuration.
Fig. 2. Experimental penetration depths e absolute values e versus the impact velocity.
P¼
m $v ar0 C0 S
In this equation, m denotes the projectile mass, a is a shape coefficient, r0 and C0 are the initial target density and bulk sound velocity, respectively, and S is the cross section of the projectile. If the projectile mass and cross section are expressed via the projectile diameter D and the projectile density rP, this may be rewritten as:
1 P 2 ¼ $v rP D 3ar0 C0 This is exactly the linear relation emerging from Fig. 3. The derivation of the above equation given in Ref. [19] is based on shock wave theory and thus valid in a regime of impact pressures significantly exceeding the target strength. This criterion is surely met here, as for a steel projectile (rP ¼ 7.85 g/cm3) and an even low impact velocity of v ¼ 100 m/s a dynamic pressure of 40 MPa results, which is 10 times the quasi-static compressive strength of the target material. Thus, this simple model provides an approach for modeling the experimental data that is an alternative to a description based on a force-law approach, as discussed in Refs.
Fig. 3. Experimental penetration depths e normalized by projectile diameter and density e versus the impact velocity.
A. Heine, M. Wickert / International Journal of Impact Engineering 75 (2015) 27e29
[17], following the spirit of [11] and there references given therein. In this context, the experimental data may also serve to refine and validate penetration formulae based on the dynamic cavityexpansion model, e.g. those from Refs. [20,21], in order to apply them to adobe. Nonetheless, the model from Ref. [19] provides already an excellent and elegant description without large complexity. 4. Summary The semi-infinite penetration of spherical projectiles into adobe was analyzed based on experimental results. In an impact velocity regime of up to 1500 m/s, rigid-body penetration persists and the experimental data are in good agreement with an analytical model. Moreover, a scale-independent description of the rigid-body penetration of projectiles in semi-infinite adobe targets is feasible. Acknowledgment The authors thank K. E. Weber for the careful acquisition and compilation of the used penetration data. Financial support by BMVg, is acknowledged. References [1] Anderson Jr CE, Morris BL, Littlefield DL. A penetration mechanics database. San Antonio, TX, USA: Southwest Research Institute; January 1992. Report 3593/001. [2] Riegel III JP. Terminal ballistics data review and analysis. In: Proceedings of the 27th international symposium on ballistics, Freiburg, Germany. Lancaster, PA, USA: DEStech Publications, Inc.; 2013. p. 1453e64. [3] Hohler V, Stilp AJ. Penetration of steel and high density rods in semi-infinite steel targets. In: 3rd international symposium on ballistics, Karlsruhe, Germany; 1977. [4] Wickert M. Penetration data for a medium caliber tungsten sinter alloy penetrator into aluminum alloy 7020 in the velocity regime from 250 m/s to 1900 m/s. In: Proceedings of the 23rd international symposium on ballistics, Tarragona, Spain; 2007. p. 1437e44. [5] Behner T, Anderson Jr CE, Holmquist TJ, Orphal DL, Wickert M, Templeton DW. Penetration dynamics and interface defeat capability of silicon carbide against long rod impact. Int J Impact Eng 2011;38:419e25.
29
[6] Young CW. Penetration equations. Sandia Contractor Report SAND97e2426. Albuquerque, NM, USA: Sandia National Laboratories; October 1997. [7] Lampert S, Jeanquartier R. Perforation of concrete targets by an eroding tungsten-alloy rod. In: Proceedings of the 22nd international symposium on ballistics, Vancouver, Canada. Lancaster, PA, USA: DEStech Publications, Inc.; 2005. p. 838e43. [8] Murphy MJ, Kuklo RM. Fundamentals of shaped charge penetration in concrete. In: Proceedings of the 18th international symposium on ballistics, San Antonio, TX, USA. Lancaster, PA, USA: Technomic Publishing Co., Inc.; 1999. p. 1057e64. [9] Rottenkolber E, Heider N. Numerical simulation of runway penetration. In: Proceedings of the 10th international symposium on ballistics, San Diego, CA, USA; 1987. [10] Riedel W. 10 years RHT: a review of concrete modelling and hydrocode applications. In: Hiermaier S, editor. Predictive modeling of dynamic processes. US: Springer; 2009. p. 143e65. [11] Allen WA, Mayfield EB, Morrison HL. Dynamics of a projectile penetrating sand. J Appl Phys 1957;28:370e6. [12] Schneider E, Stilp A. Projectile penetration into low density media. In: Proceedings of the 8th international symposium on ballistics, Orlando, FL, USA; 1984. [13] Dwivedi SK, Teeter RD, Felice CW, Gupta YM. Two dimensional mesoscale simulation of projectile instability during penetration in dry sand. J Appl Phys 2008;104(083502). [14] Bless SJ, Berry DT, Pedersen B, Satapathy S, Warren T, Lawhorn W. High-speed projectile penetration in sand. In: Proceedings of the 25th international symposium on ballistics, Beijing, China; 2010. p. 1339e46. [15] Meyer CS. Modeling experiments of hypervelocity penetration of adobe by spheres and rods. In: Proceedings of the 12th hypervelocity impact symposium, Baltimore, MD, USA; 2013. Procedia Engineering; 58: 138e46. [16] Heine A, Weber KE, Wickert M. Perforation of adobe targets by projctiles e effects of projectile shape. In: Proceedings of the 15th international symposium on the interaction of the effects of munitions with structures, Potsdam, Germany; 2013. [17] Heine A, Weber KE, Wickert M. Experimental Investigation of the penetration and perforation of building materials by projectiles. In: Proceedings of the 26th international symposium on ballistics, Miami, FL, USA. Lancaster, PA, USA: DEStech Publications, Inc.; 2011. p. 1669e78. [18] Chanteret PY, Reck B. Shaped charge jet penetration into masonry materials. In: Proceedings of the 27th international symposium on ballistics, Freiburg, Germany. Lancaster, PA, USA: DEStech Publications, Inc.; 2013. p. 1516e27. bert D, Hallo L, Chevalier J-M, Guillet F, Berthe L, et al. Pene[19] Seisson G, He tration and cratering experiments of graphite by 0.5-mm-diameter steel spheres at various impact velocities. Int J Impact Eng 2014;70:14e20. [20] Chen XW, Li QM. Deep penetration of a non-deformable projectile with different geometrical characteristics. Int J Impact Eng 2002;27:619e37. [21] Li QM, Chen XW. Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile. Int J Impact Eng 2003;28: 93e116.
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Short communication
Scale-independent description of the rigid-body penetration of spherical projectiles into semi-infinite adobe targets A. Heine*, M. Wickert Fraunhofer EMI, Eckerstr. 4, 79104 Freiburg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 June 2014 Received in revised form 18 July 2014 Accepted 19 July 2014 Available online 31 July 2014
In impact experiments, the penetration depth of spherical projectiles into semi-infinite adobe targets has been investigated. In total, 20 tests were performed with steel projectiles of diameters 7.5, 10.0 and 13.5 mm in the velocity regime from 240 to 3230 m/s. In addition, 2 single experiments with aluminum and titanium projectiles were conducted at impact velocities of 830 and 920 m/s, respectively. After an appropriate normalization, all penetration depth vs. velocity data up to impact velocities of around 1500 m/s can be described by one linear curve. This is the velocity regime where projectile fragmentation does not yet occur. The observed linear relation is discussed in the context of a recently published analytical model and shows the scalability of penetration tests also for the considered adobe target material of low strength and low density. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Penetration Adobe
1. Introduction The penetration depth of a projectile in a semi-infinite target is the simplest standard test in ballistics. It also depicts probably the kind of tests that mostly provides information on the target rather than on the projectile material, especially in the case of non-fragmenting projectiles and symmetrical impact conditions. Immense data exists for that kind of experiments, especially for traditional metallic armor materials such as steel or aluminum alloys, e.g. Refs. [1e4]. The semi-infinite penetration has also been investigated for a variety of other materials including ceramics, e.g. Refs. [5], soil [6], concrete [7e10], or sand [11e14]. A material of recent interest regarding the experimental and numerical analysis of the involved penetration processes is adobe, which is a building material characterized by a low strength and a low density. Interesting penetration phenomena have been observed for this material [15e18], requiring a deeper scientific investigation. In this context, aspects of the semi-infinite penetration of non-deforming projectiles and the scalability of ballistic tests are the most obvious questions. 2. Experimental data The performed tests are summarized in Table 1. Except for in two tests, all projectiles were steel spheres with diameters D of * Corresponding author. Tel.: þ49 761 2714 435; fax: þ49 761 2714 1435. E-mail address: [email protected] (A. Heine). http://dx.doi.org/10.1016/j.ijimpeng.2014.07.009 0734-743X/© 2014 Elsevier Ltd. All rights reserved.
7.5, 10.0 or 13.5 mm, i.e. there is a factor of 1.8 between the dimensions of the smallest and the largest projectiles used. The investigated impact velocities were 690e2040 m/s for the D ¼ 7.5 mm projectiles (4 tests), 260e3230 m/s for the D ¼ 10.0 mm projectiles (6 tests), and 240e2040 m/s for the D ¼ 13.5 mm projectiles (10 tests). In two additional tests with D ¼ 10.0 mm aluminum and titanium spheres, the impact velocities were 830 and 920 m/s. With these parameter ranges for projectile diameter and velocity, the full regime of rigid-body penetration could be covered, as for the highest velocity investigated for each diameter, projectile fragmentation already occurred. Furthermore, the two tests with non-steel projectiles allow for an analysis of the influence of the projectile material (density) within the rigid-body penetration regime. A subset of the test data has already been presented in Ref. [17]. The laboratory targets were composed of adobe bricks in front of a steel backing, laterally confined by a steel frame as shown schematically in Fig. 1. The target thickness was in the range of 140 to 280 mm and at least about two times the measured depth of penetration in each test. The typical lateral dimensions of the adobe package inside the steel frame were 120 mm by 240 mm for the adobe material. This means that the penetration crater was far away from any target edges, thus the target was effectively semi-infinite. The adobe material is characterized by a density of 1.8 g/cm3 and a quasi-static compressive strength of 4 N/mm2.
28
A. Heine, M. Wickert / International Journal of Impact Engineering 75 (2015) 27e29
Table 1 Summary of all experiments (semi-infinite penetration data). Test No.
Projectile material
Projectile D [mm]
Projectile v [m/s]
Penetration P [mm]
Projectile fragmented?
12756 12754 12758 12759 12760 12750 12753 12751 12752 12138 12757 12749 11938 11937 12027 12029 11939 12035 12038 11949 12136 12137
Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Steel Aluminum Titanium
7.5 7.5 7.5 7.5 10.0 10.0 10.0 10.0 10.0 10.0 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 10.0 10.0
690 1290 1530 2040 260 670 680 1190 1460 3230 240 480 610 900 1150 1200 1460 1510 1950 2040 830 920
41 75 79 77 16 63 61 90 104 67 23 58 61 90 126 129 125 157 131 125 24 42
No No No Yes No No No No No Yes No No No No No No Yes No Yes Yes No No
3. Analysis and interpretation The experimental penetration depths are plotted in Fig. 2 versus the impact velocity v. In order to guide the analysis, all penetration depths P were divided by the diameter D and the density rP of the used projectiles in Fig. 3. Several observations are made from Fig. 3. There is a maximum for the normalized penetration depth of 1.5 cm3/g that occurs for an impact velocity of around 1500 m/s. If the velocity is increased above 1500 m/s, the depth of penetration does not further increase, but drops below the already achieved level. Such phenomena have been observed for different target materials, e.g. for aluminum alloys [4], sand [12], or graphite [19]. The onset of such a change in behavior typically characterizes a change in the penetration mechanics, such as from rigid-body penetration to projectile failure. Also in the present case, the transition observed around 1500 m/s is correlated with a fracturing of the projectiles, see Table 1. Furthermore, in the velocity regime below 1500 m/s impact velocity, i.e. in the velocity regime below the transition from rigidbody penetration to projectile failure, as a result of the normalization all data coalesce on one straight line. Qualitatively, this indicates that the involved processes are scalable. In addition, this result may be interpreted through the application of a model description given in Ref. [19] for a graphite target. According to this, the penetration depth can be calculated as:
Fig. 1. Schematic target configuration.
Fig. 2. Experimental penetration depths e absolute values e versus the impact velocity.
P¼
m $v ar0 C0 S
In this equation, m denotes the projectile mass, a is a shape coefficient, r0 and C0 are the initial target density and bulk sound velocity, respectively, and S is the cross section of the projectile. If the projectile mass and cross section are expressed via the projectile diameter D and the projectile density rP, this may be rewritten as:
1 P 2 ¼ $v rP D 3ar0 C0 This is exactly the linear relation emerging from Fig. 3. The derivation of the above equation given in Ref. [19] is based on shock wave theory and thus valid in a regime of impact pressures significantly exceeding the target strength. This criterion is surely met here, as for a steel projectile (rP ¼ 7.85 g/cm3) and an even low impact velocity of v ¼ 100 m/s a dynamic pressure of 40 MPa results, which is 10 times the quasi-static compressive strength of the target material. Thus, this simple model provides an approach for modeling the experimental data that is an alternative to a description based on a force-law approach, as discussed in Refs.
Fig. 3. Experimental penetration depths e normalized by projectile diameter and density e versus the impact velocity.
A. Heine, M. Wickert / International Journal of Impact Engineering 75 (2015) 27e29
[17], following the spirit of [11] and there references given therein. In this context, the experimental data may also serve to refine and validate penetration formulae based on the dynamic cavityexpansion model, e.g. those from Refs. [20,21], in order to apply them to adobe. Nonetheless, the model from Ref. [19] provides already an excellent and elegant description without large complexity. 4. Summary The semi-infinite penetration of spherical projectiles into adobe was analyzed based on experimental results. In an impact velocity regime of up to 1500 m/s, rigid-body penetration persists and the experimental data are in good agreement with an analytical model. Moreover, a scale-independent description of the rigid-body penetration of projectiles in semi-infinite adobe targets is feasible. Acknowledgment The authors thank K. E. Weber for the careful acquisition and compilation of the used penetration data. Financial support by BMVg, is acknowledged. References [1] Anderson Jr CE, Morris BL, Littlefield DL. A penetration mechanics database. San Antonio, TX, USA: Southwest Research Institute; January 1992. Report 3593/001. [2] Riegel III JP. Terminal ballistics data review and analysis. In: Proceedings of the 27th international symposium on ballistics, Freiburg, Germany. Lancaster, PA, USA: DEStech Publications, Inc.; 2013. p. 1453e64. [3] Hohler V, Stilp AJ. Penetration of steel and high density rods in semi-infinite steel targets. In: 3rd international symposium on ballistics, Karlsruhe, Germany; 1977. [4] Wickert M. Penetration data for a medium caliber tungsten sinter alloy penetrator into aluminum alloy 7020 in the velocity regime from 250 m/s to 1900 m/s. In: Proceedings of the 23rd international symposium on ballistics, Tarragona, Spain; 2007. p. 1437e44. [5] Behner T, Anderson Jr CE, Holmquist TJ, Orphal DL, Wickert M, Templeton DW. Penetration dynamics and interface defeat capability of silicon carbide against long rod impact. Int J Impact Eng 2011;38:419e25.
29
[6] Young CW. Penetration equations. Sandia Contractor Report SAND97e2426. Albuquerque, NM, USA: Sandia National Laboratories; October 1997. [7] Lampert S, Jeanquartier R. Perforation of concrete targets by an eroding tungsten-alloy rod. In: Proceedings of the 22nd international symposium on ballistics, Vancouver, Canada. Lancaster, PA, USA: DEStech Publications, Inc.; 2005. p. 838e43. [8] Murphy MJ, Kuklo RM. Fundamentals of shaped charge penetration in concrete. In: Proceedings of the 18th international symposium on ballistics, San Antonio, TX, USA. Lancaster, PA, USA: Technomic Publishing Co., Inc.; 1999. p. 1057e64. [9] Rottenkolber E, Heider N. Numerical simulation of runway penetration. In: Proceedings of the 10th international symposium on ballistics, San Diego, CA, USA; 1987. [10] Riedel W. 10 years RHT: a review of concrete modelling and hydrocode applications. In: Hiermaier S, editor. Predictive modeling of dynamic processes. US: Springer; 2009. p. 143e65. [11] Allen WA, Mayfield EB, Morrison HL. Dynamics of a projectile penetrating sand. J Appl Phys 1957;28:370e6. [12] Schneider E, Stilp A. Projectile penetration into low density media. In: Proceedings of the 8th international symposium on ballistics, Orlando, FL, USA; 1984. [13] Dwivedi SK, Teeter RD, Felice CW, Gupta YM. Two dimensional mesoscale simulation of projectile instability during penetration in dry sand. J Appl Phys 2008;104(083502). [14] Bless SJ, Berry DT, Pedersen B, Satapathy S, Warren T, Lawhorn W. High-speed projectile penetration in sand. In: Proceedings of the 25th international symposium on ballistics, Beijing, China; 2010. p. 1339e46. [15] Meyer CS. Modeling experiments of hypervelocity penetration of adobe by spheres and rods. In: Proceedings of the 12th hypervelocity impact symposium, Baltimore, MD, USA; 2013. Procedia Engineering; 58: 138e46. [16] Heine A, Weber KE, Wickert M. Perforation of adobe targets by projctiles e effects of projectile shape. In: Proceedings of the 15th international symposium on the interaction of the effects of munitions with structures, Potsdam, Germany; 2013. [17] Heine A, Weber KE, Wickert M. Experimental Investigation of the penetration and perforation of building materials by projectiles. In: Proceedings of the 26th international symposium on ballistics, Miami, FL, USA. Lancaster, PA, USA: DEStech Publications, Inc.; 2011. p. 1669e78. [18] Chanteret PY, Reck B. Shaped charge jet penetration into masonry materials. In: Proceedings of the 27th international symposium on ballistics, Freiburg, Germany. Lancaster, PA, USA: DEStech Publications, Inc.; 2013. p. 1516e27. bert D, Hallo L, Chevalier J-M, Guillet F, Berthe L, et al. Pene[19] Seisson G, He tration and cratering experiments of graphite by 0.5-mm-diameter steel spheres at various impact velocities. Int J Impact Eng 2014;70:14e20. [20] Chen XW, Li QM. Deep penetration of a non-deformable projectile with different geometrical characteristics. Int J Impact Eng 2002;27:619e37. [21] Li QM, Chen XW. Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile. Int J Impact Eng 2003;28: 93e116.