FINITE ELEMENT ANALYSIS OF IMPACT AND PENETRATION OF POLYCARBONATE PLATE BY A RIGID SPHERICAL PROJECTILE C.T. Tsai Department of Mechanical Engineering Florida Atlantic University Boca Raton, FL 33431, USA Arnold Mayer Air Force Research Laboratory AFRL/VAVS WPAFB, OH 45433-7653, USA Abstract: The objective of this paper is to perform finite element analysis of the impact and penetration of polycarbonate thermoplastic plates by a rigid spherical projectile. An explicit finite element code, ABAQUS, has been used for this analysis. Four different thicknesses of polycarbonate plates with 20.32 cm in diameter are struck by rigid carbon steel spherical projectile. The diameter of the projectile is 1.27 cm. A bilinear elastoplastic constitutive equation for polycarbonate is used to include the plastic deformation in this analysis. Various impact velocities of projectile are used to strike the polycarbonate plate. The numerical results shown that the curves of residual velocity ratio (residual velocity/impact velocity) versus impact velocity ratio (impact velocity/ballistic limit) are almost coincidence for four polycarbonate plates regardless of their thickness. This result agrees well with the experimental discovery. Key Words: Penetration, Impact, Polycarbonate plate, finite element analysis. 1 Introduction Polycarbonate plate is used as a lightweight transparent armor for protection against projectiles in the subordnance velocity range. Its applications range from safety goggles, industrial machine guards, police riot shields to aircraft windshields. Polycarbonate is a polymer that has an unusually high yield strain and ductility. When combining with a significant amount of strain hardening, polycarbonates display superior impact and perforation resistance compared with other polymers, or even with some structural metals. Due to the extensive use of polycarbonate in impact and safety application, Wright et al. [1,2] have conducted a theoretical and experimental investigation of ballistic impact of polycarbonate to understand the penetration and perforation resistance of the material. The characterizations of the deformation and fracture behavior of the polycarbonate have also been investigated by Fleck et al. [3] and Wright [4]. U.S. Air Force Research Laboratory [5] also conducted a series of ballistic experiments on polycarbonate plates to study their penetration and perforation resistance. However, very little work has done on the numerical study of the penetration and perforation resistance of polycarbonate. This report presents the investigation of the ballistic impact of polycarbonate plate struck by rigid steel projectiles using finite element model. 2 Finite Element Modeling of Ballistic Impact ABAQUS is an explicit finite element software for solving the equation of dynamic equilibrium:
T Mu& = F ext ε σdv, V where M is the mass matrix, u& & is the acceleration, F ext is the external force, ε is the strain tensor and σ is the stress tensor in the domain V. ABAQUS/Explicit solves these equations using an explicit time integration scheme at a very small time step. The polycarbonate plate is 20.32 cm in diameter with four different thicknesses, 0.2032 cm, 0.4572 cm, 0.6096 cm, and 0.9144 cm. The diameter of the rigid spherical steel projectile is 1.27 cm. The mass density of steel is 7850 Kg/m 3. For polycarbonate, the mass density is1200 Kg/m 3, Young s modulus is 2.7x10 9 Pa, Poisson s ratio is 0.42, yield stress is 7x10 7 Pa, the tangent modulus of the linear curve after yield is 7.2x10 6 Pa, and the equivalent plastic strain at failure is 1.5. These material constants indicate that the elastoplastic constitutive behavior of polycarbonate is represented by a bilinear curve. The polycarbonate will be broken at the position where the equivalent plastic strain is exceeding 1.5. The mesh of the finite element model is shown in Fig. 1. Since the diameter of the projectile is much smaller than the diameter of the polycarbonate plate and the impact velocity is high (from 43 m/sec to 375 m/sec), the dominated deformation behavior is expected to be occurred near the contact area between the projectile and polycarbonate plate. Therefore, the diameter of the plate used in the finite element model can be chosen as 6.35 cm by using infinite elements around the outside edge of the model. The infinite elements are used to model the constrained boundary conditions around the outside edge of the 20.32 cm diameter plate through a 6.35 cm diameter model. The projectile strikes the center of the polycarbonate at a perpendicular angle. The size of the finite element model can be further reduced by half due to the symmetry of the problem. Different impact velocities are used to strike the polycarbonate plates with various thicknesses. The results are shown in Table 1. The ballistic limits are 57, 89, 102, and 125 m/sec for thickness of 0.2032, 0.4572, 0.6096, and 0.9144 cm, respectively. The ballistic limit of a ploycarbonate plate stroke by a projectile is defined as the initial impact velocity that will result a zero residual velocity after the strike. The ballistic limit will increase with the increase in the thickness. A curve of ballistic limit versus thickness is plotted in Fig. 2. This curve can be approximated by a power law: 0.516 V B = 39.7h, where V B is the ballistic limit (m/sec) and h is the thickness (mm) of polycarbonate plate. 2.1 Case 1: Polycarbonate thickness is 0.2032 cm The ballistic limit for this case is 57 m/sec as shown in Table 1. The deformation plots are shown in Fig. 3(a) for 0.0012 sec. The plots clearly show the penetration of the projectile through the plate. As the impact velocity decreases to 56 m/sec that is very close to the ballistic limit, the deformation plots are shown in Fig. 3(b) for 0.0012 sec. Even though the projectile is rebounded at 8.3 m/sec, the center of the plate is already broken. As the impact velocity of the projectile further decreases to 43 m/sec (75% of the ballistic limit), the projectile is rebounded at 10.7 m/sec. The plate has only shown denting on the location of impact as shown in Fig. 3(c) for 0.0012 sec. As the impact velocity increases to a large value of 171 m/sec (3 times of the ballistic limit), the projectile penetrates the plate very quickly at the residual velocity of 160 m/sec as shown in Fig. 3(d) for 0.0002 sec. It clearly shows that the impact velocity decreases only 6.4% after penetration. The residual velocity ratio (residual velocity V2/impact velocity V1) versus impact velocity is shown in Fig. 4. It clearly shows that the ratio approaches to one as the impact velocity increases to a large value. 2.2 Case 2: Ploycarbonate thickness is 0.4572 cm The ballistic limit for this case is 89 m/sec as shown in Table 1. The deformation plots are shown in Fig. 5(a) for 0.0006 sec. The plots clearly show the penetration of the projectile through the plate. As the impact velocity decreases to 88 m/sec that is very close to the ballistic limit, the deformation plots are shown in Fig. 5(b) for 0.0006 sec. Even though the projectile is rebounded at 13.7 m/sec, the center of the plate is already broken into small debris. As the impact velocity of the projectile further decreases to 67 m/sec (75% of the ballistic limit), the projectile is rebounded at 16.6 m/sec. The plate has only shown denting on the location of impact as shown in Fig. 5(c) for 0.0012 sec. As the impact velocity increases to a large value of 267 m/sec (3 times of the ballistic limit), the projectile penetrates the plate very quickly at a residual velocity of 249 m/sec as shown in Fig. 5(d) for 0.0001 sec. It clearly shows that
the impact velocity decreases only 6.7% after penetration. The residual velocity ratio (residual velocity V2/impact velocity V1) versus impact velocity is shown in Fig. 4. It clearly shows that the ratio approaches to one as the impact velocity increases to a large value. 2.3 Case 3: Ploycarbonate thickness is 0.6096 cm The ballistic limit for this case is 102 m/sec as shown in Table 1. The deformation plots are shown in Fig. 6(a) for 0.0006 sec. The plots clearly show the penetration of the projectile through the plate. As the impact velocity decreases to 101 m/sec that is very close to the ballistic limit, the deformation plots are shown in Fig. 6(b) for 0.0005 sec. Even though the projectile is rebounded at 4.3 m/sec, the center of the plate is already broken into small debris. As the impact velocity of the projectile further decreases to 75 m/sec (75% of the ballistic limit), the projectile is rebounded at 19.5 m/sec. The plate has only shown denting on the location of impact as shown in Fig. 6(c) for 0.0012 sec. As the impact velocity increases to a large value of 303 m/sec (3 times of the ballistic limit), the projectile penetrates the plate very quickly at a residual velocity of 281.8 m/sec as shown in Fig. 6(d) for 0.000075 sec. The debris are broken not only below the plat but also above the plate. It clearly shows that the impact velocity decreases only 7% after penetration. As the plate thickness increases to 0.6096 cm, the plate does not deflect as much as those shown in cases 1 and 2 for smaller plate thickness of 0.4572 and 0.2032 cm. The residual velocity ratio (residual velocity V2/impact velocity V1) versus impact velocity is shown in Fig. 4. It clearly shows that the ratio approaches to one as the impact velocity increases to a large value. 2.4 Case 4: Ploycarbonate thickness is 0.9144 cm The ballistic limit for this case is 125 m/sec as shown in Table 1. The deformation plots are shown in Fig. 7(a) for 0.0002 sec. The plots clearly show the penetration of the projectile through the plate. As the impact velocity decreases to 124 m/sec that is very close to the ballistic limit, the deformation plots are shown in Fig. 7(b) for 0.0005 sec. Even though the projectile is rebounded at 11 m/sec, the center of the plate is already broken into small debris. As the impact velocity of the projectile further decreases to 94 m/sec (75% of the ballistic limit), the projectile is rebounded at 22.3 m/sec. The plate has shown not only denting on the location of impact but also broken into small debris as shown in Fig. 7(c) for 0.0002 sec. This result is significantly different with the previous 3 cases where the plate has not been broken into debris at the impact velocity of 75% of the ballistic limit. As the impact velocity increases to a large value of 375 m/sec (3 times of the ballistic limit), the projectile penetrates the plate very quickly at a residual velocity of 346 m/sec as shown in Fig. 7(d) for 0.0001 sec. It clearly shows that the impact velocity decreases only 7.7% after penetration. As the plate thickness increases to 0.9144 cm, the plate only have small deflection with more debris during the penetration since the both impact velocity and plate thickness in this case is larger than the previous 3 cases. The residual velocity ratio (residual velocity V2/impact velocity V1) versus impact velocity is shown in Fig. 4. It clearly shows that the ratio approaches to one as the impact velocity increases to a large value. It is interested to see that the four curves shown in Fig. 11 have the same shape. If the impact velocity in the horizontal axis is divided by the ballistic limit, the curves of residual velocity ratio (residual velocity/impact velocity) versus impact velocity ratio (impact velocity/ballistic limit) are almost coincidence regardless of their thickness as shown in Fig. 8. 3 Conclusions The curves of the ballistic limit versus plate thickness and the residual velocity ratio (residual velocity/impact velocity) versus impact velocity ratio (impact velocity/ballistic limit) have shown the similar trend observed from the experimental tests. The material model used in this analysis is a simple elasto-plastic bilinear constitutive equation. A better constitutive equation, which includes a complete elaso-plastic nonlinear relationship and heat generation due to plastic deformation caused by the impact, can be included in the future finite element analysis to obtain more accurate results. The size of the projectile and the angle of impact also have significant effects on the penetration and perforation resistance of the polycarbonate plates and should be an important topic of future research.
Reference: [1] S.C. Wright, N.A. Fleck and W.J. Stronge, Ballistic Impact of Polycarbonate an Experimental Investigation, Int. J. Impact Eng. Vol. 13, No. 1, pp. 1-20, 1993. [2] S.C. Wright, Y. Huang and N.A. Fleck, Deep penetration of polycarbonate by a cylindrical indenter, Mechanics of Materials 13, pp. 277-284, 1992. [3] N.A. Fleck, W.J. Stronge and J.H. Liu, High strain rate shear response of polycarbonate and polymethyl methacrylate, Proc. Roy. Soc. Lond. A429, pp. 459-479, 1990. [4] S.C. Wright, High strain rate response and ballistic impact of Polycarbonate, Ph.D. thesis, Cambridge University, 1991. [5] A.H. Mayer, J. Sun, D. Hui, and Z. Jaeger, Ballistic response scaling derived from composite material experiments and a critical phase transition viewpoint, presented at 14 th U.S. Army Symposium on Solid Mechanics, 16-18 October 1996, Myrtle Beach, South Carolina. Thickness (cm) Impact Speed (m/sec) Residual Speed (m/sec) 0.2032-171 -160.8-95 -73.2-80 -53.7-67 -34.1-57 -6.7-56 8.3-52 9.7-48 10.2-43 10.7 0.4572-267 -249-130 -97.9-115 -68.5-100 -42.1-89 -1.7-88 13.7-83 15.7-75 15.6-67 16.6 0.6096-303 -281.8-145 -94.7-130 -79.2-115 -51.9-102 -14.5-101 4.3-95 10.6-85 18.8-75 19.5 0.9144-375 -346-175 -121.8-155 -85.5-135 -42.5-125 -7.7-124 11-115 12.4-105 23.7-94 22.3 Table 1. Impact velocity and the corresponding residual velocity for various plate thicknesses..
Fig. 1. Typical finite element mesh for a spherical projectile strikes polycarbonate plate.
140 120 100 Ballistic Limit (m/sec) 80 60 40 20 FEM Results V B =39.7h 0.516 0 0 1 2 3 4 5 6 7 8 9 10 Thickness (mm) Fig. 2. Ballistic limit (V B ) versus plate thickness (h).
Fig. 3. Deformation plots of case 1 for impact velocities of (a) 57, (b) 56, (c) 43, and (d) 160 m/sec.
Residual velocity ratio versus impact velociity 1.2 1 Residual velocity ratio V2/V1 0.8 0.6 0.4 0.2 0-0.2 Thickness=0.9144 cm Thickness=0.6096 cm Thickness=0.4572 cm Thickness=0.2032 cm 0 50 100 150 200 250 300 350 400-0.4 Impact velocity V1 (m/sec) Fig. 4. Residual velocity ration (V2/V1) versus impact velocity (V1).
Fig. 5. Deformation plots of case 2 for impact velocities of (a) 89, (b) 88, (c) 67, and (d) 267 m/sec.
Fig. 6. Deformation plots of case 3 for impact velocities of (a) 102, (b) 101, (c) 75, and (d) 303 m/sec.
Fig. 7. Deformation plots of case 4 for impact velocities of (a) 125, (b) 124, (c) 94, and (d) 375 m/sec.
Residual velocity ratio versus impact velocity ratio 1.2 1 Residual velocity ratio V2/V1 0.8 0.6 0.4 0.2 0-0.2 Thickness=0.9144 cm Thickness=0.6096 cm Thickness=0.4572 cm Thickness=0.2032 cm 0 0.5 1 1.5 2 2.5 3 3.5-0.4 Impact velocity ratio V1/Ballistic limit Fig. 8. Residual velocity ratio (V2/V1) versus impact velocity ratio (V1/Ballistic limit).