Ballistic impact behaviour of thick composites: Parametric studies

Ballistic impact behaviour of thick composites: Parametric studies N.K. Naik *, A.V. Doshi Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Abstract Ballistic impact behaviour of typical woven fabric E-glass/epoxy thick composites is presented in this paper. Specifically, energy absorbed by different mechanisms, ballistic limit velocity and contact duration are determined. The studies are carried out using the analytical method presented for the prediction of ballistic impact behaviour of thick composites in our earlier work [Naik NK, Doshi AV. Ballistic impact behaviour of thick composites: analytical formulation. AIAA J 2005;43(7):1525 36.]. The analytical method is based on wave theory and energy balance between the projectile and the target. The inputs required for the analytical method are: diameter, mass and velocity of the projectile; thickness and material properties of the target. Analytical predictions are compared with typical experimental results. A good match between analytical predictions and experimental results is observed. Further, effect of incident ballistic impact velocity on contact duration and residual velocity, effect of projectile diameter and mass on ballistic limit velocity and effect of target thickness on ballistic limit velocity and contact duration are studied. It is observed that shear plugging is the major energy absorbing mechanism. Keywords: Ballistic impact; Thick composite; Parametric studies 1. Introduction Composite materials are finding increasing uses in general engineering applications along with high performance aerospace and defence applications during the last few decades because of their high specific strength and high specific stiffness. This led to usage of more thick section composites for different applications. Such composite structures undergo different loading conditions during their service life. Impact/ballistic impact is one of the typical loading conditions. Thick composites behave differently compared with thin composites under impact/ballistic impact loading conditions. When an impact load is applied to a body, instantaneous stresses are produced. But the stresses are not immediately transmitted to all parts of the body. The remote portions of the body remain undisturbed for sometime. The stresses progress in all directions through the body in the form of disturbances of different types. In other words, stresses (and their associated deformations or strains) travel through the body at specific velocities. These velocities are functions of the material properties. Regardless of the method of application of impact load, the disturbances generated have identical properties based only on the target material properties. During an impact event the stress wave propagation takes place in all the directions. Generally, this problem is analyzed using 1D, 2D or 3D approaches. In 1D and 2D studies, the wave propagation through the thickness direction is not considered [2 4]. When these approaches are used to analyze structures, isotropic as well as orthotropic, it is assumed that the deformation behaviour along the thickness direction of the target is the same along the entire thickness. Such an assumption can be made for targets of lower thickness or, in other words, can be used for thin plates. If the thickness of the plate is increased the deformation and the induced stress behaviour of the plate would be different at different locations along the thickness direc-

448 Nomenclature A,A p A ql d dc i d h E E bb E cf E csy E dl E fr E hg E mc E mt E p E rb E sp E tf E Total F F c F i G G IIcd h h l h lc h p K KE p KE p0 l m m 0 n lfs,n s n lft n lsc cross-sectional area of the projectile quasi-lemniscate area reduction factor diameter of the projectile deceleration of the projectile during a given time interval diameter of the hole energy/young s modulus energy absorbed due to bulge formation on the back face of the target energy absorbed due to compression of the target directly below the projectile: Region 1 energy absorbed due to compression of the yarns in the surrounding region of the impacted zone: Region 2 energy absorbed due to delamination energy absorbed due to friction energy absorbed due to heat generated energy absorbed due to matrix cracking energy absorbed by matrix cracking per unit volume kinetic energy of the projectile at exit energy absorbed due to reverse bulge formation on the front face of the target energy absorbed due to shear plugging of the yarns energy absorbed due to tension in the yarns in a layer total kinetic energy lost by the projectile total force/contact force compressive force inertial force shear modulus critical strain energy release rate in Mode II thickness of the target thickness of each layer thickness of a layer after compression length of the plug numerical constant (depends on the shape of the projectile) kinetic energy of the projectile at a particular time interval kinetic energy of the projectile: incident length of the projectile mass of the projectile initial mass of the projectile number of layers failed due to shear plugging number of layers failed in tension number of layers strained in compression P d percent delaminating layers P m percent matrix cracking S sp shear plugging strength t time/contact duration V incident impact velocity/velocity V BL ballistic limit velocity V f fibre volume fraction V i velocity of the projectile at the ith instant V m matrix volume fraction V R residual velocity V zl velocity of compressive stress wave in z-direction V zt velocity of shear stress wave in z-direction x d distance upto which damage has reached x l distance the longitudinal wave has traveled in x-direction x ll,r p distance the longitudinal wave has traveled in x-direction in a layer x llc change in length of the yarn after tension x t distance the transverse wave has traveled in x-direction x tl,r t distance the transverse wave has traveled in x-direction in a layer z total depth the projectile has penetrated/projectile displacement z i Depth of projectile penetrated during a given time interval z l distance the compressive stress wave has traveled in z-direction z pl distance by which a layer has moved in forward direction from its original position z t distance the shear stress wave has traveled in z-direction Dh lc thickness by which a layer is compressed Dt increment in time interval e cz compressive strain along the thickness direction e max ultimate strain et tensile strain along the radial direction e txl tensile strain in x-direction in a layer c shear strain m Poisson s ratio q density of the target material r cz compressive stress along the thickness direction r max ultimate stress r t tensile stress along the radial direction r tx tensile stress along the radial direction s shear stress tion. For such cases the analysis is based on also considering the wave propagation along the thickness direction. There are typical studies on ballistic impact behaviour of thick composites [5 8]. Ballistic impact behaviour of typical woven fabric E-glass/epoxy thick composites is presented in this paper. First, typical experimental results are presented. Then, analytical results are presented based on the method presented

449 in [1]. Analytical studies are based on considering the projectile is rigid, cylindrical and flat-ended. The studies are carried out to evaluate energy absorbed by different mechanisms, ballistic limit velocity, contact duration and damage shape and size. Analytical predictions are compared with typical experimental results. The analytical method [1] used is presented in brief in Appendix A. 2. Experimental studies Typical experimental studies were carried out. The projectile and target information is given below: Projectile: Cylindrical, flat-ended hardened steel projectile of diameter, d = 6.33 mm; mass, m = 5.84 gm and length, l = 24 mm. Target: Woven fabric E-glass/epoxy, unsupported area of the target 125 mm 125 mm, thickness 4 7 mm. The experimental ballistic limit velocity for the case of 5 mm thickness was 148 m/s and for the case of 4 mm thickness it was 137 m/s. When the experiment was carried out with the target thickness of 7 mm, complete penetration did not take place even with the incident ballistic impact velocity of 173 m/s. Gellert et al. [9] conducted ballistic impact experimental studies on woven fabric E-glass/vinylester composites with target thickness of 19, 14.5 and 9 mm using cylindrical, flatended steel projectiles. The experimental results are given in Table 1. 3. Input data necessary for the analytical predictions of ballistic impact behaviour Ballistic impact behaviour of typical woven fabric E- glass/epoxy composites is studied. Mechanical properties of the target plate and the other details are given in Table 2 for a typical woven fabric E-glass/epoxy composite studied. The same set of mechanical properties is used in the present study to predict the ballistic impact behaviour of the woven fabric E-glass/epoxy laminates. Also, the same set of mechanical properties is used for the prediction of ballistic impact behaviour of composites used by Gellert et al. [9] and Kumar and Bhat [10]. Compressive stress strain data along the thickness direction for woven fabric E-glass/epoxy composite at high strain rate is given in Appendix B. Tensile stress strain data along the warp direction for woven fabric E-glass/ epoxy composite at high strain rate is given in Appendix C. Details regarding load displacement behaviour under quasi-static loading for the study of frictional resistance are provided in Appendix D. The plot obtained under quasi-static loading is used for the calculation of frictional energy during the ballistic impact event. The frictional resistance is offered by the target for the movement of the projectile during the later stage of the ballistic impact event. During this stage, the velocity of the projectile is less. Considering this, the plot obtained under quasi-static loading is used for the calculation of frictional energy during the ballistic impact event. 3.1. Comparison of predicted and experimental results Ballistic limit velocity is predicted for different target thicknesses and projectile parameters using the analytical method and the mechanical and other properties presented in Table 2. The results are presented in Table 1 and compared with the experimental results. It can be seen that there is a good match between the analytically predicted and experimentally obtained ballistic limit velocities. 4. Ballistic impact behaviour of thick composites Ballistic impact behaviour of thick composites is evaluated using the analytical method. The energy absorbed by different energy absorbing mechanisms, projectile kinetic energy, projectile velocity, contact force, distance traveled by the projectile and longitudinal and shear waves along the thickness direction are evaluated as a function of time. Further, tensile strain at the failure of the layer is obtained as a function of time and target thickness. Distances upto which longitudinal and transverse waves have traveled Table 1 Ballistic impact test results and analytical predictions for typical woven fabric E-glass/epoxy composites Projectile mass, m (gm) Projectile diameter, d (mm) Target thickness, h (mm) Predicted V BL (m/s) Expt. V BL (m/s) Expt. results Remarks 3.84 6.35 19 550.894 563 Penetrated Gellert et al. [9], 2000 3.84 6.35 14.5 452.546 473 Penetrated Gellert et al. [9], 2000 3.33 4.76 19 508.80 505 Penetrated Gellert et al. [9], 2000 3.33 4.76 14 392.98 389 Penetrated Gellert et al. [9], 2000 3.33 4.76 9 269.03 293 Penetrated Gellert et al. [9], 2000 5.84 6.33 7 174.29 173 Not penetrated Present study 5.84 6.33 5 142.50 148 Penetrated Present study 5.00 6.00 25 559.112 547 Penetrated Kumar and Bhat [10], 1998 Cylindrical projectiles with flat-end.

450 Table 2 Input parameters required for the analytical predictions of ballistic impact behaviour Reference Present study Projectile details (cylindrical) Mass (gm) 5.84 Shape Flat ended Diameter (mm) 6.33 Target details Material E-glass/epoxy V f (%) 50 Thickness (mm) 5 No. of layers 19 Density (kg/m 3 ) 1850 Tensile failure strain (%) 3.5 Compressive failure strain (%) 13.5 Shear plugging strength (MPa) 90 Other details Compressive stress strain curve Fig. 20 Tensile stress strain curve Fig. 21 Quasi-lemniscate factor 0.9 Delamination percentage 100 Matrix crack percentage 100 Strain energy release rate: mode II (J/m 2 ) 1000 Matrix cracking energy (MJ/m 3 ) 0.9 Calculation of frictional energy Fig. 22 Fig. 1. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. along the radial direction until the failure of the layer and damage variation along the thickness direction are also presented. 4.1. Case I Figs. 1 5 are with the following data:

451 Fig. 2. Energy absorbed by different energy absorbing mechanisms and tensile strain at failure of layer during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Fig. 3. Projectile velocity and contact force during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Target material: Woven fabric E-glass/epoxy, h = 19 mm. Projectile parameters: d = 6.35 mm, m = 3.84 gm, V = 550.894 m/s. In this case, complete perforation of the target by the projectile took place, i.e., the tip of the projectile was at the back face of the target at the end of the ballistic impact event. Hence, V = 550.894 m/s is the ballistic limit velocity.

452 Fig. 4. Distance traveled by projectile and waves along thickness direction during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Fig. 5. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction during ballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. The projectile kinetic energy decreases during the ballistic impact event. This is because the energy is absorbed by the target by different mechanisms. The ballistic impact event can be subdivided into three stages as given in Appendix A. During the first two stages, damage is taking place upto the back face of the target and the energy is

453 absorbed by different mechanisms. As can be seen from Fig. 1, for the case considered, the second stage ends at 8.1 ls. During the third stage, energy is absorbed only because of the frictional resistance offered due to the movement of the projectile. For this case, at the end of the ballistic impact event, i.e., at time 359.43 ls, the velocity of the projectile is zero (Fig. 3). The contact force increases during the first and second stages and decreases during the third stage. The major energy absorbing mechanism is shear plugging. Significant energy is absorbed by compression of the target in the region directly below the projectile (Region 1) and due to friction. Energy absorbed by matrix cracking, delamination, tension in the yarns and compression in the surrounding region of the impacted zone (Region 2) is relatively less (Fig. 2). Percentage energy absorbed by different mechanisms is given in Table 3. Tensile strain in the yarns at failure of different layers is shown in Fig. 2. It can be seen that the tensile strain exceeds the permissible tensile strain only in the last few layers. It indicates that only last few layers fail in tension. The other layers fail due to shear plugging. Fig. 2 presents the time interval when a particular layer fails and it is plotted as a function of thickness. Among those layers which fail in tension, it can be seen that lower layers fail earlier than the layers above them. This is because of bulging effect on the back face of the target. Projectile displacement during the ballistic impact event, i.e., distance traveled by the projectile is shown in Fig. 4. At the end of the second stage the distance traveled by the projectile is 3.1 mm. At this stage all the layers have failed either due to tension in the yarns/layers due to shear plugging or due to the combined effect of both the mechanisms. After this, during the third stage, the projectile along with the plug formed would be moving further towards the back face of the target. Since the velocity of the projectile is less during this stage, the time taken for the projectile to move further is significantly higher. The distance traveled by through-the-thickness longitudinal and shear waves is also shown in Fig. 4. This would be only upto the end of second stage since the yarns/layers fail by the end of second stage. Distance upto which transverse and longitudinal waves have traveled along the radial direction is shown in Fig. 5. The radial distance indicates the distance up to which the waves have reached when the corresponding layers fail either due to tension or due to shear plugging or due to the combined affect of both the mechanisms. When the induced stress exceeds the permissible strength, the fracture of the yarns would take place. But much before that, clear damage zone can be seen within the layers whenever the induced stress exceeds the damage initiation threshold stress. This damage is either due to delamination or matrix cracking. 4.2. Case II Results are presented in Figs. 6 10 for the same target and projectile parameters (Table 2) with V = 600 m/s. This velocity is above the ballistic limit velocity. In this case, the projectile is exiting with velocity equal to 303.9 m/s. In other words, the entire kinetic energy of the projectile is not absorbed by the target. In this case, the second stage of the ballistic impact event is over at 7.1 ls. The projectile exits the target at 59.1 ls (Fig. 6). In this case also, only a few layers fail in tension near the back face of the target. The other layers fail in shear plugging. The contact force increases during the first and second stages and decreases during the third stage. Projectile displacement is 3.4 mm at the end of the second stage. Further movement of the projectile would be during the third stage. The total contact duration is less in this case as compared to case I. This is because of the higher velocity of the projectile. Damage shape is nearly identical to that for case I. 4.3. Case III Results are presented in Figs. 11 15 for the same target and projectile parameters (Table 2) with V = 500 m/s. This velocity is below the ballistic limit velocity. In this case, the projectile did not completely penetrate the target. The velocity of the projectile was zero at time equal to 7.1 ls. During the ballistic impact event, shear wave has traveled along the thickness direction to a distance of 12 mm. From Fig. 12, it can be seen that all the layers upto a distance of Table 3 Percentage energy absorbed by different mechanisms with different incident ballistic impact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate Incident ballistic impact velocity (m/s) 600 550.894 500 Energy (%) Total kinetic energy of projectile, KE po 100.0 100.0 100.0 Energy absorbed due to compression: Region 1, E cf 7.1 8.4 8.0 Energy absorbed due to compression: Region 2, E csy 2.2 2.5 3.1 Energy absorbed due to tension in yarns, E tf 1.4 2.3 0.8 Energy absorbed due to shear plugging, E sp 58.0 76.4 87.0 Energy absorbed due to matrix cracking, E mc 3.0 2.9 0.7 Energy absorbed due to delamination, E dl 0.4 0.5 0.1 Energy absorbed due to friction, E fr 6.0 7.3 0 Kinetic energy of projectile at exit, E p 22.1 0 0

454 Fig. 6. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Fig. 7. Energy absorbed by different energy absorbing mechanisms and tensile strain at failure of layer during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. 12 mm have failed due to shear plugging. The remaining layers have not failed. Projectile displacement is 2.4 mm when the velocity of the projectile reaches to zero. Since complete failure has not taken place, and the projectile velocity is zero at a distance of 2.4 mm, projectile does not move further. Hence, frictional energy is not present

455 Fig. 8. Projectile velocity and contact force during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/ epoxy laminate. Fig. 9. Distance traveled by projectile and waves along thickness direction during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.

456 Fig. 10. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction during ballistic impact event, V = 600 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Fig. 11. Energy absorbed by different energy absorbing mechanisms during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. in this case. The contact force builds up at the end of the ballistic impact event when the velocity of the projectile reaches to zero. This can lead to delamination at that particular interface upto which shear plugging has taken place. The damage shape is similar to those obtained for the cases I and II, but the size is smaller.

457 Fig. 12. Tensile strain at failure of layer during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Fig. 14. Distance traveled by projectile and waves along thickness direction during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Comparison of ballistic impact parameters with different incident ballistic impact velocities is presented in Table 4.It can be noted that the total contact duration is more at ballistic limit velocity compared to other velocities. The contact force is higher at velocities lower than ballistic limit velocity. 5. Parametric studies 5.1. Incident ballistic impact velocity Contact duration between the projectile and the target as a function of incident ballistic impact velocity is presented in Fig. 16. The contact duration can be defined as follows: Partial penetration: The time interval starting from when the projectile just hits the target to when the velocity of the projectile becomes zero. Complete penetration: The time interval starting from when the projectile just hits the target to when the projectile tip reaches to the back face of the target. Fig. 13. Projectile velocity and contact force during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. At ballistic limit velocity, when the projectile tip just reaches to the back face of the target, the velocity of the projectile would be zero. For the case considered (Fig. 16), the ballistic limit velocity is 550.894 m/s. The contact duration behaviour as given in Fig. 16 and Table 4 can be divided into two parts: Part 1: Incident ballistic impact velocity is less than the ballistic limit velocity. As the incident ballistic impact velocity increases the contact duration increases. At lower velocities, the energy of the projectile would be lower and the time taken for the projectile to reach to zero velocity would be lower.

458 Fig. 15. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness direction during ballistic impact event, V = 500 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate. Table 4 Comparison of ballistic impact parameters with different incident ballistic impact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate Incident ballistic impact velocity, V (m/s) 500 550.894 600 Ballistic impact parameters Contact duration, t (ls) Upto 2nd stage 8.1 7.1 Total 7.1 359 59 Peak contact force, F (N) 708 356 179 Projectile displacement, z (mm) Upto 2nd stage 2.4 3.1 3.4 Residual velocity, V R (m/s) 0 304 As the velocity increases, the time required for the projectile to reach to zero velocity would be more. Also, at higher velocities some energy is absorbed due to friction. During this phase, the deceleration of the projectile is very slow leading to larger contact duration. The maximum contact duration is with ballistic limit velocity. Part 2: Incident ballistic impact velocity is more than the ballistic limit velocity. As the incident ballistic impact velocity increases the contact duration decreases. This is because, as the incident ballistic impact velocity increases, the exit velocity of the projectile increases. This also means that the velocity of the projectile within the target would be higher with higher incident ballistic impact velocity. Residual velocity, V R of the projectile as a function of incident ballistic impact velocity is presented in Fig. 17. This plot is for the case with h = 19 mm, projectile mass 3.84 gm and projectile diameter 6.35 mm. For this case, ballistic limit is 550.894 m/s. This indicates that complete perforation would not take place upto the incident ballistic impact velocity of 550.89 m/s. If the projectile velocity is more than ballistic limit velocity, there would be complete perforation and the projectile would be exiting with certain velocity. It is interesting to note the plot of residual velocity versus incident ballistic impact velocity. As the incident ballistic impact velocity is increased beyond the ballistic limit, the corresponding residual velocity of the projectile also increases. But the increase is very steep just above the ballistic limit. Just to give an example, complete perforation does not take place with incident ballistic impact velocity of 550.89 m/s. But with incident ballistic impact velocity of 552 m/s, complete perforation takes place with the residual velocity of 67 m/s. Similar observations were made by Zhu et al. [11], Jenq et al. [12] and Potti and Sun [13] during their experimental studies. The residual velocity of the projectile is increased as the incident ballistic impact velocity is increased above the ballistic limit. It was clearly observed that the increase in residual velocity was very steep immediately after the ballistic limit velocity was reached.

459 Fig. 16. Contact duration as a function of incident ballistic impact velocity, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/ epoxy laminate. Fig. 18. Ballistic limit velocity as a function of projectile mass and projectile diameter, h = 19 mm, woven fabric E-glass/epoxy laminate. Effect of projectile diameter and mass on ballistic limit velocity is presented in Fig. 18. With the same mass of the projectile, as the diameter of the projectile increases, ballistic limit velocity increases. For the case considered the relationship is nearly linear. Further, with the same diameter of the projectile, as the mass of the projectile increases, ballistic limit velocity decreases. But as the mass increases, the rate of decrease in ballistic limit velocity increases. 5.3. Target thickness Ballistic limit velocity and contact duration at ballistic limit velocity as a function of target thickness are shown in Fig. 19. The plots are for the case of same mass and same Fig. 17. Residual velocity as a function of incident ballistic impact velocity, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/ epoxy laminate. 5.2. Projectile diameter and mass Fig. 19. Ballistic limit velocity and contact duration at ballistic limit velocity as a function of target thickness, m = 3.84 gm, d = 6.35 mm, woven fabric E-glass/epoxy laminate.

460 diameter of the projectile. As the target thickness increases, the ballistic limit velocity increases. But as the thickness of the target increases, the rate of increase in ballistic limit velocity decreases. Also, as the target thickness increases, contact duration increases. As the thickness of the target increases, the increase in contact duration is nearly linear initially. But the rate of increase decreases later. 6. Conclusions Studies have been carried out on ballistic impact behaviour of typical woven fabric E-glass/epoxy thick composites. Effects of different target and projectile parameters are investigated. Specific observations are: The major energy absorbing mechanism is shear plugging. Compression of the region directly below the projectile (Region 1) and friction between the projectile and the target also absorb significant amount of energy. Energy absorbed by matrix cracking, delamination, tension in the yarns and compression in the surrounding region of the impacted zone (Region 2) is not significant. Contact duration depends upon incident ballistic impact velocity. For the case when incident ballistic impact velocity is less than ballistic limit velocity, contact duration increases with the increase in incident ballistic impact velocity. The maximum contact duration is obtained with ballistic limit velocity. For the case when incident ballistic impact velocity is more than ballistic limit velocity, contact duration decreases with the increase in incident ballistic impact velocity. Beyond the ballistic limit, as the incident ballistic impact velocity increases, the residual velocity of the projectile increases. But the rate of increase in residual velocity is very significant just above ballistic limit. For the same mass of the projectile, as the diameter of the projectile increases, ballistic limit velocity increases. The behaviour is nearly linear. For the same diameter of the projectile, as the mass of the projectile increases, ballistic limit velocity decreases. But as the mass increases, the rate of decrease in ballistic limit velocity increases. For the same mass and diameter of the projectile, as the thickness of the target increases, ballistic limit velocity increases. But as the thickness of the target increases, the rate of increase in ballistic limit velocity decreases. For the same mass and diameter of the projectile, as the thickness of the target increases, contact duration increases. As the thickness of the target increases, the increase in contact duration is nearly linear initially. But the rate of increase decreases later. Appendix A. Analytical formulation Stress waves are generated within the target when it is impacted by a projectile. Due to transverse impact, compressive wave and shear wave propagate along the thickness direction and tensile wave and shear wave propagate along the radial direction. The projectile applies forces on to the target. The forces acting are the compressive force, inertial force and frictional force. These forces act at various stages of the impact event. Different damage and energy absorbing mechanisms during ballistic impact are: compression of the target directly below the projectile (Region 1), possible reverse bulge formation on the front face, compression in the surrounding region of the impacted zone (Region 2), tension in the yarns, shear plugging, bulge formation on the back face, delamination and matrix cracking, friction between the target and the projectile and heat generation due to impact. Impacted materials can fail in a variety of ways. The actual mechanisms depend on such variables as projectile size, shape, mass and velocity, and target material properties and geometry and relative dimensions of the projectile and the target. Generally, the ballistic impact event can be sub-divided into three stages. During the first stage, the projectile strikes to the target and compression of the target takes place directly below the projectile face. As the compression progresses, the material would flow predominantly along the thickness direction. Material flow can also be in the radial direction as well as towards the front face of the target. This stage would continue until through-the-thickness compressive wave reaches to the back face of the target. During this stage, compressive stresses are generated within the target directly below the projectile. As a result of this the surrounding region would be under tension along the radial direction. Additionally, because of the impact force, shear stresses are generated within the target around the periphery of the projectile. Any of these stresses could lead to failure of the target. As the projectile moves further, the yarns in Region 2 of the upper layers exert pressure on the yarns in Region 2 of the lower layers. In other words, compressive deformation of the yarns takes place in Region 2 also. Because of the compression of the layers, as well as possible failure of the target by different modes, the projectile moves further. This leads to bulge formation on the back face. This is the second stage of impact. Along with bulge formation on the back face, failure of the yarns/layers would take place by different mechanisms in the upper layers of the target. This process continues and the projectile moves further. A clear plug formation can take place in front of the projectile. Also, the yarns/layers can fail in tension on the back face because of bulge formation. During the entire ballistic impact event, inplane matrix cracking and delamination between the layers can also take place. During the third stage, the projectile moves further and the plug and the projectile exit from the back face of the

461 target. As the projectile penetrates into the target and starts moving further, frictional forces act between the projectile and the target. Heat can also be generated during this process. The total kinetic energy of the projectile lost during the ballistic impact event is equal to the total energy absorbed by the target till that time interval. It is given by the following relation: E Totali ¼ E cfi þ E rbi þ E csyi þ E tfi þ E spi þ E bbi þ E dli þ E mci þ E fri þ E hgi : ð1þ The following assumptions are made in the analytical formulation: Projectile impact is normal to the surface of the target. The projectile is cylindrical with a flat-end and perfectly rigid. The compressive stress is experienced along the thickness direction only within those layers through which the compressive wave has traveled. Also, compressive strain is uniform within those layers. The shear plugging stress is experienced along the thickness direction only within those layers through which the shear wave has traveled. Also, shear plugging stress is uniform within those layers. The plug formed is freely moving forward, i.e., resistance is not offered by the target except for the frictional resistance. The peak tensile strain within the yarns is near the periphery of the projectile. Hence, the tensile failure of the yarns would take place near the periphery of the projectile. The deceleration of the projectile remains constant during each time interval. The analytical formulation for the three stages is presented below: The total force acting on the projectile is given by F ¼ F i þ F c ; ð2þ where, inertial force, F i ¼ 1 2 qkav 2 and, compressive force, F c ¼ r cz A: ð4þ For the cylindrical projectile with a flat-end, K =1 [14]. These forces are acting on the effective mass of the projectile. The effective mass of the projectile includes the material of the target displaced by the projectile moving with it. The effective mass of the projectile used in the equation is, therefore, m 0 + qaz. This is based on considering that the projectile has moved by a distance z. The equation of motion for the penetration process is ð3þ d dt ðmv Þ¼F i þ F c d dm ðmv Þ¼V dt dt þ m dv dt ¼ 1 ð5þ 2 KqAV 2 þ r cz A The rate of change of effective mass of the projectile is dm dz ¼ qa dt dt ¼ qav dv dt ¼ dv dz dz dt ¼ V dv ð6þ dz Substituting Eq. (6) into Eq. (5), following relation is obtained: qav 2 þðm 0 þ qazþv dv dz ¼ 1 2 KqAV 2 þ r cz A: ð7þ The above equation is solved by separation of variables to calculate the velocity [14] and the expression for velocity is obtained as V ðzþ ¼ V 2 r cz ðzþ i þ qð1 þ 0:5KÞ m 0 =qa ðm 0 =qaþþz 2þK r cz ðzþ qð1 þ 0:5KÞ! 1=2 : ð8þ The time required for the projectile to penetrate distance z is calculated by the following expression: Z z 1 t ¼ 0 V ðzþ dz Z z! 2þK 1=2 ¼ V 2 r cz ðzþ m 0 =qa r cz ðzþ i þ dz: 0 qð1 þ 0:5KÞ ðm 0 =qaþþz qð1 þ 0:5KÞ ð9þ The velocity calculated by this method is used only for the first time interval. After the calculation of the velocity, the energy absorbed by different energy absorbing mechanisms during the first time interval is calculated. Knowing the initial kinetic energy of the projectile and the energy absorbed during the first time interval, the velocity of the projectile for the next time interval is calculated. By knowing the velocity, various parameters such as displacement of the projectile, strain, contact force and energy absorbed by different mechanisms are calculated for the given time interval. This procedure is continued until the compressive wave reaches to the back face of the target. The shear wave follows the compressive wave along the thickness direction. The wave velocities along the thickness direction are given by sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dr cz V zl ¼ k ; ð10þ q de cz sffiffiffiffiffiffiffiffiffi 1 ds V zt ¼ k ; ð11þ q dc sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 tþ k ¼ : ð12þ ð1 þ tþð1 2tÞ For the elastic waves, E and G are used instead of the local slopes as given in Eqs. (10) and (11). The distance traveled by compressive wave and shear wave along the thickness direction at any instant of time is

462 z l ¼ V zl t; z t ¼ V zt t: ð13þ ð14þ The number of layers through which the compressive wave has traveled is n lsc ¼ z l : ð15þ h l As the projectile is impacted onto the target, tensile and shear waves are also generated along the radial direction in the target. The wave velocities can be calculated using Eqs. (10) and (11). But, in this case, the material properties are with respect to radial direction. The compressive strain in each layer at any instant of time is given by e cz ¼ z z l : ð16þ The calculation of tensile strain in a particular yarn/ layer is given in [1]. Considering the strain variation as linear with maximum strain at the periphery of the projectile and zero at a distance upto which the longitudinal radial wave has reached at that particular time, the maximum strain is calculated as below: e max txl ¼ 2e txl : ð17þ In the beginning of the ballistic impact event, all the energy is in the form of kinetic energy of the projectile. During the impact event, the target absorbs some of the energy. The energy balance at the end of ith time interval is obtain as KE p0 ¼ KE pi þ E cfði 1Þ þ E rbði 1Þ þ E csyði 1Þ þ E tfði 1Þ þ E spði 1Þ þ E bbði 1Þ þ E dlði 1Þ þ E mcði 1Þ þ E frði 1Þ þ E hgði 1Þ : ð18þ Rearranging the terms in the above equation 1 2 m 0V 2 Xi 1 E i 1 ¼ 1 2 m iv 2 i i¼1 1 or 2 m i 1V 2 i 1 E i 1 ¼ 1 2 m iv 2 i Here, E ði 1Þ ¼ E cfði 1Þ þ E rbði 1Þ þ E csyði 1Þ þ E tfði 1Þ þ E spði 1Þ þ E bbði 1Þ þ E dlði 1Þ þ E mcði 1Þ þ E frði 1Þ þ E hgði 1Þ : ð19þ ð20þ The terms on the right-hand side of Eq. (20) can be calculated for each time interval. Hence, the velocity of the projectile for the next time interval is calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m 2 i 1V 2 i 1 E ði 1Þ V i ¼ 1 2 m i : ð21þ The deceleration of the projectile during ith time interval is obtained as dc i ¼ V i 1 V i : ð22þ Dt The distance traveled by the projectile during ith time interval is obtained as z i ¼ V i 1 Dt 1 2 dc iðdtþ 2 : ð23þ The total distance traveled by the projectile is equal to the summation of the distance the projectile has traveled in each time interval. The force resisted by the target against the motion of the projectile is given by F i ¼ m i dc i : ð24þ This force is used to determine if the shear plugging is taking place or not. The above-mentioned procedure is continued during all the three stages of the ballistic impact event. Energy absorbed due to compression of the target in Region 1 Z ecz E cf ¼ A p r cz ðe cz Þde z l : ð25þ e c¼0 Energy absorbed due to compression of the yarns in the surrounding region of impacted zone in Region 2 Z E csy ¼ 2ph Xn lsc xt Z e¼ecz r cz ðe cz Þde xdx: ð26þ j¼nlf d=2 e¼0 Energy absorbed due to tension in yarns X n Z lsc xl Z etxl E tf ¼ A y r tx ðe txl Þde dx: j¼nlf 0 e¼0 ð27þ The total energy absorbed due to tension is sum of the energy absorbed by different yarns. Energy absorbed by shear plugging during a time interval is given by the product of thickness of the target that is sheared, shear plugging strength and the area over which shear stress is acting. It is given by DE spi ¼ n s h l S sp pdh: ð28þ Energy absorbed by shear plugging by the end of ith time interval is given by E sp ¼ Xi DE spi : ð29þ n¼1 Energy absorbed by delamination and matrix cracking during ith time interval is given by E dli ¼ P d px 2 d A qlg IIcd ðn 0 1Þ; E mci ¼ P m px 2 d A ð30þ qle mt hðv m Þ Frictional resistance offered by the target towards the movement of the projectile is a material property. It depends upon projectile diameter, length and surface condition. It also depends upon the diameter of the hole formed within the target due to ballistic impact, hole surface condition and the target material properties. Frictional resistance is determined experimentally.

463 Fig. 20. Compressive stress strain curve along thickness direction for woven fabric E-glass/epoxy laminates at high strain rate. Appendix B. Stress strain data at high strain rates: compressive loading The compressive stress strain curve along the thickness direction for the woven fabric E-glass/epoxy composite at high strain rate is given in Fig. 20. Typical parameters for this stress strain curve are as follows: e max = 13.5%. r max = 1350 MPa. Area under the stress strain curve = 127.57 MPa. The compressive stress strain curve is represented by the equation, y ¼ 0:0496x 4 þ 2:0004x 3 31:956x 2 þ 288:86x: Fig. 21. Tensile stress strain curve along radial direction for woven fabric E-glass/epoxy laminates at high strain rate. Here, y indicates stress in MPa and x indicates strain in percentage. Appendix C. Stress strain data at high strain rates: tensile loading The tensile stress strain curve along the radial direction for the woven fabric E-glass/epoxy composite at high strain Fig. 22. Load displacement plot under quasi-static loading: penetration of a cylindrical projectile into a composite target with a hole, h = 19 mm, d = 8 mm, d h = 8 mm, woven fabric E-glass/epoxy laminate.

464 rate is given in Fig. 21. Typical parameters for this stress strain curve are as follows: e max = 3.5%. r max = 560 MPa. Area under the stress strain curve = 13.72 MPa. The tensile stress strain curve is represented by the equation, y ¼ 1:0884x 4 þ 16:003x 3 106:09x 2 þ 381:9x: Here, y indicates stress in MPa and x indicates strain in percentage Appendix D. Load displacement plot under quasi-static loading for the study of frictional resistance Typical experiments were conducted to study the frictional resistance offered during the movement of the projectile within the target with a hole. Load displacement plot under quasi-static loading condition to study the penetration behaviour of a cylindrical, flat-ended hardened steel projectile into a composite target with a hole is presented in Fig. 22. The figure shows the distance moved by the projectile within the hole and the corresponding load. Based on this, work done to move the projectile over a distance and the corresponding energy absorbed due to friction are calculated. This information is made use for the calculating the frictional energy during the ballistic impact event. It can be seen from the figure that the behaviour is nonlinear and slope of the curve decreases as the projectile moves further. The behaviour depends upon the surface condition of the hole formed during the ballistic impact event, diameter of the hole, thickness of the target, diameter of the projectile and the material of the target. For the present study, the plot given in Fig. 22 is used for calculating frictional energy absorbed during ballistic impact event References [1] Naik NK, Doshi AV. Ballistic impact behavior of thick composites: analytical formulation. AIAA J 2005;43(7):1525 36. [2] Naik NK, Shrirao P. Composite structures under ballistic impact. Compos Struct 2004;66:579 90. [3] Naik NK, Shrirao P, Reddy BCK. Ballistic impact behaviour of woven fabric composites: parametric studies. Mater Sci Eng A 2005;412:104 16. [4] Naik NK, Shrirao P, Reddy BCK. Ballistic impact behaviour of woven fabric composites: formulation. Int J Impact Eng 2006;32:1521 52. [5] Wen HM. Penetration and perforation of thick FRP laminates. Compos Sci Technol 2001;61(8):1163 72. [6] Gillespie Jr JW, Monib AM, Carlsson LA. Damage tolerance of thick-section S-2 glass fabric composites subjected to ballistic impact loading. J Compos Mater 2003;37(23):2131 47. [7] Cheng WL, Langlie S, Itoh S. High velocity impact of thick composites. Int J Impact Eng 2003;29:167 84. [8] Gama BA, Islam SMW, Rahman M, et al. Punch shear behavior of thick-section composites under quasi-static, low velocity and ballistic impact loading. SAMPE J 2005;41(4):6 13. [9] Gellert EP, Cimporeu SP, Woodward RL. A study of effect of target thickness on the ballistic perforation of glass fiber reinforced plastic composites. Int J Impact Eng 2000;24(5):445 56. [10] Kumar KS, Bhat TB. Response of composite laminates on the impact of high velocity projectiles. Key Eng Mater 1998;141:337 48. [11] Zhu G, Goldsmith W, Dharan CH. Penetration of laminated Kevlar by projectiles I, experimental investigation. Int J Solids Struct 1992;29(4):339 420. [12] Jenq ST, Jing HS, Chung C. Predicting the ballistic limit for plain woven glass/epoxy composite laminate. Int J Impact Eng 1994;15:451 64. [13] Potti SV, Sun CT. Prediction of impact induced penetration and delamination in thick composite laminates. Int J Impact Eng 1997;19(1):31 48. [14] Awerbuch J, Bodner SR. Analysis of the mechanics of perforation of projectiles in metallic plates. Int J Solids Struct 1974;10(6):671 84.