Ballistic impact behaviour of woven fabric composites: Parametric studies

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1 Ballistic impact behaviour of woven fabric composites: Parametric studies N.K. Naik, P. Shrirao, B.C.K. Reddy Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai , India Abstract Polymer matrix composites undergo various loading conditions during their service life. For the effective use of such materials for high performance applications, their behaviour under impact/ballistic impact loading should be clearly understood. This is because the polymer matrix composites are susceptible to impact/ballistic impact loading. Ballistic impact behaviour of composite targets depends upon target geometrical and material parameters as well as projectile parameters. In the present study, investigations on the ballistic impact behaviour of two-dimensional woven fabric E-glass/epoxy composites are presented as a function of projectile and target parameters. The ballistic impact behaviour predictions are based on the analytical method presented in our earlier work [N.K. Naik, P. Shrirao, Composite structures under ballistic impact, Compos. Struct. 66 (2004) 579]. Firstly, perforation and partial penetration of the projectile into the target and cone formation on the exit side of the target are studied. Further, effect of stress wave transmission factor is studied. The other parameters considered are: thickness of the target and mass and diameter of the projectile. Ballistic limit, contact duration at ballistic limit and residual velocity are presented as a function of different target and projectile parameters. Keywords: Ballistic impact; Woven fabric composite; Prediction; Projectile parameters; Target parameters; Residual velocity 1. Introduction Polymer matrix composites are finding increasing uses in high performance applications. One of the important applications is that they can be used to provide effective protection for ballistic impact. Such materials can absorb significant kinetic energy of the projectile and are also characterized by high specific strength and specific stiffness. In recent years textile composites are used for composites for protection applications. This is because of their higher energy absorption characteristics and through the thickness stiffness and strength properties. Summary of investigations on ballistic impact and an analytical model for the prediction of ballistic impact behaviour of woven fabric composites are presented in our earlier work [1]. Summary of the analytical model is presented in Appendix A. There are three basic approaches to analyze ballistic impact problems. They are: (1) Empirical prediction models, which require lot of experimental tests and results. (2) Prediction models, which require typical ballistic impact experimental data as input. (3) Analytical models, which take only mechanical and fracture properties and geometry of the target and projectile parameters as input. The analytical method presented in [1] is based on energy transfer between the projectile and the target. The method requires mechanical and fracture properties and geometry of the target and projectile parameters as input. Possible effects of various parameters on the ballistic impact behaviour of the targets are presented in this study using the analytical method presented in [1] and summarized in Appendix A. Firstly, ballistic impact behaviour with perforation and possible partial penetration of the projectile into the target is studied. Then, development of cone on the exit side of the target is studied. Further, the effect of stress wave transmission factor, target thickness and diameter and mass of the projectile are studied.

2 105 Nomenclature a yarn width A cross-sectional area of fibre/yarn A di damaged area at time (t i ) A ql quasi-lemniscate area reduction factor b stress wave transmission factor c e elastic wave velocity c p plastic wave velocity c t transverse wave velocity d projectile diameter dc i deceleration of the projectile during ith time interval E Di energy absorbed by deformation of secondary yarns till time (t i ) E DLi energy absorbed by delamination till time (t i ) E Fi energy absorbed by friction till time (t i ) E mt energy absorbed by matrix cracking per unit volume E MCi energy absorbed by matrix cracking till time (t i ) E KEi kinetic energy of the moving cone at time (t i ) E SPi energy absorbed by shear plugging till time (t i ) E TFi energy absorbed by tension in primary yarns till time (t i ) E TOTALi total energy absorbed by the target till time (t i ) F i contact force during ith time interval G IIcd critical dynamic strain energy release rate in mode II h target thickness h l layer thickness KE op initial kinetic energy of the projectile KE pi kinetic energy of the projectile at time (t i ) m p mass of the projectile M Ci mass of the cone at time (t i ) N number of layers being shear plugged in a time interval N 0 number of layers P d percent delaminating layers P m percent matrix cracking r di radius of the damage zone at time (t i ) r pi distance covered by plastic wave till time (t i ) r ti surface radius of the cone at time (t i ), distance covered by transverse wave till time (t i ) S SP shear plugging strength t c contact duration of the projectile during ballistic impact event t i ith instant of time t duration of time interval that t c is divided into V BL, V 50 ballistic limit velocity V f fibre volume fraction V i projectile velocity at time (t i ) V I, V O incident ballistic impact velocity matrix volume fraction V m V R z i residual velocity height/depth of the cone at time (t i ), distance traveled by projectile at time (t i ) Greek letters ε strain ε 0 maximum strain in a yarn/fibre at any moment ε d damage threshold strain ε i maximum tensile strain in primary yarns at time (t i ) ε p plastic strain ε py strain in primary yarns ε sy strain in secondary yarns ρ density of the target σ stress σ p plastic stress stress in secondary yarns σ sy 2. Damage mechanisms Ballistic impact is normally a low-mass high-velocity impact by a projectile propelled by a source onto a target. Since the ballistic impact is a high velocity event, the effects on the target can be only near the location of impact. During the ballistic impact, energy transfer takes place from the projectile to the target. Based on the target geometry, material properties and projectile parameters the following are possible. (1) The projectile perforates the target and exits with a certain velocity. This indicates that the projectile initial kinetic energy was more than the energy that the target can absorb. (2) The projectile partially penetrates the target. This indicates that the projectile initial kinetic energy was less than the energy that the target can absorb. Based on the target material properties, the projectile can either be stuck within the target or rebound. (3) The projectile perforates the target completely with zero exit velocity. For such a case, the initial velocity of the projectile of a given mass is referred to as the ballistic limit. For this case the entire kinetic energy of the projectile is just absorbed by the target. For the complete understanding of the ballistic impact of composites, different damage and energy absorbing mechanisms should be clearly understood. Possible energy absorbing mechanisms are [1]: cone formation on the back face of the target, deformation of secondary yarns, tension in primary yarns/fibres, delamination, matrix cracking, shear plugging and friction between the projectile and the target. For different materials like carbon, glass or Kevlar, different mechanisms can dominate. Also, the reinforcement architecture can influence the energy absorbing mechanisms. The total energy absorbed by the target till a particular time interval is given in Appendix A. E TOTALi = E KEi + E SPi + E Di + E DLi + E MCi + E Fi (1) The following assumptions are made in the analytical model used:

3 106 (4) Energy absorption due to primary yarn/fibre breakage and deformation of the secondary yarns are treated independently. (5) Longitudinal and transverse wave velocities are the same in all the layers. (6) Projectile is in contact with the target during the ballistic impact event. (7) Projectile displacement and the cone height formed are the same at any instant of time. The assumptions are valid for relatively thin plates. The results presented are for a typical two-dimensional plain weave fabric E-glass/epoxy composite. The input parameters are given in Table Perforation and partial penetration of the projectile into the target Fig. 1. Energy absorbed by different mechanisms during ballistic impact event, V I = 158 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. (1) Projectile is perfectly rigid and remains un-deformed during the ballistic impact. (2) Projectile motion is uniform during penetration within each time interval. (3) Yarns/fibres in a layer act independently. Figs. 1 and 2 present ballistic impact behaviour at incident ballistic impact velocity (V I ) of 158 m/s, whereas Figs. 3 and 4 are with V I = 159 m/s. It may be noted that with V I = 158 m/s, complete penetration is not taking place whereas, with V I = 159 m/s, complete perforation is taking place with certain residual velocity. Strictly speaking, ballistic limit is between 158 and 159 m/s. From the practical point of view, 158 m/s can be taken as the ballistic limit. The main energy absorbing mechanisms are: deformation of secondary yarns and fracture of primary yarns. Significant amount of the kinetic energy of the projectile is transferred as the kinetic energy of the moving cone during ballistic impact event. Fig. 2. Projectile velocity, strain and contact force variation during ballistic impact event, V I = 158 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.

4 107 Table 1 Input parameters required for the analytical predictions of ballistic impact behaviour Projectile details (cylindrical) Mass Shape Diameter 2.8 g Flat ended 5 mm Target details Material Woven E-glass/epoxy V f 50% Thickness 2 mm Number of layers 6 Density 1750 kg/m 3 Other details Stress-strain Time-step 1 s Quasi-lemniscate area reduction factor 0.9 Stress wave transmission factor Delamination percent 100% Matrix crack percent 100% Shear plugging strength Mode II dynamic critical strain energy release rate 1000 J/m 2 Matrix cracking energy 0.9 MJ/m 3 Fibre failure energy 28 MJ/m 3 Fig. 3. Energy absorbed by different mechanisms during ballistic impact event, V I = 159 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. With V I = 154 m/s as shown in Figs. 5 and 6, the maximum strain in the bottom layer is lower than the ultimate strain limit. Hence, the lower layers do not fail because of the ballistic impact. The kinetic energy of the moving cone would be zero at the end of ballistic impact event. Most of the energy is absorbed because of the deformation of the secondary yarns. The energy absorbed due to deformation of secondary yarns has two components: strain energy stored within the secondary yarns and the energy absorbed because of possible matrix cracking and delamination. After the projectile velocity reaches to zero, there can be rebounding of the projectile from the target. This is because of Fig. 4. Projectile velocity, strain and contact force variation during ballistic impact event, V I = 159 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.

5 108 Fig. 5. Energy absorbed by different mechanisms during ballistic impact event, V I = 154 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. the release of strain energy stored in the secondary yarns. This energy would be converted into kinetic energy of the projectile. Some energy is absorbed in the form of yarn/fibre tensile failure. In this particular case, layers in the upper half have failed in the form of fibre tensile failure. Layers in the lower half have not failed. The energy absorbed due to matrix cracking and delamination is marginal. Fig. 7. Energy absorbed by different mechanisms during ballistic impact event, V I = 100 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. Figs. 7 and 8 present the ballistic impact behaviour with V I = 100 m/s. It may be noted that, in this case, yarns/fibres have not failed anywhere throughout the thickness of the composite plate. This is because the induced strain is lower than the ultimate strain limit. In this case the projectile would not penetrate Fig. 6. Projectile velocity, strain and contact force variation during ballistic impact event, V I = 154 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.

6 109 Fig. 8. Projectile velocity, strain and contact force variation during ballistic impact event, V I = 100 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. into the target. There can be marginal indentation because of compression of the target. 4. Cone formation on exit side of the target During the ballistic impact event, cone formation takes place on the exit side of the target just below the point of impact [1 3]. This is because of the transverse wave propagation. The surface radius of the cone increases with time. Also, the cone moves along with the projectile. With this, the height of the cone increases. It may be noted that the projectile displacement at any instant of time and the cone height formed would be the same. Variation of cone surface radius with time is presented in Fig. 9. Initially, transverse wave velocity increases significantly and then remains nearly constant during the remaining period of ballistic impact event. Variation of cone surface radius is nearly linear with respect to time. Cone height variation with time is non-linear as shown in Fig. 9. The rate of increase of Fig. 9. Cone surface radius, transverse wave velocity and projectile displacement variation during ballistic impact event, V I = 158 m/s, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.

7 110 cone height/depth decreases with time. This depends upon the velocity variation of the projectile during the ballistic impact event. 5. Contact duration The contact duration between the projectile and the target is a function of incident ballistic impact velocity as presented in Fig. 10. The contact duration can be defined as follows: - Partial penetration: The time interval starting from the point when the projectile just hits the target to the point when the velocity of the projectile becomes zero. - Complete perforation: The time interval starting from the point when the projectile just hits the target to the point when the projectile exits from the back surface of the target. At ballistic limit, the projectile would be exiting the target with zero velocity. From Fig. 10, it is seen that as the incident ballistic impact velocity increases, the contact duration decreases. It may be noted that at ballistic limit and above, the drop in contact duration is significant. Contact duration at ballistic limit velocity (V BL ) as a function of target thickness is presented in Fig. 11. Generally, it is observed that as the target thickness increases, the contact duration at ballistic limit decreases. 6. Effect of stress wave transmission factor During the ballistic impact event, as the longitudinal stress wave travels through the yarns, stress wave attenuation takes place. The stress wave transmission factor indicates the fraction of the stress wave that is transmitted as the longitudinal stress wave propagates further. Higher the transmission factor, there would be less stress attenuation, in turn, more stress transmission [1,10]. Fig. 11. Contact duration at ballistic limit velocity as a function of target thickness, m p = 2.8 g, d = 5 mm, woven fabric E-glass/epoxy laminates. If the stress wave transmission factor is higher, corresponding stress and, in turn, strain would be higher. This would lead to possible larger damage area and hence, higher energy absorption. This would lead to possible higher V BL. Fig. 12 presents ballistic limit velocity as a function of stress wave transmission factor. Higher the stress wave transmission factor, higher is V BL. For woven fabric composites, stress wave transmission factor is a function of the geometry of woven fabric composite as well as the material properties. For the given material properties of the yarns of the woven fabric composite, stress wave transmission Fig. 10. Contact duration variation as a function of incident ballistic impact velocity, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. Fig. 12. Ballistic limit velocity variation as a function of stress wave transmission factor, m p = 2.8 g, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate.

8 111 Fig. 13. Ballistic limit velocity variation as a function of projectile diameter, m p = 2.8 g, h = 2 mm, woven fabric E-glass/epoxy laminate. Fig. 14. Ballistic limit velocity variation as a function of projectile mass, h = 2 mm, d = 5 mm, woven fabric E-glass/epoxy laminate. factor can be controlled by controlling the geometry of the woven fabric composite. Parameter b used in this study is based on stress wave attenuation characteristics for each composite [10]. It is a material property and depends upon geometry of the fabric as well as mechanical and physical properties of the reinforcing material and the matrix. This parameter is determined for each material separately. In the present case, stress wave transmission factor, b = (Appendix A). 7. Effect of projectile and target parameters 159 m/s, 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 158 m/s. But with incident ballistic impact velocity of 159 m/s, complete perforation takes place with the residual velocity of 54 m/s. Variation of V BL as a function of projectile diameter is presented in Fig. 13 for the same mass of the projectile. As the projectile diameter increases, V BL increases. It may be noted that V BL increase is not significant in this case. Variation of V BL as a function of projectile mass is presented in Fig. 14 for the same diameter of the projectile. As the mass increases, V BL decreases and the behaviour is nearly linear. Variation of V BL as a function of target thickness with the same projectile mass and diameter is presented in Fig. 15. As the target thickness increases, V BL also increases and the behaviour is nearly linear. At higher incident impact velocities, contact duration would decrease. This effect can be seen in Fig Residual velocity Residual velocity (V R ) of the projectile as a function of incident ballistic impact velocity is presented in Fig. 16. This plot is for the case with target thickness, h = 2 mm; projectile mass, m p = 2.8 g and projectile diameter, d = 5 mm. For this case ballistic limit is 158 m/s. This indicates that complete perforation would not take place up to the incident ballistic impact velocity of 158 m/s. If the projectile velocity is equal to or more than Fig. 15. Ballistic limit velocity variation as a function of target thickness, m p = 2.8 g, d = 5 mm, woven fabric E-glass/epoxy laminates.

9 112 Studies have been carried out on ballistic impact behaviour of a typical plain weave fabric E-glass/epoxy composite. Effect of various target geometrical and material parameters and projectile parameters has been investigated. The specific observations are: Fig. 16. Residual velocity variation as a function of incident ballistic impact velocity, m p = 2.8 g, d = 5 mm, h = 2 mm, woven fabric E-glass/epoxy laminate. Similar observation was made by Zhu et al. [12], Jenq et al. [13], Potti and Sun [14] and Jenq et al. [15] during their experimental studies. They observed that, as the incident ballistic impact velocity is increased beyond the ballistic limit, the corresponding residual velocity of the projectile increased. But the increase was very steep just above the ballistic limit. This can be explained considering the projectile displacement and the increase in cone surface radius during the ballistic impact event. As shown in Fig. 6, at certain incident ballistic impact velocity, the upper layers would be failing whereas the lower layers may not be failing. The strain in the lower layers would be increasing initially and then start decreasing. This is because of the geometry of the cone formed, which is governed by projectile displacement, i.e., change in cone height and the cone surface radius at any instant of time. The complete perforation would take place if the induced strain exceeds the permissible strain before it starts decreasing as shown in Fig. 6. If complete perforation does not take place at this incident ballistic impact velocity, e.g., for the present case it is 158 m/s, the incident ballistic impact velocity will have to be increased for complete perforation. The necessary increase in incident ballistic impact velocity above 158 m/s could be so small that in a practical sense 158 m/s can be taken as the ballistic limit. The geometry of the cone formed, and in turn, the induced strain is governed by projectile displacement, i.e., change in cone height (z i ) and the cone surface radius (r ti ) at any instant of time. The trend given in Fig. 16 is because of the combined effect of r ti and z i. 9. Conclusions - During the ballistic impact event the kinetic energy of the projectile is transferred to the target. Significant amount of this energy is transferred as the kinetic energy of the moving cone during ballistic impact event. - At the end of ballistic impact event, major energy absorbing mechanisms are: deformation of secondary yarns and fracture of primary yarns. - Increase in cone surface radius is nearly linear with respect to time. - The rate of increase of cone depth/height decreases with time. - The contact duration between the projectile and the target decreases as the incident ballistic impact velocity increases. Beyond the ballistic limit velocity the decreases in contact duration is significant. - As the stress wave transmission factor increases, the ballistic limit velocity increases. - For the same mass of the projectile, as the diameter of the projectile increases, the ballistic limit velocity increases. - For the same diameter of the projectile, as the mass of the projectile increases, the ballistic limit velocity decreases. - For the same mass and diameter of the projectile, as the target thickness increases, the ballistic limit velocity increases. - 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 the ballistic limit. Appendix A. Analytical formulation On striking a target, energy of the projectile is absorbed by various mechanisms like kinetic energy of the moving cone E KE, shear plugging E SP, deformation of secondary yarns E D, tension in primary yarns E TF, delamination E DL, matrix cracking E MC, frictional energy E F along with other energy dissipation mechanisms. Here, an analytical model is presented for the above-mentioned mechanisms, which dominate during ballistic impact of two-dimensional woven fabric composites [1]. A.1. Analytical formulation Cone formation was observed on the back face of the target as shown in Fig. 17 when struck by a projectile [2,3]. Shear plugging on the front face and cone formation on the back face start taking place depending on the target material properties during the ballistic impact event. It is assumed that the velocity of the moving cone is the same as the velocity of the projectile. Initially, the moving cone has velocity equal to that of the projectile and has zero mass. As time progresses, the mass of the moving cone increases and velocity of the cone decreases. As the cone formation takes place, the yarns/fibres deform and absorb some energy. The primary yarns, which provide the resistive force to the projectile motion, are strained the most, thus leading to their failure. When all the primary yarns fail, the projectile exits the target. Tensile failure of the yarns thus absorbs some energy

10 113 Fig. 17. Conical deformation during ballistic impact on the back face of the composite target. of the projectile. Even before the failure of the primary yarns, there would be some energy absorption because of tension in the yarns. During the ballistic impact event, delamination and matrix cracking take place in the laminate area, which forms the cone. The total kinetic energy of the projectile that is lost during ballistic impact is the total energy that is absorbed by the target till that time interval and is given by, E TOTALi = E KEi + E SPi + E Di + E TFi + E DLi + E MCi + E Fi (A.1) An analytical model to predict the ballistic impact behaviour is presented below. Firstly, modelling of a single yarn is presented. The model is further extended to the woven fabric composites. The analytical model is for the case when the projectile impacts the target normally. A.2. Modelling of a single yarn subjected to transverse ballistic impact When a yarn is impacted by a projectile transversely, longitudinal strain wavelets propagate outward along the filament [4 6]. The outermost wavelet, called the elastic wave, propagates at velocity, c e = ( ) 1 dσ ρ dε ε=0 (A.2) The innermost longitudinal wavelet, called the plastic wave, propagates at velocity [4], ( ) 1 dσ c p = (A.3) ρ dε ε=ε p Here, ε p denotes the strain at which plastic region starts. As the strain wavelets pass a given point on the yarn, material of the filament flows inward towards the impact point. The material in the wake of the plastic wave front forms itself into a transverse wave, shaped like an inverted tent with the impact point at its vertex. In a fixed coordinate system, the base of the tent spreads outward with the transverse wave velocity. The transverse wave velocity is given by [4], c t = (1 + ε p )σ εp p ρ 0 1 ρ ( ) dσ dε dε (A.4) If the complete impact event is divided into a number of small instants, then at ith instant, the time is given by t i. By that time the transverse wave has traveled to a distance r ti and the plastic wave has traveled to a distance of r pi. The projectile has moved through a distance z i. Radii r ti and r pi after time t i = i t is given by, n=i r ti = c tn t n=0 n=i r pi = c pn t n=0 A.3. Modelling of woven fabric composites (A.5) (A.6) Woven fabric composites consist of warp and fill yarns, interlaced in a regular sequence [7 9]. As the projectile impacts on to the woven fabric composite, there can be many yarns beneath the projectile. In the present analysis, all the warp and fill yarns are treated separately. Behaviour of each yarn is analyzed as explained in the previous section. Overall behaviour of the woven fabric composite is presented considering the combined effect of all the individual yarns within a layer. A.4. Strain and deceleration history: single yarn of a woven fabric composite When a projectile impacts a woven fabric composite, a cone would be formed on the back face of the target. The shape of the cross-section of the cone formed on the surface of the target plate would be quasi-lemninscate. Quasi-lemniscate shape is because of difference in elastic properties and critical strain energy release rates along warp/fill directions and the other directions. For simplicity, the cone formed can be considered to be of circular cross-section, rather than quasi-lemniscate. The base of the cone spreads with a transverse wave velocity as given in Eq. (A.4). The plastic wave velocity is given in Eq. (A.3). After time t i the transverse wave travels to a distance r ti and the plastic wave travels to a distance of r pi. The projectile moves

11 114 through a distance z i. Radii r ti and r pi after time t i = i t are given in Eqs. (A.5) and (A.6), respectively. Because of stress wave attenuation, stress in the yarn and hence, strain in the yarn varies with the distance from the point of impact. Stress and in turn, strain is maximum at the point of impact and decreases along the length of the yarn. At ith time interval, the strain at the point of impact is given by [10], (d/2) + (r ti (d/2)) 2 + z 2 i ε i = + (r pi r ti ) r pi b (rpi/a) 1 { } ln b (A.7) a Studies have been carried out on stress wave attenuation of longitudinal waves during ballistic impact on typical woven fabric composites based on wave reflection/transmission at the interfaces [10]. Normalized stress can be represented as, y = f (x) = b x/a (A.8) where b is a constant of magnitude less than 1, x the distance, a the yarn size and y the normalized stress. It may be noted that the constant b represents the transmitted component of the stress wave. Hence, b is referred to as stress wave transmission factor. Parameter b is obtained from stress wave attenuation studies for each composite. It is a material property and depends upon geometry of the fabric as well as mechanical and physical properties of the reinforcing material and the matrix. For the target analyzed, b = As a result of stress wave attenuation from the point of impact to the point up to which the longitudinal stress wave has reached, there would be strain variation also. The maximum strain would be at the point of impact. The strain would be decreasing with distance from the point of impact to the point up to which longitudinal stress wave has reached. Once, the strain variation in a yarn/fibre is known, the energy absorbed by various mechanisms such as kinetic energy of the moving cone, deformation energy absorbed by the secondary yarns, tension in primary yarns, delamination and matrix cracking can be calculated. At the beginning of the first time interval of impact, entire energy is in the form of kinetic energy of the projectile. Later, this energy is dissipated into energy absorbed by various damage mechanisms and the kinetic energy of moving cone and projectile. Considering the energy balance at the end of ith time interval, KE P0 = KE pi + E KE + E SP(i 1) + ED (i 1) + E TF + E DL(i 1) + E MC(i 1) + E F(i 1) (A.9) Rearranging the terms in the above equation, 1 2 m pv0 2 E i 1 = 2 1 (m p + M Ci )Vi 2 (A.10) where E i 1 = E SP(i 1) + E D(i 1) + E TF(i 1) + E DL(i 1) + E MC(i 1) + E F(i 1) (A.11) The terms on the right hand side of Eq. (A.11) are known at (i 1)th instant of time. From this, the velocity of the projectile at the end of ith time interval can be obtained as, 1/2(m p V0 2 V i = E i 1) (A.12) 1/2(m p + M Ci ) If the projectile velocity is known at the beginning and the end of ith time interval, then the deceleration of the projectile during that time interval can be found out. It is given by, dc i = V i 1 V i (A.13) t It may be noted that the velocity at the beginning of the ith time interval is the same as the velocity at the end of (i 1)th time interval. Distance traveled by the projectile z i up to ith time interval is given by, n=i z i = z n n=0 z i = V i 1 t 1 2 dc i( t) 2 (A.14) (A.15) Utilizing dc i, the deceleration of the projectile during ith time interval, the force resisting the projectile motion can be calculated. It is given by, F i = m p dc i (A.16) The magnitude of the force on the projectile is the same as the magnitude of the force applied on the target, by the projectile. Hence, this force can be used to determine whether shear plugging takes place or not. The above process is repeated until all the primary yarns in the target fail, i.e., the complete perforation takes place. The velocity at the end of time interval, during which all the yarns are broken, is the residual velocity of the projectile. On the other hand, if the numerator in Eq. (A.12) becomes zero, then the projectile does not penetrate the target completely with the given initial velocity. Thus by repetition of the above procedure with various velocities so as to get complete perforation with zero residual velocity, the ballistic limit of the target laminate can be obtained. A.5. Kinetic energy of the moving cone formed The cone formed on the back face of the target absorbs some energy. By the end of ith time interval the surface radius of the cone formed is given by Eq. (A.5). Mass of the cone formed is, M Ci = πr 2 ti hρ (A.17)

12 115 The velocity of the cone formed is equal to V i, the velocity of the projectile at the end of ith time interval. So the energy of the cone formed at the end of ith time interval is, E KEi = 1 2 M CiV 2 i A.6. Energy absorbed due to shear plugging (A.18) When the target material is impacted by the projectile, shear plugging stress in the material near projectile periphery rises. As and when the shear plugging stress exceeds shear plugging strength, shear plugging failure occurs. As a result, plug formation takes place. This phenomenon is generally observed for carbon/epoxy composites. If at the beginning of the ith time interval, shear plugging stress exceeds shear plugging strength, then the energy absorbed by shear plugging during that time interval is given by the product of distance sheared, shear plugging strength and the area over which shear plugging stress is applied. It is given by, E SPi = Nh l S SP π dh (A.19) where N indicates the number of layers shear plugged during ith time interval and S SP denotes shear plugging strength. The energy absorbed by shear plugging by the end of ith time interval is given by, n=i E SPi = (A.20) E SPn n=0 A.7. Energy absorbed due to deformation of secondary yarns The secondary yarns experience different strains depending on their position. The yarns, which are close to the point of impact experience a strain, equal to the strain in the outermost primary yarn, whereas those yarns, which are away from the impact point, experience less strain. The energy absorbed in the deformation of all the secondary yarns can be obtained by the following integration [10], rti ( εsyi ) E Di = σ sy (ε sy )dε sy d/ 2 h 0 { ( )} d 2πr 8r sin 1 dr 2r (A.21) this phenomenon is stress wave attenuation. When the strain in yarns/fibres exceeds failure strain, it fails and some energy is absorbed due to tensile failure. For a yarn/fibre of cross-section area A it is given by, ( x ) ε=ε0 b x/a E TF = A σ(ε)dε dx (A.22) 0 ε=0 where ε 0 is the ultimate strain limit. If during ith time interval N numbers of yarns/fibres are failing, then the right hand side of the above expression is multiplied by N. It may be noted that during the movement of the cone formed, kinetic energy is converted into strain energy within yarns up to the point of failure. It retards the projectile. At the point of failure of the strained yarns, the strain energy stored in the yarns is dissipated. A.9. Energy absorbed due to delamination and matrix cracking Delamination and matrix cracking absorb some part of the initial kinetic energy of the projectile. A part of the conical area undergoes delamination and matrix cracking. The extent to which composite has delaminated and the matrix has cracked till (i + 1)th time instant can be calculated on the basis of strain profile in the composite at that time interval. From the results derived in the earlier section it can be observed that the strain at the impact point is the highest and it decreases along the length of the yarn. The area in which strain is more than the damage initiation threshold strain ε d, undergoes damage in the form of matrix cracking and delamination. However, complete matrix cracking may not take place. Evidence for this phenomenon is provided by the fact that after ballistic impact, matrix is still attached to the fibres and does not separate from the reinforcement completely. Due to matrix cracking, the interlaminar strength of the composite decreases. As a result, further loading and deformation causes delamination. This delamination is of mode II type. Again, delamination may not occur at all the lamina interfaces. Towards the end of ballistic impact event, when only a few nondelaminated layers are left, these non-delaminated layers are more likely to bend rather than delaminate. The area undergoing delamination and matrix cracking in the conical region is of quasi-lemniscate shape, which is taken to be A ql percent of the corresponding circular area. During (i + 1)th time interval the area of delamination and matrix cracking is given by, A.8. Energy absorbed due to tension in primary yarns π(r 2 d(i+1) r2 di )A ql (A.23) The yarns directly below the projectile, known as the primary yarns, fail in direct tension. All the primary yarns within one layer do not fail at one instant of time. As and when the strain of a particular yarn reaches the dynamic failure strain in tension, the yarn fails. It may be noted that the length of yarns/fibres failing in tension is twice the distance covered by the longitudinal wave. Also, the complete length of a primary yarn is not strained to the same extent as explained earlier. The reason for where r di indicates the radius up to which the damage has propagated until ith time interval. So the respective energies absorbed by delamination and matrix cracking during this time interval are given by, E DLi = P d π(r 2 d(i+1) r2 di )A qlg IIcd (N 0 1) E MCi = P m π(r 2 d(i+1) r2 di )A qle mt hv m (A.24) (A.25)

13 116 The factors P d and P m stand for percentage delamination and percentage matrix cracking. Hundred percent delamination indicates that complete delamination has taken place along all the interfaces within the damaged area. Hundred percent matrix cracking indicates that entire matrix within the damaged area has cracked. Energy absorbed till the ith time interval is given by, E DLi = E MCi = n=i E DLn n=1 n=i E MCn n=1 A.10. Other possible energy absorbing mechanisms (A.26) (A.27) Other possible energy absorbing mechanisms are: bending strain around the hinges at the edge of the contact patch, bending strain around the hinges at the edge of the cone formed and radial compression around the penetrating projectile. Target penetration takes place when, either all the fibres fail due to tension or all the layers fail due to shear plugging or due to the combined effect of both the mechanisms. Even after tensile failure of all the yarns or shear plug formation, the projectile has to overcome frictional resistance provided by the damaged laminate [11]. In cases when the projectile has just enough energy to fracture all the yarns but not enough energy to overcome the frictional resistance, it may get stuck up in the target. The frictional resistance depends on the type of fit between the projectile and the damaged target. And accordingly, it can result in local temperature rise. The present method is mainly for relatively thin and flexible target plates. Hence, the energy absorbing mechanisms as given in this section are not considered. Effect of matrix properties has been taken into account by considering matrix cracking energy and the energy absorbed by delamination. A.11. Calculation of contact duration The impact event starts when the projectile touches the front face of the target. End of the impact event is taken to be when the entire primary yarns fail or when the velocity of the projectile becomes zero. Total time from the start of the impact event till the end of the event is called as contact duration. If the end of the event occurs during nth time interval, then the contact duration is obtained as, t c = n t References (A.28) [1] N.K. Naik, P. Shrirao, Compos. Struct. 66 (2004) 579. [2] G. Zhu, W. Goldsmith, C.K. Dharan, Int. J.Solids Struct. 29 (1992) 399. [3] S.S. Morye, P.J. Hine, R.A. Duckett, D.J. Carrr, I.M. Ward, Compos. Sci. Technol. 60 (2000) [4] J.C. Smith, F.L. McCrackin, H.F. Schiefer, Textile Res. J. (1958) 288. [5] D. Roylance, A. Wilde, G. Tocci, Textile Res. J. (1973) 34. [6] B. Parga-Landa, F. Hernandez-Olivares, Int. J. Impact Eng. 16 (1995) 455. [7] N.K. Naik, V.K. Ganesh, Compos. Sci. Technol. 45 (1992) 135. [8] N.K. Naik, P.S. Shembekar, J. Compos. Mater. 26 (1992) [9] P.S. Shembekar, N.K. Naik, J. Compos. Mater. 26 (1992) [10] N.K. Naik, P. Shrirao, B.C.K. Reddy. Int. J. Impact Eng., in press, available online at [11] N.K. Naik, A.V. Doshi, AIAA J. 43 (2005) [12] G. Zhu, W. Goldsmith, C.K. Dharan, Int. J. Solids Struct. 29 (1992) 421. [13] S.T. Jenq, H.S. Jing, C. Chung, Int. J. Impact Eng. 15 (1994) 451. [14] S.V. Potti, C.T. Sun, Int. J. Impact Eng. 19 (1997) 31. [15] S.T. Jenq, J.T. Kuo, L.T. Sheu, Key Eng. Mater. I (1998) 349.