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, energyabsorbed by different mechanisms, ballistic limit velocity and contact duration are determined. The studies are carried out using the ana-lytical 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 onwave theory and energy balance between the projectile and the target. The inputs required for the analytical method are: diameter, massand velocity of the projectile; thickness and material properties of the target. Analytical predictions are compared with typical experi-mental results. A good match between analytical predictions and experimental results is observed. Further, effect of incident ballisticimpact velocity on contact duration and residual velocity, effect of projectile diameter and mass on ballistic limit velocity and effectof target thickness on ballistic limit velocity and contact duration are studied. It is observed that shear plugging is the major energyabsorbing mechanism.

Keywords: Ballistic impact; Thick composite; Parametric studies

1. Introduction

Composite materials are finding increasing uses in gen-eral engineering applications along with high performanceaerospace and defence applications during the last few dec-ades because of their high specific strength and high specificstiffness. This led to usage of more thick section compositesfor different applications. Such composite structuresundergo different loading conditions during their servicelife. Impact/ballistic impact is one of the typical loadingconditions. Thick composites behave differently comparedwith thin composites under impact/ballistic impact loadingconditions.

When an impact load is applied to a body, instanta-neous stresses are produced. But the stresses are not imme-diately transmitted to all parts of the body. The remoteportions of the body remain undisturbed for sometime.

The stresses progress in all directions through the body inthe form of disturbances of different types. In other words,stresses (and their associated deformations or strains) tra-vel through the body at specific velocities. These velocitiesare functions of the material properties. Regardless of themethod of application of impact load, the disturbancesgenerated have identical properties based only on the targetmaterial properties.

During an impact event the stress wave propagationtakes place in all the directions. Generally, this problemis analyzed using 1D, 2D or 3D approaches. In 1D and2D studies, the wave propagation through the thicknessdirection is not considered [2–4]. When these approachesare used to analyze structures, isotropic as well as orthotro-pic, it is assumed that the deformation behaviour along thethickness direction of the target is the same along the entirethickness. Such an assumption can be made for targets oflower thickness or, in other words, can be used for thinplates. If the thickness of the plate is increased the defor-mation and the induced stress behaviour of the plate wouldbe different at different locations along the thickness direc-

Nomenclature

A,Ap cross-sectional area of the projectileAql quasi-lemniscate area reduction factord diameter of the projectiledci deceleration of the projectile during a given time

intervaldh diameter of the holeE energy/Young’s modulusEbb energy absorbed due to bulge formation on the

back face of the targetEcf energy absorbed due to compression of the tar-

get directly below the projectile: Region 1Ecsy energy absorbed due to compression of the

yarns in the surrounding region of the impactedzone: Region 2

Edl energy absorbed due to delaminationEfr energy absorbed due to frictionEhg energy absorbed due to heat generatedEmc energy absorbed due to matrix crackingEmt energy absorbed by matrix cracking per unit

volumeEp kinetic energy of the projectile at exitErb energy absorbed due to reverse bulge formation

on the front face of the targetEsp energy absorbed due to shear plugging of the

yarnsEtf energy absorbed due to tension in the yarns in a

layerETotal total kinetic energy lost by the projectileF total force/contact forceFc compressive forceFi inertial forceG shear modulusGIIcd critical strain energy release rate in Mode IIh thickness of the targethl thickness of each layerhlc thickness of a layer after compressionhp length of the plugK numerical constant (depends on the shape of the

projectile)KEp kinetic energy of the projectile at a particular

time intervalKEp0 kinetic energy of the projectile: incidentl length of the projectilem mass of the projectilem0 initial mass of the projectilenlfs,ns number of layers failed due to shear pluggingnlft number of layers failed in tensionnlsc number of layers strained in compression

Pd percent delaminating layersPm percent matrix crackingSsp shear plugging strengtht time/contact durationV incident impact velocity/velocityVBL ballistic limit velocityVf fibre volume fractionVi velocity of the projectile at the ith instantVm matrix volume fractionVR residual velocityVzl velocity of compressive stress wave in

z-directionVzt velocity of shear stress wave in z-directionxd distance upto which damage has reachedxl distance the longitudinal wave has traveled in

x-directionxll, rp distance the longitudinal wave has traveled in

x-direction in a layerxllc change in length of the yarn after tensionxt distance the transverse wave has traveled in

x-directionxtl, rt distance the transverse wave has traveled in

x-direction in a layerz total depth the projectile has penetrated/projec-

tile displacementzi Depth of projectile penetrated during a given

time intervalzl distance the compressive stress wave has

traveled in z-directionzpl distance by which a layer has moved in forward

direction from its original positionzt distance the shear stress wave has traveled in

z-directionDhlc thickness by which a layer is compressedDt increment in time intervalecz compressive strain along the thickness directionemax ultimate strainet tensile strain along the radial directionetxl tensile strain in x-direction in a layerc shear strainm Poisson’s ratioq density of the target materialrcz compressive stress along the thickness directionrmax ultimate stressrt tensile stress along the radial directionrtx tensile stress along the radial directions shear stress

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tion. For such cases the analysis is based on also consider-ing the wave propagation along the thickness direction.There are typical studies on ballistic impact behaviour ofthick composites [5–8].

Ballistic impact behaviour of typical woven fabricE-glass/epoxy thick composites is presented in this paper.First, typical experimental results are presented. Then, ana-lytical results are presented based on the method presented

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in [1]. Analytical studies are based on considering the pro-jectile is rigid, cylindrical and flat-ended. The studies arecarried out to evaluate energy absorbed by different mech-anisms, ballistic limit velocity, contact duration and dam-age shape and size. Analytical predictions are comparedwith 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 pro-jectile and target information is given below:

• Projectile: Cylindrical, flat-ended hardened steel projec-tile of diameter, d = 6.33 mm; mass, m = 5.84 gm andlength, l = 24 mm.

• Target: Woven fabric E-glass/epoxy, unsupported areaof the target 125 mm · 125 mm, thickness 4–7 mm.

The experimental ballistic limit velocity for the case of5 mm thickness was 148 m/s and for the case of 4 mmthickness it was 137 m/s. When the experiment was carriedout with the target thickness of 7 mm, complete penetra-tion did not take place even with the incident ballisticimpact velocity of 173 m/s.

Gellert et al. [9] conducted ballistic impact experimentalstudies on woven fabric E-glass/vinylester composites withtarget thickness of 19, 14.5 and 9 mm using cylindrical, flat-ended steel projectiles. The experimental results are givenin 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 propertiesof the target plate and the other details are given in Table2 for a typical woven fabric E-glass/epoxy composite stud-ied. The same set of mechanical properties is used in thepresent study to predict the ballistic impact behaviour ofthe woven fabric E-glass/epoxy laminates. Also, the sameset of mechanical properties is used for the prediction of

Table 1Ballistic impact test results and analytical predictions for typical woven fabric

Projectile mass,m (gm)

Projectile diameter,d (mm)

Target thickness,h (mm)

PredictedVBL (m/s)

3.84 6.35 19 550.8943.84 6.35 14.5 452.5463.33 4.76 19 508.803.33 4.76 14 392.983.33 4.76 9 269.035.84 6.33 7 174.295.84 6.33 5 142.505.00 6.00 25 559.112

Cylindrical projectiles with flat-end.

ballistic impact behaviour of composites used by Gellertet al. [9] and Kumar and Bhat [10].

Compressive stress–strain data along the thicknessdirection for woven fabric E-glass/epoxy composite at highstrain rate is given in Appendix B. Tensile stress–straindata 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 underquasi-static loading for the study of frictional resistanceare provided in Appendix D. The plot obtained underquasi-static loading is used for the calculation of frictionalenergy during the ballistic impact event. The frictionalresistance is offered by the target for the movement of theprojectile during the later stage of the ballistic impactevent. During this stage, the velocity of the projectile is less.Considering this, the plot obtained under quasi-static load-ing is used for the calculation of frictional energy duringthe ballistic impact event.

3.1. Comparison of predicted and experimental results

Ballistic limit velocity is predicted for different targetthicknesses and projectile parameters using the analyticalmethod and the mechanical and other properties pre-sented in Table 2. The results are presented in Table 1and compared with the experimental results. It can beseen that there is a good match between the analyticallypredicted and experimentally obtained ballistic limitvelocities.

4. Ballistic impact behaviour of thick composites

Ballistic impact behaviour of thick composites is evalu-ated using the analytical method. The energy absorbed bydifferent energy absorbing mechanisms, projectile kineticenergy, projectile velocity, contact force, distance traveledby the projectile and longitudinal and shear waves alongthe thickness direction are evaluated as a function of time.Further, tensile strain at the failure of the layer is obtainedas a function of time and target thickness. Distances uptowhich longitudinal and transverse waves have traveled

E-glass/epoxy composites

Expt.VBL (m/s)

Expt. results Remarks

563 Penetrated Gellert et al. [9], 2000473 Penetrated Gellert et al. [9], 2000505 Penetrated Gellert et al. [9], 2000389 Penetrated Gellert et al. [9], 2000293 Penetrated Gellert et al. [9], 2000173 Not penetrated Present study148 Penetrated Present study547 Penetrated Kumar and Bhat [10], 1998

Table 2Input parameters required for the analytical predictions of ballistic impact behaviour

Reference Present study

Projectile details (cylindrical) Mass (gm) 5.84Shape Flat endedDiameter (mm) 6.33

Target details Material E-glass/epoxyVf (%) 50Thickness (mm) 5No. of layers 19Density (kg/m3) 1850Tensile failure strain (%) 3.5Compressive failure strain (%) 13.5Shear plugging strength (MPa) 90

Other details Compressive stress–strain curve Fig. 20Tensile stress–strain curve Fig. 21Quasi-lemniscate factor 0.9Delamination percentage 100Matrix crack percentage 100Strain energy release rate: mode II (J/m2) 1000Matrix cracking energy (MJ/m3) 0.9Calculation 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.

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along the radial direction until the failure of the layer anddamage variation along the thickness direction are alsopresented.

4.1. Case I

Figs. 1–5 are with the following data:

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 fabricE-glass/epoxy laminate.

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• 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 theprojectile took place, i.e., the tip of the projectile was atthe back face of the target at the end of the ballistic impactevent. Hence, V = 550.894 m/s is the ballistic limit velocity.

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 duringballistic impact event, V = 550.894 m/s, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.

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The projectile kinetic energy decreases during the ballisticimpact event. This is because the energy is absorbed bythe target by different mechanisms. The ballistic impact

event can be subdivided into three stages as given inAppendix A. During the first two stages, damage is takingplace upto the back face of the target and the energy is

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absorbed by different mechanisms. As can be seen fromFig. 1, for the case considered, the second stage ends at8.1 ls. During the third stage, energy is absorbed onlybecause of the frictional resistance offered due to the move-ment of the projectile. For this case, at the end of the bal-listic impact event, i.e., at time 359.43 ls, the velocity of theprojectile is zero (Fig. 3). The contact force increases dur-ing the first and second stages and decreases during thethird stage.

The major energy absorbing mechanism is shear plug-ging. Significant energy is absorbed by compression ofthe target in the region directly below the projectile(Region 1) and due to friction. Energy absorbed by matrixcracking, delamination, tension in the yarns and compres-sion in the surrounding region of the impacted zone(Region 2) is relatively less (Fig. 2). Percentage energyabsorbed by different mechanisms is given in Table 3.

Tensile strain in the yarns at failure of different layers isshown in Fig. 2. It can be seen that the tensile strainexceeds the permissible tensile strain only in the last fewlayers. It indicates that only last few layers fail in tension.The other layers fail due to shear plugging. Fig. 2 presentsthe time interval when a particular layer fails and it is plot-ted as a function of thickness. Among those layers whichfail in tension, it can be seen that lower layers fail earlierthan the layers above them. This is because of bulging effecton the back face of the target.

Projectile displacement during the ballistic impactevent, i.e., distance traveled by the projectile is shownin Fig. 4. At the end of the second stage the distancetraveled by the projectile is 3.1 mm. At this stage allthe layers have failed either due to tension in theyarns/layers due to shear plugging or due to the com-bined effect of both the mechanisms. After this, duringthe third stage, the projectile along with the plug formedwould be moving further towards the back face of thetarget. Since the velocity of the projectile is less duringthis stage, the time taken for the projectile to move fur-ther is significantly higher. The distance traveled bythrough-the-thickness longitudinal and shear waves isalso shown in Fig. 4. This would be only upto the endof second stage since the yarns/layers fail by the end ofsecond stage.

Table 3Percentage energy absorbed by different mechanisms with different incidentfabric E-glass/epoxy laminate

Incident ballistic impact velocity (m/s)

Energy (%)

Total kinetic energy of projectile, KEpo

Energy absorbed due to compression: Region 1, Ecf

Energy absorbed due to compression: Region 2, Ecsy

Energy absorbed due to tension in yarns, Etf

Energy absorbed due to shear plugging, Esp

Energy absorbed due to matrix cracking, Emc

Energy absorbed due to delamination, Edl

Energy absorbed due to friction, Efr

Kinetic energy of projectile at exit, Ep

Distance upto which transverse and longitudinal waveshave traveled along the radial direction is shown inFig. 5. The radial distance indicates the distance up towhich the waves have reached when the corresponding lay-ers fail either due to tension or due to shear plugging or dueto the combined affect of both the mechanisms. When theinduced stress exceeds the permissible strength, the fractureof the yarns would take place. But much before that, cleardamage zone can be seen within the layers whenever theinduced stress exceeds the damage initiation thresholdstress. This damage is either due to delamination or matrixcracking.

4.2. Case II

Results are presented in Figs. 6–10 for the same targetand projectile parameters (Table 2) with V = 600 m/s. Thisvelocity is above the ballistic limit velocity. In this case, theprojectile is exiting with velocity equal to 303.9 m/s. Inother words, the entire kinetic energy of the projectile isnot absorbed by the target. In this case, the second stageof the ballistic impact event is over at 7.1 ls. The projectileexits the target at 59.1 ls (Fig. 6). In this case also, only afew layers fail in tension near the back face of the target.The other layers fail in shear plugging. The contact forceincreases during the first and second stages and decreasesduring the third stage. Projectile displacement is 3.4 mmat the end of the second stage. Further movement of theprojectile would be during the third stage. The total contactduration is less in this case as compared to case I. This isbecause of the higher velocity of the projectile. Damageshape is nearly identical to that for case I.

4.3. Case III

Results are presented in Figs. 11–15 for the same targetand projectile parameters (Table 2) with V = 500 m/s. Thisvelocity is below the ballistic limit velocity. In this case, theprojectile did not completely penetrate the target. Thevelocity of the projectile was zero at time equal to 7.1 ls.During the ballistic impact event, shear wave has traveledalong the thickness direction to a distance of 12 mm. FromFig. 12, it can be seen that all the layers upto a distance of

ballistic impact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven

600 550.894 500

100.0 100.0 100.07.1 8.4 8.02.2 2.5 3.11.4 2.3 0.8

58.0 76.4 87.03.0 2.9 0.70.4 0.5 0.16.0 7.3 0

22.1 0 0

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.

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12 mm have failed due to shear plugging. The remaininglayers have not failed. Projectile displacement is 2.4 mmwhen the velocity of the projectile reaches to zero. Since

complete failure has not taken place, and the projectilevelocity is zero at a distance of 2.4 mm, projectile doesnot move further. Hence, frictional energy is not present

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.

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Fig. 10. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness directionduring 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.

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in this case. The contact force builds up at the end of theballistic impact event when the velocity of the projectilereaches to zero. This can lead to delamination at that par-

ticular interface upto which shear plugging has taken place.The damage shape is similar to those obtained for the casesI and II, but the size is smaller.

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 fabricE-glass/epoxy laminate.

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 fabricE-glass/epoxy laminate.

Fig. 14. Distance traveled by projectile and waves along thicknessdirection 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.

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Comparison of ballistic impact parameters with differentincident ballistic impact velocities is presented in Table 4. Itcan be noted that the total contact duration is more at bal-listic limit velocity compared to other velocities. The con-

tact force is higher at velocities lower than ballistic limitvelocity.

5. Parametric studies

5.1. Incident ballistic impact velocity

Contact duration between the projectile and the targetas a function of incident ballistic impact velocity is pre-sented in Fig. 16. The contact duration can be defined asfollows:

• Partial penetration: The time interval starting fromwhen the projectile just hits the target to when the veloc-ity of the projectile becomes zero.

• Complete penetration: The time interval starting fromwhen the projectile just hits the target to when the pro-jectile tip reaches to the back face of the target.

At ballistic limit velocity, when the projectile tip justreaches to the back face of the target, the velocity of theprojectile would be zero. For the case considered(Fig. 16), the ballistic limit velocity is 550.894 m/s. Thecontact duration behaviour as given in Fig. 16 and Table4 can be divided into two parts:

Part 1: Incident ballistic impact velocity is less than theballistic limit velocity.

As the incident ballistic impact velocity increases thecontact duration increases. At lower velocities, the energyof the projectile would be lower and the time taken forthe projectile to reach to zero velocity would be lower.

Fig. 15. Distance upto which transverse and longitudinal waves have traveled along radial direction and damage variation along thickness directionduring 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 4Comparison of ballistic impact parameters with different incident ballisticimpact velocities, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabricE-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 7.1Total 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, VR (m/s) – 0 304

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As the velocity increases, the time required for the projec-tile to reach to zero velocity would be more. Also, at highervelocities some energy is absorbed due to friction. Duringthis phase, the deceleration of the projectile is very slowleading to larger contact duration. The maximum contactduration is with ballistic limit velocity.

Part 2: Incident ballistic impact velocity is more thanthe ballistic limit velocity.

As the incident ballistic impact velocity increases thecontact duration decreases. This is because, as the incidentballistic impact velocity increases, the exit velocity of theprojectile increases. This also means that the velocity of

the projectile within the target would be higher with higherincident ballistic impact velocity.

Residual velocity, VR of the projectile as a function ofincident ballistic impact velocity is presented in Fig. 17.This plot is for the case with h = 19 mm, projectile mass3.84 gm and projectile diameter 6.35 mm. For this case,ballistic limit is 550.894 m/s. This indicates that completeperforation would not take place upto the incident ballisticimpact velocity of 550.89 m/s. If the projectile velocity ismore than ballistic limit velocity, there would be completeperforation and the projectile would be exiting with certainvelocity.

It is interesting to note the plot of residual velocity ver-sus incident ballistic impact velocity. As the incident ballis-tic impact velocity is increased beyond the ballistic limit,the corresponding residual velocity of the projectile alsoincreases. But the increase is very steep just above the bal-listic limit. Just to give an example, complete perforationdoes not take place with incident ballistic impact velocityof 550.89 m/s. But with incident ballistic impact velocityof 552 m/s, complete perforation takes place with the resid-ual velocity of 67 m/s.

Similar observations were made by Zhu et al. [11], Jenqet al. [12] and Potti and Sun [13] during their experimentalstudies. The residual velocity of the projectile is increasedas the incident ballistic impact velocity is increased abovethe ballistic limit. It was clearly observed that the increasein residual velocity was very steep immediately after theballistic limit velocity was reached.

Fig. 16. Contact duration as a function of incident ballistic impactvelocity, m = 3.84 gm, d = 6.35 mm, h = 19 mm, woven fabric E-glass/epoxy laminate.

Fig. 17. Residual velocity as a function of incident ballistic impactvelocity, 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 andprojectile diameter, h = 19 mm, woven fabric E-glass/epoxy laminate.

Fig. 19. Ballistic limit velocity and contact duration at ballistic limitvelocity as a function of target thickness, m = 3.84 gm, d = 6.35 mm,woven fabric E-glass/epoxy laminate.

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5.2. Projectile diameter and mass

Effect of projectile diameter and mass on ballistic limitvelocity is presented in Fig. 18. With the same mass of theprojectile, as the diameter of the projectile increases, ballis-tic limit velocity increases. For the case considered the rela-

tionship is nearly linear. Further, with the same diameter ofthe projectile, as the mass of the projectile increases, ballisticlimit velocity decreases. But as the mass increases, the rateof decrease in ballistic limit velocity increases.

5.3. Target thickness

Ballistic limit velocity and contact duration at ballisticlimit velocity as a function of target thickness are shownin Fig. 19. The plots are for the case of same mass and same

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diameter of the projectile. As the target thickness increases,the ballistic limit velocity increases. But as the thickness ofthe target increases, the rate of increase in ballistic limitvelocity decreases. Also, as the target thickness increases,contact duration increases. As the thickness of the targetincreases, the increase in contact duration is nearly linearinitially. But the rate of increase decreases later.

6. Conclusions

Studies have been carried out on ballistic impact behav-iour of typical woven fabric E-glass/epoxy thick compos-ites. Effects of different target and projectile parametersare investigated. Specific observations are:

• The major energy absorbing mechanism is shear plug-ging. Compression of the region directly below the pro-jectile (Region 1) and friction between the projectileand the target also absorb significant amount ofenergy. Energy absorbed by matrix cracking, delamina-tion, tension in the yarns and compression in the sur-rounding region of the impacted zone (Region 2) isnot significant.

• Contact duration depends upon incident ballistic impactvelocity.– For the case when incident ballistic impact velocity is

less than ballistic limit velocity, contact durationincreases with the increase in incident ballistic impactvelocity.

– The maximum contact duration is obtained with bal-listic limit velocity.

– For the case when incident ballistic impact velocity ismore than ballistic limit velocity, contact durationdecreases with the increase in incident ballistic impactvelocity.

• Beyond the ballistic limit, as the incident ballistic impactvelocity increases, the residual velocity of the projectileincreases. But the rate of increase in residual velocityis very significant just above ballistic limit.

• For the same mass of the projectile, as the diameter ofthe projectile increases, ballistic limit velocity increases.The behaviour is nearly linear.

• For the same diameter of the projectile, as the mass ofthe projectile increases, ballistic limit velocity decreases.But as the mass increases, the rate of decrease in ballisticlimit velocity increases.

• For the same mass and diameter of the projectile, asthe thickness of the target increases, ballistic limitvelocity increases. But as the thickness of the targetincreases, the rate of increase in ballistic limit velocitydecreases.

• For the same mass and diameter of the projectile, as thethickness of the target increases, contact durationincreases. As the thickness of the target increases, theincrease 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 isimpacted by a projectile. Due to transverse impact, com-pressive wave and shear wave propagate along the thick-ness direction and tensile wave and shear wave propagatealong the radial direction. The projectile applies forces onto the target. The forces acting are the compressive force,inertial force and frictional force. These forces act at vari-ous stages of the impact event.

Different damage and energy absorbing mechanismsduring ballistic impact are: compression of the targetdirectly below the projectile (Region 1), possible reversebulge formation on the front face, compression in the sur-rounding region of the impacted zone (Region 2), tensionin the yarns, shear plugging, bulge formation on the backface, delamination and matrix cracking, friction betweenthe target and the projectile and heat generation due toimpact. Impacted materials can fail in a variety of ways.The actual mechanisms depend on such variables as projec-tile size, shape, mass and velocity, and target materialproperties and geometry and relative dimensions of theprojectile and the target.

Generally, the ballistic impact event can be sub-dividedinto three stages. During the first stage, the projectilestrikes to the target and compression of the target takesplace directly below the projectile face. As the compressionprogresses, the material would flow predominantly alongthe thickness direction. Material flow can also be in theradial direction as well as towards the front face of the tar-get. This stage would continue until through-the-thicknesscompressive wave reaches to the back face of the target.During this stage, compressive stresses are generated withinthe target directly below the projectile. As a result of thisthe surrounding region would be under tension along theradial direction.

Additionally, because of the impact force, shear stressesare generated within the target around the periphery of theprojectile. Any of these stresses could lead to failure of thetarget. As the projectile moves further, the yarns in Region2 of the upper layers exert pressure on the yarns in Region2 of the lower layers. In other words, compressive deforma-tion of the yarns takes place in Region 2 also.

Because of the compression of the layers, as well as pos-sible failure of the target by different modes, the projectilemoves further. This leads to bulge formation on the backface. This is the second stage of impact. Along with bulgeformation on the back face, failure of the yarns/layerswould take place by different mechanisms in the upper lay-ers of the target. This process continues and the projectilemoves further. A clear plug formation can take place infront of the projectile. Also, the yarns/layers can fail in ten-sion on the back face because of bulge formation. Duringthe entire ballistic impact event, inplane matrix crackingand delamination between the layers can also take place.

During the third stage, the projectile moves further andthe plug and the projectile exit from the back face of the

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target. As the projectile penetrates into the target and startsmoving further, frictional forces act between the projectileand the target. Heat can also be generated during thisprocess.

The total kinetic energy of the projectile lost during theballistic impact event is equal to the total energy absorbedby the target till that time interval. It is given by the follow-ing relation:

ETotali ¼ Ecf i þ Erbi þ Ecsyi þ Etf i þ Espi þ Ebbi þ Edli

þ Emci þ Efri þ Ehgi: ð1Þ

The following assumptions are made in the analyticalformulation:

• 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 thick-

ness direction only within those layers through whichthe compressive wave has traveled. Also, compressivestrain is uniform within those layers.

• The shear plugging stress is experienced along the thick-ness direction only within those layers through whichthe shear wave has traveled. Also, shear plugging stressis uniform within those layers.

• The plug formed is freely moving forward, i.e., resis-tance is not offered by the target except for the frictionalresistance.

• The peak tensile strain within the yarns is near theperiphery of the projectile. Hence, the tensile failure ofthe yarns would take place near the periphery of theprojectile.

• The deceleration of the projectile remains constant dur-ing each time interval.

The analytical formulation for the three stages is pre-sented below: The total force acting on the projectile isgiven by

F ¼ F i þ F c; ð2Þwhere, inertial force,

F i ¼1

2qKAV 2 ð3Þ

and, compressive force,

F c ¼ rczA: ð4ÞFor the cylindrical projectile with a flat-end, K = 1

[14].These forces are acting on the effective mass of the pro-

jectile. The effective mass of the projectile includes thematerial of the target displaced by the projectile movingwith it. The effective mass of the projectile used in the equa-tion is, therefore, m0 + qAz. This is based on consideringthat the projectile has moved by a distance z. The equationof motion for the penetration process is

d

dtðmV Þ ¼ F i þ F c

d

dtðmV Þ ¼ V

dmdtþ m

dVdt¼ 1

2KqAV 2 þ rczA

ð5Þ

The rate of change of effective mass of the projectile is

dmdt¼ qA

dzdt¼ qAV

dVdt¼ dV

dzdzdt¼ V

dVdz

ð6Þ

Substituting Eq. (6) into Eq. (5), following relation isobtained:

qAV 2 þ ðm0 þ qAzÞV dVdz¼ 1

2KqAV 2 þ rczA: ð7Þ

The above equation is solved by separation of variablesto calculate the velocity [14] and the expression for velocityis obtained as

V ðzÞ ¼ V 2i þ

rczðzÞqð1þ 0:5KÞ

� �m0=qA

ðm0=qAÞ þ z

� �2þK

� rczðzÞqð1þ 0:5KÞ

!1=2

:

ð8ÞThe time required for the projectile to penetrate distance

z is calculated by the following expression:

t ¼Z z

0

1

V ðzÞ dz

¼Z z

0

V 2i þ

rczðzÞqð1þ 0:5KÞ

� �m0=qA

ðm0=qAÞ þ z

� �2þK

� rczðzÞqð1þ 0:5KÞ

!�1=2

dz:

ð9ÞThe 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 mecha-nisms during the first time interval is calculated. Knowingthe initial kinetic energy of the projectile and the energyabsorbed during the first time interval, the velocity of theprojectile for the next time interval is calculated. By know-ing the velocity, various parameters such as displacementof the projectile, strain, contact force and energy absorbedby different mechanisms are calculated for the given timeinterval. This procedure is continued until the compressivewave reaches to the back face of the target. The shear wavefollows the compressive wave along the thickness direction.

The wave velocities along the thickness direction aregiven by

V zl ¼ k

ffiffiffiffiffiffiffiffiffiffiffiffiffi1

qdrcz

decz

s; ð10Þ

V zt ¼ k

ffiffiffiffiffiffiffiffiffi1

qdsdc

s; ð11Þ

k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� tÞ

ð1þ tÞð1� 2tÞ

s: ð12Þ

For the elastic waves, E and G are used instead of thelocal slopes as given in Eqs. (10) and (11). The distancetraveled by compressive wave and shear wave along thethickness direction at any instant of time is

462

zl ¼ V zlt; ð13Þzt ¼ V ztt: ð14Þ

The number of layers through which the compressivewave has traveled is

nlsc ¼zl

hl

: ð15Þ

As the projectile is impacted onto the target, tensile andshear waves are also generated along the radial direction inthe target. The wave velocities can be calculated using Eqs.(10) and (11). But, in this case, the material properties arewith respect to radial direction.

The compressive strain in each layer at any instant oftime is given by

ecz ¼zzl

: ð16Þ

The calculation of tensile strain in a particular yarn/layer is given in [1].

Considering the strain variation as linear with maximumstrain at the periphery of the projectile and zero at a dis-tance upto which the longitudinal radial wave has reachedat that particular time, the maximum strain is calculated asbelow:

emaxtxl ¼ 2etxl: ð17Þ

In the beginning of the ballistic impact event, all theenergy is in the form of kinetic energy of the projectile.During the impact event, the target absorbs some of theenergy. The energy balance at the end of ith time intervalis obtain as

KEp0 ¼ KEpi þ Ecfði�1Þ þ Erbði�1Þ þ Ecsyði�1Þ þ Etfði�1Þ

þ Espði�1Þ þ Ebbði�1Þ þ Edlði�1Þ þ Emcði�1Þ

þ Efrði�1Þ þ Ehgði�1Þ: ð18Þ

Rearranging the terms in the above equation

1

2m0V 2 �

Xi�1

i¼1

Ei�1 ¼1

2miV 2

i

or1

2mi�1V 2

i�1 � Ei�1 ¼1

2miV 2

i

ð19Þ

Here,

Eði�1Þ ¼ Ecfði�1Þ þ Erbði�1Þ þ Ecsyði�1Þ þ Etfði�1Þ

þ Espði�1Þ þ Ebbði�1Þ þ Edlði�1Þ þ Emcði�1Þ

þ Efrði�1Þ þ Ehgði�1Þ: ð20Þ

The terms on the right-hand side of Eq. (20) can be cal-culated for each time interval. Hence, the velocity of theprojectile for the next time interval is calculated as

V i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12mi�1V 2

i�1 � Eði�1Þ� �

12mi

s: ð21Þ

The deceleration of the projectile during ith time intervalis obtained as

dci ¼V i�1 � V i

Dt: ð22Þ

The distance traveled by the projectile during ith timeinterval is obtained as

zi ¼ V i�1Dt � 1

2dciðDtÞ2: ð23Þ

The total distance traveled by the projectile is equal tothe summation of the distance the projectile has traveledin each time interval. The force resisted by the targetagainst the motion of the projectile is given by

F i ¼ midci: ð24ÞThis force is used to determine if the shear plugging is

taking place or not. The above-mentioned procedure iscontinued during all the three stages of the ballistic impactevent.

Energy absorbed due to compression of the target inRegion 1

Ecf ¼ Ap

Z ecz

ec¼0

rczðeczÞde

� �zl: ð25Þ

Energy absorbed due to compression of the yarns in thesurrounding region of impacted zone in Region 2

Ecsy ¼ 2phXnlsc

j¼nlf

Z xt

d=2

Z e¼ecz

e¼0

rczðeczÞde

� �xdx: ð26Þ

Energy absorbed due to tension in yarns

Etf ¼ Ay

Xnlsc

j¼nlf

Z xl

0

Z etxl

e¼0

rtxðetxlÞde

� �dx: ð27Þ

The total energy absorbed due to tension is sum of theenergy absorbed by different yarns.

Energy absorbed by shear plugging during a time inter-val is given by the product of thickness of the target that issheared, shear plugging strength and the area over whichshear stress is acting. It is given by

DEspi ¼ nshlSsppdh: ð28ÞEnergy absorbed by shear plugging by the end of ith

time interval is given by

Esp ¼Xi

n¼1

DEspi: ð29Þ

Energy absorbed by delamination and matrix crackingduring ith time interval is given by

Edli ¼ P dpx2dAqlGIIcdðN 0 � 1Þ;

Emci ¼ P mpx2dAqlEmthðV mÞ

ð30Þ

Frictional resistance offered by the target towards themovement of the projectile is a material property. Itdepends upon projectile diameter, length and surface con-dition. It also depends upon the diameter of the holeformed within the target due to ballistic impact, hole sur-face condition and the target material properties. Fric-tional resistance is determined experimentally.

Fig. 20. Compressive stress–strain curve along thickness direction forwoven fabric E-glass/epoxy laminates at high strain rate.

Fig. 21. Tensile stress–strain curve along radial direction for woven fabricE-glass/epoxy laminates at high strain rate.

463

Appendix B. Stress–strain data at high strain rates:

compressive loading

The compressive stress–strain curve along the thicknessdirection for the woven fabric E-glass/epoxy composite athigh strain rate is given in Fig. 20. Typical parametersfor this stress–strain curve are as follows:

• emax = 13.5%.• rmax = 1350 MPa.• Area under the stress strain curve = 127.57 MPa.• The compressive stress–strain curve is represented by the

equation,

y ¼ �0:0496x4 þ 2:0004x3 � 31:956x2 þ 288:86x:

Fig. 22. Load–displacement plot under quasi-static loading: penetration ofd = 8 mm, dh = 8 mm, woven fabric E-glass/epoxy laminate.

Here, y indicates stress in MPa and x indicates strain inpercentage.

Appendix C. Stress–strain data at high strain rates: tensile

loading

The tensile stress–strain curve along the radial directionfor the woven fabric E-glass/epoxy composite at high strain

a cylindrical projectile into a composite target with a hole, h = 19 mm,

464

rate is given in Fig. 21. Typical parameters for this stress–strain curve are as follows:

• emax = 3.5%.• rmax = 560 MPa.• Area under the stress strain curve = 13.72 MPa.• The tensile stress–strain curve is represented by the

equation,

y ¼ �1:0884x4 þ 16:003x3 � 106:09x2 þ 381:9x:

• Here, y indicates stress in MPa and x indicates strain inpercentage

Appendix D. Load–displacement plot under quasi-static

loading for the study of frictional resistance

Typical experiments were conducted to study the fric-tional resistance offered during the movement of the pro-jectile within the target with a hole. Load–displacementplot under quasi-static loading condition to study the pen-etration behaviour of a cylindrical, flat-ended hardenedsteel projectile into a composite target with a hole is pre-sented in Fig. 22. The figure shows the distance movedby the projectile within the hole and the correspondingload. Based on this, work done to move the projectile overa distance and the corresponding energy absorbed due tofriction are calculated. This information is made use forthe calculating the frictional energy during the ballisticimpact event.

It can be seen from the figure that the behaviour is non-linear and slope of the curve decreases as the projectilemoves further. The behaviour depends upon the surfacecondition of the hole formed during the ballistic impactevent, diameter of the hole, thickness of the target, diame-ter of the projectile and the material of the target. For the

present study, the plot given in Fig. 22 is used for calculat-ing frictional energy absorbed during ballistic impact event

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