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International Journal of Impact Engineering 38 (2011) 181e197

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Finite element modeling of impact, damage evolution and penetration ofthick-section composites

Bazle A. Gama a,*, John W. Gillespie Jr. a,b,c

aUniversity of Delaware Center for Composite Materials (UD-CCM), Newark, DE 19716, USAbDepartment of Materials Science & Engineering, University of Delaware, Newark, DE 19716, USAcDepartment of Civil & Environmental Engineering, University of Delaware, Newark, DE 19716, USA

a r t i c l e i n f o

Article history:Received 13 October 2009Received in revised form3 November 2010Accepted 5 November 2010Available online 18 November 2010

Keywords:Ballistic impactComposite damage modelingPenetration mechanicsThick-section composites

* Corresponding author. Tel.: þ1 302 831 0248; faxE-mail address: gama@udel.edu (B.A. Gama).

0734-743X/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.ijimpeng.2010.11.001

a b s t r a c t

Impact, damage evolution and penetration of thick-section composites are investigated using explicitfinite element (FE) analysis. A full 3D FE model of impact on thick-section composites is developed. Theanalysis includes initiation and progressive damage of the composite during impact and penetration overa wide range of impact velocities, i.e., from 50 m/s to 1000 m/s. Low velocity impact damage is modeledusing a set of computational parameters determined through parametric simulation of quasi-staticpunch shear experiments. At intermediate and high impact velocities, complete penetration of thecomposite plate is predicted with higher residual velocities than experiments. This observation revealedthat the penetration-erosion phenomenology is a function of post-damage material softening parame-ters, strain rate dependent parameters and erosion strain parameters. With the correct choice of theseparameters, the finite element model accurately correlates with ballistic impact experiments. The vali-dated FE model is then used to generate the time history of projectile velocity, displacement andpenetration resistance force. Based on the experimental and computational results, the impact andpenetration process is divided into two phases, i.e., short time Phase I e shock compression, and longtime Phase II e penetration. Detailed damage and penetration mechanisms during these phases arepresented.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Impact, damage and penetration modeling of thick-sectioncomposites are of great importance to many industrial, auto-motive, aerospace and defense applications. A large number ofmaterial properties and model parameters are required indamage modeling of composites using finite element analysis(FEA) techniques. A systematic model-experiment methodologyis required to validate the finite element model (FEM) from staticto impact loading conditions. A validated FEM should predict theevolution of impact damage, the rebound velocity of theprojectile, and the impact-contact or resistance force for non-penetrating projectiles. For penetrating projectiles, the predictionof projectile residual velocity and displacement, evolution ofdamage, and penetration resistance force on the projectile withsignificant accuracy is needed to predict the post-ballisticresidual strength of the composite laminate and multi-hit

: þ1 302 831 8525.

All rights reserved.

performance. For low velocity impact experiments one canmeasure the impact-contact force as a function of time to vali-date the FE model. On the other hand, high speed flash-X-Rayand photography can be used to measure the impact and residualvelocities of the projectiles, ejection velocities of debris, and thedynamic deflection of the impact plate. However, in mostcommercial ballistics facilities, the impact and residual velocitiesof the projectile are measured at a minimum. It is thus importantto validate a FEM which can accurately model the impact andresidual velocities of the projectiles over a wide range ofprojectile impact velocities. The validated FEM then can be usedto predict the time histories of projectile velocity and penetrationresistance force with confidence. This is the main goal of thepresent study.

A fair amount of work can be found in literature addressingdifferent aspects of composite impact and damage modeling underlow velocity impact [1e8] and high velocity impact on fabrics[9e17], soft laminates [18,19], and fiber reinforced composites[20e34]. This includes our original work [34] on the developmentof a quasi-static punch shear test (QS-PST) methodology to studythe quasi-static penetration mechanics behavior of thick-section

Nomenclature

Subscripts & Superscripts1 principal material direction 12 principal material direction 23 principal material direction 312 principal plane 1231 principal plane 3132 principal plane 32C compression, compositeFC fiber crushFS fiber shearI impactP projectileR residualS support spanT tension, transferred

Roman lettersCrate1 rate parameter for strengthsCrate2 rate parameter for axial moduliCrate3 rate parameter for shear moduliCrate4 rate parameter for through-thickness modulusD diametereLimit element eroding limit axial straineCrush element eroding relative volumetric compression

eExpn element eroding relative volumetric expansionE axial modulus, energyF penetration resistance forceG shear modulusH projectile displacement, heightm1 softening parameter for fiber damage in material

direction 1m2 softening parameter for fiber damage in material

direction 2m3 softening parameter for fiber crush and punch shearm4 softening parameter for matrix crack and delaminationM massP exponent of Lambert equation, pressureS strengthSFFC residual compression strength factorSDelam delamination scale factorSPR support span-to-punch/projectile diameter ratioV projectile velocityV50 ballistic limit velocityW work

Greek lettersr density4 Coulomb friction anglen Poisson’s ratioumax maximum admissible modulus reductionb parameter of Lambert equation

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197182

composites. This methodology established a series of model vali-dation experiments to determine rate independent mechanicalproperties and damage parameters for quasi-static penetrationmodeling of plain-weave 814 gsm (gm/m2) (24 oz/yd2, 5 � 5 tows/inch) S-2 glass/SC15 laminates. Our modeling parameters wereused to successfully model impact, damage and penetration ofrelatively thin composite plates [20e23,35]. Advanced theoreticalmodels which explain experiments on fabrics or body armor existsin literature [36e38], however, similar models are not availablefor thick-section composites. Theoretical impact models treatfabrics as a membrane and consider transverse impact induced 1Delasticeplastic stress wave propagation along the fiber axis, anduses tensile failure criteria for the fiber. Membrane models havebeen further modified to model impact on thin-composites [39],however, neglects the through-thickness stress wave propagation.In our previous work [35], we have shown that impact damage, andpenetration of thick-section composites occurs under differentphases and damage modes, e.g. (a) shock compression (b) com-pressioneshear (c) tensioneshear (d) and structural vibration.While theoretical models for different phases of penetrationwill bepresented elsewhere, the present work focuses on FE analysis togain insight.

The present work builds upon our previous quasi-static pene-tration model of ballistic penetration [35] and extends it to developa combined experimental and computational methodology formodeling the high velocity impact, damage and penetrationmechanics of thick-section composites under a wide range ofimpact velocities. In the present FEA work, the Lagrangian explicitfinite element analysis code LS-Dyna 971 is used in conjunctionwith the progressive composite damage model used in ourprevious studies to model the rate sensitive progressive damageand penetration behavior of thick-section composites. Validated FEmodel parameters are then used for detailed analyses of impact,stress wave propagation, damage, and penetration mechanics ofcomposites plates of different thickness and impact velocities.

2. Modeling progressive damage and penetration of thick-section composites

Computational penetration mechanics modeling of compositesis a relatively complex subject, which requires defining several fiberand matrix dominated damage modes, rate effects on materialproperties, and one or more element erosion criteria. One suchprogressive composite damage model is MAT_COMPOSI-TE_DMG_MSC (MAT162), developed by Materials Sciences Corpo-ration and implemented in LS-Dyna 971 [36]. MAT162 is speciallydesigned to model ballistic penetration of thick-section compositesin three dimensions (3D). The detailed formulation can be found inRef. [40,41]; however, a brief summary is presented in Appendix Afor readers not familiar with this damage model. MAT162 requiresa total of 34 material properties and computational modelingparameters to describe the response of orthotropic uni-directionaland/or woven composites. There are nine elastic constants (threeYoung’s moduli, three shear moduli and three Poisson’s ratios) andten failure strength properties (S0s) which can be determined fromstandard ASTM test methods or can be predicted using compositelaminate mechanics theories with the exception of fiber crushstrength (SFC) and fiber shear strength (SFS).

A set of quadratic failure functions (Appendix A) is used todefine the initiation of different damage mechanisms related tofiber fracture, fiber crush, fiber shear, in-plane matrix crack anddelamination [40]. These failure functions are also used to monitorthe progressive growth of damage of the composite. Following thesuggestion by Matzenmiller et al. [42], the post-damage softeningbehavior of the composite is modeled by an exponential functionwith four parameters: m1 for fiber damage in material direction 1,m2 for fiber damage in material direction 2, m3 for fiber crush andpunch shear damage, and m4 for matrix crack and delaminationdamage. The effect of post-damage softening parameters on thestressestrain response of a unit single element under differentloading conditions can be found in Appendix A [43]. In order to

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 183

determine the softening parameters and the element erosionparameters for practical applications, quasi-static punch shear tests(QS-PST) were simulated to match the damage modes and pene-tration load as a function of punch displacement in Ref. [35] whilesetting the rate parameters as zero. The properties of plain-weaveS-2 Glass/SC15 epoxy composites used in Ref. [35] are presented inTable 1.

The rate independent post-damage softening parameters whichprovided good correlation to QS-PST experiments at varioussupport spans were found to be: m1 ¼ m2 ¼ 2.0, m3 ¼ 0.5, andm4 ¼ 0.20e0.50 [35]. In the present study, m4 ¼ 0.20 is used. Inmodeling ballistic impact, damage and penetration it is alsoimportant to show that a set of ‘m’ values can be used to predict theexperimentally determined residual velocities of the projectileafter complete penetration for different impact velocities.

Delamination damage between two laminas of different orien-tation is computed using the delamination criteria (Appendix A,Eq. A.7), allowing the user to define a finite number of delaminationplanes through the thickness of a composite laminate. A delami-nation scale factor (SDelam) can be used as a parameter to control theinitiation and growth of delamination. Pahr et al. [44] found thatthe stress concentration at the free edges of a short beam shearspecimen approaches 1.21, and based on this observation a value of1.2 was used for SDelam in Ref. [35]. Among the 10 material strengthparameters, SFC and SFS cannot be determined using standard ASTMtest methods. Materials Sciences Corporation has developeda Laterally Confined Transverse Compression (LCTC) testmethod fordetermining SFC & SFS [45,46]. The QS-PSTmethod at span-to-punchdiameter ratio, SPR¼ 0.00 and 1.00 was also found to be suitable fordetermining SFC & SFS [34,35]. The residual strength parameterunder in-plane compression (SFFC) should be estimated fromexperiments or determined through parametric studies. A value ofSFFC¼ 0.3 is used in our earlier modeling of quasi-static punch sheartests [35] and the same is used for ballistic impact simulations. TheMohreCoulomb friction parameter (4) is used to include the effectof compressive stress on the shear strengths and can be determinedthrough an out-of-plane off-axis compression test methodology asdescribed in Ref. [47]. The quasi-static properties for plain-weaveS-2 glass/SC15 laminates in conjunction with the MAT162 damagemodels has provided good correlation between the prediction ofdamage and penetration resistance of laminates over a wide rangeof thicknesses and spans under quasi-static loading.

Rate effects on strength and stiffness are modeled using loga-rithmic functions. The current implementation uses four rateparameters: Crate1 for all strength values, Crate2 for in-plane Young’smoduli, Crate3 for all shearmoduli, and Crate4 for transversemodulus(Appendix A). Since the rate dependence of all strength values(Appendix A) is modeled with one rate parameter, Crate1, it isconsidered for simplicity an average computational parameter forall strength values. In order to determine an acceptable set of rate

Table 1MAT162 input properties & parameters for plain-weave (5 � 5 tows/inch) S-2 glass/SC15 composites [35].

MID r, kg/m3 E1, GPa E2, GPa E3, GPa n21 n31 n32162 1850.00 27.50 27.50 11.80 0.11 0.18 0.18G12, GPa G23, GPa G31, GPa2.90 2.14 2.14S1T, MPa S1

C, MPa S2T, MPa S2

C, MPa S3T, MPa SFC, MPa SFS, MPa S12, MPa

604.00 291.00 604.00 291.00 58.00 850.00 300.00 75.00S23, MPa S31, MPa SFFC 4, deg eLimit SDelam58.00 58.00 0.300 10 0.200 1.200umax eCrush eExpn Crate1 m10.999 0.001 4.500 0.030 2.00m2 m3 m4 Crate2 Crate3 Crate42.00 0.50 0.20 0.000 0.030 0.030

parameters, we will choose a set of baseline rate parameters basedon available data in literature, and then examine the effect of rateparameters on the penetration resistance behavior by conductingadditional parametric simulations. Finally, correlation with ourballistic impact testing will determine whether this assumptionand procedure is adequate for our material system.

In Ref. [48] the rate parameter of plain-weave (15 � 15 tows/inch) S-2 glass/SC15 composites for axial compressive strengths(SC1 ¼ SC2) is reported to be Crate1jSC1 ¼ 0:1860 and the rateparameter for through-thickness strength (SC3zSFC) to beCrate1jSFC ¼ 0:0273. In our recent dynamic punch shear tests therate parameter for fiber shear strength (SFS) is found to beCrate1jSFS ¼ 0:0268. Since the experimentally determined rateconstants for different strength values are different, and the ballisticpenetration is predominantly compressioneshear strength domi-nated (SFC and SFS), we have chosen the baseline value ofCrate1¼0.03. Rate dependency of in-plane axialmoduli (E1 and E2) ofplain-weave composites is not easily found in the literature.Moreover, the in-plane axial moduli are fiber dominated propertiesand can be considered as rate insensitive. We will choose thebaseline value of Crate2¼ 0.00 and subsequently study its parametriceffect on penetration resistance. From two independent references[49,50] and from one available data point in each reference at shearstrain rates 1000/s and 460/s, the rate parameter of transverse shearmodulus of E-Glass/Epoxy composites are estimated to beCrate3jG31

¼ 0:048 and 0.096, respectively. However, rate depen-dence of in-plane shear modulus was not found in literature. Sincethe experimental rate constant for transverse shear moduli variesover a wide range and the rate dependence of in-plane shearmodulus is unavailable, we chose the baseline value of the rateparameter for shear moduli to be Crate3 ¼ 0.03. Finally, the rateconstant for through-thickness modulus of plain-weave (15 � 15)S-2 Glass/SC15 is found to be Crate4 ¼ 0.023e0.0.039 [48], and wewill choose Crate4 ¼ 0.03 as the baseline. These rate dependentparameters for ballistic impact simulations are presented in Table 1.

Penetration modeling of continuum solids using finite elementanalysis requires element erosion criteria to remove a damagedelement with excessive deformation. The removal of an elementgenerates traction-free surfaces and allows the penetrator toprogress through the material. If an element with significant strainenergy is eroded, the nodes on the traction-free surface becomeunstable. Thus it is recommended that the strain energy of anelement be forced down to a minimum value before erosion.Premature erosion of an element can under predict the penetrationresistance of a continuum. Failing to erode an element with zerostrain energy or with too small or too large a volume can artificiallyincrease the penetration resistance of the material and increasecomputational time exponentially. Element erosion is essential forpenetration modeling using FEA. However, penetration-erosionphenomenology is empirical in nature and requires parametriccomputations and correlations with model-experiments. In thepresent MAT162 formulation, three maximum strain criteria areused for element erosion: the axial strain limit eLimit, relative volu-metric compression eCrush, and relative volumetric expansion eExpn.Since the ‘m’ parameters control the post-yield softening behaviorof composites, one can set the element eroding axial strain (eLimit)and the relative volumetric expansion (eExpn) to a large value so thatthe strain energy in the element is negligible at the time of elementerosion. It has been shown that the quasi-static punch shear damagecan be simulated by setting eLimit to a large value (e.g., 0.2) and bysetting eCrush to a reasonably small value (e.g., 0.001). One value ofthe relative volume expansion erosion parameter, eExpn ¼ 4.5, wasfound to be suitable for all the quasi-static punch shear tests. Thesevalues of erosion strains will be used in the impact simulations ofthe ballistic experiments discussed in the next section.

Fig. 1. Ballistic experimental set-up and the VR w VI data for the baseline 22L S-2 glass/SC15 composites.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197184

3. Ballistic experiments

The impact, damage and penetration modeling will be per-formed by simulating the ballistic experiments on baseline twentytwo layer (22L) plain-weave S-2 Glass/SC15 composite plates ofdimension 178 mm � 178 mm, and thickness HC ¼ 13.2 mm andimpacted with a right circular cylinder of mass, diameter andheight of MP ¼ 13.8 gm, DP ¼ 12.7 mm and HP ¼ 14.02 mm,respectively [34]. Test specimens were bolted between a steelsupport plate of dimension 178 mm � 178 mm and thickness50.8 mm and a cover plate of thickness 12.7-mm both havinga circular hole of diameter DS ¼ 101.6 mm (Fig. 1a). The supportspan to projectile diameter ratio (SPR) for ballistic experiments wasSPR ¼ DS/DP ¼ 8.0. The impact velocity (VI) and residual velocity(VR) of the projectiles were measured using Flash-X-Ray equip-ment. Ballistic tested specimens were sectioned through the impactcenter using a wet saw with diamond cutting blade, dipped into anink-alcohol solution and optically photographed. Fig. 1b shows theVR w VI data for the baseline 22L composites.

The ballistic limit velocity, V50, was calculated by fitting theexperimental data to the Lambert Equation [51] and found to beV50 ¼ 367 m/s

VR ¼ b�VpI � Vp

50

�1=p (1)

Fig. 2. Ballistic impact damage and pene

Additional ballistic experiments on thirty three layer (33L) S-2Glass/SC15 composites (20.6-mm thick) are conducted bymeasuring only the impact velocity of the projectile (50 cal FSP, 35�

chisel head, MP ¼ 13.5-gm, DP ¼ 12.57-mm, and HP ¼ 14.88-mm;MIL-DTL-46593B (MR)) using the same ballistic test fixture. Theballistic limit velocity is calculated by averaging the impact veloc-ities corresponding to the highest incomplete (IP) and the lowestcomplete penetration (CP) velocities following MIL-STD-662, and isfound to be V50 ¼ 540 m/s Fig. 2 shows the ballistic impact damageand penetration of 22L and 33L S-2 Glass/SC15 composites. Whilethe VR

�w�VI data for the baseline 22L composites will be used formodel validation, the ballistic limit velocity of the 33L compositewill be predicted using finite element model of the test with thesame input properties.

4. Finite element modeling of ballistic impact, damage andpenetration

In order to accurately simulate the ballistic experiments pre-sented in Figs. 1b and 2a, a square composite plate of dimension178 mm � 178 mm � 13.2-mm is modeled with 753k solidelements in 3D. A full 3D model is used instead of a quarter-symmetric model to avoid computational difficulties related toelement erosion at the symmetric boundaries. A total of 36

tration of 22L and 33L composites.

Fig. 3. 3D finite element model of ballistic impact on thick-section composite.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 185

elements are placed in the through-thickness direction in 12material layers with 11 predefined delamination interface defini-tions. The right circular cylindrical projectile is modeled using 34ksolid elements with a mesh density comparable to the center of thecomposite plate to ensure good contact between the plate and theprojectile during penetration. The square support plate and thecover plate are modeled using relatively coarser elements with aninner diameter of 101.6 mm and thickness of 50.8 mm and25.4 mm, respectively. The complete model consists of about 810kone point integration brick elements. The composite plate is con-strained through contact with the support and cover plate withhigh coefficient of friction to mimic no-slip boundary condition. Allfour edges of the cover plate and the support plate are constrainedin all degrees of freedom. A single surface contact definition is usedincluding all materials of the model. The in-plane material direc-tions of the composite laminate 1 & 2 are aligned with global X & Yaxes, and the through-thicknessmaterial direction 3 is aligned withthe global Z axis. Fig. 3 shows different aspects of the full 3D finiteelement model of the ballistic experiment.

Elastic properties of steel (r¼ 7850 kg/m3, E¼ 210 GPa, n¼ 0.29)are used to model the projectile and the fixture (cover plate andsupport plate). Since recovered projectiles from ballistic experi-ments showed negligible deformation [34], plastic deformation ofthe projectile is not considered. The baseline S-2 glass/SC15material properties provided in Table 1 are used to modelthe composite plate. The full 3D model is solved using 4 CPU’s withLS-Dyna 971 Release 3.2. In the present study, the elements usedunder the projectile have in-plane dimensions 0.56mm� 0.56mm,and two different element thicknesses 0.56 mm & 0.28 mm,respectively. LS-Dyna calculates the time step for explicit compu-tation using the minimum dimension of the element, and the

Fig. 4. LS-Dyna simulation of low velocity impact. Blue color represents intact material, and(For interpretation of the references to colour in this figure legend, the reader is referred t

computational time steps are found to vary between 0.001 and0.01 ms. The average runtime for 500 micro-seconds (at 400 m/sspeed, the projectile will travel 200-mm in space in 500 micro-seconds) of computation is about 24 h.

4.1. Model validation with ballistic experiments

Rate independent material properties and parameters (with allCrate parameters equal to zero) presented in Table 1 [35] are firstused to establish a numerical baseline. In a subsequent step, ratedependence will be included and the differences in impactresponse due to rate will be identified. Correlation to ballisticexperiments will provide validation of the input properties andnumerical models. Low velocity numerical impact experiments inthe impact velocity range 0 < VI < 300 m/s are conducted. Ata certain time of the impact, the projectile velocity became zero, thedynamic displacement of the laminate reached a maximum, andthe projectile rebounded from the composite laminate. Fig. 4 showsthe delamination and matrix damage predicted by LS-Dyna asa function of impact velocity at the time of maximum dynamicdisplacement. Delamination development through the entirethickness is observed at impact velocities higher than 50 m/s. Nosignificant penetration of the projectile into the composite lami-nate is observed below an impact velocity of 200 m/s. At an impactvelocity of 300 m/s significant projectile penetration and sometensile fiber fracture in the rear side of the composite laminate isobserved.

Numerical experiments without rate dependence are conductedat higher impact velocities up to 1000 m/s and the residualprojectile velocities are measured from the time histories ofprojectile velocity (Fig. 5a). The computational time for each impact

red color represents inter-laminar delamination. For more on color maps, see Fig. 10c.o the web version of this article.)

Fig. 5. LS-Dyna simulation of low velocity impact to high velocity ballistic impact.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197186

velocity is chosen such that the projectile velocity becomesconstant after rebound (negative) or after complete penetration(positive). The rigid body velocity of the projectile is found todecrease to a constant value in the impact velocity range360 < VI < 1000 m/s, signifying complete penetration. However,a double curvature in the velocity time history is observed forimpact velocity ranging from 300 m/s to 340 m/s in the time range0 < t < 100 ms, indicating that impact damage and penetrationmechanisms near ballistic limit velocity are different than thehigher velocity impact cases. This may also be associated with thelarger extent of delamination, reduction is flexural stiffness, andwave reflections from clamped boundaries.

The impact velocity and corresponding residual velocity (orrebound velocity for incomplete penetration) predicted from FEsimulations are presented in Fig. 5b for Crate ¼ Crate1 ¼Crate3 ¼ Crate4 ¼ 0.00 & Crate2 ¼ 0.00 and eExpn ¼ 4.5. With rateindependent model parameters, complete penetration is predictedat VI ¼ 330 m/s (VR ¼ 23.2 m/s) and incomplete penetration ispredicted at VI ¼ 320 m/s (Rebound Velocity VR ¼ �81.7 m/s). Theballistic limit velocity is approximated by linear interpolation of theresidual velocities corresponding to the above-mentioned impact

Fig. 6. LS-Dyna simulation of low velocity impact to high velocity bal

velocities, i.e., V50jCrate¼0:00 ¼ 328 m=s, which is about 10% less thanthe experimental value of V50 ¼ 367 m/s. For impact velocities lessthan 600 m/s the residual velocities of the projectile are over-predicted, and for impact velocities higher than 600 m/s theresidual velocities are under-predicted.

An analysis of the effect of strain rate on the ballistic penetrationresistance is conducted using the full set of properties given inTable 1(Crate ¼ Crate1 ¼ Crate3 ¼ Crate4 ¼ 0.03 & Crate2 ¼ 0.00). With thesechoicesof ratedependentparameters,numerical impactexperimentsare conducted in the impact velocity range 300<VI<

�1000m/s. Timehistories of the projectile velocities are presented in Fig. 6a. Theimpact and residual velocities (or rebound velocity for incompletepenetration) of the projectile are plotted on Fig. 5b with the legendCrate ¼ 0.03 and eExpn ¼ 4.5. The numerical prediction of residualvelocities shows good correlation with ballistic experiments in theimpact velocity range 380m/se500m/s. The numerical ballistic limitvelocity is calculated fromtheVR

�w�VIplotwhereVR¼0, and is foundto be V50jCrate¼0:03¼ 376 m=s, which is 2.3% higher than the experi-mental value (VR ¼ 367 m/s). The numerical VR�w�VI data is alsocurve-fittedwith theLambertequationandthevalueofV50 is foundtobe370m/s,which isabout1%higher thantheexperimentalvalue. The

listic impact. Crate ¼ Crate1 ¼ Crate3 ¼ Crate4 ¼ 0.03 & Crate2 ¼ 0.00.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 187

residual velocities in the impact velocity range 500 m/se1000 m/sshow linear behavior like Eq. (1). In this range of impact velocities(500 < VI < 1000 m/s), the model prediction of residual velocities islower than the experiment; thus the prediction of penetrationresistance is higher than the experiment. The choice of baseline rateparameters provided the prediction of V50 with sufficient accuracy,and the prediction of residual velocities with good accuracy up to500 m/s impact velocity.

Projectile velocity as a function of projectile displacement ispresented in Fig. 6b for different initial impact velocities. Fig. 6bshows that above the ballistic limit velocity (VI � 400 m/s), theresidual velocity of the projectile remains constant with respectto projectile displacement. Below the ballistic limit velocity(VI¼ 360m/s), the projectile rebounds in the backward direction. Byplotting the projectile velocity as a function of projectile displace-ment, the transition between incomplete penetration and completepenetration can be clearly identified. Following the MIL-STD-662,successive bi-section of the impact velocity range between anincomplete and complete penetration can lead to a quickernumerical prediction of the ballistic limit velocity, V50, and will beused in this study to predict the ballistic limit for a 11L and a 33Lcomposite plate.

Parametric determination of VR�w�VI plots (Fig. 5b) as a function

of different parameters is computationally extensive. Instead,parametric simulations are performed at a constant impact velocityclose to V50 at VI ¼ 380 m/s to investigate the sensitivity of pene-tration resistance to rate parameters. Using Eq. (1), the change inV50 as a function of change in residual velocity (VR) at constantimpact velocity can be expressed as:

DV50 ¼ �DVR

b

��VI

V50

�p

�1�p�1

p

(2)

where, DV50 ¼ VX50 � VBaseline

50 , and DVR ¼ VXR � VBaseline

R . Eq. (2)will allow one to investigate the effect of rate parameters on theballistic limit velocity by predicting the residual velocities at oneconstant impact velocity. Eq. (1) is empirical in nature and is alsovery non-linear near V50, thus the change in V50 predicted by Eq. (2)is qualitative. Fig. 7 shows six different test cases: (A) No RateEffects (B) Rate Effect on Strengths (C) Rate Effects on Strengths andShear Moduli (D) Rate Effects on Strengths and Through-Thickness(T-T) Modulus (E), Rate Effects on Strengths, Shear Moduli, andThrough-Thickness (T-T) Modulus (Baseline), and (F) Rate Effects onStrengths, In-Plane Moduli, Shear Moduli, and T-T Modulus (All).

-100

-50

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

A - No Rate EffectsB - StrengthsC - Strengths & Shear ModuliD - Strengths & T-T ModulusE - BaselineF - All

Time, t, µs.

Proj

ectil

e Ve

loci

ty, V

, m/s

. T

(AC(BC(CC(DC(EC(FC

Fig. 7. Effect of rate parameters

From results presented earlier, we found that the residualvelocity for the Baseline test case E is the closest to the experiments,i.e., VBaseline

R ¼ VER ¼ 38:3 m=s with VBaseline

50 ¼ VE50 ¼ 367 m=s. If

rate effects are not considered (Test case A), higher residual velocity(VA

R ¼ 134:3 m=s) and 10% lower penetration resistance(VA

50 ¼ 330 m=s) is predicted. Adding the rate effects on strengths(Test case B) reduces the residual velocity to be negative (i.e.,rebound velocity, VB

R ¼ �19:9 m=s) and increases the penetrationresistance (VB

50 ¼ 389 m=s) 6% more than the Baseline (E). Addingrate effects on shear moduli on top of the strengths (Test case C)brings the residual velocity (VC

R ¼ �9:3 m=s) a little bit closer to theBaseline but the overall time history of velocities are similar to testcase B (VC

50 ¼ 385 m=s, 4.9% increase). On the other hand, addingrate effects on T-T modulus on top of the strengths (Test case D)brings the residual velocity (VD

R ¼ 69:6 m=s) closest to the Baselinewith a 3.4% less prediction of VD

50 ¼ 355 m=s. When rate effects onboth shear moduli and T-T modulus are considered together withthe strengths, the Baseline test case E is obtained and these baselinerate parameters are used in the present simulations. In the Baselinetest case, rate effects on in-plane moduli are not considered. If therate effects on in-plane moduli are added with the baseline (Testcase F), a small reduction in residual velocity (VF

R ¼ 34:6 m=s) ascompared to the baseline is predicted, signifying a 0.4% increase inpenetration resistance (VF

50 ¼ 368 m=s). This small differencejustifies the baseline rate parameters used in the present study.

4.2. Numerical prediction of penetration resistance forces andenergies

The penetration resistance force on the projectile is calculated asthe contact force between the projectile and the composite plate andis presented in Fig. 8a with respect to projectile displacement forsimulations with baseline rate parameters. The impact and penetra-tion event can be divided into two different phases: Phase I e shockcompression, and Phase II e penetration. Impact-contact, through-thickness stress wave propagation, and local punch shear failure arethe major events during Phase I, which appears as a quick rise inforceedisplacement to a maximum level and a plateau at themaximum force. During the Phase II penetration process, com-pressioneshear and tensioneshear failure are the dominant damagemechanisms, and the penetration resistance forceedisplacementcurves show an exponential decay behavior. The Phase I/Phase IIbehaviors are common to all impact velocities at and above the

EST CASES X

RV ,

m/s

X

50V , m/s

(% Baseline)

). No Rate Effects rate1 = Crate2 = Crate3 = Crate4 = 0.00

134.29 330

(90%) ). Rate Effect on Strengths

rate1 = 0.03, Crate2 = Crate3 = Crate4 = 0.00 - 19.88

389 (106%)

). Rate Effects on Strengths & Shear Moduli rate1 = Crate3 = 0.03, Crate2 = Crate4 = 0.00

- 9.26 385

(104.9%) ). Rate Effects on Strengths & T-T Modulus

rate1 = Crate4 = 0.03, Crate2 = Crate3 = 0.00 69.58

355 (97.6%)

). Baseline rate1 = Crate2 = Crate3 = 0.03, Crate2 = 0.00

38.30 367

(100%)). Rate Effects on All (Strengths & Moduli)

rate1 = Crate2 = Crate3 = Crate4 = 0.03 34.62

368 (100.4%)

X = A, B, C, D, E, F

on penetration resistance.

Fig. 8. LS-Dyna prediction of penetration resistance force and energy. Crate ¼ Crate1 ¼ Crate3 ¼ Crate4 ¼ 0.03, Crate2 ¼ 0.00.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197188

ballistic limit. Fig. 8a shows a straight line separating the Phase I/Phase II forceedisplacement behavior.

The integral of the penetration resistance forceedisplacementcurve represents the work done by the projectile during thepenetration process

WP ¼ZHF

0

FPdH (3)

where HF is the projectile displacement at which the penetrationresistance force FP approaches zero. Fig. 8b shows different energiesas a function of impact velocities. The impact energyðEI ¼ MPV2

I =2Þ, residual energy ðER ¼ MPV2R=2Þ, energy trans-

ferred to the composite plate ðET ¼ EI � ERÞ, and the work done bythe projectile (WP) are calculated from the simulation results andfrom Eq. (1) fit to the experimental data. The energy transferred tothe composite plate (ET) is almost equal to the impact energy (EI) upto the ballistic limit velocity (designated by X on Fig. 8b) since theresidual energy (ER) is comparatively small. Beyond the ballisticlimit velocity, the energy transferred to the composite plateðET ¼ EI � ERÞ increases as a function of impact velocity and isalmost equal to thework done by the projectile (WP). The difference

Fig. 9. Time history and spatial distribution of energ

between the energy transferred to the composite plate calculatedfrom LS-Dyna simulations and from experiment beyond 600 m/simpact velocity is related to the failure of the LS-Dyna simulation topredict the correct residual velocities of the projectile. This problemremains, and will be addressed elsewhere.

At any impact velocity, the total energy should be equal to theinitial kinetic energy of the projectile. Fig. 9a shows differentcomponent of energies computed for impact velocity, VI ¼ 380 m/s,just above the V50. The total energy (TE) of the system at any time isthe sum of the kinetic energy (KE) of the projectile (KEP), thecomposite laminate (KEC), and the support structure (KES), the totalstrain energy or internal energy of the composite (IEC), the frictionalsliding energy (SLE) and the hourglass energy (HGE) of thecomputational system.

TE ¼ KEP þ KEC þ KES þ IEC þ ðSLE þ HGEÞ (4)

The slidingenergy is calculatedbetween twoparts in contactwithfriction by integrating the friction force over displacement [52].Hourglass energy (HGE) is added to the computation to prevent zeroenergy deformation modes of eight node brick elements with oneintegrationpoint. IdeallyHGE shouldbeas small as zero, however, themaximumHGE at time t¼ 200 ms is found to be 5%of the total energy,

ies. Crate1 ¼ Crate3 ¼ Crate4 ¼ 0.03, Crate2 ¼ 0.00.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 189

ensuring 95% confidence of computational energy balance. Theinternal energy (IEC¼ EESEþ ED) of the composite includes the elasticstrain energy (EESE), and the energy dissipated (ED) by differentdamage equations (Eqs. A.1eA.7 in Appendix). Unfortunately, thepresent LS-Dyna formulation doesn’t allow the decomposition of theIEC into its components, however, allows the decomposition bymaterial. The kinetic energy of the projectile decreases while thekinetic energy and internal energy of the composite laminateincrease upon impact. At time t ¼ 28 ms, the kinetic energy of thecomposite laminate reaches its maximum value (KEC ¼ 18% of TE),while the kinetic energy of the projectile reduces to 25% of its initialvalue. The internal energy of the composite increases to 49% of TE atthis time. The kinetic energy of the composite plate converts intostrain energy and the kinetic energy of the composite plate become5% of TE at time t¼ 200 ms, while projectile kinetic energy reduces to2% of TE, and the internal energy of the composite plate becomes 78%of the TE. At this point of time, the frictional sliding energy reaches itsmaximum (6% of TE). The internal energy (IE) of the compositelaminate is calculated from twelve material layers defined throughthe thickness. The IE of the composite plate also represents the totalenergy that can be dissipated bydifferent energy dissipating damagemechanisms. The percent internal energy (%IE) of each discretematerial layer through the thickness and the cumulative %IE fromimpact face to the back face is presented in Fig. 9b at time t¼ 200 ms.

From the impact face, about 40% of total internal energy is evenlydistributed over the thickness range 0.00<H/HC< 0.50; 15% of totalIE is evenly distributed in the thickness range 0.50 < H/HC < 0.75;

Fig. 10. Short time phase I damage mech

and about 45% of total IE is linearly distributed in the thickness range0.75 < H/HC < 1.00. The time history and spatial distribution ofenergies, which cannot be obtained without computational simu-lation, is essential in understanding the role of materials for opti-mized energy dissipation.

4.3. Prediction of ballistic damage and penetration

Observation of ballistic experimental damage below V50 and thepresent finite element analysis reveals that there exist two distinctphases of ballistic penetration: (i) short time Phase I shockcompression, and (ii) long time Phase II penetration. We willinvestigate the ballistic impact damage during these two phases atimpact velocities near the ballistic limit.

4.3.1. Short time phase I damage mechanismsThe development of a shock front and compressive strain/stress

wave propagation are responsible for damage development andpropagation in Phase I. Fig. 10 shows the axial strain wave (3x ¼ 31)propagation, through-thickness strain wave (3Z ¼ 33) propagation,and inter-laminar delamination damage development as a functionof time for an impact velocity of 360 m/s. Upon the impact-contactof the projectile with the composite plate, the material under theprojectile undergoes shock compression, the stress exceeds thefiber crush stress (SFC), and the damaged material moves ahead ofthe projectile, developing a shock front. Ahead of the shock front,

anisms, VI ¼ 360 m/s, VI/V50 ¼ 0.98.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197190

an elastic precursor of the compressive strain/stress wave propa-gates through the thickness of the composite laminate.

At time t ¼ 0.99 ms, the projectile-composite contact is estab-lished, a 3D strain or stress wave front develops and propagatesfrom the impact-contact site. The through-thickness compressivestrain wave propagated 6/24th of composite thickness.

At time t ¼ 2.19 ms, the projectile penetrates about 1/24th ofcomposite thickness (HC/24), the T-T strain wave propagates about12HC/24, and initiation of delamination predicted by Eq. (A.7) occursdue to the transverse shear deformation around the projectileperimeter in thefirst three predefineddelamination interfaces. Axialstrain under the projectile is tensile in nature, however, a sphericalcompressive axial strainwave is preceding the axial tensionwave upto the time t ¼ 9.68 presented in Fig. 10, at an average speed of3.95 km/s. However, the maximum amplitude of the axial strain issmall (�0.001) as compared to themaximum amplitude of TT-strain(�0.05). The amplitude of axial stress corresponding to themaximum axial strain is, 27.5 GPa � �0.001 ¼ �27.5 MPa. Theamplitude of T-T stress corresponding to the maximum T-T strain is,11.8 GPa � �0.05 ¼ �590 MPa.

At time t ¼ 3.6 ms, projectile penetrates about 2HC/24, the T-Tstrain wave propagates about 20HC/24 with an average speed of2.75 km/s, and initiation of delamination continues around theprojectile perimeter up to the first six predefined delaminationinterfaces.

At time t z 4.8 ms (not shown in the Fig. 10) the compressiveelastic strain wave reaches the back surface of the laminate andreflects back as a tensile stress wave. The traction-free boundarycondition on the back surface doubles the particle velocity of theincident wavewithout altering its direction, which causes (at a latertime greater than 4.8 ms) delamination initiation at the last inter-face and continues in the subsequent layers in the directionopposite to projectile movement. The incoming compressive strainwave and reflected tensile strainwave cancels each other, and thereremains a zero strain/stress state from the back surface to theboundary between the compressive and tensile wave fronts.

At time t ¼ 5.29 ms, the projectile penetrates about 3HC/24, thelocation of the boundary between the compressive and tensilewave front is at about 6HC/24 from the back face, the last twodelamination interfaces show delamination, and delaminationgrowth in the in-plane direction is observed.

At time t ¼ 7.29 ms, the projectile penetrates about 4HC/24, thelocation of the boundary between the compressive and tensilewave front is at about 14HC/24 from the back face, delaminationinitiation at all interfaces is almost complete, delamination growthin the in-plane direction continues, and dynamic bulging of theback face of the composite laminate and the formation of the shearcone is visually distinct.

The end of Phase I and the beginning of Phase II occurs at about2.7 mm of projectile displacement/penetration which correspondsto the time t ¼ 8.6 ms. Between t ¼ 8 ms and t ¼ 10 ms, the projectilecontinues penetrating through the composite, the dynamic bulgeor the shear cone formation continues, and delamination continuesto grow in the in-plane direction.

Fig. 11. Phase I damage mechanisms as a function of impact velocity. t ¼ 10 ms. Blue color reFig. 10c. (For interpretation of the reference to colour in this figure legend, the reader is re

The sequence of evolution of delamination through the thick-ness is the same for all impact velocities higher than 50 m/s duringPhase I; however, the depth of projectile penetration increases withincreasing impact velocity (Fig. 11). The present delaminationmodel (Eq. A.7) can predict the initiation and propagation ofdelamination in both the through-thickness and in-plane direc-tions of the composite laminate. Other options for delaminationmodeling e.g., tie-break interface and cohesive elements inconjunction with MAT162 has also been reported [53].

4.3.2. Long time phase II damage mechanismsEvolution of damage as a function of time during long time

Phase II penetration processes is presented in Fig. 12 for twodifferent impact velocities, i.e., at 360 m/s and at 400 m/s. Experi-mental ballistic damage at these impact velocities are also pre-sented in Fig. 12. At an impact velocity of 360 m/s the projectile wasstopped by the composite laminate and rebounded from thecomposite plate with a rebound velocity of 80 m/s. At an impactvelocity of 400 m/s the projectile completely penetrated thecomposite plate with a residual velocity of 121 m/s.

At time t¼ 20 ms, penetration of the projectile continueswith thegrowth of the dynamic bulge/damage cone and in-plane delami-nation due to transverse shear deformation. At time t ¼ 40 ms,penetration continues with large tensioneshear deformation andthe individual delaminated layers undergo localized bending. Initi-ation of the plug formation is observed in addition to the tensilefailure of delaminated layers ahead of the projectile and in the backof the laminate at time t ¼ 60 ms. Penetration continues andcomplete shear plug is formed during time t ¼ 60 ms to t ¼ 100 ms.

At impact velocity 360 m/s, one-fourth of the composite platefrom the back face remains intact and undergoes large deformation.We have discussed earlier that the laminas in the thickness range0.75 < H/HC < 1.00 develops 45% of total internal energy, and thesedynamic deformation revels that this part of internal energy istensioneshear dominated. Even though no significant tensilefailure in this delaminated rear part was observed in the ballisticexperiment, LS-Dyna simulation showed partial tensile failure butstopped the projectile completely and then rebounded back in thetime range t ¼ 200 ms to t ¼ 400 ms. On the other hand, for 400 m/simpact velocity, the projectile and the plug ahead are ejected fromthe composite laminate at time t ¼ 200 ms. The ballistic cross-section of the composite plate shows similar damage mechanisms,but the final state of the ballistic damage after spring back matcheswith simulations only qualitatively. The delamination model inMAT162 for finite thickness elements can predict delaminationdamage qualitatively at much lower computational cost thaninterface elements; a comparison of such techniques will beaddressed elsewhere.

4.3.3. Prediction of delamination in 3DDelamination initiation and propagation through the thickness

of the composite over a short time scale is presented in Fig.10 whileits growth in the in-plane direction as a function of time over a longtime scale is presented in Fig.12. Both these figures show the extent

presents intact material, and red color represents inter-laminar delamination. Also seeferred to the web version of this article.)

Fig. 12. Long time phase II damage mechanisms. Blue color represents intact material, and red color represents inter-laminar delamination. Also see Fig. 10c (For interpretation ofthe reference to colour in this figure legend, the reader is referred to the web version of this article.)

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 191

and growth of delamination on the XZ plane passing through theorigin. The evolution of delamination in the composite in 3D cannotbe visualized in these 2D sections. Fig. 13 shows the evolution ofdelamination in 3D at different times for impact velocity

Fig. 13. Evolution of delamination in 3D, VI ¼ 360 m/s, VI/V50 ¼ 0.98. Half

VI ¼ 360 m/s for half of the composite plate with the XZ planepassing through the origin.

As discussed earlier, delamination initiates and grows though thethickness in the time range 2 < t < 7 ms. In-plane growth of

of the composite plate with the XZ plane passing through the origin.

Fig. 14. LS-Dyna prediction of projectile velocity as a function of projectile displacement.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197192

delamination in the bottom part of the composite plate is observedin the time range 10 < t < 16 ms. Since the composite is made fromplain-weave (5� 5 tows/inch) fabricswith a [0/90] lay-up, the initialin-plane growth of delamination appears to be oriented in theprincipal directions (1, X & 2, Y); however, for the longer time scalethe growth appears to be circular. For the longer time scale,20< t<95 ms, the laminate undergoes large deformation and the in-planegrowthof delamination in the top layers is observed.However,the extent of delamination in the top layers appears to be smallerthan the bottom layers, which gives the overall delamination theclassic conical shape. Delamination reduces the bending stiffness ofthe laminate allowing local large deformation under the projectile,and is directly relatedwith the decay in penetration resistance forceof the projectile presented in Fig. 8a, and correlates well withballistic experiments presented in Fig. 2a. The strength of numericalprediction is that it provides the evolution and growth of delami-nation in both time and space, which otherwise cannot be capturedwith any other experimental techniques.

4.4. Prediction of ballistic limit of composites using validated model

In this section, we use the parameters in Table 1 to predict thepenetrationbehaviorof laminatesof different thickness toassess thepredictive capability of the numerical model. Two composite

Fig. 15. Comparison of LS-Dyna predictio

laminates with eleven layers (11L) and thirty three layers (33L) areconsidered. Knowing theballistic limit velocity of a 22L composite tobe V50j22L ¼ 367 m=s, we chose an impact velocity of 200 m/s forthe 11L composite plate as the first trial and the projectile is stopped(Fig. 14a). Complete and incomplete penetration is denoted in thelegend by C and I (Fig. 14a). The impact velocity is raised to 400 m/sand the projectile penetrates through. The impact velocity rangebetween an incomplete and a complete penetration is bisected todetermine the subsequent impact velocities, i.e., 300 m/s, 250 m/s,and 225m/s. The same procedure is followed for the 33L composite.The LS-Dyna predictions of projectile velocity and penetrationresistance force as a function of projectile displacement for 33Lcomposite are presented in Fig. 14b.

The impact and residual velocities for the 11L and 33Lcomposites are then plotted along with the 22L baseline data andare presented in Fig. 15a. The ballistic limit velocity of the 11L and33L composites are calculated from the VR w VI plots at VR ¼ 0. Theballistic limit velocity of the 11L composite is found to beV50j11L ¼ 226 m=s. For the 33L composite a ballistic limit velocity ofV50j33L ¼ 523 m=s is calculated, which is about 3% less than theexperimental ballistic limit velocity, V50jExp33L ¼ 540 m=s. Thepenetration resistance force as a function of projectile displace-ment for 11L, 22L, and 33L composites near the ballistic limit iscompared in Fig. 15b.

ns for 11L, 22L, and 33L composites.

Fig. 16. LS-Dyna damage predictions for 11L and 33L composites.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 193

Prediction of damage near the end of penetration for 11L and 33Lcomposite is presented in Fig. 16. The damage of 33L compositequalitatively correlates well with the experimental damage pre-sented in Fig. 2b (VI ¼ 528 m/s). These numerical examples give ussufficient confidence that one can use a set of validated materialmodel parameters forMAT162 to predict theballistic limit velocity of

Fig. 17. Axial strain (3x: �0.022 to 0.022) contours at different projectile ve

thick-section composites with sufficient engineering accuracywithin the impact velocity range forwhich themodel is validated for.

Dynamics of deformation and stress wave propagation in thelaminate can be better visualized if investigations are performedat impact velocities lower than the ballistic limit velocity, e.g.,VI/V50 ¼ 0.90. Fig. 17 shows contours of axial strain (3x) for three

locity ratios (V/VI) for the 11L, 22L, & 33L composites at VI/V50 ¼ 0.90.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197194

different laminates studied (i.e., 11L, 22L, & 33L) at differentinstantaneous projectile velocity to initial impact velocity ratios,i.e., at V/VI ¼ 0.75, 0.50, 0.25, and 0.00. Red and blue color contoursrepresent the strain range (|0.0132|< 3x< |0.0220|) under tension &compression, respectively.

The contour of axial strain (3x) presented in Fig.17 is with respectto the laboratory frame of coordinate and does not represent thestrain in originalmaterial coordinate. Since the laminas have rotatedmore than 30�, the original fibers along geometric X axis should beunder tensile stretch, however, such stretch cannot be visualizedfrom the laboratory frame of coordinate. Thus the axial compressivestrain inside the damage cone calculated by the data processing toolhas less significance. The areal-density parameter (ratio ofcomposite mass in front of the projectile to the projectile mass,G0 ¼ pD2

PrCHC=4MP) for these three composites plates are calcu-lated to be, G0;11L ¼ 0:112; G0;22L ¼ 0:224; G0;33L ¼ 0:336:

At the early stages of penetration (Fig.17a, V/VI¼ 0.75), through-thickness stress wave propagation, crush under the projectile, andpenetration are the dominant mechanisms. In this early phase, thedamage cone also develops due to the transverse shear deformation.Since the current projectile velocity is 75% of the initial velocity (V/VI ¼ 0.75), 25% of the initial momentum is transferred to thecomposite plate. Since G0;11L ¼ 0:112 for the 11L laminate, addi-tional mass outside the one projectile diameter is involved inmomentum exchange in the direction of projectile movementthrough transverse shear deformationmechanism, and the damagecone is fully formed. On the other hand, for the 33L composite,G0;33L ¼ 0:336, not all the mass of the composite under theprojectile has attained the projectile velocity, and thus a completedamage cone is yet not formed. For the 22L, since G0;22L ¼ 0:224,and themomentum transfer is 25%, the damage cone is fully formedwith visible dynamic deflection of the outermost layer of thecomposite plate. This observation tells us that, the damage cone isfully formedwhen the fraction of the projectilemomentum transferto the composite plate approximately equals to G0.

Fig. 17b, V/VI ¼ 0.50, represents 50% momentum transfer to thecomposite plate. Damage cone is fully developed in all the compositeplates. Axial tensile strain at yield is visible as the red color contour,while the blue color contour represents the compressive axial strainin the zoneof high transverse sheardeformation. Failure is predictedby the quadratic tensioneshear failure criterion (Eqs. A.1 & A.2) andthe element is deleted to mimic fracture.

In Fig. 17, the membrane tensile failure of the outermostcomposite layer is visible at different time for different laminatethickness. At V/VI ¼ 0.25, momentum transfer to the composite is75%, and at V/VI ¼ 0.00 is 100%. The 11L (thinner) laminate showedhigher dynamic deformation as compared to the 33L (thicker)laminate. At all impact velocity ratios, the blue color contourapproximately shows the zone of transverse shear deformation atthe corresponding time. Penetration of the projectile in to thecomposite laminate at V/VI ¼ 0.00 is more pronounce in the thickerlaminates (22L & 33L). From these observations, we can concludethat the through-thickness compression and local transverse ten-sioneshear deformation around the projectile are the dominantenergy dissipating deformation mechanisms for ballistic penetra-tion of thick-section composites. Membrane tensile failure of theoutermost composite laminas is the additional energy dissipatingdamage mechanism.

5. Summary and conclusions

Modeling the ballistic impact, damage, and penetration of thick-section composites is complex in nature, and requires modelingand tracking the evolution of different composite damage modes intime and space. We present here the explicit dynamic finite

element analysis (FEA) technique, where the equations of motionare solved in incremental time steps and the time history ofdeformations, tractions, and damages are calculated based on theconditions of earlier time steps. We use the state-of-the-art solverLS-Dyna, and the progressive composite damage model MAT162.MAT162 models the composite material as linear-elastic till theinitiation of damage, and non-linear post-damage softening usingdifferent rate sensitive failure equations and parameters.

A full 3D finite element model of ballistic impact on a thick-section composite is developed and validated with ballistic exper-iments over a wide range of impact velocities; i.e., 50 m/se1000 m/s. It has been found that the rate independent model parametersdetermined by simulating quasi-static punch shear tests [35] can beused with rate dependent parameters to simulate the ballisticimpact, damage and penetration resistance behavior of thick-section composites. The baseline rate parameters are found toprovide a sufficiently accurate prediction of V50 and good predictionof residual velocities within a certain range of impact velocities. Thevalidated FE model is then used to study the ballistic impactresponses of the baseline S-2 Glass/SC15 composites.

It has been shown that the projectile velocity vs. displacementresponse for incomplete and complete penetration is significantlydifferent and that it is computationally extensive to determine theballistic limit velocity numerically. It is relatively easier to identifya range of impact velocities betweenwhich the ballistic limit velocityexists and, by conducting additional numerical simulations in thisrange, predict the ballistic limit velocity by interpolating the numer-ical data. The penetration resistance force vs. displacement andworkdone by the projectile are found to be a function of impact velocity.

Two major phases of ballistic penetration are identified: i.e. (i)short time Phase I shock compression and (ii) long time Phase IIpenetration. The LS-Dyna analysis captured both of these penetra-tion phases. During short time Phase I penetration, through-thick-ness shock/stress wave propagation and initiation of delaminationmechanisms are identified as the main ballistic processes and arefound to exist at impact velocities at and above the ballistic limitvelocity. The dynamics of through-thickness and in-plane growth ofdelamination as a function of time are explained from modelpredictions, both quantitatively and qualitatively. Shock compres-sion of material under the projectile is also identified as the pene-tration damage mechanism, the depth of which increases asa function of impact velocity.

Evolution of damage and penetration during the Phase IIpenetration process are investigated at two different impactvelocities around the ballistic limit velocity and observed tocorrelate well with experimental ballistic damage. Tensionesheardominated damagemodes and large deformation of the rear part ofthe composite laminate are identified as the dominant damagemechanisms at impact velocities near V50. The projectile dynamicsand the penetration resistance force as a function time have beenpresented for different impact velocities. Energy statistics of theprojectile-composite system as a function of time have revealedthat 75% of the kinetic energy of the projectile is transferred to thecomposite laminate in the first 25e30 ms of time. At around V50,about 80% of the impact energy is converted into the strain energyor internal energy of the composite laminate and the rest of theenergy is converted into sliding energy and kinetic energy of thesystem. It has also been shown that 45% of the total internal energyis dissipated in the rear quarter of the composite plate.

Finally, the validated model parameters are used to predict thepenetration resistant behavior of a thinner (11L) and thicker (33L)laminate and good correlation is obtained with a 33L ballisticexperiment within the range of impact velocities for which themodel is validated for. Numerical penetration mechanics models of11L, 22L, and 33L laminates provided insight into the deformation

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 195

dynamics and identified the through-thickness compression andlocal tensioneshear deformation as the dominant energy dissi-pating deformation mechanisms for ballistic penetration of thick-section composites. This validated computational model can beextended to study the penetration mechanics of various compositestructures and to optimize the laminate structure for maximumenergy dissipation.

Acknowledgments

Research was sponsored by the Army Research Laboratory andwas accomplished under Cooperative AgreementNumberW911NF-06-2-011. The views and conclusions contained in this document arethose of the authors and should not be interpreted as representingthe official policies, either expressed or implied, of the ArmyResearch Laboratoryor theU.S. Government. TheU.S. Government isauthorized to reproduce and distribute reprints for Governmentpurposes notwithstanding any copyright notation hereon.

Appendix A. Composite damage modeling using LS-DYNAMAT162

A.1. Failure criteria

The following failure criteria are used in MAT162 for plain-weave composites. Positive values of failure function f denote theinitiation of specific damage modes.

Damage mode Failure criteria

Fiber tensioneshear along direction af6 ¼

hs1iST1

!2

þ�s212 þ s231

�S21FS

� 1 ¼ 0 (A.1)

Fiber tensioneshear along direction b f7 ¼ hs2iST2

!2

þ�s212 þ s232

�S22FS

� 1 ¼ 0 (A.2)

In-plane compression along direction a f8 ¼ D

s=1

ESC1

!2

�1 ¼ 0; s=1 ¼ �s1 þ h�s3i (A.3)

In-plane compression along direction b f9 ¼ D

s=2

ESC2

!2

�1 ¼ 0; s=2 ¼ �s2 þ h�s3i (A.4)

Crush failure under compressive pressure f10 ¼�hpiSFC

�2�1 ¼ 0; p ¼ �s1 þ s2 þ s3

3(A.5)

In-plane matrix crack f11 ¼�s12S12

�2

�1 ¼ 0 (A.6)

Through-thickness matrix crack ordelamination

f12 ¼ S2Delam

( hs3iST3

!2

þ�s23S23

�2

þ�s31S31

�2)

� 1 ¼ 0 (A.7)

A.2. Progressive damage

The reduction in modulus is defined by an exponential functiongiven by:

6 ¼ 1� e1m

�1��

33y

�m�(A.8)

where 6 is the modulus reduction parameter, m is the softeningparameter, and 3y is the yield strain. The post-yield modulus E andstress s can then be expressed as:

E ¼ ð1�6ÞE0 ¼ E0e1m

�1�� 33y

�m�(A.9)

s ¼ E3 ¼ E03e1m

�1�� 33y

�m�(A.10)

where E0 is the undamaged modulus. Dimensionless stress up toand beyond the yield point can then be expressed as:

s

sy¼ 3

3yfor

3

3y� 1 (A.11a)

s

sy¼ 3

3ye

1m

�1�� 33y

�m�for

3

3y> 1 (A.11b)

Figure A.1a shows the dimensionless stress as a function ofdimensionless strain of a unit single element (USE) defined by Eq.(A.11) for differentm values. For a very high positive value ofm, e.g.,m¼ 100, the post-yield stressestrain behavior can be considered asbrittle failure. For a near zero value of m, e.g., m ¼ 0.01, the post-yield behavior is almost perfectly plastic. Values of m in the range0 < m < 100 show different degrees of post-yield softeningbehavior. Fig. 1 also shows the dimensionless strain energy fordifferentm values, where strain energy form¼ 100 is considered as

the baseline elastic-brittle failure behavior. As the m value isreduced from 100 to 0.40, the strain energy is increased by a factorof 13.1 as compared to the elastic-brittle material response. Thisformulation of post-damage softening is applied to define tension,compression and shear in all three principal material coordinates.The effect of post-damage softening parameters on the stresse-strain response of a unit single element under different loadingconditions can be found in Ref. [43]. Fig. A.1b shows the stresse-strain response of a unit single element (USE) under differentloading conditions.

Figure A.1: Stressestrain responses of an USE.

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197196

A.3. Strain rate effects

The rate equation for strength values is given by:

S ¼ fSRTgfS0g

¼ 1þ Crate1lnf_3g_30

(A.12)

where

fSRTg ¼nST1 SC1 ST2 SC2 SFC SFS

oT(A.13)

f_3g ¼nj_31jj_31jj_32jj_32jj_33j

�_3231 þ _3232

�1=2oT(A.14)

and {S0} are the reference values of strengths {SRT} at reference strainrate _30. Fig. A.2a shows the dimensionless rate dependent strengthvalue,S, as a functionof strain rategivenbyEq. (A.12)with _30 ¼ 1 s�1.Fig. A.2b shows stressestrain behavior of a unit single element (USE)under different loading conditions at the strain rate of 10,000/s.

The rate equation for elastic moduli is similar to Eq. (A.12):

Figure A.2: Effect of strain rate on

X ¼ fERTgfE0g

¼ 1þ fCrateg lnf_3g_30

(A.15)

where

fERTg ¼ f E1 E2 E3 G12 G31 G32 gT (A.16)

f_3g ¼ �j_31jj_32jj_33jj_312jj_331jj_332j

T (A.17)

fCrateg ¼ fCrate2 Crate2 Crate4 Crate3 Crate3 Crate3 gT(A.18)

strength given by Eq. (A.12).

B.A. Gama, J.W. Gillespie Jr. / International Journal of Impact Engineering 38 (2011) 181e197 197

References

[1] Tabiei A, Aminjikarai SB. A strain-rate dependent micro-mechanical modelwith progressive post-failure behavior for predicting impact response ofunidirectional composite laminates. Composite Structures 2009;88:65e82.

[2] Bouvet C, Castanié B, Bizeul M, Barrau J. Low velocity impact modelling inlaminate composite panels with discrete interface elements. InternationalJournal of Solids and Structures 2009;46:2809e21.

[3] Minak G, Ghelli D. Influence of diameter and boundary conditions on lowvelocity impact response of CFRP circular laminated plates. Composites: Part B2008;39:962e72.

[4] Donadona MV, Iannuccib L, Falzonb BG, Hodgkinsonb JM, de Almeidaa SFM. Aprogressive failure model for composite laminates subjected to low velocityimpact damage. Computers and Structures 2008;86(11e12):1232e52.

[5] Lv L, Gu B. Transverse impact damage and energy absorption of three-dimensional orthogonal hybrid woven composite: experimental and FEMsimulation. Journal of Composite Materials 2008;42:1762e86.

[6] Ji C, Sun B, Qiu Y, Gu B. Impact damage of 3D orthogonal woven compositecircular plates. Applied Composites Material 2007;14:343e62.

[7] Hufenbach W, Ibraim FM, Langkamp A, Böhm R, Hornig A. Charpy impact testson composite structures e an experimental and numerical investigation.Composite Structures 2007;68:2391e400.

[8] Williams KV, Vaziri R. Application of a damage mechanics model for pre-dicting the impact response of composite materials. Computers and Structures2001;79:997e1011.

[9] Rao MP, Duan Y, Keefe M, Powers BM, Bogetti TA. Modeling the effects of yarnmaterial properties and friction on the ballistic impact of a plain-weave fabric.Composite Structures 2009;89:556e66.

[10] Talebi H, Wong SV, Hamouda AMS. Finite element evaluation of projectilenose angle effects in ballistic perforation of high strength fabric. CompositeStructures 2009;87:314e20.

[11] Novotny WR, Cepu�s E, Shahkarami A, Vaziri R, Poursartip A. Numericalinvestigation of the ballistic efficiency of multi-ply fabric armours during theearly stages of impact. International Journal of Impact Engineering 2007;34:71e88.

[12] Barauskas R, Abraitiene A. Computational analysis of impact of a bullet againstthe multilayer fabrics in LS-DYNA. International Journal of Impact Engineering2007;34:1286e305.

[13] Duan Y, Keefe M, Bogetti TA, Cheeseman BA, Powers BM. A numericalinvestigation of the influence of friction on energy absorption by a high-strength fabric subjected to ballistic impact. International Journal of ImpactEngineering 2006;32:1299e312.

[14] Duan Y, Keefe M, Bogetti TA, Powers BM. Finite element modeling of trans-verse impact on a ballistic fabric. International Journal of Mechanical Sciences2006;48:33e43.

[15] Tan VBC, Ching TW. Computational simulation of fabric armour subjected toballistic impacts. International Journal of Impact Engineering 2006;32:1737e51.

[16] Zeng XS, Shim VPW, Tan VBC. Influence of boundary conditions on theballistic performance of high-strength fabric targets. International Journal ofImpact Engineering 2005;32:631e42.

[17] Lim CT, Shim VPW, Ng YH. Finite-element modeling of the ballistic impact offabric armor. International Journal of Impact Engineering 2003;28:13e31.

[18] Grujicic M, Arakere G, He T, Bell WC, Cheeseman BA, Yen C-F, Scott B. Aballistic material model for cross-plied unidirectional ultra-high molecular-weight polyethylene fiber-reinforced armor-grade composites. MaterialsScience and Engineering A 2008;498:231e41.

[19] Tan VBC, Shim VPW, Tay TE. Experimental and numerical study of theresponse of flexible laminates to impact loading. International Journal ofSolids and Structures 2003;40:6245e66.

[20] Sheikh AH, Bull PH, Kepler JA. Behavior of multiple composite plates subjectedto ballistic impact. Composites Science and Technology 2009;69:704e10.

[21] Deka LJ, Bartus SD, Vaidya UK. Multi-site impact response of S2-glass/epoxycomposite laminates. Composites Science and Technology 2008;69(6):725e35.

[22] Deka LJ, Bartus SD, Vaidya UK. Damage evolution and energy absorption of E-glass/polypropylene laminates subjected to ballistic impact. Journal of Mate-rials Science 2008;43:4399e410.

[23] Silva MAG, Cismasiu C, Chiorean CG. Numerical simulation of ballistic impacton composite laminates. International Journal of Impact Engineering2005;31:289e306.

[24] Sevkat E, Liaw B, Delale F, Raju BB. A combined experimental and numericalapproach to study ballistic impact response of S2-glass fiber/toughened epoxycomposite beams. Composites Science and Technology 2009;69:965e82.

[25] Tham CY, Tan VBC, Lee HP. Ballistic impact of a KEVLAR helmet: experimentand simulations. International Journal of Impact Engineering 2008;35:304e18.

[26] Gower HL, Cronin DS, Plumtree A. Ballistic impact response of laminatedcomposite panels. International Journal of Impact Engineering 2008;35:1000e8.

[27] He T, Wen HM, Qin Y. Finite element analysis to predict penetration andperforation of thick FRP laminates struck by projectiles. International Journalof Impact Engineering 2008;35:27e36.

[28] Chan S, Fawaz Z, Behdinan K, Amid R. Ballistic limit prediction usinga numerical model with progressive damage capability. Composite Structures2005;77:466e74.

[29] Fawaz Z, Zheng W, Behdinan K. Numerical simulation of normal and obliqueballistic impact on ceramic composite armours. Composite Structures2004;63:387e95.

[30] Cheng WL, Langlie S, Itoh S. High velocity impact of thick composites. Inter-national Journal of Impact Engineering 2003;29:167e84.

[31] Nandall D, Williams K, Vaziri R. Numerical simulation of the ballistic responseof GRP plates. Composites Science and Technology 1998;58:1463e9.

[32] Sibeaud J-M, Thamié L, Puillet C. Hypervelocity impact on honeycomb targetstructures: experiments and modeling. International Journal of Impact Engi-neering 2008;35:1799e807.

[33] Wicklein M, Ryan S, White DM, Clegg RA. Hypervelocity impact on CFRP:Testing, material modeling, and numerical simulation. International Journal ofImpact Engineering 2008;35(12):1861e9.

[34] Gama BA, Gillespie Jr JW. Punch shear based penetration model of ballisticimpact of thick-section composites. Composite Structures 2008;86:356e69.

[35] Xiao JR, Gama BA, Gillespie Jr . Progressive damage and delamination in plainweave S-2 glass/SC-15 composites under quasi-static punch-shear loading.Composite Structures 2007;78(No. 2):182e96.

[36] Phoenix SL, Porwal PK. A new membrane model for ballistic impact responseand V50 performance of multi-ply fibrous systems. International Journal ofSolids and Structures 2003;Vol. 40:6723e65.

[37] Porwal PK, Phoenix SL. Modeling system effects in ballistic impact into multi-layered fibrous materials for soft body armor. International Journal of Fracture2005;Vol. 135:217e49.

[38] Cunniff PM. Dimensional parameters for optimization of textile-based bodyarmor systems, Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas.Lanchaster, PA, USA: Technomic Publishing Co; 1999. p. 1303e1310.

[39] Naik NK, Shrirao P, Reddy BCK. Ballistic impact behavior of woven fabriccomposites: formulation. International Journal of Impact Engineering 2006;32:1521e52.

[40] Keyword user’s manual, LS-Dyna, version 971. Livermore, CA: LivermoreSoftware Technology Corporation; June 2007.

[41] http://www.ccm.udel.edu/Tech/MAT162/Intro.htm.[42] Matzenmiller A, Lubliner J, Taylor RL. A constitutive model for anisotropic

damage in fiber-composites. Mechanics of Materials 1995;20:125e52.[43] Gama BA, Bogetti TA, Gillespie Jr JW. Progressive damage modeling of plain-

weave composites using LS-Dyna composite damage model MAT162, 7thEuropean LS-DYNA Conference, Austria. Germany: DYNAmore GmbH; May14e15, 2009.

[44] Pahr DH, Rammerrstorfer FG, Rosenkranz P, Humer K, Weber HW. A study ofshort-beam-shear and double-lap-shear specimens of glass/fiber/epoxycomposites. Composites: Part B 2002;33(No. 1):125e32.

[45] Materials Science Corporation (MSC). Laterally constrained transversecompressive tests of glass fiber composites; 2003.

[46] Yen C-F, Cheeseman BA, Hoppel CPR, Gama BA, Gillespie Jr JW. Modeling thecompressive failure of plain weave composites. ASC/ASTM-D30 Joint 19thAnnual Technical Conference, Atlanta, GA. Dayton, OH, USA: American Societyfor Composites; October 17–20, 2004.

[47] Gillespie JW, Gama BA, Cichanowski CE, Xiao JR. Interlaminar shear strengthof plain weave S2-Glass/SC79 composites subjected to out-of-plane highstrain rate compressive loading. Composites Science and Technology 2005;65:1891e908.

[48] Gama, B. A., Hopkinson Bar Experimental Technique: Experiments, Analysesand Applications, University of Delaware, 2004, PhD Dissertation.

[49] Leber H, Lifshitz JM. Interlaminar shear behavior of plain-weave GRP at staticand high rates of strain. Composites Science and Technology 1996;56:391e405.

[50] Naik NK, Asmelash A, Kavala VR, Ch V. Interlaminar shear properties ofpolymer matrix composites: strain rate effect. Mechanics of Materials2007;29:1043e52.

[51] Lambert JP. MD: Ballistic Research Laboratory, Aberdeen Proving Ground;1978. ARBRL-MR-02828.

[52] LS-Dyna theory manual. Livermore, CA: LSTC; March 2006. p. 26.16.[53] Kang S-G, Gama BA, Yarlagadda S, Gillespie JW. Finite element modeling of

delamination in thick-section composites using LS-Dyna. CD Proceeding, 2009ASC/CACSMA Conference, Newark, DE, USA. Dayton, OH, USA: AmericanSociety for Composites; September 15e16, 2009.