Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 2, pp. 263-272, Warsaw 2015 DOI: 10.15632/jtam-pl.53.2.263 SIMULATION OF BALLISTIC IMPACT ON POLYMER MATRIX COMPOSITE PANELS Du Long Hongdu Aviation Industry Group, Nanchang, Jiangxi, China e-mail: dragon7519 paper@sina.com Composite panels, for example fuselage skins or wing skins of military aircraft, may be subjected to a variety of ballistic impacts. It is meaningful to understand the mechanical response, damage evolution and residual velocity of a projectile for panels under ballistic impact. In this paper, the numerical simulation method of ballistic impact on a laminated composite panel is considered. A rate-dependent material model based on the continuum damage mechanics concept is developed for polymer matrix composite materials. A full three-dimensional finite element model implemented with the material model is built up using material subroutine. The ballistic impact behavior is simulated employing an explicit dynamic finite element analysis technique. The effects of projectile size and velocity, lay- up of the composite panel, and failure criteria used on the ballistic impact response are investigated. Keywords: composite materials, ballistic impact, material model, finite element modeling 1. Introduction Fiber-polymer composites are ideal for a wide range of military and civil applications, where the weight is a very important aspect. For example, composite laminates have beenwidely used to produce aircraft fuselage or wing skins. Composite skins undergo various impact loadings during the service life: low velocity impact such as a tool dropped on the surface, high velocity impact such as bird strike. For the utilization of composites as military aircraft or protective structures, the resistance to ballistic impact is very important. Investigation on the projectile impact behavior, such as impact damage initiation and propagation as well as ballistic limit, are the foundation of personnel and structural survivability analysis. The support conditions have important influence on the impact response in low velocity impact regime, but the response of the structural element is generally independent of its support conditions in high velocity impact (Naik andShrirao, 2004). So far, there aremany experimental studies on the ballistic impact of composite materials (Mouritz, 2001; Hosur et al., 2004; Ubeyli et al., 2007; Wiśniewski and Pacek, 2013). Tests confirmed that the ballistic limit of polymer composite plates increases with decreasing strength of the matrix shear (Karthikeyan et al., 2013). There is a direct proportionality between velocity and trace depth for polymer-based composites (Tarim et al., 2002).Themajor energydissipationmechanismsduringballistic impact include elastic deformation of composites, failure of fibers, delamination and matrix cracking (Morye et al., 2000). There have been several theoretical models that describe the ballistic impact on composites, see rg. Phoenix and Porwal (2003). Niak et al. (2006) presented an analytical method to iden- tify different damage and energy absorbing mechanisms during ballistic impact based on wave theory. The analysis results show that shear plugging is themajor energy absorbingmechanism (Naik and Doshi, 2008). Shirley et al. (2012) developed a nondimensional formulation to inve- stigate the ballistic behavior of composite plates. The energy absorbed by shear plugging and 264 D. Long friction are ignored in this model. Among the current ballistic impactmodels, many hypotheses are proposed, or the parameters should be determined by experimental data. Currently, finite element method (FEM) has been used to simulate high velocity impact behavior of composites by some researchers. Gower et al. (2008) simulated the ballistic impact response of laminated Kevlar composite panels used LS-DYNA. It is observed that the results are significantly influen- ced by the shape of projectile nose. Sheikh et al. (2009) usedFEM to investigate the behavior of multiple laminated panels subjected to projectile impact of a compressed air gun.The laminates were modeled with quadrilateral shell elements. In the present study, FEM is employed to study the ballistic impact behavior of polymer composite panels. Three-dimensional failure criteria and damage constitutive including strain rate effects are used to simulate thedamage of composite laminates.The contact andpenetration process as well as damage evolution mechanisms are investigated. Possible effects of various parameters on the ballistic impact behavior of composite panels are presented. 2. Materials model Different from the conventional plane-stress assumption, the out-of-plane stresses have great influence in impact problems. On the other hand, there are various types of damage occurring duringaballistic impact, andthegrowthof thesedamages is coupled.Therefore, it is necessary to develop an appropriate three-dimensionalmaterial model to accurately simulate the progressive damage of composite panels under the ballistic impact. Suppose the stress tensor in each material point is σt at time t. From time t to t+∆t, the stress increment is ∆σt. So the stress tensor at the time t+∆t can be calculated as σ t+∆t =σt+∆σt (2.1) In the micro-time range∆t, the stress increment∆σt is given by ∆σtij =C t ijkl∆ε t kl (2.2) Here∆εtkl denotes the strain increment, andC t ijkl is the stiffness coefficient at the time t. Based on continuumdamagemechanics, three damage variablesDf,Dm andDi are introdu- ced to describe the fiber failure, matrix damage and interlaminar fracture modes, respectively. Then the damaged stiffness coefficients at the time t have the forms Ct1111 =(1−Df)E t 1 1−µ23µ32 C0 Ct2222 =(1−Dm)E t 2 1−µ13µ31 C0 Ct3333 =(1−Di)E t 3 1−µ12µ21 C0 Ct1122 =(1−Df)(1−Dm)E t 1 µ21+µ31µ23 C0 Ct2211 =C t 1122 C t 1133 =(1−Df)(1−Di)E t 1 µ31+µ21µ32 C0 Ct3311 =C t 1133 C t 2233 =(1−Dm)(1−Di)E t 2 µ32+µ31µ12 C0 Ct3322 =C t 2233 C t 2323 =(1−Df)(1−Dm)G t 12 Ct3131 =(1−Dm)(1−Di)G t 23 C t 1212 =(1−Df)(1−Di)G t 13 C0 =1−µ12µ21−µ13µ31−µ23µ32−2µ21µ32µ13 (2.3) where Dk =1− (1−DkT)(1−DkC) k= f,m,i (2.4) Simulation of ballistic impact on polymer matrix composite panels 265 Here,DkT andDkC (k= f,m,i) denote the damage factors corresponding to fiber, matrix and interlaminar tensional failure and compression failure, respectively. During ballistic impact, the loading on the composite structure is dynamic. Much previous research efforts related to high strain rate loading has indicated that the properties of polymer composite materials are strain rate-dependent – the elastic modulus and strength change with the applied rate of loading (Hosur et al., 2001; Okenwa, 2001; Amos et al., 2002; Naik et al., 2007; Tarfaoui et al., 2008). So the strain rate effect should be considered in thematerial model for ballistic impact simulation of composite panels. Here, it is thought that all of the moduli vary linearly with the logarithm of strain rate as follows (Daniel et al., 2011): Et(ε̇)=E(ε̇0) [ me log ( ε̇t ε̇0 ) +1 ] (2.5) where E represents the elastic modulus or shear modulus, me is the correction coefficient of strain-rate hardening, ε̇t is the strain rate at the time t, ε̇0 denotes the reference strain rate. There aremany criteria to predict failure initiation of compositematerials (Huang andZhou, 2011). Hashin’s criterion can identify differentmodes of failure: tensile fiber failure, compressive fiber failure, tensile matrix failure and compressive matrix failure. It may be implemented in a finite element program to carry out the progressive failure analysis of composite panels under ballistic impact. The three-dimensional Hashin failure criteria are expressed as follows: i) fiber failure modes (σ11 Xt T )2 +α (σ12 St12 )2 +α (σ13 St13 )2 ­ 1 if σ11 ­ 0 (σ11 Xt C )2 ­ 1 if σ11 < 0 (2.6) ii) matrix damage modes (σ22+σ33 Y t T )2 + σ223−σ22σ33 St223 + (σ12 St12 )2 + (σ13 St13 )2 ­ 1 if σ22+σ33 ­ 0 (σ22+σ33 2St23 )2 + [( YC 2St23 )2 −1 ]σ22+σ33 Y tC + 1 St223 (σ223−σ22σ33)+ (σ12 St12 )2 + (σ13 St13 )2 ­ 1 if σ22+σ33 < 0 (2.7) The Quads criteria (Camanho and Davila, 2002) are adopted to predict the initiation and subsequent propagation of interlaminar damage (σ33 ZtT )2 + (σ13 St13 )2 + (σ23 St23 )2 ­ 1 if σ33 ­ 0 (σ13 St13 )2 + (σ23 St23 )2 ­ 1 if σ33 < 0 (2.8) In the above equations,XtT ,X t C, Y t T , Y t C,Z t T ,Z t C,S t 12,S t 13 andS t 23 are the different strength components at the time t. Here, all of the strength components are functions of the strain rate Ft(ε̇)=F(ε̇0) [ mf log ( ε̇t ε̇0 ) +1 ] (2.9) wheremf is the correction coefficient of strain-rate hardening. 266 D. Long 3. Finite element model A finite element model has been created for a composite panel under ballistic impact as shown in Fig. 1. A quarter panel of dimension 100mm×100mm×1.5mm ismodeled with 36.3k 8-node brick elements. Themechanical constitutive behavior and failure theory of the three-dimensional composites element, which have been previously described, are definedwithmaterial subroutine complied by Intel Fortran Compiler (ver 9.1). The mechanical properties of T300/NY9200GA epoxy composites used in the analysis are presented in Table 1. Fig. 1. Finite element model of ballistic impact on the composite panel Table 1.Mechanical properties of T300/NY9200GA epoxy composites E1 E2 E3 G12 G13 G23 µ12 µ13 µ23 [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] 124000 7500 7500 4600 4600 3500 0.31 0.31 0.36 Xt XC Yt YC Zt ZC S12 S13 S23 [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] 1420 913 35 130 45 180 60 60 35 DfT DfC DmT DmC DiT DiC 0.9 0.9 0.8 0.6 0.8 0.6 The round cone ended projectile is modeled with 8.294k 8-node brick elements, and the freedoms except the displacement along impact direction are fixed. The density of the projectile is 7.81g/cm3, Young’s modulus is 210GPa and Poisson’s ratio is 0.3. The nodes along the exterior sides of the panel are fixed in the x, y, z directions, and symmetric boundary conditions are applied along the planes of symmetry. The penalty contact algorithm is used to model the contact behavior between the projectile and panel. One surface is defined as the “master” surface and the other one as the “slave” surface. Thismethod applies virtual springs to surfaces in contact to calculate the restorative force at the point of penetration between the two bodies. The contact force is in direct proportion to penetration depth and materials stiffness. An explicit dynamic analysis technique is employed to simulate the ballistic impact beha- vior of the composite panel. The equations of motion are integrated using the explicit central- difference integration rule. 4. Numerical analysis results and discussions Figure 2 shows the damage evolution of the composite laminates under ballistic perforation as a function of time.The stacking sequence of laminates is [45◦/−45◦/0◦/90◦/−45◦/45◦]S (layup I), and the radius of nose is 10mm.Theprojectile completely goes across the composite panelswith Simulation of ballistic impact on polymer matrix composite panels 267 Fig. 2. Damage evolution process of the composite panel under ballistic impact 268 D. Long an impact velocity of 200m/s.When the projectile contacts the composite panel, the layer under the projectile undergoes shock compression at once, the stress exceeds the crushing strength, and the damage occurs. Different from thick panels, the damage develops and propagates to the reverse side of the laminates instantly for thin panels. Delamination and tensile or bending failure appear in the rear part very shortly from the time the contact is formed. The damage region does not extend if the nose penetrates the composite panel. The final damage regions near the upper and lower surfaces are bigger than those in themiddle layers. Different types of damage couple and competitively and unstably grow for the composite panel under ballistic impact. Figure 3 presents variation of the damage area with time for com- posite layup I.Thearea corresponding to thematrix damage is bigger than that of delamination, and the fiber failure area is smaller than that of delamination during the initial period of time from 0ms to 0.01ms. Fig. 3. Variation of the damage area versus time Figure 4 illustrates variation of the contact force with time for the constant projectile nose radius (10mm) and composite layup (layup I). The contact force reaches amaximum in a short time and decreases to zero when the projectile perforates the panel. Three different velocities of the projectile, 200m/s, 250m/s, 300m/s are used in these analyses. Themaximal contact forces duringpenetration are 2.06kN, 2.31kN, and 3.27kN, respectively. It indicates that themaximal contact force nonlinearly increases with velocity of the projectile. Fig. 4. Variation of the contact force versus time Figure 5 displays variation of the contact area versus time. Here, the projectile nose radius is 10mm, and the initial impact velocity is 200m/s. Different layups are investigated in these Simulation of ballistic impact on polymer matrix composite panels 269 analyses. The stacking sequence of layup II is [45◦/−45◦/0◦/90◦/0◦/0◦]S. Layup I and II have the same laminate thickness but different ply proportions. The distribution and propagation of damages depend on the laminates thickness, ply angle and stacking sequence. The layup has obvious influence on the contact behavior between the projectile and panel. The fluctuation of the contact area reflects the durative initiation and expansion of various damages. Fig. 5. Variation of the contact area versus time The stress histories of the reference element for three different radii of the projectile noses are shown in Fig. 6. The stress is partly but strongly dependent on the nose radius of the projectile. For the projectiles with 6mm and 8mm radius noses, the stress variation trends are similar. It is different from that of projectile with a 10mm radius nose. The reference element fails soon under ballistic impact because it is near the projectile nose if the 10mm radius nose is used. Fig. 6. Variation of the stress versus time for the reference element The maximum strain failure criterion has also been used to model the failure behavior of compositematerials under ballistic impact. Figure 7 shows variation of the kinetic energy versus time for different failure criteria.More kinetic energy loss is observedwhen themaximum strain failure criterion is used, although the difference only is 3%. For the fixed initial velocity at 200m/s, the residual velocities of projectiles have been inve- stigated for different cases. The results are shown in Fig. 8. The residual velocities of projectiles are 197.95m/s, 197.84m/s and 197.44m/s for 10mm, 8mmand 6mm radius nose, respectively. It is proved that the residual velocity of projectile increases nonlinearly with nose radius. The residual velocity of the projectile with 10mm radius nose for layup II is 197.79m/s. It indicates that the capability against ballistic impact for layup II is slightly better than that for layup I. 270 D. Long Fig. 7. Variation of the kinetic energy versus time Fig. 8. Comparison of the residual velocity of projectile/time curves Figure 9 shows the stress along the neutral axis, the intersecting line of the neutral plane and symmetrical plane of panel at different times. The elements close to the impact point have failed, so the stress values are zero. The stress reaches themaximum value sharply and then gradually declines to a constant value along the neutral axis. At the times of 2ms, 2.5ms and 3ms, the maximum value of stress is increased from 123.5MPa to 244.6MPa, and then to 330.1MPa. Fig. 9. Variation of Mises stress along the neutral axis Figure 10 shows the deformation of the projectile nose vertex versus time. Besides using a steel projectile, aluminium alloy projectile (ρ=2.7g/cm3,E=72GPa, ν =0.3, σy =390MPa) Simulation of ballistic impact on polymer matrix composite panels 271 has also been considered in this demonstration. It can be seen that the changes in the shape of the projectile nose during penetration is very small for the thin composite panel, which is generally used as aircraft skin. This is different from penetration of thick ductile composites for example thick-section aramid fabrics. Fig. 10. Deformation of the vertex of the projectile nose versus time 5. Conclusions In this paper, a rate-dependent material model based on the continuum damage mechanics concept for failure mechanism of polymer matrix composites has been developed. A three- -dimensional finite element model implemented with the material model has been developed tomodel laminated composite panel. The explicit dynamic finite element analysis technique has been employed to simulate the ballistic impact behavior. 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