Gen-lin Mo .Qian-wen Ma .Yong-xi Jin .Wen-min Yan .Zhong-xin Li ,Zhi-lin Wu ,*

a Institute of Advanced Manufacturing.Jiangsu University.Zhenjiang.212013.China

b Key Laboratory of Transient Shock.Beijing,102202.China

c Zhenjiang Hospital of Traditional Chinese Medicine.212013.China

d School of Mechanical Engineering.Nanjing University of Science and Technology.Nanjing.210094.China

ABSTRACT Cross-ply unidirectional laminates made of ultra-high molecular weight polyethylene fibers are widely used as components of bullet-proof vests.To simulate the delamination process of the material under penetration.we constructed a numerical mechanical model.which was validated by tests using 7.62 × 39 mm rifle bullets penetrating laminates of different thicknesses.The results show that the delamination region is essentially diamond-shaped.The simulated delamination region is in good agreement with the experimental data.It is found that the delamination region increases with the compression modulus along the fiber direction.The delamination region increases when the shear strength between the fabric layers decreases;However,it is little influenced by the normal strength.The delamination region of the front face of the laminate is little influenced by the failure strain of the material and the initial velocity of the bullet.The delamination region of the back face increases with the failure strain and decreases with the initial velocity.

1.Introduction

Ultra-high molecular weight polyethylene (UHMWPE) fiber is one of the most commonly used ballistic materials due to its superior mechanical properties and low mass density(0.97 g/L)[1,2].It is usually fabricated as cross-ply unidirectional (UD) resin laminates (UHMWPE laminates for short).as experiments show that their ballistic limit velocity is higher than that of 2D plain-woven and 3D single-ply orthogonal woven fabrics [3].To enhance the protective performance of UHMWPE laminates.the interactions between them and projectiles have been extensively studied[4-13].

Relevant experimental studies are as follows: Vargas-Gonzalez compared the ballistic limit and back face deformation (BFD) of UHMWPE laminates under impact of steel spheres.The laminates were prepared with various material combinations and architectures[4].Cwik compared the ballistic limit and the BFD of Dyneema HB26 and Spectra 3124 laminates subjected to the impact of copper and steel fragment-simulating projectiles [5].Furthermore.Zhang studied the effects of the boundary conditions(clamped-corners.clamped-edges.and free boundary conditions)on the BFD using spherical fragments [6,7].Last.Karthikeyan investigated the effect of the lay-up angle,the in-plane anisotropy parameter.and the areal density on the ballistic limit using steel spheres [8,9].

To investigate the influence of the mechanical properties on the ballistic performance of UHMWPE laminates.a great deal of research has been devoted to modeling.Lassig built a finite element method(FEM)model of hypervelocity(2000-7000 m/s)aluminum spheres penetrating the UHMWPE laminate,in which the laminate was treated as a whole.The relationship between the impact velocity and the residual velocity of the incident sphere was obtained[10].To improve the prediction of the numerical model for spheres penetrating the laminate at low velocities (＜1000 m/s).Ngugen modified the FEM model by dividing the laminate into multiple sub-laminates [11].However.the numerical model still failed topredict the ballistic limit of projectiles penetrating thick UHMWPE laminates (＞50 mm).Ngugen further divided the penetration process of the projectiles into two stages:the shear plugging stage and the bulging stage.An analytical model was built based on the energy and the momentum conservation laws [12].Within the study of Langston.another analytical model of projectiles penetrating the laminate was constructed.In the model.the shearing phenomena and the bulging phenomena happen simultaneously in the laminate [13].

The above-mentioned models show that delamination is an important failure mechanism in UHMWPE laminates under highvelocity impact and has a great influence on the ballistic performance [11-13].However.current models mainly focus on the ballistic performance of the laminate.They cannot adequately predict the delamination process of the laminate.

Delamination is one kind of crack in the laminate.Currently,different approaches have been reported in the open literature for predicting the crack initiation and propagation.They can be broadly divided into two categories: the finite element methods and the mesh-free methods.The finite element methods include the cohesive element method [14,15].the virtual crack closure technique[16,17],the extended finite element method[18-20]and the continuum damage mechanics approaches [21,22].The mesh-free methods include the element free Galerkin method [23,24].the smoothed particle hydro-dynamics [25].the reproducing kernel particle method [26].the peridynamics [27-29]and the radial point interpolation method [30].Comparing with other methods,the cohesive zone method is very effective and the boundary conditions can be easily applied.Thus.it is employed in this paper to simulate the delamination of the laminate under penetration.

The aim of this paper is to develop an efficient numerical model for predicting the delamination of the UHMWPE laminates under penetration.The main novelties are as follows:1)The influences of the compression modulus along the fiber direction of the laminate to the delamination are discussed by using a material model with damage.2) The influences of the initial velocity of the bullet.the failure strain of the laminate and the strengths of the interfaces of the laminate to the delamination are revealed with the numerical model.

2.Tests

The cross-ply UHMWPE laminates (Dyneema HB26) were composed of fibers with an ultimate elongation of 2.8%.The laminates were comprised of 20 layers of[0°/90°]weft-free fabrics and were formed by hot-pressing and coated with polyurethane for an hour at a 20 MPa pressure and a temperature of 110°C.The average density of the laminates was measured to be 0.93 g/cm3using the mass volume method.The dimensions of the laminate were 300 × 300 × 3.7 mm.Their structure is shown in Fig.1.

Fig.1.Structure of the UHMWPE laminate.

In the ballistic tests,the 7.62×39 mm rifle bullets,each with a total mass of 7.9 g,were launched using a ballistic gun.The bullet is made of three parts: a steel core.a lead sleeve.and a steel jacket.The structure is shown in Fig.2.

Three laminate targets,T1-T3were tested in the experiment.The targets are composed of different numbers of the laminate plates as shown in Table 1.Adjacent laminates were placed close together in T2and T3.The experimental setup is shown in Fig.3.The laminate target stands on a supporting desk facing the ballistic gun.The top of the target is sustained with a rod such that an approximately free boundary condition is obtained.The initial velocity of the bullet was measured by the circuit targets speed measuring system as shown in Table 1.It was found that all the laminates were perforated in the tests.

3.Finite element model

3.1.Bullet

The finite element model was built on the LS-DYNA platform.The meshes used for the bullets are shown in Fig.4.The bullets were meshed with the constant stress hexahedron solid elements and the one point tetrahedron solid elements.The elements are both one point integration elements.The time spent determining stresses with these elements could be largely reduced compared with other element formations [31].

The interactions between different parts of the bullet (the jacket,the sleeve and the core)are describe as the surface to surface contact.The frictional coefficient was assumed to be zero.The initial velocity of the bullet in the model was set according to the tests.The boundaries of the bullet were set as free.No forces and constraints were applied on the bullet in the model.

The Johnson-Cook model was adopted to describe the projectile materials with the Gruneisen equation of state (EoS) [32].The strength of the metal is defined as:

where A,B,C,n,and M are material constants,εpis the plastic strain,and ˙ε*is the effective strain rate normalized by the quasi-static threshold rate:

In Eq.(2).˙ε is the effective strain rate.ε0= 1/s.and T*is the homologous temperature defined as:

Fig.2.Structure of the bullet in the ballistic test.

Table 1 Ballistic test of the laminate targets.

Fig.3.Schematic of the experimental setup.

The variable T represents the temperature.T0the room temperature.and Tmeltthe melting temperature.The parameters of steel and lead are listed in Table 2.

The Gruneisen equation of state defines the pressure p in compressed materials as:

and for expanded materials as:

where ρ0is the initial density,Csis the wave speed,S1-S3,γ0and a are coefficients,and μ is the relative change in volume[32].The EoS parameters of the projectile materials are shown in Table 3.

Table 2 Johnson-Cook parameters of the bullet materials [33].

Table 3 EoS parameters of the bullet materials [33,34].

3.2.UHMWPE laminate target

The meshes used for the each laminate target are shown in Fig.5.Each laminate was modeled using 5 sublayers in order to reduce the computational resources required to complete the simulation.The sub-laminates were meshed with fully integrated hexahedron elements to achieve more accurate simulation results.The volume integration of the element is an 8-point integration[31].

Delamination of the laminate is simulated using the cohesive zone method by defining a tiebreak failure criterion at the interface between the sublayers.Adjacent sublayers were initially tied together.The tied contact fails when the following failure criterion is satisfied:

where NFLS is the tensile strength of the interface.and SFLS is the shear strength [32].The testing results of them with different methods differ quite a lot [10,11,35].Thus.their values were set according to the simulation results NFLS=SFLS=30 MPa.σnand σsare the tensile stress and shear stress,which were computed in the simulation.

The gravity and the supporting force from the desk and the rod on the laminate was ignored in the model.No constraint was applied on the laminate elements meaning the boundaries of the laminate were free.The initial velocity of the laminate was set zero before the penetration event.The interactions between two adjacent laminates were defined as the surface to surface contact after the delamination.The frictional force of the contact was ignored.The contact between the bullet and the laminate targets was defined as the eroding contact.Thus when the exterior elements of the bullet or the targets fail and get deleted.the interior element can get into contact with other elements.

The UHMWPE laminates are modeled with the material type 221 (MAT_ORTHOTROPIC_SIMPLIFIED_DAMAGE).which corresponds to an orthotropic material with damage [32].The stressstrain relationship in this case is:

Fig.4.Meshes of each component of the bullet.

Fig.5.Meshes of the UHMWPE laminate.

The 11- and 22-directions are along the fiber direction in the plane of the laminate.The 33-direction is along the thickness.The Yong’s moduli.shear moduli and Poisson’s ratios were tested in Ref.[10]as shown in Table 4.

The variables d1t.d2t.d3t.d1c.d2cand d3c.represent the tension(subscript t) and compression (subscript c) damage.which are considered separately.d23,d13and d12represent the shear damage.At the microscopic level,fibers would bend and curl rather than be compressed along their axes when the laminate is compressed in the fiber direction [36].This makes the macro-compression modulus in the fiber direction much smaller than the tension modulus.In our model,the compression modulus is assumed to be one hundredth of the tension modulus.Thus.d1c= d2c= 0.99.When the laminate is compressed or tensiled in the 33-direction,the fibers’ deformation mode does not change.Thus thecompression modulus is assumed to be the same with tension modulus,leading to d3c=0.The damage in the tension modulus and in the shear modulus is ignored.as no softening phenomena are observed in the stress-strain curve [10].Thus,d1t = d2t = d3t = d23 = d13 = d12 = 0.

Table 4 Material parameters of the UHMWPE laminate.

The laminate material is assumed to fail when the shear strain in the thickness direction reaches the failure strain εf:

When the failure criterion is satisfied at all the integration points in an element,the element would then be deleted from the model.According to the simulation results,εfis set to 0.5.

3.3.Model verification

The components of the UHMWPE laminate,fibers and the resin are both elastic materials.thus the resulting laminate should also be elastic.However.after penetration.there are obvious plastic deformations left in the laminate.It is thought to be caused by the delamination between the fabric layers.Thus.the plastic deformation region of the laminate should be similar to the delamination region.

Fig.6.Comparison of the delamination region of T1.

Fig.7.Comparison of the delamination region of T2.

Fig.8.Comparison of the delamination region of T3.

Fig.9.Diamond shape of the delamination region.

Figs.6-8 are comparisons between the simulated delamination regions and the plastic regions of the front face and the back face in tests T1-T3.It can be seen that the shapes of the plastic regions are similar to the ones of the delamination areas.approximately diamond-shaped,as shown in Fig.9.The shape can be described by two variables.L1and L2.Comparisons of the dimensions of the delamination region are listed in Tables 5 and 6.It can be seen that the relative errors are all not larger than 25%except L1of the front face of T1.However,the absolute error of L1of the front face of T1is a small value.The simulation results of L2is smaller than the experimental results.The reason is thought to be that the splitting effect is not considered in the current model (section 4.1).

Table 5 Comparison of L1 of the delamination region.

Table 6 Comparison of L2 of the delamination region.

Fig.10.Delamination of T1 laminate through the thickness.

Table 7 Dimensions of L1 of T1 through the thickness.

Delamination of the laminate T1through the thickness is shown in Fig.10.The laminate was first cut using a water-jet cutting machine in the section plane(45°with the fiber direction)as shown in Fig.6 b) (1).Then the plane was painted with red oil ink.The delaminated region sucked more ink and seemed darker.The figure shows that the width of the delamination region does not change linearly with the thickness.This phenomenon is in accordance with the simulation results.The simulated L1with depth is shown in Table 7.This evidence shows that the numerical model can reveal the tendencies of the delamination through the thickness.

4.Discussion

4.1.Splitting effect

The plastic deformation of the UHMWPE laminate can be attributed to two reasons.The first is the aforementioned delamination between the fabric layers.The second is the observed splitting effect within the layer along the fiber direction,as shown in Figs.6-8 a) (1) and b) (1).This splitting happens both on the front and the back surface of the laminate.As the arrangement of the fibers in each layer of the laminate is similar,the splitting effect presumably happens inside the laminate as well.The splitting effect would make L2 of the delamination region larger than expected.Currently,the splitting effect has not been considered in the numerical model.Thus,most of the L2of the simulation are smaller than the experimental ones as shown in Table 7.As there is a larger simulation error in L2.the delamination region of the laminate in the following discussion is mainly indicated using L1.

4.2.Mesh size

The influences of the mesh size of the laminate to the penetration process are analyzed by using three kinds of meshes: the regular mesh.the finer mesh and the finest mesh as shown in Fig.11.The regular mesh is the mesh shown in Fig.5.The finer mesh means that the layers of the laminate is doubled(ten sublayers),but the mesh in the laminate plane is the same with the regular mesh.The total elements of the finer mesh are twice the regular mesh.The finest mesh means that the laminate has ten sublayers and the mesh in the plane is doubled along the sides.The total elements are eight times the regular mesh.

Changes of the delamination region of the front face and the back of T1with different meshes are shown in Table 8.It can be seen that the delamination region of the front face is much more influenced by the mesh size than the back face.However,the simulation errors of L1of the front face are all very small(experiment,18 mm).

The influence of the mesh size to the deflection of the laminate is shown in Fig.12.The tendency of the deflection with time is similar for different meshes.T1reaches its highest deflection at about 50 μs.The height is around 3.5 mm.

The changes of the velocity of the bullet are shown in Fig.13.As the laminate T1is only 3.7 mm in thickness,there’s little change of the bullet’s velocity during the penetration.The penetration process lasts less than 20 μs.For different mesh sizes,difference of the residual velocity is within 1 m/s.

The energy ratio of the finite element model is shown in Fig.14.The total energy drops down quickly during the first 20-40 μs.Then it changes little during the later phase.Fig.14 also tell us that the lost energy becomes smaller when the element size becomes smaller.The calculation results show that the lost energy is mainly the internal energy.It is attributed to the erosion of the laminate material.

The main purpose of this paper is to simulate the delamination of the UHMWPE laminates under penetration.As the mesh size isnot important for the delamination and the deflection of the target,the regular mesh is used to construct the model.However.when the velocity of the bullet is very low.the residual velocity of the bullet and the eroded energy may become more important to the delamination process.In this circumstances.the meshes of the laminate should be refined.

Fig.11.Meshes of the center of the laminate.

Table 8 Delamination region of T1 with different meshes.

Fig.12.Deflection of T1 with different meshes.

4.3.Determination of εf, NFLS and SFLS

There are three unknown mechanical parameters in the numerical model.εfin Eq.(7).NFLS and SFLS in Eq.(8).They are determined according to the following sensitivity analysis.

Let εf=0.5,the influences of NFLS and SFLS on the delamination of the laminate T1are shown in Table 9.It can be seen that the delamination region decreases with both the shear strength.SFLS and the normal strength,NFLS.According to the experimental data,SFLS is set to 30 MPa.As the delamination region is not sensitive to NFLS.it is set equal to SFLS.

Fig.13.Velocity of the bullet with different meshes of T1.

Fig.14.Energy ratio of the model with different meshes of T1.

Let NFLS=SFLS=30 MPa.The influence of the failure strain,εfis shown in Table 10.It can be seen that the delamination region on the front face is little influenced by εfand the thickness of the target.The delamination region on the back face increases with εf.and the thickness of the target.

Table 9 Influence of NFLS and SFLS on the delamination region of T1.

Table 10 Influence of εf on the delamination region.

According to the above analysis,L1of the front face of the target is only influenced by SFLS.As the experimental results show that L1of the front face of the targets is about 20 mm.SFLS should be not smaller than 30 MPa.If SFLS is larger than 30 MPa.the simulated delamination region of the back face would get smaller as shown in Table 9.To simulate the delamination region of the back face.εfshould be larger than 0.5.This indicate there are many combinations of εf.SFLS and NFLS that could satisfy the numerical simulation.Thus.the true values of them need further experimental studies.

4.4.Influence of the initial velocity to the delamination

The influence of the initial velocity to the delamination of the laminate target is shown in Table 11.The initial velocity has little influence on the delamination of the front face.The delamination region of the back face decreases with the initial velocity.

4.5.Influence of the compression modulus to the delamination

In the numerical model.the compression modulus of the laminate along the fiber direction is assumed to be 1%of the tension modulus by setting d1c= d2c= 0.99.The influence of the compression modulus to the delamination of the laminate is shownin Table 12.It shows that the delamination region of both the front face and the back face grows with the compression modulus,and L2grows faster than L1.This indicates the assumption of the damage of the compression modulus along the fiber direction is necessary for the delamination simulation of the laminate.Otherwise.the simulation results would be much larger than the experimental ones.

Table 11 Delamination of T1 with different initial velocities.

Table 12 Delamination of T1 with different compression modulus.

5.Conclusion

In the present paper,a numerical model for bullets penetrating the UHMWPE laminates is proposed.The delamination within the laminate is reproduced with the model and the following conclusions can be drawn:

1) The difference between the compression modulus and the tension modulus of the laminate needs to be considered in the numerical model to reveal the delamination in the laminate.The delamination region increases dramatically with the compression modulus.

2) The tiebreak failure criterion of the fabric interface is appropriate to describe the delamination in the laminate.The delamination region becomes larger when the shear strength becomes smaller.However.it is little infulenced by the normal strength.

3) The delamination region of the front face of the laminate is little influenced by the failure strain of the material and the initial velocity of the bullet.The delamination region of the back face increases with the failure strain and decreases with the initial velocity.

Acknowledgements

The work is supported by the Research Foundation for Advanced Talents of Jiangsu University (15JDG038).the Foundation of National Laboratory(601010417) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(19KJB130003).