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Numerical analysis of a ballistic impact on textile fabric
International Journal of Mechanical Sciences 69 (2013) 32–39
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Numerical analysis of a ballistic impact on textile fabric Cuong Ha-Minh a,b,c, Abdellatif Imad a,n, Toufik Kanit a, Franc- ois Boussu b a b c
LMT-Cachan (ENS Cachan, CNRS UMR 8535, UPMC, PRES UniverSud Paris), 61 av. du Pre´sident Wilson, 94235 Cachan Cedex, France Laboratory of Mechanics of Lille, CNRS UMR 8107, Ecole Polytech’Lille, University of Lille1, Cite´ Scientifique, Avenue Paul Langevin, 59655 Villeneuve d’Ascq, France ENSAIT-GEMTEX, 2 alle´e Louise et Victor Champier, BP 30329, 59056 Roubaix Cedex, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 March 2011 Received in revised form 13 February 2012 Accepted 14 January 2013 Available online 29 January 2013
Textile fabric is a very useful material for protection against ballistic projectiles. A numerical analysis, performed using the finite element method, has been carried out to understand better physical and mechanical phenomena during a ballistic impact on 2D Kevlar KM2s plain-woven fabric. The study is two-fold: macroscopic and mesoscopic, with shell elements in the explicit scheme of the finite element method. The numerical problems, dealing with the influence of the number of shell elements in a cross section for having correct models are analyzed. The damage mechanisms of the fabric during penetration time of the projectile are discussed and compared to literature experimental results. The agreement with experience results in damage mechanisms and residual velocities show a good validation of this study. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Fabric/textiles Impact behavior Finite element analysis (FEA) Damage mechanics Mesoscopic model
1. Introduction Nowadays, fabrics woven with high strength yarns (aramid, HMWPE, etc.) are widely used for the ballistic protection because of their light weight, facility for forming any shape when combined with resins, and high performance against impact [1]. However, numerical studies for ballistic impact tests of woven fabrics are not so numerous because of the complexity of the fabric structures behavior under impact and the failure mechanisms at high velocity. The results are usually limited to 2D fabrics [2–4] as the modeling of 3D fabrics is particularly difficult owing to structural complexities. Duan et al. [5,6] investigated experimentally and numerically the friction effects on the ballistic performance of a Kevlar KM2s plainwoven fabric having 50.6 50.6 mm2 dimensions with different velocities. A frictional contact is defined for yarn–yarn, and fabric– projectile contacts. These authors also published their numerical results in a series of papers [5,6]. Rao et al. [7] used their numerical model to study the influence of friction and material properties on ballistic performance. Like most of the numerical models of ballistic impact [2,5–12], their model was created in LS-DYNA software [10]. In order to simulate the impact response, they chose 3D solid elements to fully describe the 2D fabrics at mesoscopic level. In this investigation, modeling results show that friction has effect on decreasing projectile velocity during the whole impact process. To reduce computation time, Barauskas and Abraitienne [2] used shell elements instead of solid ones in order to achieve a numerical simulation. A mesoscopic model for a full 2D fabric is always so
n
Corresponding author. E-mail address: [email protected] (A. Imad).
0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.01.014
much slower in computation time than a macroscopic one. However, the prediction of full mesoscopic models is more realistic than macroscopic ones for the ballistic behavior of the fabric. For example, Lim et al. [12] showed that the macroscopic model predicts the formation of a perfect cone during projectile penetration but in reality, it assumes pyramidal form. Recently, we presented a numerical tool that permits one to form mesoscopic models of all woven fabrics [13–15]. Using this type of modeling, we analyzed numerically damage phases of a 3D woven fabric submitted to an impact of a steel sphere. For a gain of computation time, we used a multi-scale model to combine macroscopic and mesoscopic models for describing the dynamic behavior of a 2D plain-woven fabric. In the present work, an impact system between a plain weave fabric and a spherical projectile is modeled in an explicit finite element code. Shell elements were used for modeling undulated yarns of the fabric. Two finite element method (FEM) models were proposed at macroscopic and mesocopic levels. In FEM analysis, the effect of the number of elements for a cross section of yarn was investigated. The study of the evolution of the projectile velocity indicates different phases during penetration time of the projectile. In order to validate the FEM simulation, a comparison of damage mechanisms of the fabric with experimental observations was also carried out.
2. Material data and damage mechanisms 2.1. Material The material used is a Kevlar KM2s plain-woven fabric 50.6 50.6 mm2. Yarns, made of thousands of fibers, are woven in this
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Fig. 2. Stress versus strain of 930 Twaron yarns (tensile test).
formula as:
F¼
Fig. 1. (a) Initial configuration of the ballistic impact system simulated in the present study, and (b) detailed illustration of Kevlar KM2 plain-woven fabric.
fabric. The yarn density is 13.4 ends/cm in both weft and warp directions, which is equivalent to a distance of 1.49 mm between yarns (Fig. 1). The mechanical properties of each yarn are assumed elastic and orthotropic along central line of yarn. Moreover, in this paper, the projectile is a steel sphere, with a diameter of 5.35 mm and a mass of 6.25 10 4 kg.
2.2. Experimental results As seen in the paper of Rao et al. [7], when using solid elements, the yarn longitudinal Young’s modulus is 62.0 GPa, the shear and transverse elastic moduli are assumed to be two and three orders of magnitude smaller than the longitudinal one, the Poisson’s ratio close to zero and 1310 kg/m3 for the yarn density. Cheng et al. in 2005 [16] presented a technique to measure the shear modulus of single fibers. The shear modulus of single Kevlar KM2s fibers is found up to 24.4 GPa by this technique. Besides, it can be assumed that yarns are elastic to failure; most authors [2–6,10,11] agree with this hypothesis. To validate this hypothesis, tensile tests on the Twaron 930 yarns were performed. The length of yarns is 250 mm and they are stretched with a velocity of 50 mm/min. The yarn behavior is compatible with the hypothesis (Fig. 2). These yarns are almost elastic to a critical strain before a complete failure. For fiber materials, it is not easy to measure the break stress; a general method commits to calculate it from measured failure force as well as strain in tensile test and the computed section of yarn. However, the section of yarn is calculated by an approximated
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 106 dtex
pr
F is the diameter of cross section of yarn (cm), dtex is the mass in grams per 100 m and r represents the material’s density in grams per cubic centimeter. Moreover, in yarn tensile test, fibers inside a yarn do not fail simultaneously; this event adds another error on yarn section calculation. Therefore, it is worth selected the strain field for controlling yarn failure criterion, because we can measure it directly from a tensile test. In general, fiber materials should have an analytical criterion of failure in terms of strain that presents elastic behavior. Indeed, in our case, the critical stress value equals to 3.4 GPa [11] corresponding to the elastic yield strain value 5.48% which is used as the failure criterion in numerical simulation. Rao et al. [7] used two static and kinetic Coulomb’s friction coefficients (ms and mk) to characterize contacts in the impact event between the steel sphere and the Kevlar KM2s plainwoven fabric. Their experiment results show that ms ¼0.23 and mk ¼0.19 were determined for the contact between Kevlar KM2s yarns. This study also indicated that mk is equal to 0.18 in the friction case between a Kevlar KM2s yarn and a steel wire. For simplification, in this paper, only static Coulomb’s coefficient (ms) is used to describe the friction in contacts yarn/yarn and yarn/ steel projectile. Basing on the results of Rao et al. [7], we can take ms ¼ 0.23 for the friction between yarns. For the friction between yarns and steel projectile, we assume ms ¼0.20 that approximates mk ¼0.18 (the experiment kinetic value found by Rao et al. [7]). Experimental result also shows that, during penetration time of the projectile, the response of the fabric can be divided into different damage zones (Fig. 3):
i) Formation of a pyramid with the top being the head of the projectile. ii) Motion of a few principle yarns (the yarns that pass across contact zone with the projectile and perpendicular to free edge) out-of the fabric plan due to non-kept by fixed edges. iii) Shrinkage of the fabric and yarn burst out of the fabric at the two free edges due to projectile penetration. iv) Localized yarn damage zone (contact zone between yarns and projectile) where principle yarns perpendicular to free edge are moved to two sides and principle yarns parallel to free edge fail by the tensile mode because of the fixation at the two ends. v) Large zones remote from the impact zone where there is no yarn damage.
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Fig. 3. Plain-woven Kevlar KM2 fabric after shooting at an impact velocity of 60.6 m/s [5].
Spherical projectile
y x uy=0, wx=0, wz=0
ux= 0, wy= 0, wz= 0
Fig. 4. Boundary conditions of the model: (a) full model and (b) quart of the model.
3. Calculation conditions
3.2. Macroscopic model
3.1. Boundary and impact conditions
A fabric is made of interlaced warp and weft yarns. Under ballistic impact, fabric response is similar to a thin homogeneous plate. This phenomenon can be easily observed at low velocities where fabrics have a global deformation or a cone is formed with the top being the impact point. In this model, the plain weave fabric is considered as a homogenous plate (Fig. 5). Therefore, shell elements can be used to represent this homogenous plate. Since cross sections of warp and weft yarns are the same and parallel yarns are located side by side in the fabric, the size of the shell elements should be equal to the distance between yarns. This choice will give the ballistic impact response of the model close to the reality where the link between yarns is much weaker than the one between fiber components in a yarn. Therefore, the shear modulus is taken close to zero as reported by Rao et al. [7] in the macroscopic model. Certainly, this model cannot describe detailed deformations of each yarn, but it can reduce the time of computation because of the large size of elements. Thus, the model can be applied to larger structures such as complete
In these studies, two edges of the fabric are fixed while the other two edges are free—termed as ‘fixed-free’ boundary conditions. It is assumed that the contact point between the fabric and the projectile is the crossover point at the fabric centre. In order to reduce computation time, only a quarter of the complete model is calculated due to the symmetry of the impact system (Fig. 4b), conditions are added on symmetrical planes. These complementary conditions fix certain displacements of the nodes on symmetry planes in corresponding normal directions and their out-of-plane rotations (Fig. 4b). In the case of a single layer plain weave fabric, projectile deformation after impact is negligible, thus, the spherical projectile is assumed infinitely rigid. In this investigation, three impact velocities are analyzed such as 60.6, 92.1, and 245.0 m/s for understanding penetration mechanisms of projectile in perforation and no perforation cases.
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ballistic armor. Logically, the fabric thickness is used as an element thickness and the mass density of each element is calculated to ensure that the mass of the whole plate, which is equal to the fabric [11]. However, since this technique does not account for the porosity of the fabric and the undulation of yarns, the calculated density for the plate ends up being a little lower than the density of the yarn material. In this case, the 2D plainwoven structure of the fabric is close to a plane, and yarn undulation is negligible; therefore, similar to other simple analytical models [17–20], this density approximation of the macroscopic model is acceptable. The number of integration points throughout the thickness of each element is equal to 2 in order to have a good approximation of deformation of plate in this direction. Since the mechanical properties of the fabric are similar to each other in both warp and weft directions, the plate can be considered as orthotropic. Similar to the density approximation above, the elastic moduli of the plate in warp and weft directions are assumed to be equal to the elastic modulus of a real yarn (see other analytical models [17–20]). Here, this model will use corresponding values of a yarn except the shear modulus as mentioned above. With a fabric held on two sides, yarn reorientation is an important phenomenon that changes the angle between warp and weft yarns in the fabric [21,22]. This reorientation results in a non-linear in-plane shear modulus for the shell elements that initially is very low and then grows larger after yarn locking. Therefore, using a non-linear value of shear modulus that varies with yarn angle will give a better prediction of the impact behavior of the fabric in this case. However, the study focuses mainly on two features: (1) compare the calculation time between the macroscopic and mesoscopic models and (2) study
Fig. 5. Macroscopic model.
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the effect of the number of elements in a yarn cross section. Hence, a constant value of shear modulus is used in this study.
3.3. Mesoscopic model This model investigates the behavior of the fabric at the yarns level, subjected to a spherical projectile. 3D shell elements are used to represent the undulated yarns (Fig. 6). An elliptic cross section is assumed constant all along the yarn path. This section is described by shell elements with different thicknesses (Fig. 7). The textile structure is very complex, at mesoscopic level, there must be a large number of elements to detail yarn undulations. Therefore, it should optimize the mesh for a single layer before the development of multi-layers. In previous works, authors used 4 [2] or 6 [5–7,11] elements for a cross section. The number of elements is always even to profit the symmetry of the impact system for calculating a quarter of the model instead of the full one. In this study, the effect of the number of shells on a cross section is investigated. When using solid elements, the number of elements can be reduced to two, but in the case of shell elements, four is the minimum to describe the elliptical form of the section. Therefore, we consider two cases: 4 and 8 elements for a section, assuming that the section area is constant (Fig. 7). The element thicknesses of a section in any of the two cases are calculated to ensure a constant value of the section area. In this model, yarns are assumed orthotropic materials with orthotropic planes for their cross section. Hence, the orthotropic material as shown above is applied to yarn thickness shell elements. The number of integration points through the thickness of elements should be equal to 2 for calculating strain in this direction and for well predicting the elastic behavior of yarns. In Ref. [2], Barauskas et al. used only one integration point for shell elements in thickness to eliminate the bending modulus of yarns. In real conditions of application, the bending modulus of yarns is very small but not equal to zero. Therefore, with two integration points through the thickness of orthotropic shell elements, prediction of bending resistance of yarns is better than with only one point of integration. Moreover, if yarns sections are modeled by only one element in thickness, physical fields are constant when using one integration point in calculations. It follows that yarns only undergo tensile strain and no shear strain at all. This is not compatible with physical laws, as shear resistance of yarns, which cannot be negligible. In this study, the optimal number of integration points through the thickness of a shell element is
Fig. 6. Modeling of the plain weave fabric using shell elements in Radioss.
Fig. 7. Modeling of yarn cross section by shell elements: (a) dimensions of a cross section of yarn; (b) model of 4 elements; and (c) model of 8 elements.
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found and equal to 2. When this value is increased beyond 2, the behavior of yarns does not change and the computation time is longer.
4. Results and discussions 4.1. Validation of the mesoscopic model Fig. 8 shows the evolution of the projectile speed in terms of time, respectively at the impact velocities of 60.6 m/s and 245.0 m/s. Before appearance of yarn failure, there is an insignificant difference between the two models: 4 elements and 8 elements. This result is similar to the one of Boisse et al. [23] who executed a numerical simulation without failure. Moreover, comparing to calculated velocity of the projectile, the values of the 8 elements model are always a very little higher than with the 4 elements. It means that the projectile penetrates with great difficulty with the model of 4 elements than 8 elements. It seems that increasing the number of shell elements in a cross section makes yarns slip more easily on themselves. Numerically, this phenomenon indicates a correct calculation with the explicit scheme. In fact, when the number of elements increases, modeled section have a good approach to the elliptic form that makes yarns smoother. Thus, the yarns motion in the fabric is easier with the enhancement of the elements number of a section. This is the reason why before the appearance of yarn failure, projectile velocity decreases more slowly when the number of elements increases.
Appearance of yarn failure t = 165 µs
Appearance of yarn failure t = 165 µs
Fig. 9. Calculation time (comparison between 4 and 8 elements).
Yarn failure appears almost at the same time for both cases 4 and 8 elements (at the moment 158 ms with impact velocity 60.6 m/s and 21 ms with impact velocity 245.0 m/s), (Fig. 8). This indicates that main modes of yarns failure, tension and transverse shearing in the fabric subjected to ballistic impact do not change when the number of elements in cross section enhances doubly. In fact, changing the number of elements without variation of the area of cross section of yarns cannot influence yarn resistances on tensile and transverse shear. The first appearance of yarn failure initiates distortions by subsequent ruptures of yarns in the zone impact between the fabric and the projectile. This causes a little larger difference on the projectile velocity—time curves between two models, 4 and 8 elements, than the time before yarn failure (Fig. 8). Therefore, the time of projectile stopped at impact velocity 60.6 m/s is 209 ms for 4 elements and 220 ms for 8 elements. Residual velocities (velocities of the projectile after impact) at the impact of 245 m/s are also distinguished for two cases, 218.1 m/s with 4 elements and 219.5 m/s with 8 elements. These differences on calculated values are insignificant but the computation time of model of 8 elements is always doubly higher than for 4 elements (Fig. 9). Moreover, it can also be found that the number of elements on a cross section of yarn influences the tightness of the fabric as discussed above. It means that each value of the number of elements gives a proper tightness of the fabric. We can use experimental results as the reference in order to determine the exact number of elements on a section following residual velocity. In both cases of 4 and 8 elements, the calculation gives a no perforation of the fabric with the 60 m/s impact and the range of the ballistic limit from 60 m/s to 92 m/s as in reality. However, in experience, with the impact velocity of 245 m/s, the residual velocity is 207 m/s, hence, one the of 4 elements on a yarn cross section are more convenient than 8 elements. Thus, in next sections, we only study and continue to validate the mesoscopic model of 4 elements on a yarn section basing on the global behavior and the damage phases of the fabric subjected to ballistic impact.
4.2. Comparison between macroscopic and mesoscopic models
Residual velocity 218.1 m/s
Fig. 8. Evolution of the projectile velocity for the impacts: (a) 60.6 m/s and (b) 245.0 m/s.
Fig. 10(a) and (b) present results of the mesoscopic and macroscopic models on damage phases of fabric submitted to a 245 m/s ballistic impact. Globally, both these models can predict main damage phases such as: yarns failure localized around impact zone; large zones without damage; narrowing of the fabric at two free edges; motion of primary yarns with two free ends. However, the macroscopic model cannot describe delicate interactions among yarns due to the homogeneity.
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t = 21µs
t = 20µs
t = 93µs t = 115µs
t = 110µs
t = 174µs
Fig. 11. Projectile velocity evolution versus time for macroscopic and mesoscopic models at impact velocities of 60.6, 92.1 and 245 m/s.
Fig. 10. Numerical results: (a) macroscopic model and (b) quart of the mesoscopic model.
An advantage of numerical models for ballistic impact is that the calculation can give us time-projectile velocity curve easily. The evolution of the projectile velocity is observed and analyzed. Thus, physical and mechanical mechanisms of projectile penetration through the fabric are cleared. Fig. 11 shows the variation of the projectile velocity at impacts of 60.6 m/s, 92.1 m/s, and 245 m/s in both the macroscopic and mesoscopic models. These curves can be divided into two different zones. The first zone starts from the initial instant of impact and ends as soon as yarn rupture initiates. This zone can be considered as the phase before rupture of yarn. The first zones in all numerical models presented in this work are parabolic. It is deduced that in this period, response consists of cone formation without yarn failure. The kinetic energy of the projectile is absorbed by the deformation of primary and secondary yarns where primary yarns are those that pass through contact zone
with projectile and secondary yarns are those that lie beyond the contact zone, i.e., the rest of the yarns. Initially, the velocity of projectile is dissipated slowly because only the de-crimping or straightening of primary yarns contributes to energy absorption. Then, the deformation constituted in primary yarns causes the propagation of strain waves to the neighboring secondary yarns and the deceleration rate of projectile increases. At the end of the parabolic zone, the deceleration of projectile velocity reaches its maximal peak because this is the point where the de-crimping of primary yarns terminates and these yarns become completely tense. It can be also observed that at the parabolic period, strong decrease of the projectile velocity of the macroscopic model is earlier than mesoscopic. In fact, there is no crimping of yarns in the macroscopic model. Hence, the fabric is submitted to tensile stress as soon as the presence of contact, the propagation of strain waves occurs immediately in this model. Comparing the couples of meso–macro curves in Fig. 11 corresponding to different impact velocities, it can be found that when the projectile velocity increases, in the parabolic period, the curve of the macroscopic model is closer to the one of the mesoscopic model. The difference between the two models on the first time of the appearance of yarn failure is also reduced (from the impact velocity 60.6–245.0 m/s). It means that for higher impact velocity, the mesoscopic model behavior is close to the macroscopic one. This result is in good agreement with experimental observations. In fact, in the higher impact velocity case, the time for strain wave propagation in the fabric decreases. With high impact velocity like that, only principle yarns contribute essentially to stop the projectile, the contribution of secondary yarns can be negligible. We can better understand this physical phenomenon with Fig. 12. Behaviors of both mesoscopic and macroscopic models come mostly from principle yarns. The second zone of velocity variation is a quasi-linear curve. It begins with the initial rupture of elements or initial failure of numerically modeled primary yarns and slip of several primary yarns out of contact zone. Due to being clamped at the two edges, primary yarns perpendicular to these edges are tensile and failed. When primary yarns failed completely, their impact resistance and contribution for the propagation of strain waves towards corresponding secondary yarns (the yarns perpendicular to them that are not fixed at both ends) are insignificant. That is why the deceleration of the projectile in this zone is slower than at the end of the first zone. This phenomenon can also be verified by making observations on the curve. The curve has a lower slope in the second zone. The successive penetrations of the projectile are determined by two mechanisms: yarn slip out of contact zone and rupture of primary yarns. The
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Fig. 12. Deformed fabric configurations with high impact velocity of 245 m/s: (a) macroscopic model and (b) mesoscopic model.
The central principle yarn pushed out of the plane of fabric
Principle yarns slipping parallel to the fixed edge
Fig. 13. Localized observation on contact zone of the mesoscopic model with impact velocity of 60.6 m/s.
Table 1 Comparison of residual velocities between model results and experimental data by Duan et al. [5]. Macroscopic model
Mesoscopic model
Experimental result of Ref. [5]
226 m/s
218.1 m/s
207 m/s
rupture mechanism occurs essentially for primary yarns perpendicular to the two fixed edges while the slippage is observed in other primary yarns parallel to these edges. This represents the physical behavior of yarns correctly as the yarns, which are under tension due to their boundary conditions fail easily. At the impact velocity of 60.6 m/s, the projectile cannot perforate the fabric. The impact ends after continuous development of slippage and rupture of a few primary yarns. In this period, the projectile is in contact with the central primary yarn parallel to fixed edges. The motion of primary yarns is depicted in Fig. 13. The central principle yarn is not slipped out of the contact zone. However, due to projectile penetration, this yarn is pulled out from the fabric plane (Fig. 13). Calculated results of both models: macroscopic and mesoscopic, show a good prediction of residual velocity of the projectile. The fabric models can stop the projectile at 60.6 m/s and not at 92.1 m/s and 245 m/s. Both models can determine residual velocities of the projectile after impact. Table 1 enlists calculated results for impact velocity of 245 m/s in order to compare them with experimental work in Ref. [5].
5. Conclusions In the present work, two FEM models have been proposed at macroscopic and mesoscopic levels for the analysis of a ballistic impact event against a 2D fabric using an explicit finite element
code. Concerning the mesoscopic model, two numbers of mesh elements in a cross section of yarns, 4 and 8, are used in this investigation. Results in terms of velocities evolution versus time are approximately the same in both cases. This presents a significant result because the computation time of model using 8 elements is always doubly higher than for 4 elements. The comparison between the model results and the experimental ones in literature shows the capability of explicit FEM modeling for ballistic events onto dry woven fabrics. Construction of a correct macroscopic model by shell elements is essential because it allows study of ballistic impact for the complete armors with a gain of computing time.
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Numerical analysis of a ballistic impact on textile fabric Cuong Ha-Minh a,b,c, Abdellatif Imad a,n, Toufik Kanit a, Franc- ois Boussu b a b c
LMT-Cachan (ENS Cachan, CNRS UMR 8535, UPMC, PRES UniverSud Paris), 61 av. du Pre´sident Wilson, 94235 Cachan Cedex, France Laboratory of Mechanics of Lille, CNRS UMR 8107, Ecole Polytech’Lille, University of Lille1, Cite´ Scientifique, Avenue Paul Langevin, 59655 Villeneuve d’Ascq, France ENSAIT-GEMTEX, 2 alle´e Louise et Victor Champier, BP 30329, 59056 Roubaix Cedex, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 March 2011 Received in revised form 13 February 2012 Accepted 14 January 2013 Available online 29 January 2013
Textile fabric is a very useful material for protection against ballistic projectiles. A numerical analysis, performed using the finite element method, has been carried out to understand better physical and mechanical phenomena during a ballistic impact on 2D Kevlar KM2s plain-woven fabric. The study is two-fold: macroscopic and mesoscopic, with shell elements in the explicit scheme of the finite element method. The numerical problems, dealing with the influence of the number of shell elements in a cross section for having correct models are analyzed. The damage mechanisms of the fabric during penetration time of the projectile are discussed and compared to literature experimental results. The agreement with experience results in damage mechanisms and residual velocities show a good validation of this study. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Fabric/textiles Impact behavior Finite element analysis (FEA) Damage mechanics Mesoscopic model
1. Introduction Nowadays, fabrics woven with high strength yarns (aramid, HMWPE, etc.) are widely used for the ballistic protection because of their light weight, facility for forming any shape when combined with resins, and high performance against impact [1]. However, numerical studies for ballistic impact tests of woven fabrics are not so numerous because of the complexity of the fabric structures behavior under impact and the failure mechanisms at high velocity. The results are usually limited to 2D fabrics [2–4] as the modeling of 3D fabrics is particularly difficult owing to structural complexities. Duan et al. [5,6] investigated experimentally and numerically the friction effects on the ballistic performance of a Kevlar KM2s plainwoven fabric having 50.6 50.6 mm2 dimensions with different velocities. A frictional contact is defined for yarn–yarn, and fabric– projectile contacts. These authors also published their numerical results in a series of papers [5,6]. Rao et al. [7] used their numerical model to study the influence of friction and material properties on ballistic performance. Like most of the numerical models of ballistic impact [2,5–12], their model was created in LS-DYNA software [10]. In order to simulate the impact response, they chose 3D solid elements to fully describe the 2D fabrics at mesoscopic level. In this investigation, modeling results show that friction has effect on decreasing projectile velocity during the whole impact process. To reduce computation time, Barauskas and Abraitienne [2] used shell elements instead of solid ones in order to achieve a numerical simulation. A mesoscopic model for a full 2D fabric is always so
n
Corresponding author. E-mail address: [email protected] (A. Imad).
0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.01.014
much slower in computation time than a macroscopic one. However, the prediction of full mesoscopic models is more realistic than macroscopic ones for the ballistic behavior of the fabric. For example, Lim et al. [12] showed that the macroscopic model predicts the formation of a perfect cone during projectile penetration but in reality, it assumes pyramidal form. Recently, we presented a numerical tool that permits one to form mesoscopic models of all woven fabrics [13–15]. Using this type of modeling, we analyzed numerically damage phases of a 3D woven fabric submitted to an impact of a steel sphere. For a gain of computation time, we used a multi-scale model to combine macroscopic and mesoscopic models for describing the dynamic behavior of a 2D plain-woven fabric. In the present work, an impact system between a plain weave fabric and a spherical projectile is modeled in an explicit finite element code. Shell elements were used for modeling undulated yarns of the fabric. Two finite element method (FEM) models were proposed at macroscopic and mesocopic levels. In FEM analysis, the effect of the number of elements for a cross section of yarn was investigated. The study of the evolution of the projectile velocity indicates different phases during penetration time of the projectile. In order to validate the FEM simulation, a comparison of damage mechanisms of the fabric with experimental observations was also carried out.
2. Material data and damage mechanisms 2.1. Material The material used is a Kevlar KM2s plain-woven fabric 50.6 50.6 mm2. Yarns, made of thousands of fibers, are woven in this
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Fig. 2. Stress versus strain of 930 Twaron yarns (tensile test).
formula as:
F¼
Fig. 1. (a) Initial configuration of the ballistic impact system simulated in the present study, and (b) detailed illustration of Kevlar KM2 plain-woven fabric.
fabric. The yarn density is 13.4 ends/cm in both weft and warp directions, which is equivalent to a distance of 1.49 mm between yarns (Fig. 1). The mechanical properties of each yarn are assumed elastic and orthotropic along central line of yarn. Moreover, in this paper, the projectile is a steel sphere, with a diameter of 5.35 mm and a mass of 6.25 10 4 kg.
2.2. Experimental results As seen in the paper of Rao et al. [7], when using solid elements, the yarn longitudinal Young’s modulus is 62.0 GPa, the shear and transverse elastic moduli are assumed to be two and three orders of magnitude smaller than the longitudinal one, the Poisson’s ratio close to zero and 1310 kg/m3 for the yarn density. Cheng et al. in 2005 [16] presented a technique to measure the shear modulus of single fibers. The shear modulus of single Kevlar KM2s fibers is found up to 24.4 GPa by this technique. Besides, it can be assumed that yarns are elastic to failure; most authors [2–6,10,11] agree with this hypothesis. To validate this hypothesis, tensile tests on the Twaron 930 yarns were performed. The length of yarns is 250 mm and they are stretched with a velocity of 50 mm/min. The yarn behavior is compatible with the hypothesis (Fig. 2). These yarns are almost elastic to a critical strain before a complete failure. For fiber materials, it is not easy to measure the break stress; a general method commits to calculate it from measured failure force as well as strain in tensile test and the computed section of yarn. However, the section of yarn is calculated by an approximated
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 106 dtex
pr
F is the diameter of cross section of yarn (cm), dtex is the mass in grams per 100 m and r represents the material’s density in grams per cubic centimeter. Moreover, in yarn tensile test, fibers inside a yarn do not fail simultaneously; this event adds another error on yarn section calculation. Therefore, it is worth selected the strain field for controlling yarn failure criterion, because we can measure it directly from a tensile test. In general, fiber materials should have an analytical criterion of failure in terms of strain that presents elastic behavior. Indeed, in our case, the critical stress value equals to 3.4 GPa [11] corresponding to the elastic yield strain value 5.48% which is used as the failure criterion in numerical simulation. Rao et al. [7] used two static and kinetic Coulomb’s friction coefficients (ms and mk) to characterize contacts in the impact event between the steel sphere and the Kevlar KM2s plainwoven fabric. Their experiment results show that ms ¼0.23 and mk ¼0.19 were determined for the contact between Kevlar KM2s yarns. This study also indicated that mk is equal to 0.18 in the friction case between a Kevlar KM2s yarn and a steel wire. For simplification, in this paper, only static Coulomb’s coefficient (ms) is used to describe the friction in contacts yarn/yarn and yarn/ steel projectile. Basing on the results of Rao et al. [7], we can take ms ¼ 0.23 for the friction between yarns. For the friction between yarns and steel projectile, we assume ms ¼0.20 that approximates mk ¼0.18 (the experiment kinetic value found by Rao et al. [7]). Experimental result also shows that, during penetration time of the projectile, the response of the fabric can be divided into different damage zones (Fig. 3):
i) Formation of a pyramid with the top being the head of the projectile. ii) Motion of a few principle yarns (the yarns that pass across contact zone with the projectile and perpendicular to free edge) out-of the fabric plan due to non-kept by fixed edges. iii) Shrinkage of the fabric and yarn burst out of the fabric at the two free edges due to projectile penetration. iv) Localized yarn damage zone (contact zone between yarns and projectile) where principle yarns perpendicular to free edge are moved to two sides and principle yarns parallel to free edge fail by the tensile mode because of the fixation at the two ends. v) Large zones remote from the impact zone where there is no yarn damage.
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Fig. 3. Plain-woven Kevlar KM2 fabric after shooting at an impact velocity of 60.6 m/s [5].
Spherical projectile
y x uy=0, wx=0, wz=0
ux= 0, wy= 0, wz= 0
Fig. 4. Boundary conditions of the model: (a) full model and (b) quart of the model.
3. Calculation conditions
3.2. Macroscopic model
3.1. Boundary and impact conditions
A fabric is made of interlaced warp and weft yarns. Under ballistic impact, fabric response is similar to a thin homogeneous plate. This phenomenon can be easily observed at low velocities where fabrics have a global deformation or a cone is formed with the top being the impact point. In this model, the plain weave fabric is considered as a homogenous plate (Fig. 5). Therefore, shell elements can be used to represent this homogenous plate. Since cross sections of warp and weft yarns are the same and parallel yarns are located side by side in the fabric, the size of the shell elements should be equal to the distance between yarns. This choice will give the ballistic impact response of the model close to the reality where the link between yarns is much weaker than the one between fiber components in a yarn. Therefore, the shear modulus is taken close to zero as reported by Rao et al. [7] in the macroscopic model. Certainly, this model cannot describe detailed deformations of each yarn, but it can reduce the time of computation because of the large size of elements. Thus, the model can be applied to larger structures such as complete
In these studies, two edges of the fabric are fixed while the other two edges are free—termed as ‘fixed-free’ boundary conditions. It is assumed that the contact point between the fabric and the projectile is the crossover point at the fabric centre. In order to reduce computation time, only a quarter of the complete model is calculated due to the symmetry of the impact system (Fig. 4b), conditions are added on symmetrical planes. These complementary conditions fix certain displacements of the nodes on symmetry planes in corresponding normal directions and their out-of-plane rotations (Fig. 4b). In the case of a single layer plain weave fabric, projectile deformation after impact is negligible, thus, the spherical projectile is assumed infinitely rigid. In this investigation, three impact velocities are analyzed such as 60.6, 92.1, and 245.0 m/s for understanding penetration mechanisms of projectile in perforation and no perforation cases.
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ballistic armor. Logically, the fabric thickness is used as an element thickness and the mass density of each element is calculated to ensure that the mass of the whole plate, which is equal to the fabric [11]. However, since this technique does not account for the porosity of the fabric and the undulation of yarns, the calculated density for the plate ends up being a little lower than the density of the yarn material. In this case, the 2D plainwoven structure of the fabric is close to a plane, and yarn undulation is negligible; therefore, similar to other simple analytical models [17–20], this density approximation of the macroscopic model is acceptable. The number of integration points throughout the thickness of each element is equal to 2 in order to have a good approximation of deformation of plate in this direction. Since the mechanical properties of the fabric are similar to each other in both warp and weft directions, the plate can be considered as orthotropic. Similar to the density approximation above, the elastic moduli of the plate in warp and weft directions are assumed to be equal to the elastic modulus of a real yarn (see other analytical models [17–20]). Here, this model will use corresponding values of a yarn except the shear modulus as mentioned above. With a fabric held on two sides, yarn reorientation is an important phenomenon that changes the angle between warp and weft yarns in the fabric [21,22]. This reorientation results in a non-linear in-plane shear modulus for the shell elements that initially is very low and then grows larger after yarn locking. Therefore, using a non-linear value of shear modulus that varies with yarn angle will give a better prediction of the impact behavior of the fabric in this case. However, the study focuses mainly on two features: (1) compare the calculation time between the macroscopic and mesoscopic models and (2) study
Fig. 5. Macroscopic model.
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the effect of the number of elements in a yarn cross section. Hence, a constant value of shear modulus is used in this study.
3.3. Mesoscopic model This model investigates the behavior of the fabric at the yarns level, subjected to a spherical projectile. 3D shell elements are used to represent the undulated yarns (Fig. 6). An elliptic cross section is assumed constant all along the yarn path. This section is described by shell elements with different thicknesses (Fig. 7). The textile structure is very complex, at mesoscopic level, there must be a large number of elements to detail yarn undulations. Therefore, it should optimize the mesh for a single layer before the development of multi-layers. In previous works, authors used 4 [2] or 6 [5–7,11] elements for a cross section. The number of elements is always even to profit the symmetry of the impact system for calculating a quarter of the model instead of the full one. In this study, the effect of the number of shells on a cross section is investigated. When using solid elements, the number of elements can be reduced to two, but in the case of shell elements, four is the minimum to describe the elliptical form of the section. Therefore, we consider two cases: 4 and 8 elements for a section, assuming that the section area is constant (Fig. 7). The element thicknesses of a section in any of the two cases are calculated to ensure a constant value of the section area. In this model, yarns are assumed orthotropic materials with orthotropic planes for their cross section. Hence, the orthotropic material as shown above is applied to yarn thickness shell elements. The number of integration points through the thickness of elements should be equal to 2 for calculating strain in this direction and for well predicting the elastic behavior of yarns. In Ref. [2], Barauskas et al. used only one integration point for shell elements in thickness to eliminate the bending modulus of yarns. In real conditions of application, the bending modulus of yarns is very small but not equal to zero. Therefore, with two integration points through the thickness of orthotropic shell elements, prediction of bending resistance of yarns is better than with only one point of integration. Moreover, if yarns sections are modeled by only one element in thickness, physical fields are constant when using one integration point in calculations. It follows that yarns only undergo tensile strain and no shear strain at all. This is not compatible with physical laws, as shear resistance of yarns, which cannot be negligible. In this study, the optimal number of integration points through the thickness of a shell element is
Fig. 6. Modeling of the plain weave fabric using shell elements in Radioss.
Fig. 7. Modeling of yarn cross section by shell elements: (a) dimensions of a cross section of yarn; (b) model of 4 elements; and (c) model of 8 elements.
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found and equal to 2. When this value is increased beyond 2, the behavior of yarns does not change and the computation time is longer.
4. Results and discussions 4.1. Validation of the mesoscopic model Fig. 8 shows the evolution of the projectile speed in terms of time, respectively at the impact velocities of 60.6 m/s and 245.0 m/s. Before appearance of yarn failure, there is an insignificant difference between the two models: 4 elements and 8 elements. This result is similar to the one of Boisse et al. [23] who executed a numerical simulation without failure. Moreover, comparing to calculated velocity of the projectile, the values of the 8 elements model are always a very little higher than with the 4 elements. It means that the projectile penetrates with great difficulty with the model of 4 elements than 8 elements. It seems that increasing the number of shell elements in a cross section makes yarns slip more easily on themselves. Numerically, this phenomenon indicates a correct calculation with the explicit scheme. In fact, when the number of elements increases, modeled section have a good approach to the elliptic form that makes yarns smoother. Thus, the yarns motion in the fabric is easier with the enhancement of the elements number of a section. This is the reason why before the appearance of yarn failure, projectile velocity decreases more slowly when the number of elements increases.
Appearance of yarn failure t = 165 µs
Appearance of yarn failure t = 165 µs
Fig. 9. Calculation time (comparison between 4 and 8 elements).
Yarn failure appears almost at the same time for both cases 4 and 8 elements (at the moment 158 ms with impact velocity 60.6 m/s and 21 ms with impact velocity 245.0 m/s), (Fig. 8). This indicates that main modes of yarns failure, tension and transverse shearing in the fabric subjected to ballistic impact do not change when the number of elements in cross section enhances doubly. In fact, changing the number of elements without variation of the area of cross section of yarns cannot influence yarn resistances on tensile and transverse shear. The first appearance of yarn failure initiates distortions by subsequent ruptures of yarns in the zone impact between the fabric and the projectile. This causes a little larger difference on the projectile velocity—time curves between two models, 4 and 8 elements, than the time before yarn failure (Fig. 8). Therefore, the time of projectile stopped at impact velocity 60.6 m/s is 209 ms for 4 elements and 220 ms for 8 elements. Residual velocities (velocities of the projectile after impact) at the impact of 245 m/s are also distinguished for two cases, 218.1 m/s with 4 elements and 219.5 m/s with 8 elements. These differences on calculated values are insignificant but the computation time of model of 8 elements is always doubly higher than for 4 elements (Fig. 9). Moreover, it can also be found that the number of elements on a cross section of yarn influences the tightness of the fabric as discussed above. It means that each value of the number of elements gives a proper tightness of the fabric. We can use experimental results as the reference in order to determine the exact number of elements on a section following residual velocity. In both cases of 4 and 8 elements, the calculation gives a no perforation of the fabric with the 60 m/s impact and the range of the ballistic limit from 60 m/s to 92 m/s as in reality. However, in experience, with the impact velocity of 245 m/s, the residual velocity is 207 m/s, hence, one the of 4 elements on a yarn cross section are more convenient than 8 elements. Thus, in next sections, we only study and continue to validate the mesoscopic model of 4 elements on a yarn section basing on the global behavior and the damage phases of the fabric subjected to ballistic impact.
4.2. Comparison between macroscopic and mesoscopic models
Residual velocity 218.1 m/s
Fig. 8. Evolution of the projectile velocity for the impacts: (a) 60.6 m/s and (b) 245.0 m/s.
Fig. 10(a) and (b) present results of the mesoscopic and macroscopic models on damage phases of fabric submitted to a 245 m/s ballistic impact. Globally, both these models can predict main damage phases such as: yarns failure localized around impact zone; large zones without damage; narrowing of the fabric at two free edges; motion of primary yarns with two free ends. However, the macroscopic model cannot describe delicate interactions among yarns due to the homogeneity.
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t = 21µs
t = 20µs
t = 93µs t = 115µs
t = 110µs
t = 174µs
Fig. 11. Projectile velocity evolution versus time for macroscopic and mesoscopic models at impact velocities of 60.6, 92.1 and 245 m/s.
Fig. 10. Numerical results: (a) macroscopic model and (b) quart of the mesoscopic model.
An advantage of numerical models for ballistic impact is that the calculation can give us time-projectile velocity curve easily. The evolution of the projectile velocity is observed and analyzed. Thus, physical and mechanical mechanisms of projectile penetration through the fabric are cleared. Fig. 11 shows the variation of the projectile velocity at impacts of 60.6 m/s, 92.1 m/s, and 245 m/s in both the macroscopic and mesoscopic models. These curves can be divided into two different zones. The first zone starts from the initial instant of impact and ends as soon as yarn rupture initiates. This zone can be considered as the phase before rupture of yarn. The first zones in all numerical models presented in this work are parabolic. It is deduced that in this period, response consists of cone formation without yarn failure. The kinetic energy of the projectile is absorbed by the deformation of primary and secondary yarns where primary yarns are those that pass through contact zone
with projectile and secondary yarns are those that lie beyond the contact zone, i.e., the rest of the yarns. Initially, the velocity of projectile is dissipated slowly because only the de-crimping or straightening of primary yarns contributes to energy absorption. Then, the deformation constituted in primary yarns causes the propagation of strain waves to the neighboring secondary yarns and the deceleration rate of projectile increases. At the end of the parabolic zone, the deceleration of projectile velocity reaches its maximal peak because this is the point where the de-crimping of primary yarns terminates and these yarns become completely tense. It can be also observed that at the parabolic period, strong decrease of the projectile velocity of the macroscopic model is earlier than mesoscopic. In fact, there is no crimping of yarns in the macroscopic model. Hence, the fabric is submitted to tensile stress as soon as the presence of contact, the propagation of strain waves occurs immediately in this model. Comparing the couples of meso–macro curves in Fig. 11 corresponding to different impact velocities, it can be found that when the projectile velocity increases, in the parabolic period, the curve of the macroscopic model is closer to the one of the mesoscopic model. The difference between the two models on the first time of the appearance of yarn failure is also reduced (from the impact velocity 60.6–245.0 m/s). It means that for higher impact velocity, the mesoscopic model behavior is close to the macroscopic one. This result is in good agreement with experimental observations. In fact, in the higher impact velocity case, the time for strain wave propagation in the fabric decreases. With high impact velocity like that, only principle yarns contribute essentially to stop the projectile, the contribution of secondary yarns can be negligible. We can better understand this physical phenomenon with Fig. 12. Behaviors of both mesoscopic and macroscopic models come mostly from principle yarns. The second zone of velocity variation is a quasi-linear curve. It begins with the initial rupture of elements or initial failure of numerically modeled primary yarns and slip of several primary yarns out of contact zone. Due to being clamped at the two edges, primary yarns perpendicular to these edges are tensile and failed. When primary yarns failed completely, their impact resistance and contribution for the propagation of strain waves towards corresponding secondary yarns (the yarns perpendicular to them that are not fixed at both ends) are insignificant. That is why the deceleration of the projectile in this zone is slower than at the end of the first zone. This phenomenon can also be verified by making observations on the curve. The curve has a lower slope in the second zone. The successive penetrations of the projectile are determined by two mechanisms: yarn slip out of contact zone and rupture of primary yarns. The
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Fig. 12. Deformed fabric configurations with high impact velocity of 245 m/s: (a) macroscopic model and (b) mesoscopic model.
The central principle yarn pushed out of the plane of fabric
Principle yarns slipping parallel to the fixed edge
Fig. 13. Localized observation on contact zone of the mesoscopic model with impact velocity of 60.6 m/s.
Table 1 Comparison of residual velocities between model results and experimental data by Duan et al. [5]. Macroscopic model
Mesoscopic model
Experimental result of Ref. [5]
226 m/s
218.1 m/s
207 m/s
rupture mechanism occurs essentially for primary yarns perpendicular to the two fixed edges while the slippage is observed in other primary yarns parallel to these edges. This represents the physical behavior of yarns correctly as the yarns, which are under tension due to their boundary conditions fail easily. At the impact velocity of 60.6 m/s, the projectile cannot perforate the fabric. The impact ends after continuous development of slippage and rupture of a few primary yarns. In this period, the projectile is in contact with the central primary yarn parallel to fixed edges. The motion of primary yarns is depicted in Fig. 13. The central principle yarn is not slipped out of the contact zone. However, due to projectile penetration, this yarn is pulled out from the fabric plane (Fig. 13). Calculated results of both models: macroscopic and mesoscopic, show a good prediction of residual velocity of the projectile. The fabric models can stop the projectile at 60.6 m/s and not at 92.1 m/s and 245 m/s. Both models can determine residual velocities of the projectile after impact. Table 1 enlists calculated results for impact velocity of 245 m/s in order to compare them with experimental work in Ref. [5].
5. Conclusions In the present work, two FEM models have been proposed at macroscopic and mesoscopic levels for the analysis of a ballistic impact event against a 2D fabric using an explicit finite element
code. Concerning the mesoscopic model, two numbers of mesh elements in a cross section of yarns, 4 and 8, are used in this investigation. Results in terms of velocities evolution versus time are approximately the same in both cases. This presents a significant result because the computation time of model using 8 elements is always doubly higher than for 4 elements. The comparison between the model results and the experimental ones in literature shows the capability of explicit FEM modeling for ballistic events onto dry woven fabrics. Construction of a correct macroscopic model by shell elements is essential because it allows study of ballistic impact for the complete armors with a gain of computing time.
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