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Finite element modeling of transverse impact on a ballistic fabric
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International Journal of Mechanical Sciences 48 (2006) 33–43 www.elsevier.com/locate/ijmecsci
Finite element modeling of transverse impact on a ballistic fabric Y. Duana,, M. Keefeb, T.A. Bogettic, B. Powersc a
Center for Composite Materials, University of Delaware, Newark, DE 19716, USA Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA c US Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
b
Received 5 August 2004; received in revised form 14 July 2005; accepted 17 September 2005 Available online 26 October 2005
Abstract A 3D finite element analysis model is created using LS-DYNA to simulate the transverse impact of a rigid right circular cylinder onto a square patch of plain-woven Kevlar fabric. The fabric is modeled to yarn level resolution and relative motion between yarns is allowed. A frictional contact is defined between yarns and between the fabric and the projectile. Three different boundary conditions are applied on the fabric: four edges left free; two opposite edges clamped; four edges clamped. Results from the modeling effort show that during initial stage of the impact, the projectile velocity drops very quickly. There exists an abrupt momentum transfer from the projectile to the local fabric at the impact zone. When the impact velocity is low, the fabric boundary condition plays an important role at later stages of the impact. It significantly affects the fabric deformation, stress distribution, energy absorption and failure modes. When the impact velocity is high enough to cause the yarns to break instantaneously, the fabric fails along the periphery of the impact zone and the fabric boundary condition does not take any effects. r 2005 Elsevier Ltd. All rights reserved. Keywords: Ballistic fabric; Transverse impact; Energy absorption; Finite element analysis
1. Introduction Weight and flexibility are two important design parameters for soft body armors that offer protection against bullets and munitions fragments. Fabrics made from highstrength fibers, also called ballistic fabrics, have the attractive properties of low density, high flexibility, high strength-to-weight ratio, and outstanding ballistic resistant property. Therefore, they have been widely used in making soft body armors since their introduction to market. The high-strength fibers used in making ballistic fabrics include aramid (Kevlar, Twaron, Technora), polyethylene (Spectra, Dyneema), and Polybenzoxazole (Zylon) [1,2]. These fibers are essentially elastic in tension and have very high tensile modulus. They have relatively low tensile failure strain and are generally not sensitive to strain rate. Hundreds of the high-strength fibers are grouped together to make a yarn and yarns are woven to produce a single layer ballistic fabric. Corresponding author. Tel.: +1 302 831 0376; fax: +1 302 831 8525.
E-mail address: [email protected] (Y. Duan). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.09.007
Typically, a soft body armor is constructed of a ballistic panel, which is assembled from multiple layers of ballistic fabric, and a carrier made from conventional garment fabric. When a projectile impacts into such a fabric structure, it is gradually slowed down and finally caught by the ballistic panel. The impact energy is converted as fabric kinetic energy, fabric strain energy, projectile deformation energy and energy dissipated in frictional sliding. The impact resistance of a ballistic panel depends on its capability to absorb energy locally at the impact zone and disperse energy quickly out of the impact zone. It is affected by a number of factors, which include fiber density, fiber tensile elastic modulus, fiber tensile failure strain, fabric weave style, fabric areal density, number of fabric layers, fabric boundary condition, projectile shape, mass and material property, impact velocity, and interfacial friction characteristics within impact system. To optimize the design of a soft body armor, in other words, increasing its ballistic resistance and at the same time keeping or even reducing its weight and rigidity, one must understand the impact behavior of the ballistic panel. To understand the impact behavior of the ballistic panel,
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one should first understand the impact behavior of its construction units such as single layer fabric, single yarn, and single fiber. During the past several decades, a lot of experiments and theoretical work have been conducted to understand the transverse impact behavior of single yarns and single layer fabrics [3–11]. Smith et al. [3], Roylance [4], Morrison [5], and Field and Sun [6] studied the response of yarns to high-speed transverse impact while Wilde et al. [7,8], Briscoe and Motamedi [9], Shim et al. [10], and Shockey et al. [11] investigated the transverse impact behavior of single layer fabrics. In this paper, a finite element analysis (FEA) model is created using LS-DYNA to simulate the transverse impact of a right circular cylinder (RCC) on a single layer plainwoven Kevlar fabric. The model allows for definition of contact between yarns and therefore takes into account the physical interaction between yarns. It more realistically describes the woven fabric structure than those computational models in which yarn crossovers are described as links, joints or bulk continuum. 2. Transverse impact on yarns 2.1. Existing theory on transverse impact on a long straight yarn As stated previously, the yarn in a ballistic fabric is composed of hundreds of high-strength fibers. It is a very complex structure. For simplicity in analysis, the interaction between fibers in a yarn is generally ignored and the yarn is assumed to be an elastic continuum. Fig. 1 shows a heavy wedge-tipped projectile transversely impacting on a long straight yarn. The yarn tensile elastic modulus is E and volumetric density is r. The impact velocity is v and it is not high enough to cause the yarn to break. According to Refs. [3,4], two mechanical waves are generated by the impact. One is a longitudinal wave, which propagates away from the impact point at the sound speed of the yarn material. The longitudinal wave speed c is given by sffiffiffiffi E c¼ . (1) r v
time = 0
v
time = t Fig. 1. A wedge-tipped projectile transversely impacts on a long straight yarn at a constant velocity of v.
Ahead of the longitudinal wave front, yarn material strain is zero; behind, a constant tensile strain is developed. The tensile strain, denoted as , is determined by the yarn tensile elastic modulus E, volumetric density r, and the impact velocity v. It is implicitly given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rv2 . (2) ð1 þ Þ 2 ¼ E The other mechanical wave generated by the impact is a transverse one, which propagates away from the impact point at a relatively lower speed. The transverse wave speed u is given by rffiffiffiffiffiffiffiffiffiffiffi . (3) u¼c 1þ
2
Across the transverse wave front, the strain of yarn material does not change; however, the motion of yarn material experiences an abrupt change. Ahead of the transverse wave front but behind the longitudinal wave front, yarn material moves longitudinally toward the impact point. Behind the transverse wave front, yarn material moves transversely in the impact direction. It can be seen from Eq. (3) that the transverse wave speed positively correlates with the yarn tensile strain. It is larger with a larger tensile strain, and vice versa. When the yarn tensile strain is zero, the transverse wave speed is also zero. The yarn kinetic energy E k and the yarn strain energy E s during the impact can be obtained from Eqs. (4) and (5), respectively. sffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 3 2 E E k ¼ At , (4) 32 1þ r sffiffiffiffiffiffi E3 E s ¼ At2 , r
(5)
where, A is the yarn cross-section area and t is the time after impact. The two formulas may be obtained from the above analysis of wave propagations in yarn material. 2.2. Comparison of FEA modeling results and predictions from the existing theory Consider an impact case: a heavy wedge-tipped projectile transversely impacts on a long straight Kevlar yarn that has a tensile elastic modulus E of 74 GPa, a volumetric density r of 1,440 kg/m3 , and a cross-section area A of 5:83 108 m2 . The impact velocity v is 200 m/s. The yarn tensile strain generated by the impact, , can be obtained from Eq. (2) by using iteration method. Substitute the values of E; r; A, and into Eqs. (4) and (5), the yarn kinetic energy and the yarn strain energy at any time t can be obtained. A 3D FEA model is created using LS-DYNA to simulate the impact. Fig. 2 shows the FEA model for the straight Kevlar yarn. The yarn is modeled as a continuum and the yarn cross-section is defined by a pair of symmetric arcs.
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Gasser et al. [12] have shown that an orthotropic elastic continuum has yarn behavior if its Poisson’s ratios are zero and the shear moduli and transverse elastic moduli are very small with respect to the longitudinal elastic modulus. In the FEA model, the yarn has locally orthotropic elastic material property. Table 1 lists the nine orthotropic elastic material data. Each element in the model defines a referential coordinate whose three axes are determined by the nodes of the element. The orthotropic elastic material data is defined in the local referential coordinates. The yarn kinetic energy and the yarn strain energy as a function of time are obtained from the FEA modeling. Fig. 3 shows a comparison of the modeling results and the analytical results obtained from Eqs. (4) and (5). It can be seen from this figure that the FEA modeling results agree well with the analytical results. The good agreement indicates that the FEA modeling approach and the orthotropic elastic material data listed in Table 1 describe well the transverse impact behavior of the Kevlar yarn.
Fig. 2. The 3D FEA model for the straight Kevlar yarn.
Table 1 Orthotropic elastic material data (GPa) for the Kevlar yarn E11
E22
E33
G12
G13
G23
n12
n13
n23
74
0.74
0.74
0.148
0.148
0.148
0
0
0
0.05 Yarn kinetic energy; FEA Yarn kinetic energy; analytical Yarn strain energy; FEA Yarn strain energy; analytical
Energy (J)
0.04 0.03 0.02 0.01 0
0
2
4
6
8
10
12
14
Time (µs) Fig. 3. A comparison of the FEA modeling results and the predictions from theory.
35
2.3. Effect of yarn ends boundary condition In the above analysis and modeling, the Kevlar yarn is assumed to be infinitely long. Therefore, the effect of yarn ends boundary condition is not taken into account. For a real impact situation, the stress/strain wave generated by the impact soon arrives at the yarn ends. The yarn ends boundary condition inevitably plays a role in the yarn impact behavior. To explore the effect of yarn ends boundary condition, two cases are modeled where a heavy wedge-tipped projectile transversely impacts at 200 m/s onto the center of a Kevlar yarn that has a length of 49 mm. In the first case, both the yarn ends are clamped, while in the second case both the yarn ends are left free. It can be seen from Table 1 that the yarn longitudinal elastic modulus is much larger than its shear moduli and transverse elastic moduli. In this situation, the value of the maximum principal stress is very close to that of the tensile stress along fiber direction. For convenient implementation in LS-DYNA, a maximum principal stress failure criterion is used in the modeling. When the maximum principal stress at a material point exceeds 2.3 GPa, the material fails and the corresponding element is deleted automatically from the mesh. The release wave resulted from deleting the element is taken into account in subsequent deformation process. The maximum principal stress failure criterion is equivalent to a maximum tensile strain failure criterion with a failure strain of 3.1%. Fig. 4 shows the yarn deformation when both of its ends are clamped, while Fig. 5 shows the yarn deformation when both of its ends are left free. It can be seen from the two figures that the yarn ends boundary condition significantly affects the yarn deformation. The yarn is broken at the impact point when its two ends are clamped while it is not broken when its two ends are left free. The different yarn behavior is a result of the different yarn ends boundary condition. As stated previously, two mechanical waves are generated by the impact: a longitudinal one and a transverse one. The longitudinal wave propagates away from the impact point at a very high speed while the transverse wave propagates at a relatively lower speed. Behind the transverse wave front, yarn material moves transversely in the impact direction. Ahead of the transverse wave front but behind the longitudinal wave front, yarn material moves longitudinally toward the impact point. When the longitudinal wave reaches the clamped ends, it is reflected back and propagates toward the impact point. Behind the reflected wave front, yarn material stops moving longitudinally and the tensile stress is doubled. The reflected longitudinal waves meet at the impact point where the tensile stress is superimposed on each other. After three reflections of the longitudinal wave, the stress at the impact point reaches the failure criterion and the yarn, as shown in Fig. 4, is broken. Similarly, when the longitudinal wave reaches the free ends, it is also reflected back and propagates toward the impact point.
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Fig. 4. Deformation of the straight Kevlar yarn at various instants of time when both of its ends are clamped; the arrows indicate the transverse wave fronts at 20 ms.
However, behind the reflected wave front, the yarn tensile stress becomes zero and the velocity of the yarn material moving longitudinally toward the impact point is doubled. As can be seen from Fig. 5, with reflection of the longitudinal wave the two ends of the yarn move longitudinally toward the impact point. The transverse wave stops propagating when the reflected longitudinal wave meets the transverse one.
Fig. 5. Deformation of the straight Kevlar yarn at various instants of time when both of its ends are left free; the arrows indicate the transverse wave fronts at 20 ms.
y RCC projectile x
3. Modeling transverse impact on a ballistic fabric Fig. 6 shows the initial geometry of an impact event: a rigid RCC projectile transversely impacts onto the center of a patch of plain-woven Kevlar fabric. The fabric is flat and aligned with the x–z plane. It is composed of 39 yarns in each of the warp direction (along the x-axis) and the weft direction (along the z-axis). The fabric maximum thickness is 0.23 mm, and both of its side length is 32.7 mm. The fabric four edges are left free. The projectile diameter is 8 mm, its mass is 2 g, and its impact velocity v is 200 m/s. During the impact, the rigid projectile can only move along the y-direction and the other five degrees of freedom are constrained. A 3D FEA model is created using LS-DYNA to simulate the aforementioned impact. The impact system has symmetry with respect to both the x–y plane and the y–z plane, therefore only a quarter of the entire system needs to
v f f f
Kevlar fabric
z
Fig. 6. A rigid RCC projectile transversely impacts onto the center of a square patch of plain-woven Kevlar fabric; the impact velocity is v.
be modeled. Fig. 7 shows a part of the 3D FEA model for the plain-woven fabric. The fabric is modeled to yarn level resolution and the yarns are modeled as continuum with locally orthotropic elastic material property. The material properties of the Kevlar yarn have been given in the
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Projectile velocity (m/s)
200 197 194 191 188 185
0
10
20
30
40
50
Time (µs)
Fig. 8. Time history of the projectile velocity for the case with four fabric edges left free and v ¼ 200 m/s. The arrow on the projectile velocity versus time curve indicates the stage of deformation shown in Fig. 9. Fig. 7. The 3D FEA model for the plain-woven Kevlar fabric.
previous section. The cross-section of the crimped yarn is the same as that of the straight yarn shown in Fig. 2. The crimped profiles of the warp and the weft yarns are identical and are defined by a series of connected arcs. The 3D FEA model for the plain-woven fabric defines yarn–yarn contact and allows for relative motion between yarns. Simple Coulomb friction is introduced between yarns and between the projectile and the fabric. A friction coefficient of 0.3, which is obtained from experiments on Kevlar fabrics, is used for both the types of friction [13]. The influence of interfacial friction on fabric impact behavior will not be discussed in this paper. However, it is worth noting that the friction between projectile and fabric, between yarn and yarn, and between fibers themselves might have significant effects on the impact behavior of ballistic fabrics, especially when the projectile is in spherical shape [9,14,15]. In order to comparatively investigate the effect of fabric boundary condition, two additional cases are modeled where all the conditions described previously are maintained except that different boundary conditions are applied on the fabric. In one case, two opposite edges of the fabric are clamped and the other two edges are left free; in the other case, all the four edges of the fabric are clamped. Finally, to explore the effect of impact velocity on the fabric ballistic performance, three cases are modeled where the impact velocity v is 400 m/s instead of 200 m/s and the three different types of boundary conditions described previously are applied on the fabric, respectively. 4. Results and discussion 4.1. Projectile-fabric interaction and energy transfer Fig. 8 shows time history of the projectile velocity for the case with four fabric edges left free and impact velocity v ¼ 200 m/s. It can be seen from this figure that within a very short period of time ð0:3 msÞ, the projectile velocity drops from 200 to 198.3 m/s. Afterwards, the projectile
gradually slows down and at 50 ms its velocity is 188 m/s. Fig. 9 depicts contour maps of the fabric transverse velocity and transverse displacement at 0:3 ms when the projectile velocity is 198.3 m/s. It can be seen from this figure that the local fabric that directly contacts the RCC projectile abruptly moves with the transverse impact. The projectile momentum is transferred to the local fabric. The momentum transfer occurs so quickly that the fabric located out of the impact zone is not affected at all. At this moment, the fabric transverse displacement coincides with the projectile-fabric contact zone and is in a pie shape. The initial momentum transfer is responsible for the abrupt drop of the projectile velocity from 200 to 198.3 m/s. After the initial momentum transfer, the local fabric in the impact zone moves together with the projectile. Due to the sudden transverse motion of the local fabric, a longitudinal wave and a transverse wave are generated in the principal yarns (those yarns that directly contact the projectile). The longitudinal wave propagates away from the impact zone at a very high speed; behind the wave front, yarn material is strained and moves longitudinally toward the impact zone. The transverse wave propagates away from the impact zone at a relatively lower speed; behind the wave front, yarn material moves transversely in the impact direction. Fig. 10 shows contour maps of the fabric resultant displacement at initial stages of the impact. It can be seen from this figure that at 0:5 ms, mainly the principal yarns are affected. With propagation of the mechanical waves, yarn–yarn interactions cause the secondary yarns (those yarns that do not directly contact the projectile) to move. At 1:0 ms, the impact-affected zone takes a square form, and with time going on it gradually expands outward. During the process, the fabric absorbs energy from the projectile and the projectile gradually slows down. At 3:0 ms when most of the fabric has been affected by the impact, the projectile velocity is 197.9 m/s. The projectile velocity is reduced by 1.7 m/s during the initial 0:3 ms while it is only reduced by 0.4 m/s during the period from 0.3 to 3 ms.
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Fig. 9. The fabric deformation at 0:3 ms. (a) Contour map of the fabric transverse velocity ð103 m/s). (b) Contour map of the fabric transverse displacement (mm).
Fig. 10. Contour maps of the fabric resultant displacement (mm) at the initial stage of the impact (v ¼ 200 m/s, four fabric edges left free).
Deformed configurations of the fabric at various instants of time are illustrated in Fig. 11 for the case with four fabric edges left free and impact velocity v ¼ 200 m/s. It can be seen from this figure that during the impact, the fabric at the impact zone conforms to the flat round nose of the RCC projectile. Lim et al [16] have observed by using highspeed camera the conformation of fabric to the nose shape of a flat-nosed projectile during transverse impact. It is found from the deformed configurations of the fabric that two warp yarns and one weft yarn are broken during the impact. Except for the three yarns, no other yarns in the fabric are broken. It can be seen from Fig. 11 that the fabric transverse displacement is in the form of a conical frustum at 10 ms. At that instant of time, the four fabric edges slightly bow toward the impact zone but most of the fabric is not affected by the transverse wave yet. Bowings of the fabric edges indicate that the longitudinal wave has been reflected back from the fabric edges. With time going on, the four fabric edges gradually bow toward the impact
zone and the fabric transverse wave gradually propagates outward. It is noted that the fabric transverse wave front evolves during the impact. It is in the form of a circle at 10 ms while it is in the form of a round-filleted square at 30 ms. The corresponding projectile velocity for each of the deformed configurations of the fabric can be found in Fig. 8. There is no external force acting on the system during the impact. The energy in the system is therefore conserved. The lost projectile kinetic energy is completely absorbed by the fabric. Fig. 12 shows time history of the energy transfer between the projectile and the fabric. It can be seen that at the initial momentum transfer, the projectile loses 0.68 J of kinetic energy, of which 80% is absorbed by the fabric in the form of yarn kinetic energy, 19% is absorbed by the fabric in the form of yarn strain energy, and the remaining 1% is dissipated as heat through friction between the projectile and the fabric and between yarns themselves. It is evident that yarn kinetic energy is the dominant energy
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Fig. 11. Top and side view of the fabric deformation at various instants of time (v ¼ 200 m/s, four fabric edges left free).
4.2. Effect of fabric boundary condition Two additional cases with different fabric boundary conditions (two opposite edges clamped; four edges clamped) are modeled to comparatively study the effect of fabric boundary condition. Fig. 13 shows the projectile velocity as a function of time for the three cases that have the same impact velocity v of 200 m/s but different fabric boundary conditions. It can be seen from this figure that within the initial 5 ms, the projectile velocity is the same for all the three cases. This result indicates that the fabric boundary conditions do not play a role in decelerating the projectile during that period of time. After 5 ms, the fabric boundary conditions take effect and the projectile velocity becomes different for the three cases; the projectile velocity drops most slowly when all the four fabric edges are left free while it drops most quickly when all of the four fabric edges are clamped. It can be seen that at 10 ms, the projectile velocity is 197.0 m/s for the case with four fabric edges left free while it is 196.1 m/s for the case with two opposite fabric edges clamped and 195.1 m/s for the case with four fabric edges clamped. Though the fabric most quickly decelerates the projectile when its four edges are
6 Loss of projectile kinetic energy Yarn kinetic energy Yarn strain energy Friction dissipated energy
5 Energy (J)
absorption mechanism during the impact. At 50 ms, around 67% of the lost projectile kinetic energy is absorbed by the fabric in the form of yarn kinetic energy, while 25% in the form of yarn strain energy and 8% in the form of friction dissipated energy.
4 3 2 1 0
0
10
20
30 Time (µs)
40
50
Fig. 12. Time history of energy transfer between the projectile and the fabric (v ¼ 200 m/s, four fabric edges left free).
clamped, it loses function at the earliest time; the projectile punches through the fabric at 12 ms and moves away at a constant velocity of 194.4 m/s. When two opposite edges of the fabric are clamped, the projectile gradually slows down until 45 ms when the fabric is punched through and loses its capability to decelerate the projectile. As can be seen, the fabric most effectively slows down the projectile when all of its four edges are left free; at 50 ms, the projectile velocity is 188.0 m/s when the four fabric edges are left free while it is 193.1 m/s when two opposite edges of the fabric are clamped and 194.4 m/s when all of the four fabric edges are clamped.
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Fig. 14 shows the maximum principal stress distribution in the fabric at 8 ms for the three impact cases. It can be seen that the maximum principal stress mainly distributes in the clamped principal yarns and it is very small in the unclamped principal yarns or in the secondary yarns. The fabric boundary condition significantly affects the stress distribution pattern in the fabric. Fig. 15 shows contour maps of the fabric transverse displacement at 10 ms for the three cases. As can be seen, the transverse wave front is in the form of a circle when all the four fabric edges are left free, while it is in the form of an ellipse with the long axis
Projectile velocity (m/s)
200
196
192 Four fabric edges left free Two opposite fabric edges clamped Four fabric edges clamped
188
184 0
10
20
30
40
50
Time (µs) Fig. 13. The projectile velocity as a function of time for the three cases that have the same impact velocity of 200 m/s but different boundary conditions.
along the clamped yarns when two opposite fabric edges are clamped and a filleted square when all the four fabric edges are clamped. The different stress distributions and the different transverse displacements are due to the different fabric boundary conditions. As stated previously, with the transverse impact, two mechanical waves originate in the impact zone. The longitudinal wave propagates away from the impact zone at a very high speed while the transverse wave propagates at a lower speed. Behind the longitudinal wave front, yarn material is tensioned and moves longitudinally toward the impact zone, while behind the transverse wave front, yarn material moves transversely. When the longitudinal wave arrives at the free fabric edges, it is reflected back and propagates toward the impact zone. Behind the reflected wave front, the yarn tensile stress fades away and the velocity of the yarn material moving toward the impact zone is doubled; the interactions between yarns gradually produce bowings along the free fabric edges. When the longitudinal wave arrives at the clamped edges, it is also reflected back and propagates toward the impact zone. However, behind the reflected wave front, the yarn tensile stress is doubled and the yarn material stops moving longitudinally toward the impact zone. At 8 ms, the longitudinal wave has been reflected by the fabric edges and propagated back to the impact zone. Therefore, the stress in the clamped principal yarns is much larger than the stress in the unclamped principal yarns and the secondary yarns. As shown by Eq. (3), the transverse wave speed in a yarn is determined by the sound speed of
Fig. 14. Distribution of maximum principal stress (103 GPa) in the fabric at 8 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
Fig. 15. Contour maps of the fabric transverse displacement (mm) at 10 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
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Fig. 16. The fabric deformation at 40 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
4.3. Effect of impact velocity Three cases with an impact velocity of 400 m/s and different boundary conditions (four fabric edges left free; two opposite fabric edges clamped; four fabric edges clamped) are modeled to comparatively study the effect of impact velocity. Fig. 17 shows time history of the projectile velocity while Fig. 18 shows the fabric deformed configurations at various instants of time for the case with four fabric edges left free. It can be seen that the projectile is decelerated very quickly at initial stage of the impact. Within 0:3 ms, the projectile velocity drops from 400 to 396.7 m/s. The abrupt drop of the projectile velocity is due to the initial momentum transfer from the projectile to the
400 Projectile velocity (m/s)
the yarn material and the tensile strain in the yarn. It is higher with a larger strain and lower with a smaller strain. Due to reflection of the longitudinal wave from the fabric edges, the transverse wave propagates much quicker along the clamped yarns than along the free yarns. Therefore, for the case with two opposite fabric edges clamped, the transverse wave front is in the form of an ellipse, with its long axis along the clamped yarns and its short axis along the free yarns. Fig. 16 illustrates the deformed configurations of the fabric at 40 ms for the three cases. It can be seen from this figure that the fabric boundary condition significantly affects the fabric deformation at later stage of the impact. For the case with four fabric edges left free, large bowings are produced along the four edges; the integrity of the fabric is maintained well and only a few yarns are broken along the periphery of the impact zone. For the case with two opposite fabric edges clamped, bowings are produced along the two free edges; most of the clamped yarns are broken. For the case with four fabric edges clamped, all the principal yarns are broken at the impact zone. As Fig. 13 shows, the fabric most effectively slows down the projectile when all its four edges are left free. The reason for the high performance of the fabric is that only few yarns are broken during the impact when all the four fabric edges are left free. Due to the local failure at the impact zone, the fabric loses capability to decelerate the projectile at later stages of the impact when two or four of its edges are clamped.
399 398 397 396 395 0
2
4
6
8
10
Time (µs) Fig. 17. Time history of the projectile velocity for the case with four fabric edges left free and v ¼ 400 m/s.
local fabric at the impact zone. The projectile velocity drops by 0.5 m/s during the period from 0.3 to 4 ms when the fabric completely loses capability to decelerate the projectile. The fabric fails along the periphery of the impact zone. The local fabric at the impact zone is punched out while most of the fabric does not move transversely. The fabric deformation is very different from that with an impact velocity of 200 m/s (see Fig. 11). The energy transfer between the projectile and the fabric is illustrated in Fig. 19 for the case with v ¼ 400 m/s and four fabric edges left free. It can be seen that yarn kinetic energy is the dominant energy absorption mechanism; it accounts for around 63% of the total absorbed energy. Fig. 20 shows the projectile velocity as a function of time for the three cases with the same impact velocity of 400 m/s but different boundary conditions, while Fig. 21 depicts contour maps of the fabric transverse displacement at 10 ms. It can be seen that the time history of the projectile velocity is the same for all the three cases and the fabric deformed configurations show little difference for the three different boundary conditions. The fabric boundary condition does not take any effect when the impact velocity is 400 m/s. The phenomena are very different from those with an impact velocity of 200 m/s (see Figs. 13 and 16). The fabric energy absorption as a function of time is shown in Fig. 22 for the six cases that have different impact velocities
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Fig. 18. Top and side view of the fabric deformation at various instants of time (v ¼ 400 m/s, four fabric edges left free).
6
400
Energy (J)
5 4
Projectile velocity (m/s)
Loss of projectile kinetic energy Yarn kinetic energy Yarn strain energy Friction dissipated energy
3 2
Four fabric edges left free Two opposite fabric edges clamped Four fabric edges clamped
399 398 397 396
1 395
0 0
2
4
6
8
10
0
Time (µs) Fig. 19. Time history of energy transfer between the projectile and the fabric (v ¼ 400 m/s, four fabric edges left free).
and different fabric boundary conditions. It can be seen from this figure that the fabric responses under 200 and 400 m/s are very different. When the impact velocity is 200 m/s, the fabric boundary condition significantly affects the fabric energy absorption. However, when the impact velocity is 400 m/s, the fabric boundary condition does not have any effect on the fabric energy absorption. Further modeling work shows that the transition takes place at around 300 m/s. The results indicate that fabric boundary condition plays an important role only when the impact velocity is low. When the impact velocity is high enough to
2
4
6 Time (µs)
8
10
Fig. 20. The projectile velocity as a function of time for the three cases that have the same impact velocity of 400 m/s but different boundary conditions.
cause yarns to break instantaneously, the fabric deformation is localized at the impact region and the fabric far field boundary condition does not take any effects. 5. Conclusions A 3D FEA model is created using LS-DYNA to simulate the transverse impact of a rigid RCC projectile on a single layer plain-woven Kevlar fabric. The fabric is modeled to yarn level resolution and relative motion between yarns is allowed. A frictional contact is defined between yarns and
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Fig. 21. Contour maps of the fabric transverse displacement (mm) at 10 ms for the three cases that have the same impact velocity of 400 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
References
Energy absorbed by fabric (J)
7 v=200 m/s, four fabric edges left free v=200 m/s, two fabric edges clamped v=200 m/s, four fabric edges clamped v=400 m/s, all boundary conditions
6 5 4 3 2 1 0 0
10
20
30
40
50
Time (µs) Fig. 22. Time history of the fabric energy absorption for the six cases with different impact velocities and boundary conditions.
between the fabric and the projectile. Three boundary conditions are applied on the fabric: four edges left free; two opposite edges clamped; four edges clamped. Modeling results show that during initial stage of the impact, the projectile velocity drops very quickly. There exists an abrupt momentum transfer from the projectile to the local fabric at the impact zone. When the impact velocity is low, the fabric boundary condition plays an important role. It significantly affects the fabric deformation, stress distribution, energy absorption, and failure modes. The fabric most effectively slows down the projectile when all its four edges are left free. The reason for the high performance of the fabric is that only few yarns are broken during the impact when all the four fabric edges are left free. When the impact velocity is high and causes yarns to break instantaneously, the fabric deformation is localized at the impact region and the fabric far field boundary condition does not take any effects on the fabric ballistic performance. Acknowledgements The support of the US Army Research Laboratory at Aberdeen Proving Ground and the Center for Composite Materials at University of Delaware (UD-CCM) during this research is gratefully acknowledged.
[1] Phoenix SL, Porwal PK. A new membrane model for the ballistic impact response and v50 performance of multi-ply fibrous system. International Journal of Solids Structure 2003;40(24): 6723–65. [2] Jacobs MJN, Van Dingenen JLJ. Ballistic protection mechanisms in personal armour. Journal of Materials Science 2001;36(13): 3137–42. [3] Smith JC, Blandford JM, Schiefer HF. Stress-strain relationships in yarns subjected to rapid impact loading. Part VI: velocities of strain waves resulting from impact. Text Research Journal 1960;30:752–60. [4] Roylance D. Ballistics of transversely impacted fibers. Text Research Journal 1977;47(10):679–84. [5] Morrison C. The mechanical response of an aramid textile yarn to ballistic impact. PhD thesis, University of Surrey, UK, 1984. [6] Field JE, Sun Q. A high speed photographic study of impact on fibers and woven fabrics. In: Garfield B, Rendell J, editors. Proceedings of the 19th international congress on high-speed photography and photonics; 1990. p. 703–12. [7] Wilde AF, Roylance DK, Rogers JM. Photographic investigation of high-speed missile impact upon nylon fabric. Part I: energy absorption and cone radial velocity in fabric. Text Research Journal 1973;43(12):753–61. [8] Wilde AF. Photographic investigation of high-speed missile impact upon nylon fabric. Part II: retarding force on missile and transverse critical velocity. Text Research Journal 1974;44(10):772–8. [9] Briscoe BJ, Motamedi F. The ballistic impact characteristics of aramid fabrics: the influence of interfacial friction. Wear 1992; 158(1–2):229–47. [10] Shim VPW, Tan VBC, Tay TE. Modeling deformation and damage characteristics of woven fabric under small projectile impact. International Journal of Impact Engineering 1995;16(4):585–605. [11] Shockey DA, Erlich DC, Simons JW. Improved barriers to turbine engine fragments. US Department of Transportation, Federal Aviation Administration, DOT/FAA/AR-99/8 III, 2001. [12] Gasser A, Boisse P, Hanklar S. Mechanical behavior of dry fabric reinforcement. 3D simulations versus biaxial tests. Composite Materials Science 2000;17(1):7–20. [13] Martinez MA, Navarro C, Cortes R, Rodriguez J, Sanchez-Galvez V. Friction and wear behaviour of Kevlar fabrics. Journal of Materials Science 1993;28(5):1305–11. [14] Duan Y, Keefe M, Bogetti TA, Cheeseman BA. Modeling friction effects on the ballistic impact behavior of a single-ply high-strength fabric. International Journal of Impact Engineering 2005;31(8): 996–1012. [15] Duan Y, Keefe M, Bogetti TA, Cheeseman CA. Modeling the role of friction during ballistic impact of a high-strength plain-weave fabric. Composite Structure 2005;68(3):331–7. [16] Lim CT, Tan VBC, Cheong CH. Perforation of high-strength doubleply fabric system by varying shaped projectiles. International Journal of Impact Engineering 2002;27(6):577–91.
International Journal of Mechanical Sciences 48 (2006) 33–43 www.elsevier.com/locate/ijmecsci
Finite element modeling of transverse impact on a ballistic fabric Y. Duana,, M. Keefeb, T.A. Bogettic, B. Powersc a
Center for Composite Materials, University of Delaware, Newark, DE 19716, USA Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA c US Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
b
Received 5 August 2004; received in revised form 14 July 2005; accepted 17 September 2005 Available online 26 October 2005
Abstract A 3D finite element analysis model is created using LS-DYNA to simulate the transverse impact of a rigid right circular cylinder onto a square patch of plain-woven Kevlar fabric. The fabric is modeled to yarn level resolution and relative motion between yarns is allowed. A frictional contact is defined between yarns and between the fabric and the projectile. Three different boundary conditions are applied on the fabric: four edges left free; two opposite edges clamped; four edges clamped. Results from the modeling effort show that during initial stage of the impact, the projectile velocity drops very quickly. There exists an abrupt momentum transfer from the projectile to the local fabric at the impact zone. When the impact velocity is low, the fabric boundary condition plays an important role at later stages of the impact. It significantly affects the fabric deformation, stress distribution, energy absorption and failure modes. When the impact velocity is high enough to cause the yarns to break instantaneously, the fabric fails along the periphery of the impact zone and the fabric boundary condition does not take any effects. r 2005 Elsevier Ltd. All rights reserved. Keywords: Ballistic fabric; Transverse impact; Energy absorption; Finite element analysis
1. Introduction Weight and flexibility are two important design parameters for soft body armors that offer protection against bullets and munitions fragments. Fabrics made from highstrength fibers, also called ballistic fabrics, have the attractive properties of low density, high flexibility, high strength-to-weight ratio, and outstanding ballistic resistant property. Therefore, they have been widely used in making soft body armors since their introduction to market. The high-strength fibers used in making ballistic fabrics include aramid (Kevlar, Twaron, Technora), polyethylene (Spectra, Dyneema), and Polybenzoxazole (Zylon) [1,2]. These fibers are essentially elastic in tension and have very high tensile modulus. They have relatively low tensile failure strain and are generally not sensitive to strain rate. Hundreds of the high-strength fibers are grouped together to make a yarn and yarns are woven to produce a single layer ballistic fabric. Corresponding author. Tel.: +1 302 831 0376; fax: +1 302 831 8525.
E-mail address: [email protected] (Y. Duan). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.09.007
Typically, a soft body armor is constructed of a ballistic panel, which is assembled from multiple layers of ballistic fabric, and a carrier made from conventional garment fabric. When a projectile impacts into such a fabric structure, it is gradually slowed down and finally caught by the ballistic panel. The impact energy is converted as fabric kinetic energy, fabric strain energy, projectile deformation energy and energy dissipated in frictional sliding. The impact resistance of a ballistic panel depends on its capability to absorb energy locally at the impact zone and disperse energy quickly out of the impact zone. It is affected by a number of factors, which include fiber density, fiber tensile elastic modulus, fiber tensile failure strain, fabric weave style, fabric areal density, number of fabric layers, fabric boundary condition, projectile shape, mass and material property, impact velocity, and interfacial friction characteristics within impact system. To optimize the design of a soft body armor, in other words, increasing its ballistic resistance and at the same time keeping or even reducing its weight and rigidity, one must understand the impact behavior of the ballistic panel. To understand the impact behavior of the ballistic panel,
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one should first understand the impact behavior of its construction units such as single layer fabric, single yarn, and single fiber. During the past several decades, a lot of experiments and theoretical work have been conducted to understand the transverse impact behavior of single yarns and single layer fabrics [3–11]. Smith et al. [3], Roylance [4], Morrison [5], and Field and Sun [6] studied the response of yarns to high-speed transverse impact while Wilde et al. [7,8], Briscoe and Motamedi [9], Shim et al. [10], and Shockey et al. [11] investigated the transverse impact behavior of single layer fabrics. In this paper, a finite element analysis (FEA) model is created using LS-DYNA to simulate the transverse impact of a right circular cylinder (RCC) on a single layer plainwoven Kevlar fabric. The model allows for definition of contact between yarns and therefore takes into account the physical interaction between yarns. It more realistically describes the woven fabric structure than those computational models in which yarn crossovers are described as links, joints or bulk continuum. 2. Transverse impact on yarns 2.1. Existing theory on transverse impact on a long straight yarn As stated previously, the yarn in a ballistic fabric is composed of hundreds of high-strength fibers. It is a very complex structure. For simplicity in analysis, the interaction between fibers in a yarn is generally ignored and the yarn is assumed to be an elastic continuum. Fig. 1 shows a heavy wedge-tipped projectile transversely impacting on a long straight yarn. The yarn tensile elastic modulus is E and volumetric density is r. The impact velocity is v and it is not high enough to cause the yarn to break. According to Refs. [3,4], two mechanical waves are generated by the impact. One is a longitudinal wave, which propagates away from the impact point at the sound speed of the yarn material. The longitudinal wave speed c is given by sffiffiffiffi E c¼ . (1) r v
time = 0
v
time = t Fig. 1. A wedge-tipped projectile transversely impacts on a long straight yarn at a constant velocity of v.
Ahead of the longitudinal wave front, yarn material strain is zero; behind, a constant tensile strain is developed. The tensile strain, denoted as , is determined by the yarn tensile elastic modulus E, volumetric density r, and the impact velocity v. It is implicitly given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rv2 . (2) ð1 þ Þ 2 ¼ E The other mechanical wave generated by the impact is a transverse one, which propagates away from the impact point at a relatively lower speed. The transverse wave speed u is given by rffiffiffiffiffiffiffiffiffiffiffi . (3) u¼c 1þ
2
Across the transverse wave front, the strain of yarn material does not change; however, the motion of yarn material experiences an abrupt change. Ahead of the transverse wave front but behind the longitudinal wave front, yarn material moves longitudinally toward the impact point. Behind the transverse wave front, yarn material moves transversely in the impact direction. It can be seen from Eq. (3) that the transverse wave speed positively correlates with the yarn tensile strain. It is larger with a larger tensile strain, and vice versa. When the yarn tensile strain is zero, the transverse wave speed is also zero. The yarn kinetic energy E k and the yarn strain energy E s during the impact can be obtained from Eqs. (4) and (5), respectively. sffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 3 2 E E k ¼ At , (4) 32 1þ r sffiffiffiffiffiffi E3 E s ¼ At2 , r
(5)
where, A is the yarn cross-section area and t is the time after impact. The two formulas may be obtained from the above analysis of wave propagations in yarn material. 2.2. Comparison of FEA modeling results and predictions from the existing theory Consider an impact case: a heavy wedge-tipped projectile transversely impacts on a long straight Kevlar yarn that has a tensile elastic modulus E of 74 GPa, a volumetric density r of 1,440 kg/m3 , and a cross-section area A of 5:83 108 m2 . The impact velocity v is 200 m/s. The yarn tensile strain generated by the impact, , can be obtained from Eq. (2) by using iteration method. Substitute the values of E; r; A, and into Eqs. (4) and (5), the yarn kinetic energy and the yarn strain energy at any time t can be obtained. A 3D FEA model is created using LS-DYNA to simulate the impact. Fig. 2 shows the FEA model for the straight Kevlar yarn. The yarn is modeled as a continuum and the yarn cross-section is defined by a pair of symmetric arcs.
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Gasser et al. [12] have shown that an orthotropic elastic continuum has yarn behavior if its Poisson’s ratios are zero and the shear moduli and transverse elastic moduli are very small with respect to the longitudinal elastic modulus. In the FEA model, the yarn has locally orthotropic elastic material property. Table 1 lists the nine orthotropic elastic material data. Each element in the model defines a referential coordinate whose three axes are determined by the nodes of the element. The orthotropic elastic material data is defined in the local referential coordinates. The yarn kinetic energy and the yarn strain energy as a function of time are obtained from the FEA modeling. Fig. 3 shows a comparison of the modeling results and the analytical results obtained from Eqs. (4) and (5). It can be seen from this figure that the FEA modeling results agree well with the analytical results. The good agreement indicates that the FEA modeling approach and the orthotropic elastic material data listed in Table 1 describe well the transverse impact behavior of the Kevlar yarn.
Fig. 2. The 3D FEA model for the straight Kevlar yarn.
Table 1 Orthotropic elastic material data (GPa) for the Kevlar yarn E11
E22
E33
G12
G13
G23
n12
n13
n23
74
0.74
0.74
0.148
0.148
0.148
0
0
0
0.05 Yarn kinetic energy; FEA Yarn kinetic energy; analytical Yarn strain energy; FEA Yarn strain energy; analytical
Energy (J)
0.04 0.03 0.02 0.01 0
0
2
4
6
8
10
12
14
Time (µs) Fig. 3. A comparison of the FEA modeling results and the predictions from theory.
35
2.3. Effect of yarn ends boundary condition In the above analysis and modeling, the Kevlar yarn is assumed to be infinitely long. Therefore, the effect of yarn ends boundary condition is not taken into account. For a real impact situation, the stress/strain wave generated by the impact soon arrives at the yarn ends. The yarn ends boundary condition inevitably plays a role in the yarn impact behavior. To explore the effect of yarn ends boundary condition, two cases are modeled where a heavy wedge-tipped projectile transversely impacts at 200 m/s onto the center of a Kevlar yarn that has a length of 49 mm. In the first case, both the yarn ends are clamped, while in the second case both the yarn ends are left free. It can be seen from Table 1 that the yarn longitudinal elastic modulus is much larger than its shear moduli and transverse elastic moduli. In this situation, the value of the maximum principal stress is very close to that of the tensile stress along fiber direction. For convenient implementation in LS-DYNA, a maximum principal stress failure criterion is used in the modeling. When the maximum principal stress at a material point exceeds 2.3 GPa, the material fails and the corresponding element is deleted automatically from the mesh. The release wave resulted from deleting the element is taken into account in subsequent deformation process. The maximum principal stress failure criterion is equivalent to a maximum tensile strain failure criterion with a failure strain of 3.1%. Fig. 4 shows the yarn deformation when both of its ends are clamped, while Fig. 5 shows the yarn deformation when both of its ends are left free. It can be seen from the two figures that the yarn ends boundary condition significantly affects the yarn deformation. The yarn is broken at the impact point when its two ends are clamped while it is not broken when its two ends are left free. The different yarn behavior is a result of the different yarn ends boundary condition. As stated previously, two mechanical waves are generated by the impact: a longitudinal one and a transverse one. The longitudinal wave propagates away from the impact point at a very high speed while the transverse wave propagates at a relatively lower speed. Behind the transverse wave front, yarn material moves transversely in the impact direction. Ahead of the transverse wave front but behind the longitudinal wave front, yarn material moves longitudinally toward the impact point. When the longitudinal wave reaches the clamped ends, it is reflected back and propagates toward the impact point. Behind the reflected wave front, yarn material stops moving longitudinally and the tensile stress is doubled. The reflected longitudinal waves meet at the impact point where the tensile stress is superimposed on each other. After three reflections of the longitudinal wave, the stress at the impact point reaches the failure criterion and the yarn, as shown in Fig. 4, is broken. Similarly, when the longitudinal wave reaches the free ends, it is also reflected back and propagates toward the impact point.
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Fig. 4. Deformation of the straight Kevlar yarn at various instants of time when both of its ends are clamped; the arrows indicate the transverse wave fronts at 20 ms.
However, behind the reflected wave front, the yarn tensile stress becomes zero and the velocity of the yarn material moving longitudinally toward the impact point is doubled. As can be seen from Fig. 5, with reflection of the longitudinal wave the two ends of the yarn move longitudinally toward the impact point. The transverse wave stops propagating when the reflected longitudinal wave meets the transverse one.
Fig. 5. Deformation of the straight Kevlar yarn at various instants of time when both of its ends are left free; the arrows indicate the transverse wave fronts at 20 ms.
y RCC projectile x
3. Modeling transverse impact on a ballistic fabric Fig. 6 shows the initial geometry of an impact event: a rigid RCC projectile transversely impacts onto the center of a patch of plain-woven Kevlar fabric. The fabric is flat and aligned with the x–z plane. It is composed of 39 yarns in each of the warp direction (along the x-axis) and the weft direction (along the z-axis). The fabric maximum thickness is 0.23 mm, and both of its side length is 32.7 mm. The fabric four edges are left free. The projectile diameter is 8 mm, its mass is 2 g, and its impact velocity v is 200 m/s. During the impact, the rigid projectile can only move along the y-direction and the other five degrees of freedom are constrained. A 3D FEA model is created using LS-DYNA to simulate the aforementioned impact. The impact system has symmetry with respect to both the x–y plane and the y–z plane, therefore only a quarter of the entire system needs to
v f f f
Kevlar fabric
z
Fig. 6. A rigid RCC projectile transversely impacts onto the center of a square patch of plain-woven Kevlar fabric; the impact velocity is v.
be modeled. Fig. 7 shows a part of the 3D FEA model for the plain-woven fabric. The fabric is modeled to yarn level resolution and the yarns are modeled as continuum with locally orthotropic elastic material property. The material properties of the Kevlar yarn have been given in the
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Projectile velocity (m/s)
200 197 194 191 188 185
0
10
20
30
40
50
Time (µs)
Fig. 8. Time history of the projectile velocity for the case with four fabric edges left free and v ¼ 200 m/s. The arrow on the projectile velocity versus time curve indicates the stage of deformation shown in Fig. 9. Fig. 7. The 3D FEA model for the plain-woven Kevlar fabric.
previous section. The cross-section of the crimped yarn is the same as that of the straight yarn shown in Fig. 2. The crimped profiles of the warp and the weft yarns are identical and are defined by a series of connected arcs. The 3D FEA model for the plain-woven fabric defines yarn–yarn contact and allows for relative motion between yarns. Simple Coulomb friction is introduced between yarns and between the projectile and the fabric. A friction coefficient of 0.3, which is obtained from experiments on Kevlar fabrics, is used for both the types of friction [13]. The influence of interfacial friction on fabric impact behavior will not be discussed in this paper. However, it is worth noting that the friction between projectile and fabric, between yarn and yarn, and between fibers themselves might have significant effects on the impact behavior of ballistic fabrics, especially when the projectile is in spherical shape [9,14,15]. In order to comparatively investigate the effect of fabric boundary condition, two additional cases are modeled where all the conditions described previously are maintained except that different boundary conditions are applied on the fabric. In one case, two opposite edges of the fabric are clamped and the other two edges are left free; in the other case, all the four edges of the fabric are clamped. Finally, to explore the effect of impact velocity on the fabric ballistic performance, three cases are modeled where the impact velocity v is 400 m/s instead of 200 m/s and the three different types of boundary conditions described previously are applied on the fabric, respectively. 4. Results and discussion 4.1. Projectile-fabric interaction and energy transfer Fig. 8 shows time history of the projectile velocity for the case with four fabric edges left free and impact velocity v ¼ 200 m/s. It can be seen from this figure that within a very short period of time ð0:3 msÞ, the projectile velocity drops from 200 to 198.3 m/s. Afterwards, the projectile
gradually slows down and at 50 ms its velocity is 188 m/s. Fig. 9 depicts contour maps of the fabric transverse velocity and transverse displacement at 0:3 ms when the projectile velocity is 198.3 m/s. It can be seen from this figure that the local fabric that directly contacts the RCC projectile abruptly moves with the transverse impact. The projectile momentum is transferred to the local fabric. The momentum transfer occurs so quickly that the fabric located out of the impact zone is not affected at all. At this moment, the fabric transverse displacement coincides with the projectile-fabric contact zone and is in a pie shape. The initial momentum transfer is responsible for the abrupt drop of the projectile velocity from 200 to 198.3 m/s. After the initial momentum transfer, the local fabric in the impact zone moves together with the projectile. Due to the sudden transverse motion of the local fabric, a longitudinal wave and a transverse wave are generated in the principal yarns (those yarns that directly contact the projectile). The longitudinal wave propagates away from the impact zone at a very high speed; behind the wave front, yarn material is strained and moves longitudinally toward the impact zone. The transverse wave propagates away from the impact zone at a relatively lower speed; behind the wave front, yarn material moves transversely in the impact direction. Fig. 10 shows contour maps of the fabric resultant displacement at initial stages of the impact. It can be seen from this figure that at 0:5 ms, mainly the principal yarns are affected. With propagation of the mechanical waves, yarn–yarn interactions cause the secondary yarns (those yarns that do not directly contact the projectile) to move. At 1:0 ms, the impact-affected zone takes a square form, and with time going on it gradually expands outward. During the process, the fabric absorbs energy from the projectile and the projectile gradually slows down. At 3:0 ms when most of the fabric has been affected by the impact, the projectile velocity is 197.9 m/s. The projectile velocity is reduced by 1.7 m/s during the initial 0:3 ms while it is only reduced by 0.4 m/s during the period from 0.3 to 3 ms.
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Fig. 9. The fabric deformation at 0:3 ms. (a) Contour map of the fabric transverse velocity ð103 m/s). (b) Contour map of the fabric transverse displacement (mm).
Fig. 10. Contour maps of the fabric resultant displacement (mm) at the initial stage of the impact (v ¼ 200 m/s, four fabric edges left free).
Deformed configurations of the fabric at various instants of time are illustrated in Fig. 11 for the case with four fabric edges left free and impact velocity v ¼ 200 m/s. It can be seen from this figure that during the impact, the fabric at the impact zone conforms to the flat round nose of the RCC projectile. Lim et al [16] have observed by using highspeed camera the conformation of fabric to the nose shape of a flat-nosed projectile during transverse impact. It is found from the deformed configurations of the fabric that two warp yarns and one weft yarn are broken during the impact. Except for the three yarns, no other yarns in the fabric are broken. It can be seen from Fig. 11 that the fabric transverse displacement is in the form of a conical frustum at 10 ms. At that instant of time, the four fabric edges slightly bow toward the impact zone but most of the fabric is not affected by the transverse wave yet. Bowings of the fabric edges indicate that the longitudinal wave has been reflected back from the fabric edges. With time going on, the four fabric edges gradually bow toward the impact
zone and the fabric transverse wave gradually propagates outward. It is noted that the fabric transverse wave front evolves during the impact. It is in the form of a circle at 10 ms while it is in the form of a round-filleted square at 30 ms. The corresponding projectile velocity for each of the deformed configurations of the fabric can be found in Fig. 8. There is no external force acting on the system during the impact. The energy in the system is therefore conserved. The lost projectile kinetic energy is completely absorbed by the fabric. Fig. 12 shows time history of the energy transfer between the projectile and the fabric. It can be seen that at the initial momentum transfer, the projectile loses 0.68 J of kinetic energy, of which 80% is absorbed by the fabric in the form of yarn kinetic energy, 19% is absorbed by the fabric in the form of yarn strain energy, and the remaining 1% is dissipated as heat through friction between the projectile and the fabric and between yarns themselves. It is evident that yarn kinetic energy is the dominant energy
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Fig. 11. Top and side view of the fabric deformation at various instants of time (v ¼ 200 m/s, four fabric edges left free).
4.2. Effect of fabric boundary condition Two additional cases with different fabric boundary conditions (two opposite edges clamped; four edges clamped) are modeled to comparatively study the effect of fabric boundary condition. Fig. 13 shows the projectile velocity as a function of time for the three cases that have the same impact velocity v of 200 m/s but different fabric boundary conditions. It can be seen from this figure that within the initial 5 ms, the projectile velocity is the same for all the three cases. This result indicates that the fabric boundary conditions do not play a role in decelerating the projectile during that period of time. After 5 ms, the fabric boundary conditions take effect and the projectile velocity becomes different for the three cases; the projectile velocity drops most slowly when all the four fabric edges are left free while it drops most quickly when all of the four fabric edges are clamped. It can be seen that at 10 ms, the projectile velocity is 197.0 m/s for the case with four fabric edges left free while it is 196.1 m/s for the case with two opposite fabric edges clamped and 195.1 m/s for the case with four fabric edges clamped. Though the fabric most quickly decelerates the projectile when its four edges are
6 Loss of projectile kinetic energy Yarn kinetic energy Yarn strain energy Friction dissipated energy
5 Energy (J)
absorption mechanism during the impact. At 50 ms, around 67% of the lost projectile kinetic energy is absorbed by the fabric in the form of yarn kinetic energy, while 25% in the form of yarn strain energy and 8% in the form of friction dissipated energy.
4 3 2 1 0
0
10
20
30 Time (µs)
40
50
Fig. 12. Time history of energy transfer between the projectile and the fabric (v ¼ 200 m/s, four fabric edges left free).
clamped, it loses function at the earliest time; the projectile punches through the fabric at 12 ms and moves away at a constant velocity of 194.4 m/s. When two opposite edges of the fabric are clamped, the projectile gradually slows down until 45 ms when the fabric is punched through and loses its capability to decelerate the projectile. As can be seen, the fabric most effectively slows down the projectile when all of its four edges are left free; at 50 ms, the projectile velocity is 188.0 m/s when the four fabric edges are left free while it is 193.1 m/s when two opposite edges of the fabric are clamped and 194.4 m/s when all of the four fabric edges are clamped.
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Fig. 14 shows the maximum principal stress distribution in the fabric at 8 ms for the three impact cases. It can be seen that the maximum principal stress mainly distributes in the clamped principal yarns and it is very small in the unclamped principal yarns or in the secondary yarns. The fabric boundary condition significantly affects the stress distribution pattern in the fabric. Fig. 15 shows contour maps of the fabric transverse displacement at 10 ms for the three cases. As can be seen, the transverse wave front is in the form of a circle when all the four fabric edges are left free, while it is in the form of an ellipse with the long axis
Projectile velocity (m/s)
200
196
192 Four fabric edges left free Two opposite fabric edges clamped Four fabric edges clamped
188
184 0
10
20
30
40
50
Time (µs) Fig. 13. The projectile velocity as a function of time for the three cases that have the same impact velocity of 200 m/s but different boundary conditions.
along the clamped yarns when two opposite fabric edges are clamped and a filleted square when all the four fabric edges are clamped. The different stress distributions and the different transverse displacements are due to the different fabric boundary conditions. As stated previously, with the transverse impact, two mechanical waves originate in the impact zone. The longitudinal wave propagates away from the impact zone at a very high speed while the transverse wave propagates at a lower speed. Behind the longitudinal wave front, yarn material is tensioned and moves longitudinally toward the impact zone, while behind the transverse wave front, yarn material moves transversely. When the longitudinal wave arrives at the free fabric edges, it is reflected back and propagates toward the impact zone. Behind the reflected wave front, the yarn tensile stress fades away and the velocity of the yarn material moving toward the impact zone is doubled; the interactions between yarns gradually produce bowings along the free fabric edges. When the longitudinal wave arrives at the clamped edges, it is also reflected back and propagates toward the impact zone. However, behind the reflected wave front, the yarn tensile stress is doubled and the yarn material stops moving longitudinally toward the impact zone. At 8 ms, the longitudinal wave has been reflected by the fabric edges and propagated back to the impact zone. Therefore, the stress in the clamped principal yarns is much larger than the stress in the unclamped principal yarns and the secondary yarns. As shown by Eq. (3), the transverse wave speed in a yarn is determined by the sound speed of
Fig. 14. Distribution of maximum principal stress (103 GPa) in the fabric at 8 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
Fig. 15. Contour maps of the fabric transverse displacement (mm) at 10 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
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Fig. 16. The fabric deformation at 40 ms for the three cases that have the same impact velocity of 200 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
4.3. Effect of impact velocity Three cases with an impact velocity of 400 m/s and different boundary conditions (four fabric edges left free; two opposite fabric edges clamped; four fabric edges clamped) are modeled to comparatively study the effect of impact velocity. Fig. 17 shows time history of the projectile velocity while Fig. 18 shows the fabric deformed configurations at various instants of time for the case with four fabric edges left free. It can be seen that the projectile is decelerated very quickly at initial stage of the impact. Within 0:3 ms, the projectile velocity drops from 400 to 396.7 m/s. The abrupt drop of the projectile velocity is due to the initial momentum transfer from the projectile to the
400 Projectile velocity (m/s)
the yarn material and the tensile strain in the yarn. It is higher with a larger strain and lower with a smaller strain. Due to reflection of the longitudinal wave from the fabric edges, the transverse wave propagates much quicker along the clamped yarns than along the free yarns. Therefore, for the case with two opposite fabric edges clamped, the transverse wave front is in the form of an ellipse, with its long axis along the clamped yarns and its short axis along the free yarns. Fig. 16 illustrates the deformed configurations of the fabric at 40 ms for the three cases. It can be seen from this figure that the fabric boundary condition significantly affects the fabric deformation at later stage of the impact. For the case with four fabric edges left free, large bowings are produced along the four edges; the integrity of the fabric is maintained well and only a few yarns are broken along the periphery of the impact zone. For the case with two opposite fabric edges clamped, bowings are produced along the two free edges; most of the clamped yarns are broken. For the case with four fabric edges clamped, all the principal yarns are broken at the impact zone. As Fig. 13 shows, the fabric most effectively slows down the projectile when all its four edges are left free. The reason for the high performance of the fabric is that only few yarns are broken during the impact when all the four fabric edges are left free. Due to the local failure at the impact zone, the fabric loses capability to decelerate the projectile at later stages of the impact when two or four of its edges are clamped.
399 398 397 396 395 0
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Time (µs) Fig. 17. Time history of the projectile velocity for the case with four fabric edges left free and v ¼ 400 m/s.
local fabric at the impact zone. The projectile velocity drops by 0.5 m/s during the period from 0.3 to 4 ms when the fabric completely loses capability to decelerate the projectile. The fabric fails along the periphery of the impact zone. The local fabric at the impact zone is punched out while most of the fabric does not move transversely. The fabric deformation is very different from that with an impact velocity of 200 m/s (see Fig. 11). The energy transfer between the projectile and the fabric is illustrated in Fig. 19 for the case with v ¼ 400 m/s and four fabric edges left free. It can be seen that yarn kinetic energy is the dominant energy absorption mechanism; it accounts for around 63% of the total absorbed energy. Fig. 20 shows the projectile velocity as a function of time for the three cases with the same impact velocity of 400 m/s but different boundary conditions, while Fig. 21 depicts contour maps of the fabric transverse displacement at 10 ms. It can be seen that the time history of the projectile velocity is the same for all the three cases and the fabric deformed configurations show little difference for the three different boundary conditions. The fabric boundary condition does not take any effect when the impact velocity is 400 m/s. The phenomena are very different from those with an impact velocity of 200 m/s (see Figs. 13 and 16). The fabric energy absorption as a function of time is shown in Fig. 22 for the six cases that have different impact velocities
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Fig. 18. Top and side view of the fabric deformation at various instants of time (v ¼ 400 m/s, four fabric edges left free).
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Energy (J)
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Loss of projectile kinetic energy Yarn kinetic energy Yarn strain energy Friction dissipated energy
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Four fabric edges left free Two opposite fabric edges clamped Four fabric edges clamped
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Time (µs) Fig. 19. Time history of energy transfer between the projectile and the fabric (v ¼ 400 m/s, four fabric edges left free).
and different fabric boundary conditions. It can be seen from this figure that the fabric responses under 200 and 400 m/s are very different. When the impact velocity is 200 m/s, the fabric boundary condition significantly affects the fabric energy absorption. However, when the impact velocity is 400 m/s, the fabric boundary condition does not have any effect on the fabric energy absorption. Further modeling work shows that the transition takes place at around 300 m/s. The results indicate that fabric boundary condition plays an important role only when the impact velocity is low. When the impact velocity is high enough to
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Fig. 20. The projectile velocity as a function of time for the three cases that have the same impact velocity of 400 m/s but different boundary conditions.
cause yarns to break instantaneously, the fabric deformation is localized at the impact region and the fabric far field boundary condition does not take any effects. 5. Conclusions A 3D FEA model is created using LS-DYNA to simulate the transverse impact of a rigid RCC projectile on a single layer plain-woven Kevlar fabric. The fabric is modeled to yarn level resolution and relative motion between yarns is allowed. A frictional contact is defined between yarns and
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Fig. 21. Contour maps of the fabric transverse displacement (mm) at 10 ms for the three cases that have the same impact velocity of 400 m/s but different boundary conditions. (a) Four fabric edges left free. (b) Two opposite fabric edges clamped. (c) Four fabric edges clamped.
References
Energy absorbed by fabric (J)
7 v=200 m/s, four fabric edges left free v=200 m/s, two fabric edges clamped v=200 m/s, four fabric edges clamped v=400 m/s, all boundary conditions
6 5 4 3 2 1 0 0
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Time (µs) Fig. 22. Time history of the fabric energy absorption for the six cases with different impact velocities and boundary conditions.
between the fabric and the projectile. Three boundary conditions are applied on the fabric: four edges left free; two opposite edges clamped; four edges clamped. Modeling results show that during initial stage of the impact, the projectile velocity drops very quickly. There exists an abrupt momentum transfer from the projectile to the local fabric at the impact zone. When the impact velocity is low, the fabric boundary condition plays an important role. It significantly affects the fabric deformation, stress distribution, energy absorption, and failure modes. The fabric most effectively slows down the projectile when all its four edges are left free. The reason for the high performance of the fabric is that only few yarns are broken during the impact when all the four fabric edges are left free. When the impact velocity is high and causes yarns to break instantaneously, the fabric deformation is localized at the impact region and the fabric far field boundary condition does not take any effects on the fabric ballistic performance. Acknowledgements The support of the US Army Research Laboratory at Aberdeen Proving Ground and the Center for Composite Materials at University of Delaware (UD-CCM) during this research is gratefully acknowledged.
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