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Analytical modeling for the ballistic perforation of planar plain-woven fabric target by projectile
Composites: Part B 34 (2003) 361–371 www.elsevier.com/locate/compositesb
Analytical modeling for the ballistic perforation of planar plain-woven fabric target by projectile Bohong Gu* College of Textiles, Dong Hua University, Shanghai 200051, People’s Republic of China Received 1 March 2003; accepted 23 November 2003
Abstract This paper presents an analytical model to calculate decrease of kinetic energy and residual velocity of projectile penetrating targets composed of multi-layered planar plain-woven fabrics. Based on the energy conservation law, the absorbed kinetic energy of projectile equals to kinetic energy and strain energy of planar fabric in impact-deformed region if deformation of projectile and heat generated by interaction between projectile and target are neglected. Then the decrease of kinetic energy and residual velocity of projectile after the projectile perforating multi-layered planar fabric targets could be calculated. Owing to fibers in fabric are under a high strain rate state when fabric targets being perforated by a high velocity projectile, the mechanical properties of the two kinds of fibers, Twaronw and Kuralonw, respectively, at strain rate from 1.0 £ 1022 to 1.5 £ 103 s21, are used to calculate the residual velocity of projectile. It is shown that the mechanical properties of fibers at high strain rate should be adopted in modeling rate-sensitivity materials. Prediction of the residual velocities and energy absorbed by the multi-layered planar fabrics show good agreement with experimental data. Compared with other models on the same subject, the perforating time in this model can be estimated from the time during which certain strain at a given strain rate is generated. This method of time estimation is feasible in pure theoretical modeling when the perforation time cannot be obtained from experiments or related empirical equations. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Fabrics; Ballistic perforation; Strain energy; Strain rate
1. Introduction Composite materials are widely used in personal and vehicle armors because of their lightweight and high bulletproof performance. But it is difficult to simulate numerically or analytically the ballistic impact behaviors of composite material because of the complexity of damage mechanism. The first step and valid way to model behaviors of composite under ballistic impact could be the study of the primary components of the composites. The fabric is undoubtedly the most important one, and an analytical model of multi-layered planar fabric based on strain rate effect of fibers is developed in this paper to calculate the decrease of kinetic energy and residual energy of projectile penetrating planar fabric target. It is complex and interesting to study the ballistic penetration and perforation of fabrics. Cunniff [1], ChocronBenloulo et al. [2], Navarro [3] and Lee et al. [4] published * Tel.: þ86-21-62373456; fax: þ86-21-62193061. E-mail address: [email protected] (B. Gu).
concise surveys of the analytical models of penetration of projectiles into woven fabrics, which covered the major works that had been published so far. Hearle et al. [5] and Leech et al. [6] analyzed the ballistic impact mechanics of other forms of fiber assemblies, such as tri-axial woven fabrics and knitting fabrics. Up to now, many theoretical models on fiber assemblies under ballistic impact were established. They could be divided into two kinds. One is finite element model or finite difference model that is based on variational principle [6 –9]. As pointed out by Cunniff [1], conclusions in these models are often contrary to experiments because of the lack of constitutive properties and failure criterion for armor materials. The other is simple analytical model by intuitive approach through qualitative understanding of impact dynamics of less complex fabric and applied to more complex systems [2,10 –12], but only the breakage of principal yarns contacted with projectile in woven fabric is concerned in these models, the absorbed energy by other yarns and kinetic energy of fabric in deformed region is not considered.
1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-8368(02)00137-3
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Nomenclature 1 10 s 1_ 1ins 1max n, ns nr c r Ulab u
strain in yarn strain of principal yarn stress in yarn strain rate instant strain in yarn caused by longitudinal wave failure strain of yarn impact velocity of projectile residual velocity of projectile longitudinal wave velocity in yarn half of yarn length between two constrained points transverse wave velocity in yarn relative to the laboratory velocity of transverse wave front relative to points on the unstrained yarn
The damage mechanism of fiber assemblies under ballistic impact could be illustrated by these models to some extent, however, the most important factor, the strain rate effect on mechanical properties of fibers, was neglected in all these models. Because the strain rate of materials under ballistic impact is often greater than 102 s21, an adequate understanding of the rate-dependence of the mechanical properties of the materials involved is required if problems associated with ballistic impact are to be properly analyzed. Our purposes here are to put forward analytical model of multi-layered planar fabric under ballistic perforation and to calculate the decrease of kinetic energy of projectile. The energy absorbed by all yarns in rhombic region when fabrics are deformed into tetrahedron shape under ballistic impact, and especially the mechanical properties of yarn at high strain rate, are taken into consideration. The determination of failure criterion is based on energy conservation law and maximum strain criterion. Finally the predictions computed and the experimental results of residual velocities of projectiles were compared.
E Tex
r m S nw, nF t Dt WKY WSY Wi WSF WKF
Young’s modulus of yarn linear density of yarn (In textile engineering: 1 Tex ¼ 1 g/1000 m) volume density of yarn mass of projectile cross-section area of yarn warp and filling (weft) density of fabric, respectively (per 10 cm) time time elapsed in ballistic perforation kinetic energy of yarn strain energy of yarn strain energy of ith yarn strain energy of fabric in rhombus region kinetic energy of fabric in rhombus region
2. Description of planar fabric deformation under normal ballistic impact Planar woven fabrics are composed of two kinds of yarns that are perpendicular to each other, named weft yarn (filling yarn) and warp yarn, respectively. Fabric deformation under normal ballistic impact has been discussed in several papers [6,13]. Leech et al. [6] predicted that, for an orthogonally woven fabric, such as planar plain-woven fabric, the transverse wave front is a rhombus with sides given by the following sets of equations t¼^
x y ^ ux uy
ð1Þ
where x and y are the spatial co-ordinates of the front at time t, and ux, uy are the propagation speeds of transverse signals in the x and y directions, respectively (Fig. 1). For plainwoven fabric in which the diameter of warp yarns equals to that of weft yarns, it can be deduced that ux ¼ uy : This wave front is also confirmed by our experimental observations.
Fig. 1. Three-dimensional tetrahedron shape and 2-dimensional rhombus region of fabric impacted by projectile.
B. Gu / Composites: Part B 34 (2003) 361–371
In Fig. 1 the yarn number n is: n ¼ ux DtnW
or
n ¼ uy DtnF
ð2Þ
From observations of fabric samples after ballistic perforation, it was found that only the yarns contacted with conically cylindral projectile (X and Y-axis in Fig. 1) would be broken, other yarns in this rhombus region would not be fractured. Because the friction between projectile and target, and the deformed energy of projectile could be regarded as negligible when the fabric target is perforated by perfectly rigid projectile [12,14], the strain energy of fabric target could be divided into two parts: one is the fracture work of yarns (which are referred as principal yarns) contacted with projectile, the other is elastic strain energy and plastic strain energy of yarns in rhombus region except principal yarns. Here the dissipation of heat energy generated in the whole process of ballistic penetration or perforation is neglected. When the sum of the two kinds of energies absorbed by fabric is calculated, the decrease of kinetic energy of projectile could be obtained based on the energy conservation law.
3. Kinetic energy absorbed by principal yarns 3.1. Unconstrained yarns
363
the yarn) is constant and can be calculated as follows [15,16] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi v ¼ c 21 1ð1 þ 1Þ 2 12 ð3Þ where n is the impact velocity of the projectile, pffiffiffiffiffi c is longitudinal wave velocity in the yarn and c ¼ E=r (E is the Young’s modulus at high strain rate and r is the volume density of the yarn). The transverse wave velocity with respect to the laboratory is pffiffiffiffiffiffiffiffiffiffiffi ulab ¼ cð 1ð1 þ 1Þ 2 1Þ ð4Þ and the angle between the line of impact and the yarn is given by 1 v ð5Þ 1þ1 u pffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ c ð1=1 þ 1Þ; u is the velocity of transverse wave front relative to points on the strained yarn (i.e. u is a Lagrangian wave speed). Because the yarn is a flexible material, the ballistic impact damage of the yarn could be attributed to tensile failure. ÐThe tensile strain energy stored in per cubic meter yarn is 10max sð1Þd1: Then the strain energy of infinite long yarn under ballistic impact after time Dt is: ð1ins WSY ¼ ScDt sð1Þd1 ð6Þ cos u ¼
0
The behavior of principal yarns under normal impact is familiar with that of long yarns under transverse impact. The theory of single yarn under transverse impact is fairly well developed by Smith in series of papers [15,16]. The theory indicates that the strain developed in an infinite long yarn under transverse impact is dependant on yarn constitutive properties and impact velocity, the magnitude of strain in the wake of the longitudinal wave is constant in the yarn for a linear-elastic materials. As shown in Fig. 2, the projectile has traveled a distance Dy after Dt. If the yarn is linearly elastic, ahead of the elastic wave front, the strain is zero, and behind, the strain 1 (generated in yarn at the instance when the projectile contacted with
If the instant strain 1ins in the principal yarn was greater than the failure strain 1max, the principal yarn would be perforated. The decrease of kinetic energy of projectile is equal to the strain energy of the principal yarn and the kinetic energy of the yarn in the transverse wave region, i.e. 1 mðv2s 2 v2r Þ 2 " ¼ 2 ScDt
ð1max 0
1 ðDt ulab t sð1Þd1 þ Tex ðvðtÞÞ2 dt 2 0 sin u
#
ð7Þ Because the v(t) is not known, here we suppose that vðtÞ ¼ ðvs þ vr Þ=2 for simplicity, i.e. v(t) is a linear descending function. If the instant strain 1ins in the principal yarn is less than the failure strain 1max, the principal yarn would not be broken. The projectile would be arrested by the principal yarn. From Eq. (6), the strain energy of principal yarns is proportional to the cross-section area and Young’s modulus of the yarn. The bigger the cross-section area and Young’s modulus, are the more kinetic energy absorbed would be. 3.2. Constrained yarn
Fig. 2. Single yarn impacted by projectile.
In this case, the behavior of yarn is different with yarn of infinite length. The most distinct feature of unconstrained is
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that the strain would not increase from 1ins as the time elapsing because the strain spread to other part of yarn at the speed of longitudinal wave. When the yarn specimen is clamped at two points equidistant from the point of impact, reflections of the longitudinal strain wave will alternately transfer between clamping points and impact point. After each reflection the local strain increases by small increments. The strain in transverse wave region will increase to the maximum strain when the projectile is arrested or rupture strain is attained. At a given striking velocity vs, the 1ins is a constant for given yarn. As for constrained yarn, the strain of yarn in transverse wave region would be increased from 1ins to 1max. This is shown in Fig. 3. In Fig. 3(a), when cDt # r; this case is the same as unconstrained yarn.
In Fig. 3(b), when cDt . r and Ulab Dt # r; the strain of yarn outside of transverse wave region is the superposition of 1ins and reflection strain 1reflect, the strain of yarn within transverse wave region would be increased from 1ins to 1max because of tensile deformation. In Fig. 3(c), when Ulab Dt . r; the whole yarn is under tensile state. The strain energy of the constrained yarn fractured at different time could be written as: WSY ¼ 2cDtS WSY ¼ 2rS
ð1max
sð1Þd1;
0 , Dt #
0
ð1ins þ1reflect
sð1Þd1 þ 2uDtS
ð1max
0
sð1Þd1;
1ins
0
r r , Dt # c Ulab ð1max WSY ¼ 2rS sð1Þd1;
r c
ð8Þ
r , Dt Ulab
The kinetic energy of the constrained yarn fractured at different time could be written as: ðDt u t vs þ vr 2 lab Tex dt; sin u 2 0 r 0 , Dt # Ulab ðDt r vs þ vr 2 WKY ¼ Tex dt; sin u 2 0
WKY ¼
ð9Þ r , Dt Ulab
4. kinetic energy absorbed by single layer planar fabric 4.1. modeling of crossed nonwoven yarns
Fig. 3. The shape of constrained yarn under normal impact at different time; (a) constrained yarn under normal impact at time t ¼ 0; (b) constrained yarn under normal impact at time t # r=c; (c) constrained yarn under normal impact at time r=c , t # r=Ulab ; (d) constrained yarn under normal impact at time r=Ulab , t:
The phenomena of strain wave propagation and spread in fabric are more complex than those in yarn when fabric is under ballistic impact because of wave transmission and reflection at interweave point. The following hypotheses are made in order to simplify modeling. (1) Planar plain-woven fabric are composed of crossed nonwoven yarns perpendicular to each other; (2) longitudinal waves only spread in principal yarn; (3) transverse wave in principal yarn only spread to perpendicular yarns, not spread to parallel yarns. From these hypotheses, the crossed nonwoven yarns would be deformed as ‘pyramid’ shape under ballistic impact. This is also shown in Fig. 1. In Fig. 1, points on the principal yarns within the transverse deflection appear to be moving faster than adjacent points on corresponding parallel yarn in Z direction. This means that yarn closer to the principal yarn is under greater strain. The strain in yarn of rhombus region except principal yarns could be simply calculated by linear
B. Gu / Composites: Part B 34 (2003) 361–371
interpolation from zero to principal yarn strain. i 1i ¼ 1 2 i ¼ 0; 1; …; n 1; n 0 Then the strain energy of ith yarn in Fig. 1 is: i ð1i r sð1Þd1; 0,t# Wi ¼ SuDt 1 2 n Ulab 0 ð1i r Wi ¼ Sri Dt sð1Þd1; ,t Ulab 0
ð10Þ
n X
Wi
ð11Þ
ð12Þ
i¼1
The length of ith yarn in transverse wave region in Fig. 1 is i Dl; i ¼ 0; 1; 2; …; n ðDlÞi ¼ 1 2 n where Dl is the length of principal yarn in transverse wave region in Fig. 1. The sum of kinetic energy of yarns from 1 to n in Fig. 1 is: n X Dli vs þ vr 2 Tex WK1 ¼ 4 sin u 2 i¼1 ¼4
n ulab Dt v þ vr 2 X i Tex s 12 sin u n 2 i¼1
The kinetic energy of principal yarn is: u Dt v þ vr 2 WK2 ¼ lab Tex s sin u 2
the whole multi-layered fabric target under ballistic impact is equal to the product of absorbed energy of single layer fabric and the number of layers.
5. determination of time interval of ballistic perforation
From Eqs. (8) and (10), the whole strain energy of rhombus region including two principal yarns is: WS ¼ 2WSY þ 4
365
ð13Þ
The time interval Dt, which from the beginning of impact to the instant when the yarn is broken or the projectile is stopped, is a key to solve equations related to Dt. Only the Dt is known, could the kinetic energy of projectile absorbed by fabric target be calculated. But this problem is not solved in a reasonable way in all other ballistic penetration models. Experimentally, the time interval Dt could be determined precisely by high-speed photographs from ballistic impact. In theoretical consideration, this method is not feasible because it losses theoretical purism. Furthermore, it is a circular argument if we adopt Dt from high-speed photographs or any other empirical formulae from experiments. Since the experiments have been finished, is it necessary to model the experiment and to get a prediction? The method to estimate Dt in this paper is from definition of strain rate. The strain rate concerning with ballistic penetration is at the range of 5 £ 102 – 15 £ 102 s21 and depends on the projectile velocity [18]. Because the strain rate is relatively more stable in ballistic perforation than in penetration (In the case of penetration, the projectile could be arrested. The strain rate is changed in wide range.), the time to failure Dt can be calculated as follows: Dt ¼
ð14Þ
Then the total kinetic energy WK of yarns in deformed region could be obtained from Eqs. (13) and (14).
D1 1_
ð16Þ
Eq. (16) is an approximate method to estimate Dt. In theoretical way, there are no better method to get a more precise result than Eq. (16).
4.2. modeling of single layer plain-woven fabric
6. Tensile tests of yarn at various strain rates
The method of modeling plain-woven fabric from perpendicular crossed nonwoven yarns herein is the same as in Refs. [12,17]. It is assumed that the longitudinal wave pffiffi velocity in single yarn should be divided by 2 in actual calculation, because the linear density of yarn along which the wave propagates is effectively doubled in a plain-woven fabric. So far, the total energy W of single layer fabric could be obtained by above-mentioned method and Eqs. (12) – (14), i.e.
The most probable failure modes of fabric under ballistic impact are tensile failure and shear failure of fibers at an adiabatic state. Due to the flexibility of fibers, the tensile failure is the main failure mode in ballistic impact. It is very easy to test the tensile property of fibers at low strain rates (corresponding to the low velocity tensile test), but owing to technical difficulties in test, there are no detailed data regarding the fibers’ tensile behavior at moderate or high strain rates. Kawata [19] and Harding [20] obtained the stress – strain curves of fabric reinforced composites and unidirectionally reinforced CFRP at the strain rate of 1 £ 103 s21 almost at the same time in 1982 by selfdesigned apparatus. It was the starting point of studying the mechanical properties of fiber-reinforced composites at high strain rate. In 1990, Xia et al. [21] first successfully performed tensile impact test on fiber bundles. More recently, a statistical model of rate-dependence of E-glass
W ¼ WS þ WK ¼
1 2
mðv2s 2 v2r Þ
ð15Þ
Then the residual velocity vr of projectile could be calculated. As for multi-layered fabrics, because the friction between fabric layers could be neglected in the analysis of ballistic impact [12], the strain energy and kinetic energy of
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Fig. 4. Schematic diagram of the fiber bundles specimen 1-supplement plate, 2-lining block, 3-fiber bundles, 4-input bar, 5-output bar.
fiber bundles and kevlar fiber bundles was set up [22 – 24], and strain rate-dependence of mechanical behavior of these fiber bundles was also discussed. The method developed by Xia et al. [21] is adopted in this paper to test tensile properties of fibers at high strain rate. 6.1. Tensile tests at low strain rate (quasi-static tensile test) The quasi-static tensile tests were performed in a MTS810.23 testing machine at strain rate 1022 s21. 6.2. Tensile tests at high strain rate 6.2.1. Experimental apparatus Tensile impact tests were conducted in rotating disk bar – bar tensile impact apparatus that is similar to split Hopkinson tension bar. This apparatus was designed by the University of Science and Technology of China. The structure of this apparatus and experimental measuring principle could be seen in Ref. [25].
Fig. 6. Typical strain gauge signals (Kuralonw).
Nobel) and Kuralonw PVA filaments (Kuralonw 7901-1, 2000 dtex/1000f, manufactured by Kuraray of Japan). The schematic diagram of the filament specimen was shown in Fig. 4. ls was the specimen gauge length, which was 8.0 mm. The number of filaments in the specimen was 18 in total. The specimens were tested at room temperature and approximately 55% relative humidity. The results at each strain rate are derived from the average values of at least three experiments. Typical experimental waves of Twaronw aramid filaments and Kuralonw PVA filaments under tensile impact loading are shown in Figs. 5 and 6, respectively, in which the X-axis gives the time and the Y-axis gives the strain signal measured by strain gauges on the input bar and output bar. 6.3. Results
6.2.2. Specimens in experiment The test materials were Twaronw aramid filaments (type: Twaronw CT1000, 1680 dtex/1000f, manufactured by Akzo
Fig. 7 shows the stress – strain curves of Twaronw fiber bundles under different loading conditions. The strain rates are 1022, 180, 480, 1000 s21, respectively. Table 1 lists the average values of E, smax and 1max. Fig. 7 and Table 1 indicate that Twaronw fiber bundles have sensitivity to strain rate. E, smax and 1max increase with increasing strain
Fig. 5. Typical strain gauge signals (Twaronw).
Fig. 7. Tensile curves of Twaronw filaments at different strain rate.
B. Gu / Composites: Part B 34 (2003) 361–371
367
Table 1 Mechanical properties of Twaronw fiber bundles versus strain rate
Table 2 Mechanical properties of Kuralonw fiber bundles versus strain rate
Strain rate (s21)
E (GPa)
smax (GPa)
1max (%)
Strain rate (s21)
E (GPa)
smax (GPa)
1max (%)
1022 180 480 1000
62 69 70 72
2.395 2.596 2.704 2.753
5.19 5.22 5.47 5.70
1022 270 600 1500
20.3 49.7 52.6 51.2
1.19 1.50 1.61 1.85
9.89 4.70 4.93 5.97
rate from 1022 to 1000 s21. Because E increases significantly from the strain rate of 1022 to 1000 s21, it is obviously not reasonable if we introduce the Young’s modulus E at quasi-state to the analytical model. Fig. 8 shows the stress – strain curves of Kuralonw fiber bundles under different strain rates. The strain rates are 1022, 270, 600, 1500 s21, respectively. Table 2 lists the average values of E, smax and 1max. Fig. 8 and Table 2 indicate that Kuralonw fiber bundles have sensitivity to strain rate. smax and 1max increase with increasing strain rate from 1022 to 1500 s21. The Young’s modulus E increases with increasing strain rate from 0.01 to 600 s21. But the Young’s modulus at strain rate of 1500 s21 is lower than that of 600 s21. It implies that PVA fibers appear to undergo a ductile –brittle – ductile transition strain rate range from 0.01 to 1500 s21 [26]. The Young’s modulus E at quasistate is far less than that of high strain rate. It is also obviously not reasonable if we introduce the Young’s modulus E at quasi-state to the analytical model.
7. Ballistic perforation tests of multi-layered planar fabric target Tests were conducted to study the energy absorbed capacity of fabric target under ballistic perforation and to verify numerical predictions [27]. Conically cylindral steel projectile of 7.62 mm diameter and 7.95 g, was used in ballistic test. Strike velocities of projectiles were controlled by adjusting the weight of gunpowder. It was observed from
the experiments that the deformation of projectile could be neglected. The deformation energy of projectile could also be ignored in the analytical model. Circular target plates of different fabric layers were clamped between two steel rings of 60 mm inner radius. Striking and residual velocities were measured, respectively, by two laser-diode pairs. Two kinds of multi-layered planar plain fabric targets, which were made of Twaronw and Kuralonw fibers, respectively, were perforated by conically cylindral steel projectile at different strike velocities. The specifications of Twaronw and Kuralonw plain-woven fabric are listed in Table 3. Experimental results are shown in Table 4.
8. Comparisons between theoretical and experimental results 8.1. Algorithm The strain energy stored in per cubic meter yarn is equal Ð to 10 sð1Þd1: The value of this integral could be obtained from numerical integral in experimental stress – strain curve of yarn. Because the data of stress and strain of yarn were sampled by computer at the time interval of 1026 s, then the value of this integral could be obtained by numerical integral methods. Another way to obtain the value of this integral is relatively simpler. The yarn was presumed as a linear-elastic material in above-mentioned modeling, the constitutive equation of yarn is sм E1: We could substitute this constitutive equation in 10 sð1Þd1: Then Eqs. (8) and (11) could be rewritten as: WSY ¼ cDtSEð1max Þ2 ;
0 , Dt #
r c
ð8aÞ
WSY ¼ rSEð1ins Þ2 þ uDtSEð1max 2 1ins Þ2 ; r r , Dt # c Ulab WSY ¼ rSEð1max Þ2 ;
r , Dt Ulab
1 i 3 ð10 2 1ins Þ2 ; Wi ¼ SuDtE 1 2 2 n Fig. 8. Tensile curves of Kuralonw filaments at different strain rate.
r 0,t# Ulab
ð11aÞ
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B. Gu / Composites: Part B 34 (2003) 361–371
Table 3 Specifications of Twaronw and Kuralonw plain-woven fabric Fabric type
Linear density (dtex)
Twaronw Kuralonw
Wi ¼
Weave density (per 10 cm)
Strength at break (N cm21)
Warp
Weft
Warp
Weft
Warp
Weft
3360 2000
3360 2000
67 198
66 194
3070 1486
3138 771
1 i 2 Sri DtE 1 2 ð10 2 1ins Þ2 ; 2 n
r ,t Ulab
Eqs. (8a) and (11a) are used in calculation program of this paper. 8.2. Numerical calculations Following parameters of Twaronw and Kuralonw fiber, plain-woven fabrics are substituted into the analytical model. 1. Cross-section area of yarn S (Twaronw: S ¼ 2:33 £ 1027 m2 ; Kuralonw: S ¼ 1:49 £ 1027 m2 ) 2. Volume density of yarn r (Twaronw: r ¼ 1:44 g=cm3 ; Kuralonw: r ¼ 1:34 g=cm3 ) 3. Tensile property of Twaronw and Kuralonw yarn at different strain rate (Tables 1 and 2) 4. Time interval Dt (the strain rate with maximum probability in ballistic impact is 5 £ 102 –15 £ 102 s21
Areal density (kg m22)
Construction
0.4824 0.4014
Plain-woven Plain-woven
[18]). Here we determined that the time interval begins at the projectile contacting with yarn and end in the time of yarn fracture, namely Dt ¼ ð1max 2 1ins Þ=1000 for Twaronw and Dt ¼ ð1max 2 1ins Þ=1500 for Kuralonw. The residual velocities of projectiles perforating the Twaronw and Kuralonw planar fabric targets at different time interval by two kinds of algorithms are given in Tables 5 and 6, respectively. The quasi-static tensile property of Twaronw and Kuralonw are also adopted in calculation to illustrate the necessity of introducing rate-dependent effect of materials. 8.3. comparison between experimental and theoretical results The decrease of kinetic energy of projectile versus area density, in experimental and theoretical, could be shown in Figs. 9 and 10. From Tables 4 – 6 and Figs. 9 and 10, it could be concluded that there are agreement between
Table 4 Experimental results of multi-layered Twaronw and Kuralonw fabrics under ballistic impact Target type Twaronw
Kuralonw
Target layers
Target area density (kg m22)
5
2.412
10
4.824
15
7.236
20
9.648
25
12.060
30
14.472
6
2.462
12
4.925
24
9.850
30
12.310
36
14.770
Strike velocity (m s21)
Residual velocity (m s21)
Kinetic energy absorbed (J)
341 375 362 404 361 363 341 367 369 379 378 350
323 358 334 382 327 326 293 319 311 324 312 278
47.51 49.53 77.46 68.74 92.98 101.33 120.97 130.89 156.77 153.69 181.02 179.73
338 374 372 360 337 342 363 397 374 386
331 367 355 343 299 299 315 353 316 333
18.61 20.62 49.13 47.51 96.07 109.56 129.36 131.18 159.08 151.48
B. Gu / Composites: Part B 34 (2003) 361–371
369
Table 5 Calculated results of multi-layered Twaronw fabrics under ballistic impact Layers
Strain rate 1000 s21 Vr (m s21)
480 s21 W (J)
Vr (m s21)
0.01 s21 W (J)
Vr (m s21)
W (J)
5
327.1 363.7
36.9 33.8
329.5 365.1
30.7 29.1
336.3 371.3
12.7 11.0
10
338.8 386.5
64.6 55.0
344.1 389.5
50.2 45.7
353.9 398.2
23.1 18.5
15
327.1 329.3
92.7 92.7
335.6 337.8
70.3 70.2
348.8 351.0
34.4 34.1
20
291.5 326.1
124.5 112.7
304.0 335.2
94.9 88.8
322.4 351.3
49.0 44.8
25
318.8 331.8
137.2 133.4
330.4 342.3
107.3 105.2
349.7 361.0
55.1 52.9
30
321.6 279.5
156.8 176.4
334.5 298.4
123.2 133.0
356.3 323.7
63.3 70.4
experimental and theoretical results. From the comparison in Tables 5 and 6, the accuracy of the analytical model at high strain rate is better than that at low strain rate because the yarns in fabric are under high strain rate in ballistic impact.
9. Discussions (1) The shape of projectile is not considered in the analytical model; instead, the projectile is simplified as a particle in the analysis. The factors, such as cone angle, diameter and length of projectile could affect the absorbed kinetic energy of projectile in the ballistic perforation. It is Table 6 Calculated results of multi-layered Kuralonw fabrics under ballistic impact Layers
Fig. 9. Comparison between kinetic energy absorbed of projectile and strain energy of fabric (Twaronw).
also shown that there are yarn breakages at the impact point and yarn slippage around the projectile in perforated fabric target. If the bullet shape is considered, the friction between bullet and fabric target must be introduced in analytical model. At the same time, the friction between warp yarns and weft yarns should also be introduced when yarn slippage is considered. (2) The mechanical properties of yarn at quasi-static state and high strain rate state were used in numerical calculation of the analytical model. The results of high strain rate are closer to experimental results because the yarns of fabric are under high strain rate state in ballistic impact. The reasons for calculated results of Kuralonw at quasi-static state closer to experimental results are that Kuralonw are ductile materials at low and high strain rate, and brittle materials at medium strain rate [26]. (3) The prediction errors, especially the errors between strain energy, kinetic energy of fabric target and the decrease of kinetic energy of the projectile are probably caused by the following factors. (i) The number of principal
Strain rate 1500 s21 Vr (m s21)
270 s21 W (J)
Vr (m s21)
0.01 s21 W (J)
Vr (m s21)
W (J)
6
331.1 368.9
18.4 15.1
335.5 372.1
6.7 5.6
328.9 367.8
24.1 18.3
12
358.2 345.3
40.1 41.2
363.2 350.9
25.7 25.7
357.3 345.4
42.6 40.9
24
301.4 304.2
90.3 97.1
308.1 310.4
74.1 81.9
300.1 306.7
93.4 91.0
30
320.6 361.2
115.2 107.9
330.9 367.1
88.5 90.8
318.5 359.2
120.5 113.6
36
328.4 343.1
127.3 124.3
332.8 347.9
115.8 111.1
326.7 339.8
131.7 133.3
Fig. 10. Comparison between kinetic energy absorbed of projectile and strain energy of fabric (Kuralonw).
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yarn that contacted with projectile in each fabric layer is probably greater than two if the shape of conically cylindral projectile is considered in analytical model. This is a main source of prediction errors. (ii) The fabric was simplified as crossed nonwoven yarns. The real structure of plain-woven fabric (for example, yarn crimp) is considered from dividing pffiffi the longitudinal wave velocity of infinite long yarn by 2; the same method also adopted by Parga-Landa et al. [12] and Roylance [17]. (iii) The strain rate of fibers is in the regime of 5 £ 102 –15 £ 102 s21 in ballistic impact when the projectile velocity is about sonic speed. For different materials and different projectile velocities, the specific strain rate is different. This will influence the accuracy of time interval estimation in ballistic perforation. Unfortunately, there are no better method so far for estimating Dt in theoretical modeling. (iv) The impact point could not always coincide with interweave point of fabric; each fabric layer often has horizontal deviation to some extent. (4) A series of analysis by Cunniff [1,28] could be conducted when the residual velocity of projectile is obtained from experiment or theoretical calculation. From the analysis of Cunniff [1,28], the ballistic limit ‘band’ of target plate of different weave construction, fiber modulus, and area density could also be deduced. In all these analysis, the residual velocity is a key element. For the Twaronw and Kuralonw fabric target plate, when the residual velocity is obtained by the analytical modeling described in this paper, these works could be finished by the Cunniff’s method. (5) The projectile used in ballistic test is conically cylindral steel projectile, not the fragment simulating projectile (FSP) described and stipulated by NIJ standard. The pity herein is that the two kinds of projectile cannot be compared for further evaluating the validity and correctness of the analytical model by the data in other literature [1,28].
10. Conclusions The multi-layered planar plain-woven fabric targets will be deformed as a ‘pyramid’ tetrahedron shape, in which the diagonal line length of rhombus bottom is the length of principal yarn in transverse wave region, when the targets are penetrated by high velocity projectile. The strain energy and kinetic energy of all the yarns in deformed region is analyzed at the unconstrained and constrained state. The energy absorbed by all yarns in rhombic region when fabrics are deformed into tetrahedron shape under ballistic impact, and especially the mechanical properties of yarn at high strain rate, are taken into consideration. The determination of failure criterion is based on maximum strain criterion. From the energy conservation law, the residual kinetic energy of projectile after ballistic impact could be obtained. The residual velocity and the decrease of kinetic energy of projectile, in experimental and theoretical, are all in agreement. It proves that the analytical model is correct and valid in prediction. From the comparison of
the calculated results between quasi-static state and high strain rate state, it is concluded that only in the condition of strain rate effect and rate-dependent properties of fibers are considered could the prediction accurately be obtained.
Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 19902016 and Foundation for University Key Teacher, Foundation of Key Technology Project, the Ministry of Education of China. Supports for this work and permission to publish by the Office of the Quartermaster General of China are also gratefully acknowledged.
References [1] Cunniff PM. An analysis of the system effects in woven fabrics under ballistic impact. Text Res J 1992;62(9):495–509. [2] Chocron-Benloulo IS, Rodriguez J, Sauchez-Galvez V. A simple analytical model to simulate textile fabric ballistic behavior. Text Res J 1997;67(7):520 –8. [3] Navarro C. Simplified modeling of the ballistic behavior of fabrics and fiber-reinforced polymeric matrix composites. Key Engng Mater 1998;141–143:383– 400. [4] Lee BL, Walsh TF, Won ST, Patts HM, Song JW, Mayer AH. Penetration failure mechanism of armor-grade fiber composites under impact. J Compos Mater 2001;35(18):1605–29. [5] Hearle JWS, Leech CM, Cork CR. Ballistic impact resistance of multi-layer textile fabrics. AD-A128064; 1981. [6] Leech CM, Hearle JWS, Mansell J. A variational model for the arrest of projectiles by woven cloth and nets. J Text Inst 1979;70(11): 469 –78. [7] Roylance D, Wang SS. In: Laible RC, editor. Ballistic materials and penetration mechanics. Amsterdam: Elsevier; 1980. p. 278. [8] Shim VP, Tan VBC, Tay TE. Modeling deformation and damage characteristics of woven fabric under small projectile impact. Int J Impact Engng 1995;16(4):585 –605. [9] Billon HH, Robinson DJ. Models for the ballistic impact of fabric armor. Int J Impact Engng 2001;25:411–422. [10] Vinson JR, Zukas JA. On the ballistic impact of textile body armor. J Appl Mech 1975;42(6):263–8. [11] Taylor WJ, Vinson JR. Modeling ballistic impact into flexible materials. AIAA J 1990;28(2):2098–103. [12] Parga-Landa B, Hernandez-Olivares F. An analytical model to predict impact behaviour of soft armours. Int J Impact Engng 1995;16(3): 455 –66. [13] Leech CM. The dynamics of flexible filaments assemblies. In: Hearle JWS, Thwaites JJ, Amirbayat J, editors. Mechanics of flexible fibre assemblies. The Netherlands: Sijthoff & Noordhoff; 1980. [14] Awerbuch J, Bodner SR. Analysis of the mechanics of perforation of projectiles in metallic plates. Int J Solids Struct 1974;10:671 –84. [15] Smith JC, McCrackin FL, Scniefer HF. Stress–strain relationships in yarns subjected to rapid impact loading. Part V. Wave propagation in long textile yarns impacted transversely. Text Res J 1958;28(4): 288 –302. [16] Smith JC, Blandford JH, Towne KH. Stress–strain relationships in yarns subjected to rapid impact loading. Part VIII. Shock waves, limiting breaking velocities, and critical velocities. Text Res J 1962; 32:67.
B. Gu / Composites: Part B 34 (2003) 361–371 [17] Roylance DK, Wilde A, Tocci G. Ballistic impact of textile structures. Text Res J 1973;43:34 –41. [18] Sierakowski RL, Chaturvedi SK. Dynamic loading and characterization of fiber-reinforced composites. New York: Wiley; 1997. p. 17. [19] Kawata K, Hashimoto S, Takeda N. Mechanical behaviours in high velocity tension of composites. In: Hayashi T, Kawata K, Umekawa S, editors. Proceedings of Fourth International Conference on Composite Materials (ICCM-4), Tokyo. 1982. p. 829 –36. [20] Harding J, Welsh LH. Impact testing of fiber reinforced composite materials. In: Hayashi T, Kawata K, Umekawa S, editors. Proceedings of Fourth International Conference on Composite Materials (ICCM4), Tokyo. 1982. p. 845 –52. [21] Dong LM, Xia YM, Yang BCh. Tensile impact testing of fiber bundles. Acta Mater Compos Sinica 1990;7(4):9 –15. in Chinese. [22] Wang Z. Experimental evaluation of the strength distribution of e-glass fibers at high strain rate. Appl Compos Mater 1995;2(4):257–64.
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[23] Wang Y, Xia YM. Strain rate effects on kevlar fibre bundles. Composites, Part A 1998;29A(6):1411–5. [24] Xia YM, Yuan JM, Yang BCh. A statistical model and experimental study of the strain-rate dependence of the strength of fibers. Compos Sci Technol 1994;52(4):499 –504. [25] Zhou Y, Xia Y. The effect of strain rate and prestress on the mechanical behavior of ALL. Composites, Part A 1998;29A: 517–24. [26] Xiong J, Gu B, Zhou G, Wang S. Investigation on the high strain rate properties of PVA fiber bundles. J China Text Univ 2000;26(3):90 –4. in Chinese. [27] Gu B, Zhao D. The ballistic impact property of multi-layered fabrics. J Text Res 2000;21(4):16– 18 (in Chinese). [28] Cunniff PM. A semiempirical model for the ballistic impact performance of textile-based personnel armor. Text Res J 1996; 66(1):45–59.
Analytical modeling for the ballistic perforation of planar plain-woven fabric target by projectile Bohong Gu* College of Textiles, Dong Hua University, Shanghai 200051, People’s Republic of China Received 1 March 2003; accepted 23 November 2003
Abstract This paper presents an analytical model to calculate decrease of kinetic energy and residual velocity of projectile penetrating targets composed of multi-layered planar plain-woven fabrics. Based on the energy conservation law, the absorbed kinetic energy of projectile equals to kinetic energy and strain energy of planar fabric in impact-deformed region if deformation of projectile and heat generated by interaction between projectile and target are neglected. Then the decrease of kinetic energy and residual velocity of projectile after the projectile perforating multi-layered planar fabric targets could be calculated. Owing to fibers in fabric are under a high strain rate state when fabric targets being perforated by a high velocity projectile, the mechanical properties of the two kinds of fibers, Twaronw and Kuralonw, respectively, at strain rate from 1.0 £ 1022 to 1.5 £ 103 s21, are used to calculate the residual velocity of projectile. It is shown that the mechanical properties of fibers at high strain rate should be adopted in modeling rate-sensitivity materials. Prediction of the residual velocities and energy absorbed by the multi-layered planar fabrics show good agreement with experimental data. Compared with other models on the same subject, the perforating time in this model can be estimated from the time during which certain strain at a given strain rate is generated. This method of time estimation is feasible in pure theoretical modeling when the perforation time cannot be obtained from experiments or related empirical equations. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Fabrics; Ballistic perforation; Strain energy; Strain rate
1. Introduction Composite materials are widely used in personal and vehicle armors because of their lightweight and high bulletproof performance. But it is difficult to simulate numerically or analytically the ballistic impact behaviors of composite material because of the complexity of damage mechanism. The first step and valid way to model behaviors of composite under ballistic impact could be the study of the primary components of the composites. The fabric is undoubtedly the most important one, and an analytical model of multi-layered planar fabric based on strain rate effect of fibers is developed in this paper to calculate the decrease of kinetic energy and residual energy of projectile penetrating planar fabric target. It is complex and interesting to study the ballistic penetration and perforation of fabrics. Cunniff [1], ChocronBenloulo et al. [2], Navarro [3] and Lee et al. [4] published * Tel.: þ86-21-62373456; fax: þ86-21-62193061. E-mail address: [email protected] (B. Gu).
concise surveys of the analytical models of penetration of projectiles into woven fabrics, which covered the major works that had been published so far. Hearle et al. [5] and Leech et al. [6] analyzed the ballistic impact mechanics of other forms of fiber assemblies, such as tri-axial woven fabrics and knitting fabrics. Up to now, many theoretical models on fiber assemblies under ballistic impact were established. They could be divided into two kinds. One is finite element model or finite difference model that is based on variational principle [6 –9]. As pointed out by Cunniff [1], conclusions in these models are often contrary to experiments because of the lack of constitutive properties and failure criterion for armor materials. The other is simple analytical model by intuitive approach through qualitative understanding of impact dynamics of less complex fabric and applied to more complex systems [2,10 –12], but only the breakage of principal yarns contacted with projectile in woven fabric is concerned in these models, the absorbed energy by other yarns and kinetic energy of fabric in deformed region is not considered.
1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-8368(02)00137-3
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Nomenclature 1 10 s 1_ 1ins 1max n, ns nr c r Ulab u
strain in yarn strain of principal yarn stress in yarn strain rate instant strain in yarn caused by longitudinal wave failure strain of yarn impact velocity of projectile residual velocity of projectile longitudinal wave velocity in yarn half of yarn length between two constrained points transverse wave velocity in yarn relative to the laboratory velocity of transverse wave front relative to points on the unstrained yarn
The damage mechanism of fiber assemblies under ballistic impact could be illustrated by these models to some extent, however, the most important factor, the strain rate effect on mechanical properties of fibers, was neglected in all these models. Because the strain rate of materials under ballistic impact is often greater than 102 s21, an adequate understanding of the rate-dependence of the mechanical properties of the materials involved is required if problems associated with ballistic impact are to be properly analyzed. Our purposes here are to put forward analytical model of multi-layered planar fabric under ballistic perforation and to calculate the decrease of kinetic energy of projectile. The energy absorbed by all yarns in rhombic region when fabrics are deformed into tetrahedron shape under ballistic impact, and especially the mechanical properties of yarn at high strain rate, are taken into consideration. The determination of failure criterion is based on energy conservation law and maximum strain criterion. Finally the predictions computed and the experimental results of residual velocities of projectiles were compared.
E Tex
r m S nw, nF t Dt WKY WSY Wi WSF WKF
Young’s modulus of yarn linear density of yarn (In textile engineering: 1 Tex ¼ 1 g/1000 m) volume density of yarn mass of projectile cross-section area of yarn warp and filling (weft) density of fabric, respectively (per 10 cm) time time elapsed in ballistic perforation kinetic energy of yarn strain energy of yarn strain energy of ith yarn strain energy of fabric in rhombus region kinetic energy of fabric in rhombus region
2. Description of planar fabric deformation under normal ballistic impact Planar woven fabrics are composed of two kinds of yarns that are perpendicular to each other, named weft yarn (filling yarn) and warp yarn, respectively. Fabric deformation under normal ballistic impact has been discussed in several papers [6,13]. Leech et al. [6] predicted that, for an orthogonally woven fabric, such as planar plain-woven fabric, the transverse wave front is a rhombus with sides given by the following sets of equations t¼^
x y ^ ux uy
ð1Þ
where x and y are the spatial co-ordinates of the front at time t, and ux, uy are the propagation speeds of transverse signals in the x and y directions, respectively (Fig. 1). For plainwoven fabric in which the diameter of warp yarns equals to that of weft yarns, it can be deduced that ux ¼ uy : This wave front is also confirmed by our experimental observations.
Fig. 1. Three-dimensional tetrahedron shape and 2-dimensional rhombus region of fabric impacted by projectile.
B. Gu / Composites: Part B 34 (2003) 361–371
In Fig. 1 the yarn number n is: n ¼ ux DtnW
or
n ¼ uy DtnF
ð2Þ
From observations of fabric samples after ballistic perforation, it was found that only the yarns contacted with conically cylindral projectile (X and Y-axis in Fig. 1) would be broken, other yarns in this rhombus region would not be fractured. Because the friction between projectile and target, and the deformed energy of projectile could be regarded as negligible when the fabric target is perforated by perfectly rigid projectile [12,14], the strain energy of fabric target could be divided into two parts: one is the fracture work of yarns (which are referred as principal yarns) contacted with projectile, the other is elastic strain energy and plastic strain energy of yarns in rhombus region except principal yarns. Here the dissipation of heat energy generated in the whole process of ballistic penetration or perforation is neglected. When the sum of the two kinds of energies absorbed by fabric is calculated, the decrease of kinetic energy of projectile could be obtained based on the energy conservation law.
3. Kinetic energy absorbed by principal yarns 3.1. Unconstrained yarns
363
the yarn) is constant and can be calculated as follows [15,16] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi v ¼ c 21 1ð1 þ 1Þ 2 12 ð3Þ where n is the impact velocity of the projectile, pffiffiffiffiffi c is longitudinal wave velocity in the yarn and c ¼ E=r (E is the Young’s modulus at high strain rate and r is the volume density of the yarn). The transverse wave velocity with respect to the laboratory is pffiffiffiffiffiffiffiffiffiffiffi ulab ¼ cð 1ð1 þ 1Þ 2 1Þ ð4Þ and the angle between the line of impact and the yarn is given by 1 v ð5Þ 1þ1 u pffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ c ð1=1 þ 1Þ; u is the velocity of transverse wave front relative to points on the strained yarn (i.e. u is a Lagrangian wave speed). Because the yarn is a flexible material, the ballistic impact damage of the yarn could be attributed to tensile failure. ÐThe tensile strain energy stored in per cubic meter yarn is 10max sð1Þd1: Then the strain energy of infinite long yarn under ballistic impact after time Dt is: ð1ins WSY ¼ ScDt sð1Þd1 ð6Þ cos u ¼
0
The behavior of principal yarns under normal impact is familiar with that of long yarns under transverse impact. The theory of single yarn under transverse impact is fairly well developed by Smith in series of papers [15,16]. The theory indicates that the strain developed in an infinite long yarn under transverse impact is dependant on yarn constitutive properties and impact velocity, the magnitude of strain in the wake of the longitudinal wave is constant in the yarn for a linear-elastic materials. As shown in Fig. 2, the projectile has traveled a distance Dy after Dt. If the yarn is linearly elastic, ahead of the elastic wave front, the strain is zero, and behind, the strain 1 (generated in yarn at the instance when the projectile contacted with
If the instant strain 1ins in the principal yarn was greater than the failure strain 1max, the principal yarn would be perforated. The decrease of kinetic energy of projectile is equal to the strain energy of the principal yarn and the kinetic energy of the yarn in the transverse wave region, i.e. 1 mðv2s 2 v2r Þ 2 " ¼ 2 ScDt
ð1max 0
1 ðDt ulab t sð1Þd1 þ Tex ðvðtÞÞ2 dt 2 0 sin u
#
ð7Þ Because the v(t) is not known, here we suppose that vðtÞ ¼ ðvs þ vr Þ=2 for simplicity, i.e. v(t) is a linear descending function. If the instant strain 1ins in the principal yarn is less than the failure strain 1max, the principal yarn would not be broken. The projectile would be arrested by the principal yarn. From Eq. (6), the strain energy of principal yarns is proportional to the cross-section area and Young’s modulus of the yarn. The bigger the cross-section area and Young’s modulus, are the more kinetic energy absorbed would be. 3.2. Constrained yarn
Fig. 2. Single yarn impacted by projectile.
In this case, the behavior of yarn is different with yarn of infinite length. The most distinct feature of unconstrained is
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that the strain would not increase from 1ins as the time elapsing because the strain spread to other part of yarn at the speed of longitudinal wave. When the yarn specimen is clamped at two points equidistant from the point of impact, reflections of the longitudinal strain wave will alternately transfer between clamping points and impact point. After each reflection the local strain increases by small increments. The strain in transverse wave region will increase to the maximum strain when the projectile is arrested or rupture strain is attained. At a given striking velocity vs, the 1ins is a constant for given yarn. As for constrained yarn, the strain of yarn in transverse wave region would be increased from 1ins to 1max. This is shown in Fig. 3. In Fig. 3(a), when cDt # r; this case is the same as unconstrained yarn.
In Fig. 3(b), when cDt . r and Ulab Dt # r; the strain of yarn outside of transverse wave region is the superposition of 1ins and reflection strain 1reflect, the strain of yarn within transverse wave region would be increased from 1ins to 1max because of tensile deformation. In Fig. 3(c), when Ulab Dt . r; the whole yarn is under tensile state. The strain energy of the constrained yarn fractured at different time could be written as: WSY ¼ 2cDtS WSY ¼ 2rS
ð1max
sð1Þd1;
0 , Dt #
0
ð1ins þ1reflect
sð1Þd1 þ 2uDtS
ð1max
0
sð1Þd1;
1ins
0
r r , Dt # c Ulab ð1max WSY ¼ 2rS sð1Þd1;
r c
ð8Þ
r , Dt Ulab
The kinetic energy of the constrained yarn fractured at different time could be written as: ðDt u t vs þ vr 2 lab Tex dt; sin u 2 0 r 0 , Dt # Ulab ðDt r vs þ vr 2 WKY ¼ Tex dt; sin u 2 0
WKY ¼
ð9Þ r , Dt Ulab
4. kinetic energy absorbed by single layer planar fabric 4.1. modeling of crossed nonwoven yarns
Fig. 3. The shape of constrained yarn under normal impact at different time; (a) constrained yarn under normal impact at time t ¼ 0; (b) constrained yarn under normal impact at time t # r=c; (c) constrained yarn under normal impact at time r=c , t # r=Ulab ; (d) constrained yarn under normal impact at time r=Ulab , t:
The phenomena of strain wave propagation and spread in fabric are more complex than those in yarn when fabric is under ballistic impact because of wave transmission and reflection at interweave point. The following hypotheses are made in order to simplify modeling. (1) Planar plain-woven fabric are composed of crossed nonwoven yarns perpendicular to each other; (2) longitudinal waves only spread in principal yarn; (3) transverse wave in principal yarn only spread to perpendicular yarns, not spread to parallel yarns. From these hypotheses, the crossed nonwoven yarns would be deformed as ‘pyramid’ shape under ballistic impact. This is also shown in Fig. 1. In Fig. 1, points on the principal yarns within the transverse deflection appear to be moving faster than adjacent points on corresponding parallel yarn in Z direction. This means that yarn closer to the principal yarn is under greater strain. The strain in yarn of rhombus region except principal yarns could be simply calculated by linear
B. Gu / Composites: Part B 34 (2003) 361–371
interpolation from zero to principal yarn strain. i 1i ¼ 1 2 i ¼ 0; 1; …; n 1; n 0 Then the strain energy of ith yarn in Fig. 1 is: i ð1i r sð1Þd1; 0,t# Wi ¼ SuDt 1 2 n Ulab 0 ð1i r Wi ¼ Sri Dt sð1Þd1; ,t Ulab 0
ð10Þ
n X
Wi
ð11Þ
ð12Þ
i¼1
The length of ith yarn in transverse wave region in Fig. 1 is i Dl; i ¼ 0; 1; 2; …; n ðDlÞi ¼ 1 2 n where Dl is the length of principal yarn in transverse wave region in Fig. 1. The sum of kinetic energy of yarns from 1 to n in Fig. 1 is: n X Dli vs þ vr 2 Tex WK1 ¼ 4 sin u 2 i¼1 ¼4
n ulab Dt v þ vr 2 X i Tex s 12 sin u n 2 i¼1
The kinetic energy of principal yarn is: u Dt v þ vr 2 WK2 ¼ lab Tex s sin u 2
the whole multi-layered fabric target under ballistic impact is equal to the product of absorbed energy of single layer fabric and the number of layers.
5. determination of time interval of ballistic perforation
From Eqs. (8) and (10), the whole strain energy of rhombus region including two principal yarns is: WS ¼ 2WSY þ 4
365
ð13Þ
The time interval Dt, which from the beginning of impact to the instant when the yarn is broken or the projectile is stopped, is a key to solve equations related to Dt. Only the Dt is known, could the kinetic energy of projectile absorbed by fabric target be calculated. But this problem is not solved in a reasonable way in all other ballistic penetration models. Experimentally, the time interval Dt could be determined precisely by high-speed photographs from ballistic impact. In theoretical consideration, this method is not feasible because it losses theoretical purism. Furthermore, it is a circular argument if we adopt Dt from high-speed photographs or any other empirical formulae from experiments. Since the experiments have been finished, is it necessary to model the experiment and to get a prediction? The method to estimate Dt in this paper is from definition of strain rate. The strain rate concerning with ballistic penetration is at the range of 5 £ 102 – 15 £ 102 s21 and depends on the projectile velocity [18]. Because the strain rate is relatively more stable in ballistic perforation than in penetration (In the case of penetration, the projectile could be arrested. The strain rate is changed in wide range.), the time to failure Dt can be calculated as follows: Dt ¼
ð14Þ
Then the total kinetic energy WK of yarns in deformed region could be obtained from Eqs. (13) and (14).
D1 1_
ð16Þ
Eq. (16) is an approximate method to estimate Dt. In theoretical way, there are no better method to get a more precise result than Eq. (16).
4.2. modeling of single layer plain-woven fabric
6. Tensile tests of yarn at various strain rates
The method of modeling plain-woven fabric from perpendicular crossed nonwoven yarns herein is the same as in Refs. [12,17]. It is assumed that the longitudinal wave pffiffi velocity in single yarn should be divided by 2 in actual calculation, because the linear density of yarn along which the wave propagates is effectively doubled in a plain-woven fabric. So far, the total energy W of single layer fabric could be obtained by above-mentioned method and Eqs. (12) – (14), i.e.
The most probable failure modes of fabric under ballistic impact are tensile failure and shear failure of fibers at an adiabatic state. Due to the flexibility of fibers, the tensile failure is the main failure mode in ballistic impact. It is very easy to test the tensile property of fibers at low strain rates (corresponding to the low velocity tensile test), but owing to technical difficulties in test, there are no detailed data regarding the fibers’ tensile behavior at moderate or high strain rates. Kawata [19] and Harding [20] obtained the stress – strain curves of fabric reinforced composites and unidirectionally reinforced CFRP at the strain rate of 1 £ 103 s21 almost at the same time in 1982 by selfdesigned apparatus. It was the starting point of studying the mechanical properties of fiber-reinforced composites at high strain rate. In 1990, Xia et al. [21] first successfully performed tensile impact test on fiber bundles. More recently, a statistical model of rate-dependence of E-glass
W ¼ WS þ WK ¼
1 2
mðv2s 2 v2r Þ
ð15Þ
Then the residual velocity vr of projectile could be calculated. As for multi-layered fabrics, because the friction between fabric layers could be neglected in the analysis of ballistic impact [12], the strain energy and kinetic energy of
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Fig. 4. Schematic diagram of the fiber bundles specimen 1-supplement plate, 2-lining block, 3-fiber bundles, 4-input bar, 5-output bar.
fiber bundles and kevlar fiber bundles was set up [22 – 24], and strain rate-dependence of mechanical behavior of these fiber bundles was also discussed. The method developed by Xia et al. [21] is adopted in this paper to test tensile properties of fibers at high strain rate. 6.1. Tensile tests at low strain rate (quasi-static tensile test) The quasi-static tensile tests were performed in a MTS810.23 testing machine at strain rate 1022 s21. 6.2. Tensile tests at high strain rate 6.2.1. Experimental apparatus Tensile impact tests were conducted in rotating disk bar – bar tensile impact apparatus that is similar to split Hopkinson tension bar. This apparatus was designed by the University of Science and Technology of China. The structure of this apparatus and experimental measuring principle could be seen in Ref. [25].
Fig. 6. Typical strain gauge signals (Kuralonw).
Nobel) and Kuralonw PVA filaments (Kuralonw 7901-1, 2000 dtex/1000f, manufactured by Kuraray of Japan). The schematic diagram of the filament specimen was shown in Fig. 4. ls was the specimen gauge length, which was 8.0 mm. The number of filaments in the specimen was 18 in total. The specimens were tested at room temperature and approximately 55% relative humidity. The results at each strain rate are derived from the average values of at least three experiments. Typical experimental waves of Twaronw aramid filaments and Kuralonw PVA filaments under tensile impact loading are shown in Figs. 5 and 6, respectively, in which the X-axis gives the time and the Y-axis gives the strain signal measured by strain gauges on the input bar and output bar. 6.3. Results
6.2.2. Specimens in experiment The test materials were Twaronw aramid filaments (type: Twaronw CT1000, 1680 dtex/1000f, manufactured by Akzo
Fig. 7 shows the stress – strain curves of Twaronw fiber bundles under different loading conditions. The strain rates are 1022, 180, 480, 1000 s21, respectively. Table 1 lists the average values of E, smax and 1max. Fig. 7 and Table 1 indicate that Twaronw fiber bundles have sensitivity to strain rate. E, smax and 1max increase with increasing strain
Fig. 5. Typical strain gauge signals (Twaronw).
Fig. 7. Tensile curves of Twaronw filaments at different strain rate.
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367
Table 1 Mechanical properties of Twaronw fiber bundles versus strain rate
Table 2 Mechanical properties of Kuralonw fiber bundles versus strain rate
Strain rate (s21)
E (GPa)
smax (GPa)
1max (%)
Strain rate (s21)
E (GPa)
smax (GPa)
1max (%)
1022 180 480 1000
62 69 70 72
2.395 2.596 2.704 2.753
5.19 5.22 5.47 5.70
1022 270 600 1500
20.3 49.7 52.6 51.2
1.19 1.50 1.61 1.85
9.89 4.70 4.93 5.97
rate from 1022 to 1000 s21. Because E increases significantly from the strain rate of 1022 to 1000 s21, it is obviously not reasonable if we introduce the Young’s modulus E at quasi-state to the analytical model. Fig. 8 shows the stress – strain curves of Kuralonw fiber bundles under different strain rates. The strain rates are 1022, 270, 600, 1500 s21, respectively. Table 2 lists the average values of E, smax and 1max. Fig. 8 and Table 2 indicate that Kuralonw fiber bundles have sensitivity to strain rate. smax and 1max increase with increasing strain rate from 1022 to 1500 s21. The Young’s modulus E increases with increasing strain rate from 0.01 to 600 s21. But the Young’s modulus at strain rate of 1500 s21 is lower than that of 600 s21. It implies that PVA fibers appear to undergo a ductile –brittle – ductile transition strain rate range from 0.01 to 1500 s21 [26]. The Young’s modulus E at quasistate is far less than that of high strain rate. It is also obviously not reasonable if we introduce the Young’s modulus E at quasi-state to the analytical model.
7. Ballistic perforation tests of multi-layered planar fabric target Tests were conducted to study the energy absorbed capacity of fabric target under ballistic perforation and to verify numerical predictions [27]. Conically cylindral steel projectile of 7.62 mm diameter and 7.95 g, was used in ballistic test. Strike velocities of projectiles were controlled by adjusting the weight of gunpowder. It was observed from
the experiments that the deformation of projectile could be neglected. The deformation energy of projectile could also be ignored in the analytical model. Circular target plates of different fabric layers were clamped between two steel rings of 60 mm inner radius. Striking and residual velocities were measured, respectively, by two laser-diode pairs. Two kinds of multi-layered planar plain fabric targets, which were made of Twaronw and Kuralonw fibers, respectively, were perforated by conically cylindral steel projectile at different strike velocities. The specifications of Twaronw and Kuralonw plain-woven fabric are listed in Table 3. Experimental results are shown in Table 4.
8. Comparisons between theoretical and experimental results 8.1. Algorithm The strain energy stored in per cubic meter yarn is equal Ð to 10 sð1Þd1: The value of this integral could be obtained from numerical integral in experimental stress – strain curve of yarn. Because the data of stress and strain of yarn were sampled by computer at the time interval of 1026 s, then the value of this integral could be obtained by numerical integral methods. Another way to obtain the value of this integral is relatively simpler. The yarn was presumed as a linear-elastic material in above-mentioned modeling, the constitutive equation of yarn is sм E1: We could substitute this constitutive equation in 10 sð1Þd1: Then Eqs. (8) and (11) could be rewritten as: WSY ¼ cDtSEð1max Þ2 ;
0 , Dt #
r c
ð8aÞ
WSY ¼ rSEð1ins Þ2 þ uDtSEð1max 2 1ins Þ2 ; r r , Dt # c Ulab WSY ¼ rSEð1max Þ2 ;
r , Dt Ulab
1 i 3 ð10 2 1ins Þ2 ; Wi ¼ SuDtE 1 2 2 n Fig. 8. Tensile curves of Kuralonw filaments at different strain rate.
r 0,t# Ulab
ð11aÞ
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B. Gu / Composites: Part B 34 (2003) 361–371
Table 3 Specifications of Twaronw and Kuralonw plain-woven fabric Fabric type
Linear density (dtex)
Twaronw Kuralonw
Wi ¼
Weave density (per 10 cm)
Strength at break (N cm21)
Warp
Weft
Warp
Weft
Warp
Weft
3360 2000
3360 2000
67 198
66 194
3070 1486
3138 771
1 i 2 Sri DtE 1 2 ð10 2 1ins Þ2 ; 2 n
r ,t Ulab
Eqs. (8a) and (11a) are used in calculation program of this paper. 8.2. Numerical calculations Following parameters of Twaronw and Kuralonw fiber, plain-woven fabrics are substituted into the analytical model. 1. Cross-section area of yarn S (Twaronw: S ¼ 2:33 £ 1027 m2 ; Kuralonw: S ¼ 1:49 £ 1027 m2 ) 2. Volume density of yarn r (Twaronw: r ¼ 1:44 g=cm3 ; Kuralonw: r ¼ 1:34 g=cm3 ) 3. Tensile property of Twaronw and Kuralonw yarn at different strain rate (Tables 1 and 2) 4. Time interval Dt (the strain rate with maximum probability in ballistic impact is 5 £ 102 –15 £ 102 s21
Areal density (kg m22)
Construction
0.4824 0.4014
Plain-woven Plain-woven
[18]). Here we determined that the time interval begins at the projectile contacting with yarn and end in the time of yarn fracture, namely Dt ¼ ð1max 2 1ins Þ=1000 for Twaronw and Dt ¼ ð1max 2 1ins Þ=1500 for Kuralonw. The residual velocities of projectiles perforating the Twaronw and Kuralonw planar fabric targets at different time interval by two kinds of algorithms are given in Tables 5 and 6, respectively. The quasi-static tensile property of Twaronw and Kuralonw are also adopted in calculation to illustrate the necessity of introducing rate-dependent effect of materials. 8.3. comparison between experimental and theoretical results The decrease of kinetic energy of projectile versus area density, in experimental and theoretical, could be shown in Figs. 9 and 10. From Tables 4 – 6 and Figs. 9 and 10, it could be concluded that there are agreement between
Table 4 Experimental results of multi-layered Twaronw and Kuralonw fabrics under ballistic impact Target type Twaronw
Kuralonw
Target layers
Target area density (kg m22)
5
2.412
10
4.824
15
7.236
20
9.648
25
12.060
30
14.472
6
2.462
12
4.925
24
9.850
30
12.310
36
14.770
Strike velocity (m s21)
Residual velocity (m s21)
Kinetic energy absorbed (J)
341 375 362 404 361 363 341 367 369 379 378 350
323 358 334 382 327 326 293 319 311 324 312 278
47.51 49.53 77.46 68.74 92.98 101.33 120.97 130.89 156.77 153.69 181.02 179.73
338 374 372 360 337 342 363 397 374 386
331 367 355 343 299 299 315 353 316 333
18.61 20.62 49.13 47.51 96.07 109.56 129.36 131.18 159.08 151.48
B. Gu / Composites: Part B 34 (2003) 361–371
369
Table 5 Calculated results of multi-layered Twaronw fabrics under ballistic impact Layers
Strain rate 1000 s21 Vr (m s21)
480 s21 W (J)
Vr (m s21)
0.01 s21 W (J)
Vr (m s21)
W (J)
5
327.1 363.7
36.9 33.8
329.5 365.1
30.7 29.1
336.3 371.3
12.7 11.0
10
338.8 386.5
64.6 55.0
344.1 389.5
50.2 45.7
353.9 398.2
23.1 18.5
15
327.1 329.3
92.7 92.7
335.6 337.8
70.3 70.2
348.8 351.0
34.4 34.1
20
291.5 326.1
124.5 112.7
304.0 335.2
94.9 88.8
322.4 351.3
49.0 44.8
25
318.8 331.8
137.2 133.4
330.4 342.3
107.3 105.2
349.7 361.0
55.1 52.9
30
321.6 279.5
156.8 176.4
334.5 298.4
123.2 133.0
356.3 323.7
63.3 70.4
experimental and theoretical results. From the comparison in Tables 5 and 6, the accuracy of the analytical model at high strain rate is better than that at low strain rate because the yarns in fabric are under high strain rate in ballistic impact.
9. Discussions (1) The shape of projectile is not considered in the analytical model; instead, the projectile is simplified as a particle in the analysis. The factors, such as cone angle, diameter and length of projectile could affect the absorbed kinetic energy of projectile in the ballistic perforation. It is Table 6 Calculated results of multi-layered Kuralonw fabrics under ballistic impact Layers
Fig. 9. Comparison between kinetic energy absorbed of projectile and strain energy of fabric (Twaronw).
also shown that there are yarn breakages at the impact point and yarn slippage around the projectile in perforated fabric target. If the bullet shape is considered, the friction between bullet and fabric target must be introduced in analytical model. At the same time, the friction between warp yarns and weft yarns should also be introduced when yarn slippage is considered. (2) The mechanical properties of yarn at quasi-static state and high strain rate state were used in numerical calculation of the analytical model. The results of high strain rate are closer to experimental results because the yarns of fabric are under high strain rate state in ballistic impact. The reasons for calculated results of Kuralonw at quasi-static state closer to experimental results are that Kuralonw are ductile materials at low and high strain rate, and brittle materials at medium strain rate [26]. (3) The prediction errors, especially the errors between strain energy, kinetic energy of fabric target and the decrease of kinetic energy of the projectile are probably caused by the following factors. (i) The number of principal
Strain rate 1500 s21 Vr (m s21)
270 s21 W (J)
Vr (m s21)
0.01 s21 W (J)
Vr (m s21)
W (J)
6
331.1 368.9
18.4 15.1
335.5 372.1
6.7 5.6
328.9 367.8
24.1 18.3
12
358.2 345.3
40.1 41.2
363.2 350.9
25.7 25.7
357.3 345.4
42.6 40.9
24
301.4 304.2
90.3 97.1
308.1 310.4
74.1 81.9
300.1 306.7
93.4 91.0
30
320.6 361.2
115.2 107.9
330.9 367.1
88.5 90.8
318.5 359.2
120.5 113.6
36
328.4 343.1
127.3 124.3
332.8 347.9
115.8 111.1
326.7 339.8
131.7 133.3
Fig. 10. Comparison between kinetic energy absorbed of projectile and strain energy of fabric (Kuralonw).
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B. Gu / Composites: Part B 34 (2003) 361–371
yarn that contacted with projectile in each fabric layer is probably greater than two if the shape of conically cylindral projectile is considered in analytical model. This is a main source of prediction errors. (ii) The fabric was simplified as crossed nonwoven yarns. The real structure of plain-woven fabric (for example, yarn crimp) is considered from dividing pffiffi the longitudinal wave velocity of infinite long yarn by 2; the same method also adopted by Parga-Landa et al. [12] and Roylance [17]. (iii) The strain rate of fibers is in the regime of 5 £ 102 –15 £ 102 s21 in ballistic impact when the projectile velocity is about sonic speed. For different materials and different projectile velocities, the specific strain rate is different. This will influence the accuracy of time interval estimation in ballistic perforation. Unfortunately, there are no better method so far for estimating Dt in theoretical modeling. (iv) The impact point could not always coincide with interweave point of fabric; each fabric layer often has horizontal deviation to some extent. (4) A series of analysis by Cunniff [1,28] could be conducted when the residual velocity of projectile is obtained from experiment or theoretical calculation. From the analysis of Cunniff [1,28], the ballistic limit ‘band’ of target plate of different weave construction, fiber modulus, and area density could also be deduced. In all these analysis, the residual velocity is a key element. For the Twaronw and Kuralonw fabric target plate, when the residual velocity is obtained by the analytical modeling described in this paper, these works could be finished by the Cunniff’s method. (5) The projectile used in ballistic test is conically cylindral steel projectile, not the fragment simulating projectile (FSP) described and stipulated by NIJ standard. The pity herein is that the two kinds of projectile cannot be compared for further evaluating the validity and correctness of the analytical model by the data in other literature [1,28].
10. Conclusions The multi-layered planar plain-woven fabric targets will be deformed as a ‘pyramid’ tetrahedron shape, in which the diagonal line length of rhombus bottom is the length of principal yarn in transverse wave region, when the targets are penetrated by high velocity projectile. The strain energy and kinetic energy of all the yarns in deformed region is analyzed at the unconstrained and constrained state. The energy absorbed by all yarns in rhombic region when fabrics are deformed into tetrahedron shape under ballistic impact, and especially the mechanical properties of yarn at high strain rate, are taken into consideration. The determination of failure criterion is based on maximum strain criterion. From the energy conservation law, the residual kinetic energy of projectile after ballistic impact could be obtained. The residual velocity and the decrease of kinetic energy of projectile, in experimental and theoretical, are all in agreement. It proves that the analytical model is correct and valid in prediction. From the comparison of
the calculated results between quasi-static state and high strain rate state, it is concluded that only in the condition of strain rate effect and rate-dependent properties of fibers are considered could the prediction accurately be obtained.
Acknowledgements This work was supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 19902016 and Foundation for University Key Teacher, Foundation of Key Technology Project, the Ministry of Education of China. Supports for this work and permission to publish by the Office of the Quartermaster General of China are also gratefully acknowledged.
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