Composite Structures 93 (2011) 3163–3174
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Computational modeling of the probabilistic impact response of flexible fabrics Gaurav Nilakantan a, Michael Keefe a,d, Eric D. Wetzel e, Travis A. Bogetti e, John W. Gillespie Jr. a,b,c,⇑ a
Center for Composite Materials, University of Delaware, DE 19716, USA Department of Materials Science and Engineering, University of Delaware, DE 19716, USA c Department of Civil and Environmental Engineering, University of Delaware, DE 19716, USA d Department of Mechanical Engineering, University of Delaware, DE 19716, USA e US Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA b
a r t i c l e
i n f o
Article history: Available online 29 June 2011 Keywords: Aramid fiber Flexible composites Woven fabrics Impact behavior Finite element analysis (FEA) Probabilistic methods
a b s t r a c t The impact response of flexible woven fabrics is probabilistic in nature and described through a probabilistic velocity response curve or V0–V100 curve. Computational impact analyses based on deterministic methods are incapable of predicting the experimentally observed probabilistic fabric impact response. To overcome this limitation we have developed a probabilistic computational framework within a finite element analysis to predict the V0–V100 response. The finite element model is a yarn-based representation of the fabric architecture, with a principal stress based failure criterion implemented uniformly within each yarn, but varying for each yarn within the fabric. For each impact simulation, individual yarn strengths are mapped from experimentally obtained yarn strength distributions, resulting in fabric models with spatially non-uniform failure conditions. Impact simulations are run for the case of a spherical projectile of diameter 5.556 mm impacting a single layer of 50.8 50.8 mm, edge-clamped, unbacked, aramid fabric. Three different yarn strength models are implemented, representing spool yarns, and yarns extracted from greige and scoured woven fabrics. Decreases in yarn strength are found to correlate to decreases in the V1, V50, and V99 velocities predicted by the simulations. The relationships between yarn strength distribution and probabilistic fabric impact response are discussed. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Polymeric, carbon, and glass fibers possessing high modulus, high strength, and high strength-to-weight ratios are often used in impact resistant protective systems, in the form of flexible woven fabrics and as structural reinforcements in composite structures. Typical applications include protective clothing for military and law enforcement personnel, spall liners in infantry vehicles, and turbine fragment barriers in airline fuselages. Some of the commonly used materials for these types of applications are Kevlar, Twaron, Zylon, Vectran, and S-2 glass. For many of these protective systems, the primary design requirement is preventing penetration by high kinetic energy projectiles. Generally, the penetration behavior of fabric systems is probabilistic, exhibiting a probability of penetration at each impact velocity. This behavior can be represented by a continuous probabilistic velocity response (PVR) curve as shown in Fig. 1, or by discrete Vn values, where V is the impact velocity at which the fabric has an n% probability of being penetrated. The V50 velocity, at which penetration occurs during 50% of impacts, requires the few⇑ Corresponding author at: Center for Composite Materials, University of Delaware, DE 19716, USA. Tel.: +1 302 831 8702; fax: +1 302 831 8525. E-mail address:
[email protected] (J.W. Gillespie). 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.06.013
est number of impact tests to determine experimentally. Therefore V50 values are often characterized and used to compare the performance of protective systems. For the design of practical protective systems, it is more preferable to determine V1 or V0.1 values, representing velocities at which penetration respectively occurs during 1% or 0.1% of impacts. The precise limiting Vn value used as a design point is chosen based on desired margins of safety and acceptable risks. However, compared to V50 characterization, considerably more experiments are required to determine these values with confidence. Fig. 1 shows notional PVRs for three different protective systems. Systems 1 and 2 exhibit identical V50 velocities, with different V1 velocities. Systems 2 and 3 exhibit identical V1 velocities, with different V50 values. These examples illustrate how V50 velocities alone can be misleading for comparing protective systems intended for penetration resistance. Instead, it is critical to characterize and consider the entire PVR curve when comparing the impact performance of different protective fabric systems. The sources of variability that contribute to the probabilistic impact nature of fabric systems consist of parameters that stochastically vary from one fabric sample to the next and from one experimental test to the next. These parameters are either difficult to control or predict in a deterministic manner. The sources can be broadly divided into two groups: intrinsic and extrinsic. Intrinsic sources refer to the geometric and material properties of the
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Fig. 1. Sample PVR curves.
filaments and yarns that comprise the fabric architecture. For example the distribution of filaments within the yarn cross section varies in arrangement from one yarn to the next, and the cross-sectional area of each of the filaments varies along the yarn length [1,29]. The material properties such as tensile modulus and tensile strength also vary from one filament to another and consequently between yarns [18]. Filament–filament and yarn–yarn frictional properties present another source of intrinsic variability. Extrinsic sources refer to all sources other than from the actual fabric system, especially those associated with the experimental impact test. These sources include, but are not limited to, variations in projectile geometry, material properties, rotation rate, obliquity, yaw, and pitch; boundary conditions, such as fabric slippage at clamped edges [20]; position of impact relative to boundaries; and position of impact relative to yarn position (at the cross-over or in-between the cross-over) [17]. A number of simulations tools have been developed for modeling the impact and penetration of fabrics, typically using a finite element analysis approach [24]. Woven fabrics have been modeled as a single homogenized membrane [8,10,28], with yarn-level architecture [4,7,31,32], and with filament-level architecture [30]. Recently multiscale models have emerged in an attempt to balance the computational requirements of the finite element model with the accuracy of predictions [3,21,22,26]. These finite element models prove very useful in investigating projectile–fabric, yarn–yarn, and fabric layer–layer interactions as well as mechanisms of deformation and energy dissipation. Parametric finite element simulations allow the effect of many factors on the impact performance to be rapidly studied such as (i) fabric architecture [16,17] (e.g. yarn geometry and undulations), (ii) yarn material properties [5,27,32] (e.g. modulus, strength, frictional coefficient), and (iii) projectile characteristics [13,25] (e.g. shape, size). The advantages of these simulations relative to experimental characterization include savings in time, labor, and material; detailed insights into the progression of stress, deformation, and failure during penetration; precise control over boundary conditions; and the ability to evaluate geometries and materials that may not be readily available for testing. A limitation of these existing fabric impact simulation tools is that, in most cases, they have been implemented in a deterministic manner that does not result in the prediction of probabilistic penetration behavior [24]. For example, it is known from experimental testing that the tensile strength of yarns follows a statistical distribution [18]. However in the yarn-level fabric model of Duan et al.
Fig. 2. Overview of the probabilistic computational framework.
[4] all yarns in the fabric have been uniformly assigned the same failure stress of 2.3 GPa, while in Grujicic et al. [7] all yarns have been uniformly assigned the same strain-to-failure of 4%. This deterministic implementation of the yarn failure model results in identical responses during multiple runs of the same impact simulation, thereby making it difficult to capture the probabilistic penetration behavior. In this paper, the effect of a single source of variability, statistical yarn tensile strength, on probabilistic penetration behavior (the PVR curve) is studied. To conduct this study, we implement a yarnlevel finite element model representing a specific fabric type, Kevlar S706. The tensile strength distributions used in our probabilistic fabric modeling were measured experimentally and reported in Nilakantan et al. [18]. Strength models are implemented that represent as-spun (spool) yarns, warp and fill yarns in a greige fabric (spin finish intact), and warp and fill yarns in a scoured fabric (mechanically washed by the weaver to remove all finishes). These strength distributions are randomly sampled and mapped into our finite element model, creating a unique strength mapping for each impact simulation. Impact velocities are strategically varied for each impact to generate a rich set of response data (penetrating or non-penetrating response). Statistical techniques are then used to estimate best-fit, approximate PVR functions for fabrics representing spool-state, greige, and scoured conditions. Fig. 2 provides an overview of the computational probabilistic framework used in this study. 2. Computational probabilistic framework 2.1. Fabric architecture The model fabric represents Kevlar S706 fabric, a plain weave fabric comprised of 600 denier Kevlar KM2 yarns, with a count of 34 yarns per inch in the warp and fill directions, and an areal density of 180 g/m2. Each yarn in the real fabric is comprised of 400 circular filaments of approximately 12 lm diameter with a density of 1.44 g/cm3. Fig. 3 displays micrograph images of the Kevlar warp and fill yarns. The warp yarns are observed to have a greater degree of undulations than the fill yarns. This feature is important to incorporate in the model. During impact these warp yarns with greater
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Fig. 3. Micrographs and corresponding finite element models of the Kevlar S706 fabric (a) warp trajectory with fill cross sections and (b) fill trajectory with warp cross sections.
crimp will take longer than the fill yarns to decrimp or straighten before they can start to dissipate energy via tensile deformation. The undulating yarn centerline is well described by the periodic cosine (or sine) function as
y¼
thickness px cos 2 span
ð1Þ
where the y coordinate is in the yarn thickness direction and the x coordinate is along the yarn length. The span for both the warp and fill yarns is 0.747 mm. The thickness of the yarns obtained by averaging the measurements of several micrographs respectively, are 0.131 mm (warp) and 0.162 mm (fill). The geometric dimensions of the yarns are input into the preprocessor DYNAFAB [19] which then directly outputs the fabric finite element model with the desired mesh density. The undulating trajectories and cross sections of the warp and fill yarns are well captured by the corresponding finite element models also shown in Fig. 3. The dynamic finite element code LS-DYNA is used for the analyses. The yarns are modeled using single integration point solid elements (LS-DYNA Type 1 [9]). Five elements are used across the yarn width and one element through the thickness. By using an enhanced assumed strain stiffness hourglass form (LS-DYNA Type 9 [9]), it becomes possible to model the yarns with only one through-thickness element instead of the usual two elements as has been used in Duan et al. [5] and Nilakantan et al. [15]. This leads to dramatic savings in the model computational requirements. The left and right sides of the yarn cross sections are truncated to prevent using degenerate elements. 2.2. Yarn elastic properties Due to the homogenized nature of the yarn model, the filament volume fraction needs to be computed in order to define material properties. The actual yarn cross sectional area (in mm2) can be computed by dividing the yarn denier by the factor (9000 q), where q is the filament density in g/cm3. Thus the actual area occupied by the filaments within the yarn is 0.0463 mm2. Note that this value accounts for the varying cross sectional area of the filaments along the yarn length and is very close to the value determined by
assuming all filaments are perfectly circular with a 12 lm diameter, which yields 0.0452 mm2. The filament volume fractions are obtained by dividing the actual cross sectional area by the finite element yarn cross sectional areas. The calculated volume fractions (vfilament) are as follows: 76.02% (warp) and 66.54% (fill). The yarns are assigned a linear orthotropic elastic material model (LS-DYNA Type 2 [9]). The yarn material properties are defined with respect to a local material coordinate system that rotates with each finite element such that the longitudinal direction (xx) is aligned with the undulating yarn centerline trajectory while the two transverse directions (yy, zz) are along the yarn thickness and width directions. The longitudinal tensile modulus of a 600 denier Kevlar KM2 yarn is 82.6 GPa [11]. Accordingly, the tensile modulus of the homogenized finite element warp and fill yarns can be obtained by scaling the actual yarn tensile modulus by the computed filament volume fractions (i.e. Exx_yarn_homogenized = vfilament Exx_yarn_actual), yielding 62.8 GPa (Ewarp) and 55.0 GPa (Efill). The three components of the shear modulus (Gxy, Gyz, Gzx) are assumed to be the same and are assigned a value of 148 MPa as reported in Duan et al. [4]. The relatively low value of shear modulus (two orders of magnitude lower than the longitudinal modulus) is justified because zerotwist or low-twist continuous filament yarns offer minimal resistance to shear deformations. Consequently a filament bundle is modeled as a homogenous yarn by assigning low shear moduli. Non-zero shear moduli are required to avoid numerical instabilities and hourglassing in the finite element analysis. The results of the simulation were also found to be insensitive to the choice of shear moduli in this range. In addition, the three Poisson ratios of the homogenous yarn (mxy, myz, mzx) are zero as reported in Duan et al. [4]. 2.3. Yarn contact An eroding surface-to-surface type contact algorithm is used between the projectile and the fabric (LS-DYNA Type 14 [9]). An eroding single-surface type contact algorithm (LS-DYNA Type 15 [9]) is used for the warp and fill yarns at the impact region where yarn failure is observed to occur, while an automatic surface-tosurface type contact algorithm (LS-DYNA Type a3 [9]) is used for
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the yarns everywhere else. A static coefficient of friction of 0.23 is used between the projectile and fabric, and 0.18 between the yarns of the fabric. The magnitudes of these frictional coefficients are similar to those reported in Rao et al. [27]. 2.4. Yarn failure model Yarn failure is incorporated using a maximum principal stress failure criterion. The impact behavior of the single layer fabric studied here is dominated by the yarn longitudinal tensile modulus and tensile strength. The magnitude of the tensile modulus is two orders of magnitude higher than the other moduli. Thus yarn failure has been incorporated using an element erosion technique based on a maximum principal stress failure criterion (LS-DYNA MAT_ADD_EROSION [9]). The yarn strength model is based on the experimental work and statistical analysis of Nilakantan et al. [18], who measured yarn strengths over a range of gage lengths for 600 denier KM2 Kevlar yarns. These yarns display length scale effects where the strength is observed to decrease with increasing gage lengths during quasi static tensile testing [18]. However yarn length scale effects at high rates is as yet unknown. For our particular simulation conditions, the first instant of yarn failure consistently occurs well after the longitudinal strain wave has reached the fabric boundaries. Therefore it is reasonable to assume that the full length of the yarns in the fabric, 50.8 mm, is involved for the entirety of the impact event, and that this gage length should be selected for assigning yarn strengths. While this simple assumption was made for lack of better data, we recognize that although the entire yarn length is uniformly loaded during quasi static tensile testing, during transverse yarn impact the yarn region underneath the projectile usually experiences higher stress levels and the rest of the yarn usually experiences lower non-uniform stress levels based on the strain wave propagations, and that these two events may constitute different failure responses. Fig. 4a displays the median ranks (color1 symbols) assigned to the experimental strength data of the 600 denier Kevlar KM2 yarns, and fitted cumulative distribution functions (CDF) (smooth color lines) of tensile yarn strengths, pertaining to a 50.8 mm gage length [18]. These CDFs were generated using either the 3-parameter Weibull or G-Gamma statistical distributions depending on whichever yielded the best fit to the experimental data. This strength data was measured at room temperature and a nominal fifty per cent relative humidity such that other environmental factors (e.g. temperature, moisture, and exposure to ultraviolet radiations [23,6]) were negligible. A few salient trends observable in Fig. 4a are that the weaving process causes strength degradations in the yarns while the scouring process further degrades these strengths, and that warp yarns undergo an increased extent of strength degradation compared to the fill yarns. Table 1 lists the parameter values of the statistical strength distributions. The list of symbols used in Table 1 is as follows: gage length (GL), G-Gamma distribution (GG), 3-parameter Weibull distribution (3P), and correlation coefficient (q). The three parameters (P1–P3) respectively represent: l, r, and k (for the GGamma distribution) and m, r0, and x (for the 3-parameter Weibull distribution) [18]. However these CDF distributions in Fig. 4a cannot be used in their current form. Recall that the yarns are modeled as homogenous continua to represent the filament level architecture. Similar to the tensile modulus, the tensile strength data also need to be scaled (i.e. r0_yarn_homogenized = vfilament r0_yarn_actual) by the respective warp and fill yarn filament volume fractions, before it
1 For interpretation of color in Figs. 1, 3–12, the reader is referred to the web version of this article.
Fig. 4. Tensile strength distributions for the 50.8 mm gage length 600 denier Kevlar KM2 yarns. (a) Experimental and (b) adjusted by the computed filament volume fractions.
can be mapped onto the model. To perform this scaling, the original experimental yarn strength data is scaled by the filament volume fraction and then assigned new median ranks as shown in Fig. 4b. New CDFs are then fit to these new median ranks, using the 3-parameter Weibull distribution for each case. Table 2 lists the best-fit parameter values (shape m, scale ro, threshold x) for these distributions, as well as their coefficients of variation (CV). The CVs indicate the extent of scatter in the adjusted strength data. Next these adjusted yarn tensile strength distributions from each respective source (spool, greige fabric, scoured fabric) are separately mapped onto the warp and fill yarns of the fabric finite element model. Each yarn is assumed to possess uniform failure strength. This results in three fabric models for the subsequent impact simulations: (i) fabric with spool based strengths (baseline) (ii) fabric with greige yarn based strengths, and (iii) fabric with scoured yarn based strengths. For the fabric case with spool based strengths, since the warp and fill yarns have different volume fractions, there are two separate distributions for the warp and fill yarns as shown in Fig. 4b. For each of the three fabric models, 30 independent mappings are created for the subsequent impact simulations. Fig. 5 displays four sample mappings using spool based strengths wherein the mapped strengths (refer to the color legend) of the warp and fill yarns pertain to the adjusted strength distributions shown in Fig. 4b. As a check that the mappings were
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GL (mm)
Type
P1
P2
P3
q
Mean strength (MPa)
Standard deviation (MPa)
Spool Greige warp Greige fill Scoured warp Scoured fill
50.8 50.8 50.8 50.8 50.8
GG 3P 3P GG GG
7.9929 3.8656 5.5071 7.7090 7.8685
0.0371 365.5862 615.3447 0.0332 0.0369
0.6890 2192.0415 2209.9266 0.6434 0.2487
0.9968 0.9890 0.9942 0.9927 0.9908
2928.57 2522.63 2778.14 2244.15 2625.47
115.99 94.05 114.54 81.70 94.17
Table 2 Statistical distribution parameters used to characterize the strength data adjusted by the filament volume fraction. Yarn source
Mapped yarn direction
GL (mm)
Type
Scale (ro) (MPa)
Threshold (x) (MPa)
q
CV (%)
Spool
Warp Fill
50.8 50.8
3P 3P
4.1072 4.0853
362.5957 316.1123
1897.2930 1661.9400
0.9913 0.9913
3.96 3.96
Greige fabric
Warp Fill
50.8 50.8
3P 3P
3.9032 5.5207
280.0207 410.2568
1664.3410 1469.6840
0.9889 0.9942
3.73 4.12
Scoured fabric
Warp Fill
50.8 50.8
3P 3P
12.7657 3.7586
692.5811 243.0083
1040.5410 1527.8750
0.9923 0.9876
3.64 3.59
Shape (m)
Fig. 5. Sample yarn strength mappings using the spool based adjusted strength distributions.
performed correctly, histograms of the warp and fill yarn mappings from each of the four mapped cases in Fig. 5 were generated and found to recreate the respective adjusted strength distributions in Fig. 4b with good fidelity. In this particular implementation all elements with a single yarn have been assigned the same strength. In real fabrics, we would expect that the yarn strength would vary along the yarn length. However, for the specific problem of center impact on an edge-clamped fabric, yarn stresses are non-uniform, with
maximum principal stresses occurring at the impact location [20]. Therefore, the location of yarn failure during impact is primarily driven by load distribution, not flaw distribution, and failure will nearly always occur at the impact location. This failure localization is also observed experimentally [20] where yarn failure always occurred underneath the projectile. This supports a yarn-level mapping approach which will always result in yarn failure at the impact region underneath the projectile since it is the highest stressed region. Other quasi-static or dynamic impact
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Fig. 6. Computational setup of the fabric impact test.
problems, such as the uniform tensile loading of a yarn where failure can initiate at any point along the yarn, may require models that implement strength variations within a yarn. For such a scenario which is primarily driven by flaw distribution, not load distribution, one possible strength mapping approach is at an elementlevel where each element within the yarn is assigned to a different strength. This will allow yarn failure to occur randomly along different regions of the same yarn. Such an element-level mapping approach is beyond the scope of this paper, and will be dealt with in later studies. 2.5. Impact conditions The simulations model a single layer of Kevlar S706 fabric with four clamped edges as shown in Fig. 6. The total in-plane
dimensions of the fabric are 50.8 mm 50.8 mm. The fabric is perfectly clamped (zero slippage) on all four sides by constraining all the nodal degrees of freedom at the fabric edges. The projectile is a 5.556 mm diameter, rigid sphere with a mass of 0.692 g, representing a 0.22 caliber stainless steel ball. The projectile is modeled using shell elements. Since the projectile was observed to undergo no deformation during comparable experimental impact testing [20], a rigid material model (LS-DYNA Type 20 [9]) was used for the projectile. All impacts are at the center of the fabric, at the exact center of a yarn–yarn cross-over location. To generate PVR curves experimentally, fabric targets are impacted over a range of velocities. Shot velocities are selected to objectively interrogate the target and generate the most statistically relevant data set for generating the desired PVR parameters. Techniques such as Langlie [12] enable efficient determination of V50 values, while more powerful approaches such as Neyer-D [14] enable estimation of the full PVR behavior. These strategies use the simple, binary outcomes or penetration responses (no penetration ‘0’ or penetration ‘1’) of previous tests to determine the next shot velocity. Thus, tests proceed sequentially one at a time, and do not directly utilize residual velocity data for penetrating impacts. For computational models, however, it is often more efficient to run multiple simulations in parallel, and residual velocity data is readily available. Therefore, the existing techniques such as Langlie and Neyer-D are not ideally suited for impact velocity selection in this computational study. In the present study, simulations were performed simultaneously to rapidly generate penetration responses. The responses of each round of testing, including residual velocity values, were used to guide the selection of the next set of simulation shot velocities. These specific velocity selections were based on first roughly locating a zone of mixed results (ZMR), further defined in Section 3, where the velocity of a non-penetrating shot is higher than the velocity of a penetrating shot. Next shot velocities both lower than the penetrating shot and higher than the non-penetrating shot
Table 3 Simulation test results. Shot #
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Spool based
Greige fabric
Scoured fabric
Vi (m/s)
Response
Vr (m/s)
Vi (m/s)
Response
Vr (m/s)
Vi (m/s)
Response
Vr (m/s)
30.00 35.00 40.00 43.00 44.00 44.10 44.20 44.50 44.80 45.00 45.00 45.00 45.00 45.10 45.20 45.50 45.80 45.90 46.00 46.20 46.40 46.70 46.80 46.90 47.00 47.00 47.20 50.00 55.00 60.00
0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
– – – – – – – – – – – 6.72 – 2.24 4.34 – 3.56 7.85 5.11 8.94 9.60 – 0.22 9.67 6.99 14.42 11.05 14.11 15.19 30.13
30.00 35.00 40.00 41.00 42.00 42.10 42.30 42.50 42.50 42.70 42.80 42.90 43.00 43.10 43.10 43.20 43.30 43.50 43.50 43.80 43.80 44.00 44.20 44.40 44.60 44.80 45.00 50.00 55.00 60.00
0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1
– – – – – – – – 0.23 – – – 5.11 – 5.08 0.71 5.08 5.66 5.89 4.83 – 3.46 4.26 9.40 6.91 5.97 6.34 11.66 19.48 36.35
30.00 35.00 38.00 39.00 39.50 40.00 40.00 40.50 40.70 40.90 41.00 41.00 41.20 41.40 41.50 41.70 41.90 42.00 42.00 42.10 42.20 42.50 42.80 43.00 43.10 43.50 45.00 50.00 55.00 60.00
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1
– – – – – – – – – – 5.19 0.35 – – – – 5.65 – 5.21 – 5.82 2.05 1.03 5.41 6.72 6.07 11.24 19.33 25.57 34.50
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were chosen at closely spaced intervals in an attempt to maximize the width of the ZMR. Residual velocity values for penetrating impacts were used heuristically to determine how much to adjust the velocity interval in subsequent shot velocities. Thirty impacts for each fabric type (spool, greige, scoured) were conducted, each with an independent strength mapping, over a range of velocities. The binary response data (0 or 1) is fit with a parametric cumulative distribution function using the maximum likelihood estimator (MLE) method. Commonly assumed distributions include the normal, lognormal, and logistic functions. The selection of one particular distribution function to represent the response data is somewhat arbitrary, as it is rare to generate sufficient experimental data to be able to confidently discern differences in fit quality between the various functions. The standard NIJ 0101.06 [2] recommends using the logistic distribution, primarily due to mathematical convenience. In this study the data set is fit to a normal distribution function. Once the parametric distribution parameters (mean l, standard deviation r) have been determined, the full estimated PVR or V0–V100 curve can be generated. Since the normal distribution is symmetric, the mean (l) corresponds to the V50 velocity. 3. Results Table 3 lists the impact velocity, response (non-penetration ‘0’ or penetration ‘1’), and residual velocity of penetrating shots, for the thirty simulations of the spool based, greige, and scoured fabrics. The shots have been arranged in an increasing order of impact velocities. Two interesting observations are immediately apparent from Table 3: (i) some of the penetrating shots have an impact velocity lower than some of the non-penetrating shots and (ii) the residual velocities of some of the penetrating shots are higher than the residual velocities of other penetrating shots with lower impact velocities. For example consider the results of the spool strength based fabric from Table 3. Shot #12 has an impact velocity of 45.00 m/s and resulted in a penetration, while shot #22 with a higher impact velocity of 46.70 m/s resulted in a non-penetration. Also, comparing penetrating shots #12 and #23, we see that shot #23 with a higher impact velocity of 46.80 m/s had a lower residual velocity of 0.22 m/s compared to the residual velocity of 6.72 m/s for shot #12. These features are signatures of probabilistic penetration behavior, and would not be predicted using deterministic models. Specifically, this data indicates the presence of a zone of mixed results (ZMR). A ZMR occurs when penetrating impacts are observed at velocities below non-penetrating impacts, and is defined as the region between the highest non-penetrating impact velocity and the lowest penetrating impact velocity. A fabric with probabilistic penetration behavior will exhibit a ZMR. A wider ZMR is an indication of a fabric with less deterministic behavior. In experimental impact testing, ZMRs are commonly observed and, in some cases, required for the acceptance of an impact test series. Fig. 7 displays the results of the thirty simulations for the three fabric models. For each tested impact velocity the recorded nonpenetration or penetration results are represented by the symbols in Fig. 7. The ZMR is clearly identifiable in each figure. This important experimental phenomenon (ZMR) has now been successfully captured by the computational probabilistic framework. The fitted normal cumulative distributions which represent the respective PVR curves are indicated by the smooth lines in Fig. 7. Table 4 lists the parameters of the three fitted normal distributions. Table 5 lists the lower and upper bounds of the mean (also V50 velocity) and standard deviation at the 95% and 99% confidence levels for the three normal distributions (or PVR curves). The estimated V50 velocity of the fabric with spool based strengths is
Fig. 7. Simulation test results and PVR curves for the fabric models with yarn strengths based on (a) spool, (b) greige, and (c) scoured.
Table 4 Normal distribution parameters used to fit the simulation test results.
# Shots MLE estimate of l (m/s) MLE estimate of r (m/s) CV = r/l (%)
Spool based
Greige fabric
Scoured fabric
30 45.40 0.87 1.92
30 43.01 0.63 1.47
30 41.72 0.83 1.99
45.40 m/s. Compared to the spool strength based model, the greige and scoured fabric models respectively show a 5.3% and 8.1% drop
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Table 5 Confidence intervals for the normal distribution’s mean (l) and standard deviation (r). Confidence (%)
llower (m/s)
lupper (m/s)
rlower (m/s)
rupper (m/s)
Spool based
95 99
44.71 44.29
46.00 46.29
0.49 0.42
2.11 3.26
Greige fabric
95 99
42.46 42.10
43.46 43.67
0.33 0.29
1.61 2.49
Scoured fabric
95 99
41.11 40.82
42.43 42.87
0.43 0.36
2.13 3.29
Table 6 Comparison of probabilistic impact performance.
Spool based Greige fabric Scoured fabric
V1 (m/s)
% drop in V1
V50 (m/s)
% drop in V50
V99 (m/s)
% drop in V99
43.38 41.54 39.78
– 4.2 8.3
45.40 43.01 41.72
– 5.3 8.1
47.43 44.48 43.65
– 6.2 8.0
in the V50 velocity as seen in Table 6. Fig. 8 compares the PVR curves for all three cases. Compared to the spool based model, a clear shift towards the left in the greige and scoured fabric PVR curves can be seen indicating the weaker strength distributions of the greige and scoured fabrics have resulted in a poorer impact performance.
Fig. 10. Drop in impact performance as a function of the drop in the average of the mean yarn strengths.
4. Discussion
Fig. 8. Comparison of PVR curves.
Fig. 9. Relating drops in mean yarn strength (r0) to drops in impact performance (V50).
Through this framework a direct relationship between the statistical yarn strength behavior and probabilistic impact performance has been established. Table 6 lists the V1, V50, and V99 velocities for the three fabric models, as well as the percentage drops in each velocity performance parameter with respect to the hypothetical spool strength based fabric which serves as the baseline. The mean experimental tensile strength of the yarns from the spool is 2929 MPa [18]. Compared to this mean strength, the warp and fill yarns from the greige and scoured fabrics show percentage strength reductions as follows: 13.9% (greige warp), 5.14% (greige fill), 23.4% (scoured warp), and 10.3% (scoured fill). These strength reductions resulted in a 5.3% drop in the V50 velocity of the greige fabric to 43.01 m/s and an 8.1% drop in that of the scoured fabric to 41.72 m/s, compared to the V50 velocity of 45.40 m/s for the spool strength based fabric. This trend is shown in Fig. 9 which displays the drops in mean yarn strengths (r0) and the corresponding drops in V50 velocities of the greige and scoured fabrics as compared to the spool strength based fabric. Based on these results, one observes that the percentage drops in V50 velocity (that fails both warp and fill yarns during penetration) are approximately one-third the mean strength reduction of the weakest yarns (warp) and comparable to the mean strength reduction of the strongest yarns (fill) for both greige and scoured fabrics. The V1 and V99 velocities (see Table 6) for the greige fabric respectively showed drops of 4.2% and 6.2% compared to the spool
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strength based fabric while those for the scoured fabric were 8.3% and 8.0% respectively. It is important to note that the magnitude of the percentage drops in V1, V50, and V99 velocities of the greige and scoured fabrics compared to the spool strength based fabrics were smaller than the respective drops in tensile yarn strengths. One reason is that the spherical projectile engages multiple warp and fill yarns at the impact region. For this fabric impact scenario where the fabric is gripped on all four sides, the primary mode of energy absorption is by tensile deformation of the yarns. The kinetic energy component of the fabric due to momentum transfer between the projectile and fabric is much smaller, while the energy dissipated by frictional sliding between the yarns is very small. Thus the tensile strength of the yarns is a predominant factor that controls the penetration event. The strength data for spool, greige, and scoured yarns showed different mean strengths, but similar degrees of variability between each other, as measured by the CV which varied between 3.59% and 4.12% (see Table 2). It is not surprising then that the PVR curves for fabrics comprised of spool, greige, and scoured yarns show different V50 values but similar CV values between each other which varied between 1.47% and 1.99% (see Table 4). Fig. 10 displays the drop in V50 velocity as a function of the drop in the average of the mean warp and mean fill yarn strengths, as compared to the spool strength based model. Here the average is taken between the mean yarn strength (r0) of the warp and fill yarns (as measured experimentally, without the vfilament scaling used in the finite element model) from the same yarn source i.e. f greige fabric or scoured fabric ððrw 0 þ r0 Þ=2Þ. For example the percentage mean experimental strength drops of the greige yarns compared to the spool yarns were as follows: (warp) 13.90% and (fill) 5.14%, which results in a percentage drop of 9.52% in the average of the mean experimental greige yarn strengths. Similarly
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the percentage drop in the average of the mean experimental scoured yarn strengths is 16.85%. The spool strength based fabric serves as the baseline and correspondingly has a zero percentage strength drop. Recall from Table 6 that the respective drops in the V50 velocities for the greige and scoured fabrics were 5.3% and 8.1%. Surprisingly a linear trend line well describes the data shown in Fig. 10. The slope of the trend line is approximately 0.5 indicating that for every 10% drop in the average of the mean warp and fill yarn strengths, there is a 5% drop in the V50 velocity. Perhaps this trend occurs because the fabric patch is square in shape and the same number of warp and fill yarns are engaged by the projectile at the impact region. Another possible reason is that the degrees of variability (or CV) in the yarn strength distributions were about the same for the spool, greige, and scoured yarns, which made it possible to consider the average of the mean yarn strengths. Had the yarn strength distributions showed wide variations between each other in the CV, or had the projectile engaged a different number of warp and fill yarns in a non-square fabric target, such a simple linear relationship between the drop in V50 velocity and the drop in the average of the mean yarn strengths as shown in Fig. 10 would probably not exist. This insight is underscored by the discussion in Section 1 where the probabilistic impact performance of a fabric must be gauged using the entire PVR curve, since it is possible that two fabric systems have the same V50 velocity but different V1 velocities, for a variety of reasons that include constituent yarn strength distributions with the same mean strength but widely varying degrees of scatter. In the same manner, the mean yarn strength cannot solely be used as an estimator of performance, rather the entire CDF of the yarn strength distributions must be considered. Fig. 11 displays rear view (non-impacted fabric face) states of fabric deformation and penetration at various time instants for
Fig. 11. Spool strength based fabric impacted at 55 m/s, rear view of deformation states at time instants of (a) 120 ls, (b) 135 ls, (c) 140 ls, and (d) 145 ls.
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the spool strength based fabric impacted at a velocity of 55.00 m/s, which is much higher than the estimated V50 velocity of 45.40 m/s. For this particular simulation run, the projectile residual velocity was 15.19 m/s. The central warp yarn was the first yarn to fail as seen in the deformation profile at time instant 120 ls, followed by the central fill yarn as seen at the 135 ls instant. After the failure of these two central yarns, the surrounding warp and fill yarns begin to fail until the projectile is able to completely penetrate through the fabric. The diameter of the projectile is 5.556 mm and this spans around 7 warp and 7 fill yarns. However as seen in Fig. 11d at the time instant of 145 ls, only
5 warp and 5 fill yarns have failed before the projectile is able to push the neighboring warp and fill yarns apart and penetrate completely through the fabric. Thus the projectile is pushing its way through a hole in the fabric smaller in size than the projectile’s diameter. Fig. 12 displays rear view states of fabric penetration at various time instants for the spool strength based fabric impacted at a velocity of 45.90 m/s which is slightly higher than the estimated V50 velocity of 45.40 m/s. For this particular simulation run, the projectile residual velocity was 7.85 m/s. Contrary to the deformation states in Fig. 11, for this case the central fill yarn fails first followed by the central warp yarn. Although the
Fig. 12. Spool strength based fabric impacted at 45.9 m/s, rear view of deformation states at time instants of (a) 155 ls, (b) 165 ls, (c) 170 ls, (d) 175 ls, (e) 180 ls, and (f) front view – 245 ls.
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mapped strengths of the warp yarns are weaker than the fill yarns as seen in Fig. 4, recall the warp yarns have greater crimp (undulations) and therefore take longer to straighten out only after which they begin to get stressed in tension. Therefore depending on the extent of imbalance in crimp and the particular strength mapping used in a simulation run, either the warp yarn or fill yarn may fail first. Once again in this set of deformation states, only 5 warp and 5 fill yarns fail before the projectile is able to push its way through the fabric. This behavior is more clearly seen in Fig. 12f at the time instant of 245 ls which provides a front view (impacted fabric face) of the deformation after the projectile has fully penetrated through the fabric. The non-failed warp and fill yarns at the periphery of the hole that were pushed aside by the projectile as it was penetrating through the fabric have now sprung back to their initial positions. The sequence of yarn failure as seen by the deformation states of Figs. 11 and 12 are different from each other since both runs had different strength mappings. In a similar manner, even if two simulation runs are conducted at the same impact velocity, the sequence of yarn failure and deformation states would vary from each other. This phenomenon which is also observed experimentally is made possible by the probabilistic computational framework. It is important to keep in mind while drawing conclusions from this study that the effects of statistical yarn strength distributions have been isolated for investigation and all other intrinsic and extrinsic sources of variability have been eliminated. This provides a systematic approach to study the probabilistic impact performance of these high-strength and high-modulus fabrics used in protective applications. In reality, there are many factors that affect the impact performance and these factors may be highly coupled. Based on the four sided perfectly clamped boundary conditions used in this study we assumed a static coefficient of yarn friction that spatially remained constant throughout the entire fabric domain. However in a fabric gripped only on two sides or at the four corners, filament and yarn level frictional sliding interactions become very important mechanisms of energy transfer and consequently the fabric response is sensitive to the setup and modeling of frictional interactions. For example the frictional coefficient may vary with the relative velocity between two sliding surfaces, the degree of transverse yarn compression (higher at the impact site), as well as the type of surface treatment or sizing applied to the yarns. As mentioned earlier, in certain situations a projectile that impacts a fabric between two yarns may result in a different response than if the projectile had directly impacted at the center of a yarn cross-over. However the extent of this effect of the projectile impact location depends, amongst other factors, on the level of filament and yarn frictional interactions demonstrating how these sources of variability may be highly coupled. Thus by first isolating each source of intrinsic and extrinsic variability for a given impact scenario and understanding its effect on the probabilistic impact performance, these highly coupled factors can then be combined together in various fashions to understand their overall effect on the probabilistic impact performance. The yarn material properties used in this study correspond to experimental data obtained from quasi static testing of these Kevlar yarns. This is a common practice adopted in the finite element modeling of fabric impact due to the lack of such data at high strain rates [24]. However using the probabilistic framework presented in this study, once such high rate data is available, its effect on the probabilistic impact performance can be studied and more importantly, it can also be compared to the predictions using quasi static based data. Such a study would also allow the effect of strain rate to be studied and can only be accomplished computationally for obvious reasons.
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5. Conclusions A probabilistic computational framework has been developed that simulates the effect of statistical yarn strength on the probabilistic penetration performance of a woven fabric. This model maps experimentally obtained yarn strengths directly onto the individual yarns of the fabric finite element model. The framework was applied to Kevlar S706 greige and scoured fabrics, as well as the corresponding fabric based on spool yarn strengths. Using the results (projectile impact velocity and penetration history) from the impact simulations, probabilistic velocity response (or V0– V100) curves were generated for fabrics based on spool, greige and scoured 600 denier Kevlar KM2 yarn strengths. The implementation of probabilistic yarn strength models directly enabled the prediction of a probabilistic fabric penetration response, a behavior that is rarely captured in traditional fabric impact simulations. The estimated V50 velocity for the fabric based on the spool strengths was 45.40 m/s while that of the greige and scoured fabrics were respectively 43.01 m/s and 41.72 m/s. Weaving and scouring processes degraded the yarn strengths by shifting the parametric CDF strength distributions to lower strengths. These strength drops had a direct effect on the PVR curves of the greige and scoured fabrics by shifting them towards the left or lower impact velocities, which culminated in a drop in the V50 velocity of the greige fabric by 5.3% and the scoured fabric by 8.1% compared to the baseline fabric based on the spool yarn strengths. For all three fabric cases studied, the degree of uprightness of the three PVR curves as indicated by the coefficient of variation of the normal distribution used to fit the simulation results remained about the same. This trend could be attributed to the similarity in scatter of the yarn strengths for the yarns from the spool, and greige and scoured fabrics, since the impact event in this study corresponded to a fabric held on four sides where yarn tensile strengths are a predominant controlling factor. By systematically isolating each source of variability, such as statistical yarn strengths and studying its effect on the probabilistic impact performance, the degree of predictability of computational woven fabric models can be significantly increased and lend insight into this complex probabilistic event that involves highly coupled factors. Further, certain investigations that could not be performed experimentally can now be accomplished through this probabilistic framework. One example is studying the effects of weaving strength degradations on the impact performance. Parameters that are difficult to experimentally control can now be precisely controlled in this computational study, such as a zero-slippage four sided boundary condition and a shot location exactly at the fabric dead center. Other factors such as the characteristics of the projectile (size and shape), fabric boundary conditions (two sides held, four corners held), shot location, yarn material properties (frictional coefficient, modulus) can now be systematically incorporated into the probabilistic computational framework to assess their individual and combined effects on the PVR curve. These factors will be dealt with in future studies. It is envisioned that the simple probabilistic computational framework presented in this study will pave the way forward for more advanced computational models that significantly improve the degree of accuracy and predictability of impact simulations over the deterministic models from the past few decades, and ultimately will reduce the dependence on destructive experimental impact testing leading to significant savings in both time and cost. A limitation of this study is that the simulations results are not compared to experimental impact data in a qualitative or quantitative way. In a previous study [20], we attempted to perform such a comparison, but found that the experimental results were highly sensitive to clamping conditions and the degree of fabric slippage
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