# Modeling and Experiments on Ballistic Impact into UHMWPE Yarns Using Flat and Saddle-Nosed Projectiles

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## Abstract

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^{®}SK76 yarns (1760 dtex), twisted to 40 turns/m, and initially tensioned to stresses ranging from 29 to 2200 MPa. Yarns were impacted, transversely, by two types of cylindrical steel projectiles at velocities ranging from 150 to 555 m/s: (i) a reverse-fired, fragment simulating projectile (FSP) where the flat rear face impacted the yarn rather than the beveled nose; and (ii) a ‘saddle-nosed projectile’ having a specially contoured nose imparting circular curvature in the region of impact, but opposite curvature transversely to prevent yarn slippage off the nose. Experimental data consisted of sequential photographic images of the progress of the triangular transverse wave, as well as tensile wave speed measured using spaced, piezo-electric sensors. Yarn Young’s modulus, calculated from the tensile wave-speed, varied from 133 GPa at minimal initial tension to 208 GPa at the highest initial tensions. However, varying projectile impact velocity, and thus, the strain jump on impact, had negligible effect on the modulus. Contrary to predictions from the classical Cole-Smith model for 1D yarn impact, the critical velocity for yarn failure differed significantly for the two projectile types, being 18% lower for the flat-faced, reversed FSP projectile compared to the saddle-nosed projectile, which converts to an apparent 25% difference in yarn strength. To explain this difference, a wave-propagation model was developed that incorporates tension wave collision under blunt impact by a flat-faced projectile, in contrast to outward wave propagation in the classical model. Agreement between experiment and model predictions was outstanding across a wide range of initial yarn tensions. However, plots of calculated failure stress versus yarn pre-tension stress resulted in apparent yarn strengths much lower than 3.4 GPa from quasi-static tension tests, although a plot of critical velocity versus initial tension did project to 3.4 GPa at zero velocity. This strength reduction (occurring also in aramid fibers) suggested that transverse fiber distortion and yarn compaction from a compressive shock wave under the projectile results in fiber-on-fiber interference in the emerging transverse wave front, causing a gradient in fiber tensile strains with depth, and strain concentration in fibers nearest the projectile face. A model was developed to illustrate the phenomenon.

## 1. Introduction

^{®}and Spectra

^{®}, or aramid fiber under trade names such as Twaron

^{®}or Kevlar

^{®}. Such yarns are available in a wide variety of linear densities (deniers), and materials processing history, which affects their axial stiffness and tensile strength.

^{3}for UHMWPE and 1.44 g/cm

^{3}for aramids) and individual fibers typically exhibit elastic behavior all the way to tensile failure with strengths exceeding 3 GPa. Also, as studied by Phoenix and Skelton [1], under transverse compression, individual fibers exhibit plastic-like yielding at stresses of approximately 40 MPa, which is almost two orders of magnitude lower than their tensile strengths, yet they can tolerate considerable, permanent distortion of their cross-sectional shape with little sacrifice in tensile strength, as shown by Cheng et al. [2] and Golovin and Phoenix [3]. Thus, during transverse compression of a yarn, void space between fibers can be squeezed out so that the transverse stiffness and resisting compressive stress becomes governed by bulk modulus properties of the fiber material. These unique features, not shared by brittle carbon or E-glass fibers, make such polymeric fibers very forgiving of large transverse loads, and thus, well suited to ballistic protection applications, as discussed by van der Werff and Heisserer [4].

#### 1.1. Fundamental Parameters Governing the Ballistic Resistance of Fibrous Systems

_{50}, as obtained from an extensive set of controlled, laboratory ballistic experiments on multi-ply fabric and panel systems. The projectiles used were standardized, right circular cylindrical (RCC) projectiles of various weights and dimensions.

_{50}ballistic performance through a two-parameter function, $\mathsf{\Phi}$, having overall mathematical structure

^{®}1000 UHMWPE yarns, Cunniff [5] found the associated UHMWPE fabrics and composites performed below expectations in that the critical value, ${V}_{\mathrm{C}}=\sqrt[3]{{U}^{*}}$, had to be artificially reduced by 16% in order for Equation (1) to be able to resolve the data. Cunniff proposed that an effective strength reduction (by about 23%) may be the result of strength loss from thermal softening connected to the relatively low melting point of UHMWPE fibers compared to the higher thermal stability of other polymeric fibers, such as aramids and virgin PBO.

^{4}/s, and thus, is 6 to 7 orders of magnitude larger than typical values of 10

^{−2}/s to 10

^{−3}/s used in quasi-static tension tests. Various methods have been used to determine Young’s modulus and sometimes the strength of such materials at high strain rates. Lim et al. [13] used a miniature Kolsky bar, and others have used a split Hopkinson bar, as discussed by Wang [14], who points out that the results take considerable experimental sophistication to interpret. The method we use involves ‘yarn shooting experiments’ whereby a single yarn, under a prescribed level of initial tension, is subjected to transverse impact by a projectile. The yarn strength and Young’s modulus are calculated from the evolving transverse and tension waves using results from classical theory for impact into a 1D string, as discussed below.

#### 1.2. Use of Yarn Shooting Experiments to Measure Fundamental Yarn Mechanical Properties

#### 1.3. Experiments on Ballistically Measured Yarn Modulus and Strength in the Literature

^{®}1000 yarns of 195 dtex and density 1000 kg/m

^{3}, and Kevlar

^{®}yarns of 938 dtex (844 denier) and density 1470 kg/m

^{3}. The tensile wave-speed in the yarn was calculated from the transverse wave-speed and the projectile velocity measured from high-speed photographs and interpreted using the Cole-Smith equations. They calculated a tensile modulus of $310\text{}\mathrm{GPa}\pm 15\%$ for the Spectra

^{®}1000 and a value $225\text{}\mathrm{GPa}\pm 15\%$ for Kevlar

^{®}. In their experiments, Prevorsek et al. [20] used various values of pretension, which in the case of Spectra

^{®}1000, ranged from 250 MPa to 1500 MPa, however, little if any effect was observed on Young’s modulus, and the same was true for Kevlar

^{®}. Many of the results in Prevorsek et al. [20] were also reported in paper by Field and Sun [27], who provided additional details.

^{®}1000 are about 110 GPa in quasi-static tension tests (other sources give values in the 120 GPa range), so the reported value of 310 GPa represented an increase by almost a factor of 3. Unfortunately, they did not mention the particular Kevlar

^{®}version used, nor was a quasi-static Young’s modulus value given (although Field and Sun [27] did indicate the use of Kevlar

^{®}29 at one point, but not elsewhere in the paper). Kevlar

^{®}129 is available as an 840 denier yarn and quasi-static modulus values of 95 GPa can be found in DuPont product literature. Their use of Kevlar

^{®}KM2 cannot be ruled out, however, since this yarn is available as an 850 denier yarn, and Cheng et al. [2] obtained a Young’s modulus value of about 85 GPa. Nonetheless, from the yarn shooting experiments of Prevorsek et al., the measured modulus of the Kevlar

^{®}fibers used exceeds these quasi-static values by a factor of at least 2.3. Indeed, the values obtained by Prevorsek et al. [20] are much higher than the high strain-rate values, 112 to 143 GPa, developed by Lim et al. [13] for Kevlar

^{®}129, using a miniature Kolsky bar (where the values vary depending on whether one assumes a tangent or secant modulus).

^{®}1000 yarn and what was said in the paper to be Kevlar

^{®}29 yarn, but using a very different projectile, namely a sabot with a razor edge attached. Interestingly, the same linear densities were quoted as those used by Prevorsek et al. [20] (one co-author appeared on both works). Perhaps there was a typographical error and the fiber used was Kevlar

^{®}129 as we have conjectured above in the Prevorsek et al. experiments, since Kevlar

^{®}29 does not appear to be available in 840 denier form. Based on results using pre-stress values ranging from about 30 MPa to 700 MPa, the projected modulus values at zero tension for Spectra

^{®}1000 and their Kevlar

^{®}were, respectively, 159 GPa and 122 GPa, which are about half the values reported by Prevorsek et al. [20] and Field and Sun [27]. Wang et al. [16] also noted a corresponding 23% and 17% increase, respectively, in these modulus values when preloads were increased from 54 to 658 MPa and 32 to 478 MPa, respectively. These authors interpreted this modulus increase with pretension as implying that the fiber has a non-linear stress-strain curve, yet the effect of impact velocity on modulus (i.e., through the tensile wave-speed) appeared to be negligible, which seems contradictory.

^{®}1000 and Kevlar

^{®}were 715 MPa and 555 MPa, respectively, or about 1/5th the values found from quasi-static tension tests. At ballistic loading rates the fiber modulus values obtained were about half those obtained by Prevorsek et al. [20] and Field and Sun [27], though were consistent with the high strain-rate values obtained by Lim et al. [13]. Later on, Kavesh and Prevorsek [28] referred to an unpublished report wherein a Young’s modulus of 230 GPa was calculated from a directly measured, tensile wave-speed in Spectra

^{®}fiber, though the exact version and level of pre-tension was not specified.

^{®}SK66) and aramid yarns (930 dtex Kevlar

^{®}129, 940 dtex Kevlar

^{®}KM2, as well as 930 dtex Twaron CT yarns) where the main purpose was to investigate the failure modes during impact failure, rather than the critical velocity and yarn failure stresses. The projectile used was a steel sphere of mass 0.68 g and diameter 5.50 mm, and impact velocities ranged from 346 to 720 m/s. Note that the fully dense yarn diameters (viewed as one large cylindrical fiber without air spaces) were approximately 0.239 mm and 0.338 mm for the Dyneema

^{®}SK66 yarns and 0.286 mm for the Kevlar

^{®}and Twaron

^{®}yarns, so accounting for roughly 80% packing of their filaments, these diameters would inflate by about 12% to 0.268, 0.378 and 0.320. These diameters are still more than an order of magnitude smaller than the projectile diameter.

^{®}SK66 yarns and generally 618 m/s for the aramid yarns. An interesting feature associated with the so-called, high velocity shear mode was that a “shear plug” consisting of a finite length of yarn would be severed and then carried along by the projectile after failure. In the lower velocity TSW mode, the usual transverse triangle wave would be observed for a short time before yarn failure, which presumably was more distributed in nature, though not stated specifically beyond mentioning “gross permanent disruption of the yarn structure in all specimens for a distance of about 40 mm”, which is about 7 projectile diameters. The appearance of these two types of failure modes will be the subject of further discussion, but here it suffices to note that the existence of a severed shear plug counters the notion that a strain concentration naturally occurs at the midpoint of spherical contact.

_{0}value, 670 m/s, as well as the deflection angle, $31\xb0$, results in a modulus value 115 GPa, so virtually identical to their 114 GPa quasi-static value.

^{®}SK76 UHMWPE (1760 dtex) and Twaron

^{®}(1760 dtex) yarns, and used a fixed 2.0 Kg weight to apply tension to the specimens. They used both an RCC flat-faced cylindrical projectile as well as a specially designed “saddle” projectile designed to capture the yarn (so it does not slide off the projectile nose) and to induce curvature in the yarn in the impact region, similar to that generated by a spherical projectile. They used the same theory as Prevorsek et al. [20] and calculated Young’s moduli of about 198 GPa and 135 GPa for their Dyneema

^{®}SK76 and Twaron

^{®}yarns, respectively, with limiting impact velocities said to be 485 m/s and 380 m/s, respectively. They did not, however, attempt to report yarn tensile strength results, possibly because of the obscuring effects of yarn fraying associated with having untwisted yarns. The moduli they calculated were said to be higher than the quasi-static values by factors of 1.5 and 1.3, respectively, but again, are much lower than those obtained by Prevorsek et al. [20] for materials of similar type.

^{®}S5705, Dyneema

^{®}SK65 and Zylon

^{®}PBO yarns. For a range of incremental impact velocities, it was determined whether or not yarns fail immediately by evaluating frames from high speed video. This gave ‘bracket’ values for the so-called critical threshold velocities causing yarn failure, which were lower than expected. Regarding the transverse wave speed, Smith’s theoretical predictions were compared with both experimental values and numerical values generated using LS-Dyna software using orthotropic continuum elements to model the yarn. A good match was found between the transverse wave speeds obtained by the three methods. The ability to accurately predict the transverse wave speed in impacted yarns is critical to being able to model impact into complex single and multi-layer woven systems for comparison to experiments.

^{®}KM2 fiber using a sufficient number of elements over the cross section to resolve a non-uniform stress state using a linear, orthotropic material model. A sensitivity study of the effect of the longitudinal shear moduli indicated that flexural waves next to the projectile are observed for increasing shear stiffness. Sockalingam et al. also showed that to model the quasi-static transverse compression of Kevlar

^{®}KM2 [41,42] and Dyneema

^{®}SK76 [43], it is crucial to include nonlinear inelastic behavior. If this approach is applied to transverse impact on 600 denier KM2 yarn with 400 individually modelled fibers (84 three-dimensional solid elements in the fiber cross section), a multi-axial stress state occurs with progressive loading through the thickness, as well as fiber squashing [44]. From the model, the predicted critical impact velocity (causing yarn failure) is about 500 m/s, while the classical value is over 900 m/s. The critical velocity is also sensitive to the longitudinal shear modulus where higher moduli result in lower velocities. In this case failure starts at the back of the curved fiber bundle.

#### 1.4. Key Issues Arising from Study of Yarn Shooting Experiments in the Literature

- Most studies report an increase in the effective fiber Young’s modulus at ballistic loading rates as compared to values calculated from quasi-static tension tests. However, the increase varied from almost none to an almost tripling of the quasi-static value.
- Studies of the effect on Young’s modulus of applying a steady yarn pre-tension prior to impact, were contradictory regarding the effect of increasing the yarn tension level.
- The tensile strengths calculated from yarn shooting experiments in the various studies fell short of quasi-static values by from 20% to as much as 80%.
- Energy absorbed by single yarns near the critical projectile impact velocity for yarn failure had very different trends for UHMWPE yarns versus aramid yarns, increasing with projectile velocity in the former and decreasing with projectile velocity in the latter.

#### 1.5. Overview of the Paper

## 2. Analysis of the Impact of a Flat-Faced Projectile on a 1-D Yarn

#### 2.1. Strain Immediately Following Impact

#### 2.2. Strain at Long Times

#### 2.3. Transverse Wave Angle at Long Times

_{i}, prior to wave collisions under the projectile. This means that the strain, $\Delta {\epsilon}_{\mathrm{i}}$ immediately after impact, and before tension waves clash under the projectile, is smaller than the eventual, steady state value, $\Delta {\epsilon}_{\infty}$. The situation is different for the short-acting, strain enhancement resulting from collision of tension waves under the projectile, as we consider next.

#### 2.4. Strain Jumps Caused by the Collision of Edge-Emitted Tension Waves Traveling under the Projectile

#### 2.5. Strain Concentrations Arising from Tensile Wave Collision under the Projectile

#### 2.6. Penalty on Critical Failure Velocity Due to Strain Concentration from Wave Collision

## 3. Experimental Apparatus, Materials, and Measurement Systems

#### 3.1. Yarn Materials

^{®}SK76 with material density ${\rho}_{0}=0.980\mathrm{g}/{\mathrm{cm}}^{3}$ (or $980\mathrm{kg}/{\mathrm{m}}^{3}$) and twisted to 40 turns per meter (tpm). The yarn had linear density, 1760 dtex, so the total cross-sectional area of the fibers in the yarn was 0.1796 mm

^{2}. When fully compacted to a fiber volume fraction of 100%, the yarn would have an effective cylindrical diameter of 0.478 mm and outer-fiber helix angle of $3.44\xb0$, whereas at 80% fiber volume fraction, the effective diameter is 0.534 mm and the helix angle to $3.85\xb0$.

^{®}SK76 and Twaron

^{®}aramid yarns, progressive fiber fraying could be seen in the growing transverse wave even at impact velocities more than 100 m/s below the critical velocity. The likely reason for the fraying is the existence of randomly distributed flaws along the individual filaments that results in both variability in their strengths and a reduction in strength with gage length, typically using Weibull statistics as discussed in Porwal et al. [47].

#### 3.2. Reverse-Fired FSP and “Saddle” Nosed Projectiles

#### 3.3. Yarn Shooting Test Apparatus and Recording Equipment

## 4. Results and Observations from Yarn Shooting Experiments

^{®}SK76 yarns under impact velocities ranging from 25 m/s to 550 m/s and tensile loads from 0.535 Kg to 40 Kg corresponding to fiber tensile stresses from 29.22 MPa to 2185 MPa. In each case, the results are for velocities ranging from the critical velocity to well below that value. Except perhaps at the very lowest load of 0.535 Kg, the tensile wave-speed measured was minimally influenced by the impact velocity by at most 5%.

#### 4.1. Yarn Young’s Modulus Versus Pre-Stress and Impact Velocity

^{®}SK76 yarn with very little or no twist, under a load of 2.012 Kg, Utomo and Broos [25] reported a value of about 196 GPa. In our case, at a load of 2 kg the measured modulus is considerably lower at about 150 GPa, and to achieve 196 GPa requires a load in our case corresponding to 20 Kg. While the difference would seem to be the result of twist in the yarn (40 tpm vs. almost none), we must also note that our modulus values were calculated directly from the measured tensile wave speed, whereas the values of Utomo and Broos [25] were calculated indirectly from the tangent of the deflection angle (projectile velocity over transverse wave-speed in ground coordinates) using the Cole-Smith equations.

^{®}but using 40 tpm. Unlike the situation with Dyneema

^{®}SK76, at a 2 Kg load we obtained a close but slightly higher value, 141 GPa, than their reported value of about 135 GPa, and interestingly this value is much larger than the value often quoted 90 GPa from standard tension tests. We note that, because of the difference in densities, the Twaron

^{®}outer helix angle was significantly less than for the Dyneema

^{®}SK76, making the twist effect smaller.

^{®}. Prevorsek et al. [20] used photographic images of the growth of the transverse wave rather than directly measuring the tensile wave-speed. The various results for Young’s modulus obtained for UHMWPE fibers are summarized in Figure 10, and what singularly stands out is the very high tensile modulus for Spectra

^{®}1000 of about 310 GPa as was mentioned in the Introduction. This value is particularly puzzling in view of the much lower values obtained by Wang et al. [16], who apparently used the same material (and shared a common co-author). It might also be noted that the Prevorsek et al. [20] value of 225 GPa for Kevlar

^{®}, which appears to have been either Kevlar

^{®}129 or Kevlar

^{®}KM2, is far higher than the value 137 GPa obtained by Utomo and Broos [25] for a similar fiber Twaron

^{®}, and the even lower value 122 GPa reported by Wang et al. [16] apparently for the same Kevlar

^{®}used by Prevorsek et al. [20].

^{®}SK76 for the same preload stress and the value obtained by Wang et al. [16] for the same material and preload stress. Prevorsek et al. [20] did not provide geometric details of the yarn specimens, especially their length, but they were suspended vertically with a clamp at one end and a hanging weight on the other. However, based on more extensive photograph sequences and figures provided in Field and Sun [27], which were not duplicates of those in Prevorsek et al. [20], we estimated that the upper clamp or guide restricted transverse motion approximately 10 cm above the impact point, which agrees with the value 0.1 m stated in the paper. The type of clamp was not described, so the potential for yarn slippage and attenuation of the tension jump upon reflection from the clamp could not be assessed (although the use of neoprene is indicated in their figure). At an impact velocity of 130 m/s, use of the last two photographic frames to determine the transverse wave angle (at about 100 $\mathsf{\mu}\mathrm{s}$ after impact) would require a gage length of about 1 m, impacted centrally, to avoid strain increases from tensile wave reflections from the extremities, which in turn would more than double the tension and lower the angle beginning at the transverse wave-front and progressing backwards. Avoiding this problem would require placing the upper clamp at 0.5 m above the impact point, so much more than the 0.1 m mentioned in the paper, and thus, yarn slippage could not mitigate the strain increases resulting from the implied multiple reflections of the tensile wave. These observations are provided to offer an explanation for the large discrepancy between the results of both Prevorsek et al. [20] and Field and Sun [27], and our measurements as well as those of Utomo and Broos [25] for Dyneema

^{®}SK76 and also those of Wang et al. [16] for Spectra

^{®}1000.

#### 4.2. Critical Velocity for Yarn Failure Versus Projectile Type and Yarn Pre-Stress Level

#### 4.3. Yarn Failure Stress and Strain at Critical Velocity Versus Tensile Pre-Stress

^{®}SK76 yarns tested here, but was also specifically noted in the experiments of Bazhenov et al. [22] on aramid yarns, and upon further study, is apparent also in most of the yarn shooting experiments mentioned above, whether using UHMWPE or aramid yarns.

^{®}1000 multi-layered fabrics. Cunniff [5] achieved V50 velocities that required a normalizing critical velocity of ${V}_{\mathrm{C}}=672\mathrm{m}/\mathrm{s}$. He originally assumed a yarn strength of ${\sigma}_{\mathrm{max}}=2.57\text{}\mathrm{GPa}$, a Young’s modulus $E=120\text{}\mathrm{GPa}$, and failure strain, ${\epsilon}_{\mathrm{max}}=0.0350$, (implying non-Hookean stress-strain behavior), and this resulted in ${V}_{\mathrm{C}}=801\text{}\mathrm{m}/\mathrm{s}$, which was too high in value. Using Equation (4) assuming the lower critical velocity ${V}_{\mathrm{C}}=672\mathrm{m}/s$ requires a lower tensile strength of ${\sigma}_{\mathrm{max}}=2.0\text{}\mathrm{GPa}$, assuming the same Young’s modulus, $E=120\text{}\mathrm{GPa}$, and scaled-back failure strain. Even this strength is higher than our experimental value of ${\sigma}_{\mathrm{max}}=1.70\text{}\mathrm{GPa}$ at zero pre-stress.

#### 4.4. Strain Concentration from Impact and Shock Wave Distortion of the Yarn Cross-Section

^{®}SK76, said earlier to have diameter ${h}_{0}\approx 0.53\text{}\mathrm{mm}$ (with 20% air voids between fibers) before impact, are estimated to flatten under the projectile during impact to approximate dimensions of perhaps ${h}_{\mathrm{c}}\approx 0.25\text{}\mathrm{mm}$ and perhaps 1.00 mm width (i.e., approximately a rectangular cross-section with aspect ratio of 4 and slightly reduced void space—see Song et al. [34] for a demonstration of this phenomenon). The effective velocity of such a compressive shockwave, is not easy to calculate for such a complex transverse yarn structure with voids, but may well be of the order of 1200 m/s, so that compaction would be complete in perhaps in as little as $0.5\text{}\mathsf{\mu}\mathrm{s}$, though still long enough for the tension wave to travel 5 to 6 mm.

## 5. Conclusions

^{®}SK76 twisted yarns with various pretension levels and using two types of projectiles, a flat-faced cylindrical projectile (reverse-fired FSP) and a special saddle-nosed projectile. Experimental data consisted of a time-sequence of high-speed, photo-images of the growth of the triangular transverse wave, as well as direct measurement of the tensile wave speed using two precisely-spaced, piezo-electric sensors. The yarn Young’s modulus calculated directly from the tensile wave-speed varied from 133 GPa at almost no initial tension (slightly higher than that from standard tension tests) to 208 GPa at the highest initial tensions. However, the actual impact velocity had little effect on the measured Young’s modulus.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**Flat surface (RCC) projectile distortion process in the impacted yarn in the first few microseconds after impact (ignoring the effect of tensile wave collision under the projectile).

**Figure A2.**Curved surface (saddle) projectile distortion process in the impacted yarn in the first few microseconds after impact, including effect of tensile wave collision under the projectile.

**Figure A3.**Stress concentration under no tension and for various distortion velocity ratios and average strain values. Initially $\eta $ is small, and $\phi $ is fairly close to one in value.

## Appendix B

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**Figure 1.**Classic diagram of a projectile impacting a 1D string. (Based on a diagram in Rakhmatulin and Dem’yanov [15] but revised and re-drawn with our notation.)

**Figure 2.**Evolution in terms of multiple frames (snapshots) over time of the colliding tension waves and subsequent reflections under a flat-faced projectile impacting a frictionless yarn. Multiple arrows also show the direction of particle flow.

**Figure 5.**Three photographs showing a sequence of exposures of transverse wave progression in three yarn shooting tests (The white bar in the right image is a photographic artifact).

**Figure 7.**Yarn Young’s modulus calculated from the results shown in Figure 6 as well as additional impact experiments at low velocity.

**Figure 8.**Young’s modulus measured at the various pretension loads and averaged over various impact velocities.

**Figure 9.**Comparison of the measured image angle and calculated angle from the Cole-Smith theory (Appendix B) and Equation (60).

**Figure 10.**Comparison of various Young’s results reported in the literature for UHMWPE fiber in connection to yarn shooting experiments.

**Figure 11.**Comparison of experimental results from the right circular cylindrical (RCC) and saddle projectiles with the theoretical results on the stress concentration from wave collision under the flat nose of an RCC.

**Figure 12.**Calculated strength versus pre-stress and critical velocity for the two projectile types. ‘sc’ denotes the flat RCC (reverse-fired fragment simulating projectile (FSP)) case while ‘no sc’ refers to the saddle projectile.

**Figure 13.**Calculated strength versus pre-stress and critical velocity for the two projectile types.

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**MDPI and ACS Style**

Phoenix, S.L.; Heisserer, U.; Van der Werff, H.; Van der Jagt-Deutekom, M.
Modeling and Experiments on Ballistic Impact into UHMWPE Yarns Using Flat and Saddle-Nosed Projectiles. *Fibers* **2017**, *5*, 8.
https://doi.org/10.3390/fib5010008

**AMA Style**

Phoenix SL, Heisserer U, Van der Werff H, Van der Jagt-Deutekom M.
Modeling and Experiments on Ballistic Impact into UHMWPE Yarns Using Flat and Saddle-Nosed Projectiles. *Fibers*. 2017; 5(1):8.
https://doi.org/10.3390/fib5010008

**Chicago/Turabian Style**

Phoenix, Stuart Leigh, Ulrich Heisserer, Harm Van der Werff, and Marjolein Van der Jagt-Deutekom.
2017. "Modeling and Experiments on Ballistic Impact into UHMWPE Yarns Using Flat and Saddle-Nosed Projectiles" *Fibers* 5, no. 1: 8.
https://doi.org/10.3390/fib5010008