# Role of Inelastic Transverse Compressive Behavior and Multiaxial Loading on the Transverse Impact of Kevlar KM2 Single Fiber

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## Abstract

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^{®}KM2 fibers exhibit a nonlinear inelastic behavior in transverse compression with an elastic limit less than 1.5% strain. The effect of this transverse behavior on a single KM2 fiber subjected to a cylindrical and a fragment-simulating projectile (FSP) transverse impact is studied with a 3D finite element model. The inelastic behavior results in a significant reduction of fiber bounce velocity and projectile-fiber contact forces up to 38% compared to an elastic impact response. The multiaxial stress states during impact including transverse compression, axial tension, axial compression and interlaminar shear are presented at the location of failure. In addition, the models show a strain concentration over a small length in the fiber under the projectile-fiber contact. A failure criterion, based on maximum axial tensile strain accounting for the gage length, strain rate and multiaxial loading degradation effects are applied to predict the single-fiber breaking speed. Results are compared to the elastic response to assess the importance of inelastic material behavior on failure during a transverse impact.

## 1. Introduction

^{®}KM2, Spectra

^{®}, and Dyneema

^{®}are widely used in ballistic impact personnel protection applications [1,2] in the form of flexible textile woven fabrics and laminates. Ballistic impact onto these materials is a complicated multiscale problem due to the hierarchical material structure, projectile geometry, anisotropic material behavior, as well as other factors. The simplest way of understanding the impact response of these materials is through the impact onto a single fiber. However, the current state-of-the-art experimental capabilities in transverse impact testing do not have the spatial resolution to monitor individual single fiber (Kevlar KM2 fiber 12.0 µm in diameter) deformations in real-time. The impact experiments are typically conducted at the yarn and fabric length scales. The yarn transverse impact experiments are, in general, focused on measuring the transverse wave velocity and the transverse wave ‘V’ angle. The transverse wave velocity from the Smith theory [3] is reported to correlate approximately with the experimental measurements [4]. However, experimentally-measured yarn breaking speeds are reported to be significantly lower (up to 40%) than the theoretical predictions [3], as shown in Table 1. The theoretical 1D solution (Equation (1)) assumes that the yarns are homogeneous (i.e., the theory does not differentiate between a single fiber and a yarn) and loaded only in uniaxial tension (i.e., the theory does not consider gradients of stresses within the fiber/yarn and does not consider projectile-fiber contact interactions that induce multi-axial loading and progressive failure). The theoretical (Equation (1)) breaking speed for the transverse impact of a 0.30 caliber fragment simulating projectile (FSP) onto an 850 denier KM2 yarn with a 4% failure strain is 926 m/s while the experimental breaking speed falls between 621 and 634 m/s [5], respectively.

## 2. Single-Fiber Transverse Impact

#### 2.1. Cylindrical Projectile Impact

#### 2.2. FSP Impact

## 3. Multiaxial Loading and Failure

#### 3.1. Cylindrical Projectile Impact

#### 3.2. FSP Impact

#### 3.3. Failure Criterion

_{0}is the reference gage length (12.7 mm) at which the scale ${\mathsf{\epsilon}}_{0}=0.044$ and shape parameters m = 13.25 are determined, and $\mathrm{P}\left(\mathsf{\epsilon},\mathrm{L}\right)$ is the cumulative probability of failure of a gage length L at a strain level $\mathsf{\epsilon}$. In general, failure strains exhibit an increasing trend with decreasing gage length and plateau at higher gage lengths. The tensile failure strain of a perfect Kevlar crystal chain calculated from molecular dynamics is about 19% [27] which may be considered as the intrinsic strength of the material. Strain rate effects may play a role in predicting the fiber failure. Sanborn and Weerasooriya [26] showed a constant 18% increase in the failure strength of single fibers at high strain rate (1200 1/s) compared to quasi-static strain rates for 2 and 5 mm gage lengths. The predicted local maximum axial strain rates in the fiber are of the order of 10

^{6}1/s. A constant 18% increase in the average failure strain due to strain rate effect is assumed for all gage lengths, as shown in Figure 11 (Weibull high rate curve). While the strain rate effects increase the tensile strength [26], multiaxial loading effects degrade the strength [28,29,30].

_{3,max}is the maximum axial tensile strain predicted by the model, ε

_{3},fail is the axial tensile failure strain, which is a function of the failure strain based on the Weibull model at a gage length equal to contact length (L

_{c}), (1 − ACr), (1 − TCr), and (1 − ILSr) are the reduction factors in the respective individual deformation modes based on the maximum levels of loading in the time history. ACr, TCr, and ILSr are the degradation percentages in the respective individual deformation modes. SR is the strain rate factor to account for increase in the failure strain at high strain rates, SR = 0.18.

_{3}(L

_{c}) used in the criterion is based on the Weibull model, where the length (L

_{c}) of the fiber subjected to the strain concentration due to projectile-fiber contact. The strain to failure at a 50% probability of failure increases from 4.24% at 12.7 mm (gage length used in the experiment) to 6.98% at a 20 µm gage length calculated for both cylindrical projectile and FSP.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Half-symmetric 3D single-fiber model (not to scale); (

**a**) cylindrical impact at the mid-span; (

**b**) fiber cross-section; (

**c**) structure of Kevlar; and (

**d**) yield stress-effective plastic strain in transverse compression.

**Figure 3.**Ratio of bounce velocity to impact velocity for a rigid 1.0 mm radius cylindrical projectile (

**a**) at 200 m/s; and (

**b**) function of impact velocity.

**Figure 4.**Cylindrical projectile impact at V = 200 m/s at 0.0784 µs: (

**a**) transverse displacement, axial stress contours in GPa; (

**b**) elastic; and (

**c**) inelastic.

**Figure 5.**Contours of transverse compressive strains at a 200 m/s impact at 0.03 µs; (

**a**) elastic; and (

**b**) inelastic.

**Figure 6.**Half-symmetric 3D single-fiber model: (

**a**) FSP impact at the mid-span; and (

**b**) fiber cross-section.

**Figure 7.**FSP impact at 250 m/s: (

**a**) bounce/impact velocity at z/lc/2 = 0; (

**b**) bounce/impact velocity at z/lc/2 = 1; and (

**c**) transverse displacement at 0.60 µs.

**Figure 8.**FSP impact at 300 m/s, inelastic fiber behavior: (

**a**) contours of axial stresses in GPa at 0.50 µs; and (

**b**) transverse compressive strains at 0.05 µs.

**Figure 9.**Cylindrical impact inelastic fiber behavior: (

**a**) evolution of average axial strains; (

**b**) evolution of axial tensile and compressive strains; (

**c**) axial strain and SCF as a function of velocity; and (

**d**) TC strain as a function of velocity.

**Figure 10.**FSP impact inelastic fiber behavior: (

**a**) evolution of average axial strains; (

**b**) evolution of axial tensile and compressive strains; (

**c**) axial strain and SCF as a function of velocity; and (

**d**) TC strain as a function of velocity.

**Figure 12.**Failure prediction: (

**a**) failure criterion for a cylindrical projectile; and (

**b**) failure criterion for FSP.

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**MDPI and ACS Style**

Sockalingam, S.; Gillespie, J.W.,.; Keefe, M.
Role of Inelastic Transverse Compressive Behavior and Multiaxial Loading on the Transverse Impact of Kevlar KM2 Single Fiber. *Fibers* **2017**, *5*, 9.
https://doi.org/10.3390/fib5010009

**AMA Style**

Sockalingam S, Gillespie JW,, Keefe M.
Role of Inelastic Transverse Compressive Behavior and Multiaxial Loading on the Transverse Impact of Kevlar KM2 Single Fiber. *Fibers*. 2017; 5(1):9.
https://doi.org/10.3390/fib5010009

**Chicago/Turabian Style**

Sockalingam, Subramani, John W., Gillespie, and Michael Keefe.
2017. "Role of Inelastic Transverse Compressive Behavior and Multiaxial Loading on the Transverse Impact of Kevlar KM2 Single Fiber" *Fibers* 5, no. 1: 9.
https://doi.org/10.3390/fib5010009