Finite-element-based Fiber-length-scale Modeling of the Ballistic-Impact Response of KEVLAR ® KM2 Fa

Textiles and Light Industrial Science and Technology (TLIST) Volume 5, 2016 doi: 10.14355/tlist.2016.05.001

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Finite-element-based Fiber-length-scale Modeling of the Ballistic-Impact Response of KEVLAR® KM2 Fabric M. Grujicic1, S. Ramaswami, J. S. Snipes, R. Galgalikar Department of Mechanical Engineering, Clemson University, Clemson SC 29634 gmica@clemson.edu

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Abstract In the present work, finite-element-based sub-yarn-level modeling of the Kevlar® KM2 ballistic-fabric is carried out in order to help establish the relationships between the fabric penetration resistance and various fiber-level phenomena such as fiber-fiber friction, fiber twist, transverse properties of the fibers and the stochastic nature of fiber strength. In particular, a series of transient non-linear dynamics finite-element analyses was carried out of the transverse impact of a test patch of Kevlar® KM2 fabric by a solid spherical projectile, in order to investigate the role of the following two important sub-yarn phenomena on the penetration resistance of this fabric: (a) fiber-transverse properties including non-linear elastic and plastic response; and (b) stochastic nature of the fiber strength.

Keywords Finite-Element Modeling; Kevlar® KM2 Fibers and Fabric; Flexible Armor; Penetration Resistance; Ballistic Performance

Introduction The present work deals with the high-performance polymeric-fiber-based fabrics and with the computational modeling of the protection offered by such fabrics against ballistic threats. Therefore, the topics most relevant to the present study include: (a) the basics of high-performance polymeric-fiber-based fabrics; and (b) an overview of prior ballistic-fabric modeling efforts. In the remainder of this section, a brief description is provided of these topics.

The Basics of High-Performance Polymeric-Fiber-Based Fabrics Military systems, in particular those supporting the U.S. ground forces, are being continuously transformed to become faster, more agile, and more mobile, so that these systems can be quickly transported to operations conducted throughout the world in order to respond to the new enemy threats and warfare tactics. Consequently, an increased emphasis is being placed on the development of improved lightweight body-armor and lightweight vehicle-armor systems as well as on the development of new high-performance armor materials. Highperformance fiber-based materials have been exploited for both body-armor (e.g. as soft, flexible fiber mats for personal-armor vests) and for the vehicle-armor systems (e.g. as reinforcements in rigid polymer matrix composites, PMCs, for lightweight vehicle-armor systems).

Flexible lightweight materials have been used, throughout history, in body-armor systems to provide protection against specified threats, at reduced weight and without compromising a person’s mobility. Early materials used included leather, silk, metal chainmail and metal plates. Replacement of metal with a nylon (poly-amide) fabric and an E-glass fiber/ethyl cellulose composite in body armor systems can be linked to the Korean War [1]. Although, primarily due to their low cost, nylon and E-glass fibers are still being used today, other non-nylon highperformance polymeric fibers (typically used in the form of woven fabrics) is now the standard in most fiberreinforced body-armor applications. These high-performance polymeric fibers enabled the development of novel flexible-armor systems with exceptional protection against bullet penetration [2].

The defining characteristics of high-performance polymeric fibers used today are their superior strength, stiffness and ballistic performance. Among these high-performance fibers the most notable are: (a) poly-aramids (e.g. Kevlar® ,

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Twaron® , Technora® ); (b) highly-oriented poly-ethylene (e.g. Spectra® , Dyneema® ); (c) poly-benzobis-oxazole, PBO (e.g.

Zylon® ), and (d) poly-pyrido-bis-imidazole, PIPD (e.g. M5® ). The present work is concerned with a particular grade of Kevlar® fibers, the Kevlar® KM2 fibers.

When tested in tension, the high-performance fibers mentioned above differ significantly from the nylon fibers, having very high absolute stiffness, extremely high specific strength (i.e. strength normalized by the mass-density), and quite low (<4%) strains-to-failure. These fibers essentially behave, in tension, as rate-independent linear elastic materials. When tested in transverse compression, however, these fibers are similar to nylon and can undergo large non-linear elastic/plastic deformation without a significant loss in their tensile load-carrying capacity. On the other hand, this behavior is quite different from that found in carbon or glass fibers, which tend to shatter under transverse compression loading conditions [3].

The ballistic behavior of high-performance polymeric fibers is, in general, characterized by their ability to: (a) absorb the projectile’s kinetic energy locally; and (b) spread out the absorbed energy fast before local conditions for the failure are met. In simple terms, the ability of high-performance fibers to absorb energy per their unit mass, Esp , is related to the fiber tenacity/failure-strength,  fail , the fiber strain-to-failure,

 fail ,

and the fiber density,  , as

Esp  0.5 fail fail  . The ability of fibers to spread out energy is governed by their speed of sound, vsound , which is

defined in terms of their axial modulus of elasticity, E , and their density as vsound  E  .

A summary of the key properties of the most-commonly used high-performance fibers is provided in Table 1. In Fig. 1, the two ballistic-performance parameters are displayed for the same set of high-performance fibers.

TABLE 1. TYPICAL MECHANICAL PROPERTIES AND MASS DENSITY OF HIGH-PERFORMANCE FIBERS

Fiber Type

Failure Strength (GPa)

Failure Strain

Axial Modulus (GPa)

Density (kg/m3)

Aramid

2.8-3.2

0.015-0.045

60-115

1,390-1,440

HMWPE

2.8-4.0

0.029-0.038

90-140

970-980

LCP

2.7-2.9

0.033-0.035

64-66

1,400-1,420

PBO

5.4-5.6

0.024-0.026

270-290

1,540-1,560

PIPD

3.9-4.1

0.011-0.013

320-340

1,690-1,710

Nylon

0.06-0.08

1.5-2.5

1.0-1.5

1,070-1,170

S-glass

4.64-4.66

0.053-0.055

82-92

2,470-2,490

FIGURE 1. SOUND SPEED VS. MASS-BASED ENERGY ABSORPTION CAPACITY FOR A NUMBER OF HIGH-PERFORMANCE FIBERS

While the attention above was focused on fibers and the associated materials, it should be recognized that flexiblearmor structures are made of fabric/textiles containing these fibers. The fabric itself possesses a hierarchical structure, i.e. fibers are arranged into bundles called yarns and yarns are assembled into fabric in specific patterns

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through weaving or braiding processes. It is important to note that fabric ballistic-penetration resistance is influenced not only by fiber physical properties, such as strength, ductility, elastic modulus, mass-density, viscosity and thermal properties, but also by fabric architecture in its as-fabricated state [4â&#x20AC;&#x201C;6].

An Overview of Prior Ballistic-Fabric Modeling Efforts Attempts were made as early as the 1960s to guide the development of new flexible-armor systems using computer-aided-engineering and modeling/simulation approaches. Nevertheless, the design and development of textile armor systems remains primarily governed by experiments and experience. Despite the fact that numerical models typically have not been used to guide the design/development of the new flexible-armor systems, these models have often been employed to quantify the effect of specific fiber/yarn/fabric parameters on the ballisticpenetration resistance of the armor [e.g. 7â&#x20AC;&#x201C;15]. It should also be noted that, while it is well established that the fabric performance is highly affected by the fabric-manufacturing process and the as-fabricated architecture, these effects are generally not incorporated into the models. Furthermore, the effect of fabric micro-structural parameters such as yarn denier, end count, yarn structures, filament spatial paths and fiber-to-fiber interactions are generally either not included, or included incompletely-incorrectly by most of the flexible-armor ballistic-penetration numerical models. Consequently, a quantitative relation among material properties, manufacturing process, fabric micro-geometry, and ballistic performance are yet to be established by these models.

It is generally recognized that the level of detail of the fabric-architecture used in the numerical model influences the resolution and the accuracy/fidelity of the numerical results obtained [16â&#x20AC;&#x201C;18]. For instance, if a fabric is considered as a continuum (rather than a discrete-yarn woven structure), details of yarn-to-yarn interaction and projectile-to-yarn interaction cannot be modeled. As a result, the effect of fabric weave pattern on ballistic resistance cannot be inferred. Likewise, if the fabric is represented as an assembly of inter-woven yarns but the yarns are represented as a continuum (rather than an assembly of engaged discrete fibers), details of fiber-to-fiber interaction and projectile-to-fiber interaction cannot be investigated. As a result, the effect of yarn microstructure on the ballistic resistance cannot be quantified.

The numerical models of the flexible armor systems reported in the open literature can be divided into two general groups: (a) those in which the fabric is modeled as a continuum; and (b) those in which the fabric structure is accounted for explicitly but the yarns are treated as a continuum. Since a detailed overview of these two groups of material models was provided in our prior work [9], the same overview will not be given here. Instead, only a few general points regarding the two groups of models will be highlighted.

Continuum-Fabric Material Models: There are a large number of models of this type, and they differ in the way the internal (not explicitly represented) architecture of the fabric and the yarn is accounted for. While the models which account for more details of the fabric architecture generally yield better results, the whole group of models does not generally give satisfactory results. This is attributed to the fact that these models do not account for yarnto-yarn and projectile-to-yarn interactions, the interactions which strongly influence the flexible-armor ballistic performance [19].

Discrete-Fabric/Continuum-Yarn Material Models: Within this type of models, an explicit account is taken of the phenomena such as yarn crimp, slip at yarn crossover points, yarn slippage at yarn/projectile contact surface, yarn weaving pattern, multiple-ply fabric architecture, etc. Since these important phenomena could not be accounted for within the continuum-fabric material models, the discrete-fabric/continuum-yarn material models yield substantially better results. However, the phenomena taking place at the fiber-length-scale are not accounted for by the discrete-fabric/continuum-yarn material models. This shortcoming is addressed in the present work [20].

Main Objective The main objective of the present work is to develop a finite-element-based model for ballistic fabric in which fabric, yarns and fibers are all explicitly accounted for and to investigate the structural (including failure) response of the fabric to ballistic impact. To validate the model, the key model predictions will be compared with their experimental counterparts [18].

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Formulation of the Computational Model In this section, a detailed description is provided of the procedure used to construct the ballistic-fabric computational model and of its application to the analysis of a simple ballistic-impact event.

Problem Formulation The basic problem analyzed in the present work involves the impact of an 8-mm-diameter steel spherical projectile, mass = 2.1 g, with a 33 mm × 33 mm square patch of a Kevlar® KM2 single-ply fabric clamped along all its four edges. Only the normal (i.e. zero obliquity angle) impact case is considered, with the projectile incident velocity varied in a 35–210m/s range.

Finite-Element Model and Analysis Modeling of the ballistic fabric and of its normal impact by a spherical steel projectile is carried out using the conventional finite-element method/analysis. Within such an analysis, a complete problem formulation requires specification of the following: (a) geometrical model; (b) meshed model; (c) computational algorithm; (d) initial conditions; (e) boundary/loading conditions; (f) contact interactions; (g) material models; (h) computational tool; and (i) computational accuracy, stability and cost. These items are briefly overviewed in the remainder of this subsection.

1) Geometrical Model The initial geometrical model for the projectile/fabric assembly is shown in Fig. 2(a). A close-up of the fabric is displayed in Fig. 2(b) in order to reveal the plain-weave fabric architecture and the sub-yarn microstructure (involving seven fibers, in the present case).

The fabric architecture and its areal density will be discussed in the subsection dealing with the material models.

The fabric test patch consists of 42 warp yarns and 42 weft yarns. Each yarn is discretized into discrete circularcross-section fibers bundled into a regular (identical) hexagonal close-packed configuration. Various yarn discretizations (containing 1, 7, 14, 19, 24 and 30 fibers) were considered in the initial portion of the work in order to identify the one (containing 7 fibers) which provides the best trade-off between the physical accuracy and computational cost. For each of the configurations considered, fiber diameter was calculated in such a way that the correct fabric areal density is obtained. Fibers are treated as discrete circular-section solid entities. Yarns are treated as bundles of collinear, transversely-interacting fibers while the fabric is treated as a plain-woven structure consisting of effectively orthogonal warp and weft yarns.

FIGURE 2. (A) A GEOMETRICAL MODEL FOR THE PROJECTILE/FABRIC INTERACTION PROBLEM ANALYZED IN THE PRESENT WORK; AND (B) A CLOSE-UP OF THE FABRIC REGION SHOWING SUB-YARN MICROSTRUCTURE AND THE YARN-WEAVE PATTERN.

2) Meshed Model Within the finite-element model each fiber is discretized into a number of three-dimensional beam elements with the beam length chosen to match the fiber diameter. Since each yarn contained seven fibers, the fabric contained approximately 400,000 beam elements.

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The spherical projectile is discretized using continuum first-order tetrahedron elements with an edge length comparable to the fiber beam-element length. Consequently, the projectile contained approximately 100,000 finite elements.

3) Computational Algorithm To ensure robustness of the computational analysis, all the calculations carried out in the present work utilized a transient, displacement-based, purely-Lagrangian, conditionally-stable, dynamic explicit finite-element algorithm.

4) Initial Conditions At the beginning of the analysis, the following initial conditions are specified: (a) the projectile is assigned a preselected negative value of the velocity z-component (the mid-plane of the fabric is placed on the xy-plane associated with z=0, while the projectile center of gravity is placed at x=y=0 and z>projectile radius); and (b) the fabric is assumed to be stress-free, deformation-free and at rest.

5) Boundary/Loading Conditions As mentioned earlier, during the simulation the fabric is fully clamped along all its four edges. As far as the projectile is concerned, its motion was left fully unconstrained.

6) Material Model for Projectile Due to the fact that the steel projectile used in the experimental studies relevant to the present work [18], did not suffer any permanent deformation during impact testing, the projectile material (steel) is treated as an isotropic linear elastic material with a mass-density of 7850 kg/m3, a Young’s modulus of 210 GPa, and a Poisson’s ratio of 0.3.

7) Material Model for Fabric Fiber Properties: Experimental investigation of Chen et al. [18], clearly established that, in the longitudinal direction, Kevlar® KM2 fiber behaves as a rate-independent inelastic material with an average longitudinal Young’s Modulus of 84.62 ± 4.18 GPa with 95% confidence. As far as the longitudinal strength,  , is concerned, it is found to be a stochastic property (due to inevitable presence of various microstructural defects/imperfections within the fibers) and to be implicitly given by a bi-modal Weibull-type failuredistribution function, F ( ) , in the form [22]:  L F ( )  1  exp    L0

mi         i 1   0 i    n

(1)

where n  2 (due to the bi-modal nature of the distribution function), L0  0.025 m (the fiber reference gauge length), L  L0 (the fiber actual gauge length),  01  3.323 GPa and  02  3.7004 GPa (the two Weibull scale parameters) and m1  18.0023 and m2  24.8803 (the two Weibull shape parameters). The bi-modal nature of the Kevlar® KM2 fiber strength is related to the distinct defect/flaw potency distributions in the core and in the skin regions of this class of fibers.

While in the transverse direction, Kevlar® KM2 fibers show non-linear and inelastic behavior, following the work of Wang and co-workers [21], the constitutive response of the fibers in the transverse direction is modeled at three different levels of physical accuracy/fidelity: (a) at the highest level of approximation, the response is treated as being linear elastic with the associated transverse modulus of 1.34 ± 0.35 GPa (with 95% confidence based on the normal distribution of this quantity [10]); (b) The response is considered to be non-linear elastic with the loading/unloading nominal stress  vs. nominal

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strain  relation given as:

  0.006e9  0.5296e9 1.2208e9 2  6.9417e9 3

(2)

where the nominal stress is defined as a ratio of the contact force and the product of the fiber diameter and the fiber length, while the nominal strain was defined as the ratio of the fiber diameter change and the fiber diameter; and (c) At the highest level of fidelity, the transverse response is considered as being elastic/plastic with Eq. (2) defining the strain hardening yielding-behavior of the fiber during loading (involving plastic deformation) while the stress during elastic unloading/reloading is defined as: 2 K  0.0646e9  7.0632e9 max  18.666e9 max

(3)

where  max is the maximum transverse normal strain experienced by the local material point of the fiber.

It should be noted that since fiber transverse stiffness provides little contribution to either fabric or yarn transverse stiffness, one cannot generally determine fiber transverse properties by testing yarns or the fabric.

Instead, testing must be done using individual fibers [21].

It should be noted, as will be discussed in more detail below, that due to the beam character of the fiber finite elements, the longitudinal properties of the fibers are specified within the material definition while the transverse properties are specified as a part of the fiber/fiber contact definition.

As established above, fiber fracture strength is a stochastic quantity. To assess the effect of the stochastic nature of the fiber fracture strength on the fabric ballistic-impact response, analyses are carried out in which fiber fracture strength is treated as a deterministic quantity, and in other analyses the same is treated as a stochastic quantity. In the latter case, a Monte Carlo procedure is employed within which a random number in a range (0, 1) is drawn (for each fiber element) from the uniform distribution and assigned to the failure probability of the element in question. Then, the Eq. (1) is solved for the unknown fracture strength of the material residing within this element. Within the ABAQUS/Explicit [23] computational environment, the most efficient way to implement the stochastically distributed fiber fracture strength is through the use of a user material subroutine VUMAT. Within this subroutine, material strength associated with a given beam-element/integration-point is treated as a material-state variable. This variable retains its initially-assigned value until the onset of fiber failure (i.e. until the axial stress becomes equal to the material strength) and then decreases monotonically to zero (in a manner consistent with the pre-assigned value of the material fracture toughness).

Yarn Properties: It should be noted that yarns are represented in the present work as a bundle of collinear discrete fibers. Hence, yarn properties are not defined as input parameters to the numerical model but rather could be assessed through the appropriate post-processing schemes. Nevertheless, it should be recognized that the yarn longitudinal properties are directly controlled by the fiber longitudinal properties. For example, the longitudinal stiffness modulus of the yarns is defined as a product of the fiber longitudinal stiffness modulus and a ( 1  F ( ) ) factor in order to account for the effect of potential fiber failure.

As far as the yarn transverse properties are concerned, they are mainly controlled by yarn microstructure, fiber/fiber friction and yarn tension.

Fabric Properties: As in the case of the yarns, fabric properties are not defined as input parameters to the numerical model but rather could be assessed through the appropriate post-processing schemes. In general, both the longitudinal and transverse properties of the fabric depend on the fabric areal density, yarn count per fabric unit length, longitudinal properties of the fibers and yarns, fabric architecture/weave pattern, fiber/fiber friction and yarn tension. The Kevlar® KM2 fabric analyzed in the present work has an areal density of 180 g/m2 and 13.4 yarns per centimeter. The fabric contains a plain-woven architecture which is obtained by “snaking” through orthogonally-oriented warp and weft yarns.

8) Contact Interactions The response of the fibers in the transverse directions, due to fiber/fiber and fiber/projectile normal interactions,

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is modeled using a contact algorithm implemented within the VUINTER user subroutine of the commercial finite-element package ABAQUS/Explicit [23]. Within this subroutine, the fiber transverse constitutive relations defined in Eqs. (2) and (3) are implemented. Within the same subroutine, tangential fiber/fiber and fiber/projectile interactions are modeled using a simple Coulomb-type friction law within which the transmission of shear stresses across the contact interfaces is defined in terms of static and kinetic friction coefficients and an upper-bound shear stress limit (a maximum value of shear stress which can be transmitted before the contacting surfaces begin to slide). Based on the results obtained in our prior work [24], the friction coefficient was set to a value of 0.1 (unless otherwise stated). This value is very consistent with its experimental counterpart as reported in Ref. [21].

9) Computational Tool All the calculations carried out in the present work were done using ABAQUS/Explicit [23] within which the problem at hand, defined in terms of a set of mass, momentum and energy conservation differential equations along with the material constitutive relations and contact, initial and boundary conditions, is solved using a finite-element approach.

10) Computational Accuracy, Stability and Cost A standard mesh sensitivity analysis was carried out (the results not shown for brevity) in order to ensure that the results obtained are accurate, i.e. insensitive to further refinements in the finite-element size used.

Due to the conditionally-stable nature of the explicit finite element analysis used, the maximum time increment during each computational step had to be lower than the attendant stable time increment. To keep the computational cost reasonable while ensuring accuracy and stability of the computational procedure, a mass scaling algorithm is used. This algorithm adaptively adjusts material density in the critical stable-timeincrement-controlling finite elements without significantly affecting the computational analysis results.

A typical 210 clock) time on a 12-core, 3.0 GHz machine with 16 GB of memory.

7-fiber yarns required 3.5 hours of (wall-

Results and Discussion The work carried out in the present study was aimed at helping answer the following two questions: (a) what role the transverse properties of the fibers play in the ballistic-penetration resistance of the fabric; and (b) what effect the stochastic nature of the fiber strength has on the ballistic-penetration resistance of the fabric. The results obtained, which are most pertinent to helping answer these two questions are presented and discussed in this section.

The Role of Fiber Transverse Properties When a projectile strikes the flexible-armor/fabric, it applies transverse loading either directly through the projectile/fiber interactions or indirectly through the interaction of the projectile-engaged fibers and other fibers in contact with them. It should be noted that when a projectile strikes flexible armor, there are two general outcomes: either (a) the projectile is defeated by the flexible armor and rebounds; or (b) the projectile penetrates the flexible armor while retaining a portion of its initial velocity (typically referred to as the residual velocity). An important question to be answered is to what extent fiber transverse properties affect the mechanical response of the fabric [including the magnitude of the projectile residual velocity (the velocity of the projectile after it has defeated the flexible armor or has been defeated by the armor), fabric transverse displacement profile, fabricâ&#x20AC;&#x2122;s ability to deform and absorb/dissipate the projectileâ&#x20AC;&#x2122;s kinetic energy, the stress magnitude/distribution within the fibers and the overall fabric-armor strength].

As mentioned earlier, three material models are used to describe the transverse behavior of the KevlarÂŽ KM2 fibers in the present work. In increasing order of complexity and computational burden, these models are: (a) linear elastic; (b) nonlinear elastic; and (c) rate-independent elastic-plastic.

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The results obtained in the present work pertaining to the radial distribution of the fabric back face nodal displacements in the projectile-propagation (z) direction for the case of projectile initial velocity of 63m/s and the post-impact time of 0.174ms are displayed in Fig. 3. Three sets of the computational results are displayed in this figure, one for each of the three (linear elastic, non-linear elastic and elastic-plastic) fiber transverse-response conditions. For comparison, the experimental results reported in Ref. [21] are also depicted in this figure.

Examination of the results displayed in this figure clearly reveals that the computational analysis in which an elastic-plastic material model was used to represent the transverse response of the fibers yields the best overall agreement with the experimental data.

Frame 001  23 Jul 2015  | | | 200

Projectile Residual Velocity, m/s

Experimental [18] Linear Elastic

150

Nonlinear Elastic Elastic-Plastic

100

50

0

-50

50

100

150

Projectile Strike Velocity, m/s

FIGURE 3. VARIATION OF THE FABRIC BACK-SURFACE DISPLACEMENT WITH THE RADIAL DISTANCE FROM THE FABRIC CENTER FOR THE CASE OF 2.1g STEEL SPHERICAL PROJECTILE IMPACTING THE KEVLAR® KM2 FABRIC AT AN INITIAL VELOCITY OF 63m/s. THE EXPERIMENTAL RESULTS FROM REF. [21] ARE SHOWN.

FIGURE 4. VARIATION OF THE PROJECTILE RESIDUAL/REBOUND VELOCITY WITH THE PROJECTILE INITIAL VELOCITY FOR THE CASE OF 2.1g STEEL SPHERICAL PROJECTILE IMPACTING THE KEVLAR® KM2 FABRIC. THE EXPERIMENTAL RESULTS FROM REF. [21] ARE ALSO SHOWN. REBOUND VELOCITIES ARE ASSIGNED NEGATIVE VALUES.

Other results generated in the present work but not shown for brevity confirm that the elastic-plastic material model for fibers in the transverse direction yields the best results relative to the magnitude of the projectile residual velocity (the velocity of the projectile after it has defeated the flexible armor or has been defeated by the armor), fabric transverse displacement profile, fabric’s ability to deform and absorb/dissipate the projectile’s kinetic energy, the stress magnitude/distribution within the fibers and the overall armor strength. Consequently, this type of material model was adopted and used in the remainder of the present work.

The Role of the Stochastic Nature of Fiber Strength Depending on the quality of the fabrication process, fibers may contain different populations of flaws and, thus fiber strength becomes a stochastic quantity. This effect is not generally taken into account and high-performance fibers are often assigned a constant stress value (set equal to the fiber mean strength). The results presented in this section should help address the question of the role of the stochastic nature of fiber strength in the ballisticpenetration resistance of the fabric. Since the extent of this role may be affected by the magnitude of the fiber/fiber friction and the fiber/projectile friction coefficients, the results will be discussed in the context of the attendant values of the friction coefficients.

Typical results pertaining to the spatial distribution of the fabric damage during impact by a 2.1g steel spherical projectile traveling at an initial velocity of 63m/s for the case of a deterministic fiber fracture strength (=3.4GPa) at four different post-impact times (75 . 5(a)-(d). It should be noted that for improved clarity, the results in this figure are displayed using the undeformed fabric mesh and the projectile is not shown. The undamaged elements are shown in blue while the damaged elements are shown in a color signifying the extent of the damage (red color denotes completely damaged and deleted elements).

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Examination of the results displayed in Fig. 5(a)-(d) shows that the fabric damage, in the deterministic fiberstrength case, is mainly localized in the region underneath the projectile. This finding is not consistent with the experimental results reported in Ref. [21], which indicate that, for the most part, fiber damage is a non-local phenomenon.

FIGURE 5. SPATIAL DISTRIBUTION OF THE FABRIC DAMAGE DURING IMPACT BY A 2.1g STEEL SPHERICAL PROJECTILE TRAVELING AT AN INITIAL VELOCITY OF 63 m/s FOR THE CASE OF A DETERMINISTIC FIBER FRACTURE STRENGTH (=3.4GPa) AT POSTELEMENTS (OTHER COLORS) ARE SHOWN USING THE UNDEFORMED FABRIC MESH, AND THE PROJECTILE IS NOT SHOWN. THE FABRIC IS VIEWED FROM THE SIDE NOT DIRECTLY IMPACTED.

Typical results pertaining to the spatial distribution of fiber axial stress under the impact conditions identical to the ones stated in conjunction with Fig. 5(a)-(d) are shown in Fig. 6(a)-(d). In this case, the deformed mesh of the fabric and the projectile are shown. Examination of the results displayed in Fig. 6(a)-(d) confirms the general experimental findings in Ref. [21] that fabric penetration is a local process. In other words, penetration of the fabric is mainly controlled by the creation and evolution of damage in the fabric region which is beneath and, hence, directly impacted by the projectile.

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FIGURE 6. SPATIAL DISTRIBUTION OF FIBER AXIAL STRESS DURING IMPACT BY A 2.1 g STEEL SPHERICAL PROJECTILE TRAVELING AT AN INITIAL VELOCITY OF 63 m/s FOR THE CASE OF A DETERMINISTIC FIBER FRACTURE STRENGTH (=3.4 GPa) AT POST-IMPACT TIMES OF: (A)75

Typical results pertaining to the spatial distribution of the fabric damage during impact by a 2.1g steel spherical projectile traveling at an initial velocity of 63m/s for the case of stochastic fiber fracture strength at four different post-(d). It should be noted that for improved clarity, the results in this figure are displayed using the undeformed fabric mesh and the projectile is not shown. The undamaged elements are shown in blue while the damaged elements are shown in a color signifying the extent of the damage (red color denotes completely damaged and deleted elements). Examination of the results displayed in Fig. 7(a)-(d) shows that the fabric damage, in the stochastic fiber failure-strength case, is not entirely localized within the fabric region directly impacted by the projectile. Rather, some of the fibers and yarns outside this region have also suffered damage and failed. This finding is fully consistent with the experimental results reported in Ref. [21].

FIGURE 7. SPATIAL DISTRIBUTION OF THE FABRIC DAMAGE DURING IMPACT BY A 2.1 g STEEL SPHERICAL PROJECTILE TRAVELING AT AN INITIAL VELOCITY OF 63 m/s FOR THE CASE OF STOCHASTIC FIBER FRACTURE STRENGTH AT POST-IMPACT TIMES OF: (A) 140 COLORS) ARE SHOWN USING THE UNDEFORMED FABRIC MESH, AND THE PROJECTILE IS NOT SHOWN. THE FABRIC IS VIEWED FROM THE SIDE NOT DIRECTLY IMPACTED.

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Typical results pertaining to the spatial distribution of fiber axial stress under the impact conditions identical to the ones stated in conjunction with Fig. 7(a)-(d) are shown in Fig. 8(a)-(d). In this case, the deformed mesh of the fabric and the projectile are shown. Examination of the results displayed in Fig. 8(a)-(d) confirms the general experimental findings reported in Ref. [21] that fabric penetration is a non-local process. In other words, fiber damage is typically initiated in the fabric region which is not directly in contact with the projectile but contains high-potency flaws/defects. The resulting redistribution of the stress within the fibers and the formation of high stress regions leads to spreading of damage towards the region impacted by the projectile. Thus, fabric penetration is no longer a local phenomenon and is highly affected by the stochastic and distributed nature of fabric failure strength.

FIGURE 8. SPATIAL DISTRIBUTION OF FIBER AXIAL STRESS DURING IMPACT BY A 2.1 g STEEL SPHERICAL PROJECTILE TRAVELING AT AN INITIAL VELOCITY OF 63 m/s FOR THE CASE OF STOCHASTIC FRACTURE STRENGTH AT POST-IMPACT TIMES OF:

It should be noted that the distribution of fiber axial stress are shown at different post-impact times in Figures 6(a)– (d) and 8(a)–(d). The reason is that the times selected were chosen to best make the point that in the case of the deterministic-fiber-strength, Figures 6(a)–(d), failure is a local phenomenon, while in the case of the stochastic fiber-strength, Figures 8(a)–(d), failure is a non-local phenomenon.

FIGURE 9. VARIATION OF PENETRATION FREQUENCY OF THE FABRIC BY THE PROJECTILE WITH THE MAGNITUDE OF THE FIBERFIBER FRICTION COEFFICIENT FOR THE CASE OF: (A) DETERMINISTIC FIBER-FRACTURE STRENGTH; AND (B) STOCHASTIC FIBERFRACTURE STRENGTH.

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The effect of the magnitude of the fiber-fiber friction coefficient on the penetration frequency of the fabric by the projectile for the case of deterministic fiber-fracture strength is displayed in Fig. 9(a). Clearly, due to the deterministic nature of the fiber strength, the results obtained under nominally identical conditions are fully repeatable. Hence, the curves shown in Fig. 9(a) are of a step-function type. The results displayed in this figure confirm the general experimental findings in Ref. [21] that: (a) a higher value of the fiber-fiber friction coefficient increases the fabricâ&#x20AC;&#x2122;s penetration resistance (as quantified by the projectile velocity at which the penetration frequency changes from zero to one); and (b) the effect of friction coefficient on the fabric penetration resistance is relatively weak.

The effect of the magnitude of the fiber-fiber friction coefficient on the penetration frequency of the fabric by the projectile for the case of stochastically distributed fiber fracture strength is displayed in Fig. 9(b). Due to the stochastic nature of the fiber strength, the penetration frequency vs. projectile strike velocity curves no longer resembles a step function. Instead, there is a range of projectile velocities between the maximum velocity at which no penetration occurs and a minimum velocity at which penetration always occurs. The results displayed in this figure confirm the general experimental findings in Ref. [21] that: (a) There is an optimal value of the fiber-fiber friction coefficient which is associated with the maximum value of the fabric penetration resistance. The existence of an optimum friction coefficient is generally rationalized in the following way. At lower values of the friction coefficient, the fibers can be readily pushed laterally by the projectile, and the fabric defeated without, or with little, damage to the fibers. At too-high values of the friction coefficient, fibers are over-constrained and get defeated by the projectile by transverse shear or transverse crushing before experiencing sufficient axial scratching (the latter deformation mode is of key importance for the ballistic performance of fabric); and (b) The effect of fiber-fiber friction coefficient on fabric penetration resistance is substantially stronger in the case of the stochastic nature of the fiber strength than in the deterministic-case.

Additional results (not shown for brevity) were also generated in this portion of the work and their analysis and comparison with the experimental results in Ref. [21] revealed: (a) Failure of a (stretched) fiber is followed by a fairly complex stress-redistribution/unloading process which can be divided into three well defined stages: (i) the propagation of the release waves from the fractured ends of the two fiber fragments towards the clamped end of these fragments. The swept region of the fiber fragments is fully relieved of the prior tensile longitudinal stress; (ii) the release waves become arrested at one of the yarn/yarn crossover points producing locally a high level of tensile stress; and (iii) transmitted release waves are emitted from the high-stress region towards the clamped end of the fiber fragments, reducing the extent of stress concentration and increasing the average level of tension in the loaded portion of the fiber fragments; (b) the distance travelled by the release waves, before their arrest, scales inversely with the magnitude of the fiberfiber friction coefficient; (c) if and when the enhanced average level of tension defined in (a),(iii) exceeds the fiber local failure strength, the fiber fracture as well as the aforementioned stress-redistribution processes reoccur; (d) in the case of the reoccurring fracture of the fibers, the average fragment length scales inversely with the magnitude of the fiber-fiber friction coefficient; (e) the deterministic vs. stochastic nature of the fiber strength affects both the magnitude of the fabric ballistic penetration resistance, the morphology of the fabric-fracture process as well as the role of the fiber-fiber friction coefficient; (f) when fiber strength is treated as a deterministic quantity, fabric failure appears to be a local phenomenon and occurs in the highest-stress region (the region which is in direct contact with the projectile tip). In this case, a lower magnitude of the friction coefficient gives rise to a longer length of the tension-free region of the fractured fiber(s).

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This enables the projectile to more readily push aside the fractured fibers and to defeat the fabric. However, due to the local nature of the fiber failure process, the effect of the magnitude of fiber-fiber friction coefficient on the fabric ballistic strength is relatively weak; (g) when fiber strength is treated as a stochastic quantity, fabric failure appears to be a non-local phenomenon and generally occurs not in the high stress region but rather in the regions associated with the most potent microstructural imperfections/flaws in the fibers. Typically, these regions are located outside the impacted fabric region and the associated release waves mentioned above can cause a spread of the fiber fracture into the fabricimpact region. In other words, the fabric fracture process becomes a non-local phenomenon in this case. Since the average fiber fragment length is controlled by the fiber-fiber friction coefficient, the magnitude of the friction coefficient tends to play a more significant role than in the deterministic case; and (h) when the projectile incident velocity substantially exceeds V50 (the incident velocity of the projectile at which the probability of the target defeat is 50%), the deterministic vs. stochastic nature of the fiber strength has little effect on the projectile residual velocity or on the projectile/fabric interaction process in general. The fabric failure in this case is a local phenomenon occurring in the fabric impacted region. That is, although in the stochastic case the fabric initial failure can occur outside the impacted region, insufficient time is available for this fracture to spread into the impact region before full penetration of the fabric takes place.

Summary and Conclusions 1. A finite-element-based fiber-length-scale model has been developed for Kevlar® KM2 ballistic fabric.

2. The model is used to reveal the mechanical response of a fabric test patch subjected to a transverse impact by a spherical projectile.

3. The model is also used to reveal the role of the transverse properties of the fibers, fiber/fiber and fiber/projectile friction coefficient and the stochastic nature of the fiber strength on the penetration-resistance of the fabric.

4. The model is validated by comparing its key predictions with the available experimental counterparts.

ACKNOWLEDGEMENTS

The material presented in this paper is based on work supported by an Army Research Office (ARO) sponsored grant entitled “Friction Stir Welding Behavior of Selected 2000-series and 5000-series Aluminum Alloys” (Contract Number W911NF-11-1-0207). The authors are indebted to Dr. Asher Rubinstein of ARO for his continuing support and interest in the present work.

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