Modeling yarn slip in woven fabric at the continuum level: Simulations of ballistic impact

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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 61 (2013) 265–292

0022-50

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n Corr

E-m

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Modeling yarn slip in woven fabric at the continuum level:Simulations of ballistic impact

Ethan M. Parsons n, Michael J. King, Simona Socrate n

Massachusetts Institute of Technology, Cambridge MA 02139, USA

a r t i c l e i n f o

Article history:

Received 17 January 2012

Received in revised form

14 May 2012

Accepted 31 May 2012Available online 6 July 2012

Keywords:

Constitutive behaviour

Dynamics

Finite elements

Woven fabric

Yarn slip

96/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jmps.2012.05.005

esponding authors. Tel.: þ1 617 324 6417.

ail addresses: [email protected] (E.M. P

a b s t r a c t

Woven fabric is used in a wide variety of military and commercial products—both in

neat form and as the reinforcement phase of composites. In many applications, yarn

slip, the relative sliding of the yarns composing the weave, is an important mode of

deformation or failure. Yarn slip can significantly change the energy absorption capacity

and yarn density of the fabric and also cause yarns to unravel from the weave. Virtually

all existing models for woven fabric that allow yarn slip are discrete in nature. They

simulate every yarn in the weave and are therefore computationally expensive and

difficult to integrate with other material models. A promising alternative to discrete

models is the mesostructure-based continuum technique. With this technique, homo-

genized continuum properties are determined from a deforming analytic model of the

fabric mesostructure at each material point. Yarn-level mechanisms of deformation are

thus captured without the computational cost of simulating every yarn in the fabric.

However, existing mesostructure-based continuum models treat the yarns as pinned

together at the cross-over points of the weave, and an operative model that allows yarn

slip has not been published. Here, we introduce a mesostructure-based continuum

model that permits yarn slip and use the model to simulate the ballistic impact of

woven fabric. In our approach, the weave is the continuum substrate on which the

model is anchored, and slip of the yarns occurs relative to the weave continuum. The

cross-over points of the weave act as the material points of the continuum, and the

evolution of the local weave mesostructure at each point of the continuum is

represented by state variables. At the same time, slip velocity fields simulate the slip

of each yarn family relative to the weave continuum and therefore control the evolution

of the yarn pitch. We found that simulating yarn slip significantly improves finite

element predictions of the ballistic impact of a Kevlars woven fabric, in particular by

increasing the energy absorbed at high initial projectile velocities. Further simulations

elucidate the micromechanisms of deformation of ballistic impact of woven fabric with

yarn slip. Our findings suggest ways to improve the performance of flexible armor and

indicate that this approach has the potential to simulate many other types of woven

fabric in applications in which yarn slip occurs.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Valued for its flexibility, formability, and high specific strength, woven fabric is an increasingly important part of manydefense and commercial systems. These systems include personal body armor, deployable structures such as air bags, sails,

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arsons), [email protected] (S. Socrate).

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E.M. Parsons et al. / J. Mech. Phys. Solids 61 (2013) 265–292266

and parachutes, and restraint systems such as seat belts and head restraints. Woven fabric also reinforces polymers orceramics in helmets, armor panels, and numerous structural applications.

Modeling woven fabric is challenging due to the complexity of the weave architecture and the multiple modes ofdeformation of the yarns (Lomov et al., 2001). At the mesoscale, a woven fabric consists of two orthogonal, interlaced setsof yarns, warp yarns and weft yarns, each typically composed of hundreds of continuous fibers. During weaving, the warpyarns are held taut in the loom and moved up and down by the heddles. Meanwhile, the weft yarns are pulled through thegaps by the shuttle. The resulting undulation of each yarn over and under the orthogonal yarns is described as crimp, andwherever the yarns cross is termed a cross-over point. The macroscopic deformation of woven fabric, decidedly nonlinearand anisotropic, is controlled by one or more of the following mechanisms: (1) yarn stretch, the elongation of the yarns;(2) uncrimping, the straightening of the initially undulating yarns; (3) in-plane shear, the relative rotation of the yarns at orbetween the cross-over points; (4) locking or ‘‘jamming,’’ the in-plane contact of the yarns with each other; (5) yarn slip,the relative sliding of the yarns at the cross-over points.

Mesostructure-based continuum models have been shown to simulate accurately and efficiently the deformation ofwoven fabric in many circumstances (Boisse et al., 1997, 2001; Tanov and Brueggert, 2003; Ivanov and Tabiei, 2004; Nadleret al., 2006; Shahkarami and Vaziri, 2007; Hamila et al., 2009; Parsons et al., 2010b; Assidi et al., 2011; Xia et al., 2011).These models assume that at an appropriate scale, sufficiently larger than the spacing of the yarns, woven fabric deformsin an affine manner and may therefore be approximated as a continuum. At each material point, the deformation ofsections of discrete yarns are simulated by the deformation of an analytic unit cell of the fabric, typically composed of pin-joined trusses or beams. The unit cell deforms with the continuum and thus tracks the orientation and, in some cases, thecrimp of the yarns. The geometry of the deformed unit cell, together with the constitutive relations of the yarns,determines the stress in the continuum. As a result, the macroscopic predictions of these models derive from the actualmechanisms of deformation of the yarns. An additional benefit of this type of model is that critical yarn-level details, suchas orientation, tension, areal density, shear angle and crimp amplitude, are also calculated. All the models referencedabove, however, approximate the fabric as slip-free, essentially by pinning the yarns together at every cross-over point ofthe unit cell.

Omitted in nearly all mesostructure-based continuum models, slip of the yarns at the cross-over points is an importantmode of deformation in many applications. During ballistic impact, yarn slip is initiated primarily by gradients of tensionin the yarns struck by the projectile. It occurs when the differential yarn tension across a given cross-over point exceedsthe friction force at that same cross-over point, causing the yarns to slide relative to one another. Yarn slip affects theenergy absorbed by the fabric (Lastnik and Karageorgis, 1982; Briscoe and Motamedi, 1992; Bazhenov, 1997) and causesunraveling at the free edges of the fabric. Furthermore, yarn slip has been shown with experiments (Godfrey and Rossettos,1998, 1999) and with micromechanical modeling (Abbott and Skelton, 1972; Popova and Iliev, 1993) to blunt the stressconcentration at the tips of cuts in woven fabric. During composite forming operations, yarn slip can be caused bygradients of shear locking forces (Zhu et al., 2007) and transverse shear loads, affecting the shape and properties of theformed part as well as potentially causing the fabric to fail.

The simulation of yarn slip at the scale of an actual fabric has largely been limited to two types of discrete model, bothwith drawbacks compared to the mesostructure-based continuum approach. One type of model is the discrete three-dimensional approach, in which every yarn (or even fiber) is modeled in three dimensions with finite elements (Shockeyet al., 1999; Duan et al., 2005, 2006a; Zhang et al., 2008; Chocron et al., 2011, among others). These finite element meshesare tedious to construct (and must be reconstructed for any change in the geometry of the fabric), computationally veryexpensive, and difficult to integrate with other material models for the simulation of fabric-reinforced composites.The second type of model is that of Roylance et al. (1973), in which the fabric is modeled as an array of point massesconnected by massless, pin-joined trusses. Roylance et al. (1995), Termonia (2004), and Zeng et al. (2006), among others,extended this approach to include yarn slip. Although more efficient than the three-dimensional approach, the point massmodels still must simulate every yarn in the fabric. In addition, with this type of model, it is difficult to include contactbetween layers in multi-layer simulations and to incorporate the matrix in the simulation of fabric-reinforced composites.Notably also, Boubaker et al. (2007) used somewhat of a hybrid technique, in which the undulating yarns are discretizedas elastic trusses connected by rotational springs, to study the effects of inter-yarn friction on yarn extension at themeso-level.

The mesostructure-based continuum approach is more efficient and versatile than both types of discrete approach. Themesostructure of the weave can be varied without remeshing in order to optimize the design of new fabrics, and othercontinuum material models can be integrated easily for the simulation of composites. In the literature, to our knowledge,the only continuum-level approach for modeling yarn slip in woven fabric is that of Nadler and Steigmann (2003) andNadler (2009). These authors proposed simulating the two yarn families as two interacting surfaces, each composed ofcontinuously distributed yarns. The material points of surface-1 are tracked explicitly by the equations of motion. In turn,in the deformed configuration, the evolving points of surface-2 that interact with the material points of surface-1 aredetermined by the relative slip between the two surfaces. The efficacy of this approach is difficult to evaluate because itappears that no operative implementations or results have been published. Because this model does not explicitly trackthe motion of the cross-over points of the weave, using it to simulate the interactions between the yarns and to describethe evolution of the mesostructure of the weave might be challenging. Furthermore, this model does not permit yarn slipat the edges of the fabric, the locations where slip is most apt to occur if the edges are not clamped tightly. In light of the

E.M. Parsons et al. / J. Mech. Phys. Solids 61 (2013) 265–292 267

scarcity of existing continuum-level models, the first objective of this work was to develop, implement, and verify withexperiments a method to simulate yarn slip with the mesostructure-based continuum approach. Requirements for thismethod included that it both track the motion of the cross-over points of the weave and allow yarn slip to occur at theedges of the fabric.

Due to the complexity of the required experiments and the dearth of suitable models, there is no consensus on theeffects of yarn slip and inter-yarn friction on the ballistic performance of woven fabric. In ballistic impact experiments onKevlars fabrics with different coatings, Briscoe and Motamedi (1992) showed the energy absorbed to decrease withdecreasing assumed coefficient of friction. On the other hand, in experiments on Kevlar-reinforced composites, Lastnik andKarageorgis (1982) reported energy absorbed to increase with increasing yarn mobility. Simulations of ballistic impact arein similar conflict. In point mass simulations of the ballistic impact of a para-aramid fabric, Termonia (2004) showed theenergy absorbed to decrease up to 50% with increasing yarn slip at a projectile velocity 33% higher than the ballistic limit.Conversely, also in point mass simulations, but of a Twarons fabric over a wide range of projectile velocities, Zeng et al.(2006) showed the energy absorbed to increase by up to 65% with decreasing inter-yarn coefficient of friction m, formZ0:1. In discrete three-dimensional simulations of a Kevlar KM2 fabric, Rao et al. (2009) showed the ballistic limit todecrease about 7% when the inter-yarn friction was decreased from m¼ 0:23 to m¼ 0. However, in discrete three-dimensional simulations of a Zylons fabric, Duan et al. (2006a) showed the energy absorbed to increase about 5% whenthe inter-yarn friction was decreased from m¼ 0:5 to m¼ 0 at a projectile velocity well above the ballistic limit. Therefore,considering the conflicting results in the literature, the second objective of this work was to increase the understanding ofthe effect of yarn slip on the ballistic impact of woven fabric.

The rest of this paper is organized as follows: Section 2 presents the woven fabric model, specific in parts to a plain-weave fabric. First, after a brief overview of the model, the analytic unit cell of the fabric is described, and the method isgiven to determine the unit cell’s deformed geometry from the deformation of the weave continuum. Then, thekinematical framework used to simulate the motion of both the weave continuum and the constituent yarns is outlined.Next, balance principles are used to develop expressions for the accelerations of the weave points and the yarns as well asexpressions for the evolution with yarn slip of the unloaded geometry of the unit cell. The description of the model iscompleted with the constitutive relations for inter-yarn friction and for each component of the unit cell. Section 2concludes with a brief summary of the finite element implementation of the model. Section 3 presents finiteelement simulations of ballistic impact experiments. It shows that allowing yarn slip to occur improves the accuracyof the simulations. The mechanics of energy absorption and the effects of inter-yarn friction and boundary conditionson yarn slip are explored. The discussion of Section 4 summarizes the results and shows that the findings are consistentwith experimental and simulation results in the literature. Section 5, the conclusion of the paper, offers perspectives onboth the simulation of yarn slip at the continuum-level and the importance of yarn slip in the deformation of wovenfabrics.

2. Continuum-level model of the deformation of woven fabric with yarn slip

The discrete structure of the woven fabric is modeled as an anisotropic ‘‘weave continuum’’ (Fig. 1). Each ‘‘materialpoint’’ in the continuum represents a section of fabric encompassing a number of yarn cross-over points. The differencebetween our approach and all previously proposed continuum-level models for woven fabric is that points in the weavecontinuum represent the cross-over points of the weave, not the locations of the constituent yarns. In the absence of yarnslip, the cross-over points are true material points because the same locations on each yarn family stay in contact.However, when yarn slip occurs at a cross-over point, yarn locations that coincide in the reference configuration no longercoincide in the deformed configuration. In this case, yarn slip is captured by additional field variables, which are functionsof gradients of yarn forces and the inter-yarn friction. The fabric continuum thus represents the structure of the weave, andwe designate points in this continuum as simply weave points.

This approach simulates the evolution of the yarn pitch of the fabric as yarn slip occurs and therefore can predict thedeformed mesostructure of the fabric. We define an analytic unit cell of the fabric to represent, on average, thedeformation in the vicinity of each weave point. In the finite element implementation of the model, the spacing of theweave points is determined by the local mesh refinement, which is chosen to capture the gradients present in the actualmesostructure of the fabric. The in-plane stretch and shear of the analytic unit cell are determined from the deformation ofthe weave continuum, characterized by the weave deformation gradient, Fw. The out-of-plane deformation of the unit cell iscalculated from conservation of momentum in the through-thickness direction. When yarn slip occurs, the amount of yarnmaterial contained within the unit cell can change, causing the unloaded geometry of the unit cell to evolve. The deformedand unloaded geometries of the unit cell, together with the constitutive relations of the yarns, determine the tensile andshear forces on the unit cell. These forces are then homogenized over the boundary of the unit cell in order to calculate thestress in the weave continuum.

The analytic unit cell and the constitutive relations were developed for Kevlars S706, a plain-weave fabric composed ofKevlar KM2 yarns. It would be straightforward to apply this approach to other plain-weave fabrics. Fabrics with differentweave patterns would require more complex unit cells, but the modeling technique would not change.

Fig. 1. Schematic of continuum-level modeling technique for woven fabric: (1) Discrete fabric with weave direction unit vectors g1 and g2 is

approximated as a continuum of weave points. (2) The deformation of the continuum, the balance of out-of-plane momentum, and the yarn slip velocity

fields determine the deformed and unloaded geometries of the analytic unit cell at each weave point. (3) Yarn forces are homogenized over the boundary

of the unit cell to calculate the stress in the continuum. (Shear and locking forces are omitted from the schematic for clarity.)

Fig. 2. Analytic unit cell of a plain-weave fabric: (a) geometry of warp and weft yarns, (b) locking trusses to simulate shear-locking and cross-locking of

the fabric.


2.1. Deformation of the analytic unit cell of a plain-weave fabric with yarn slip

The analytic unit cell approximates the mesostructure of a plain-weave fabric with trusses and springs. Woven fabric isexceptionally uniform, and therefore the unit cell can be considered a representative volume element of the fabric.The unit cell of a plain-weave fabric consists of two crossing sections of yarn (Fig. 2). It simulates the geometry andmesoscale modes of deformation that underlie the macroscopic response of the fabric. Owing to the symmetry of thestructure, only one-quarter of the actual repeating unit cell needs to be considered. The warp yarn (k¼1) and the weft yarn(k¼2) are each approximated as two pin-joined elastic trusses with the following geometric properties in the deformedconfiguration (Fig. 2a):

(1)
yarn quarter-wavelength (half-pitch), pðkÞ
(2)
yarn half-length or simply ‘‘length,’’ ‘ðkÞ


(3)

Fig.with

yarn crimp angle, bðkÞ

(4)
yarn crimp amplitude, AðkÞ
(5)
in-plane angle between yarns, y, or ‘‘shear angle’’, g¼ p=2�y
These properties are not independent, because any two of pðkÞ, ‘ðkÞ, bðkÞ, and AðkÞ are sufficient to determine the geometry of
yarn-k. The trusses are extensible but infinitely stiff in bending. The compliant bending response of each yarn is lumped ina torsional spring (termed a ‘‘bending spring’’) at the hinge connecting the two trusses of each yarn section. The two yarnsare themselves connected by a ‘‘cross-over spring,’’ which provides both torsional resistance to yarn rotation andresistance to transverse compression (compaction) of the yarns. Four locking trusses resist both shear-locking and cross-locking of the yarns. Each locking truss extends from a yarn end at the boundary of the unit cell to the perpendicular pointon the yarn of the opposite family, and it has the following geometric properties (Fig. 2b):
(1)
locking truss length, dðkÞ
(2)
angle between locking truss and plane of the fabric, aðkÞ
The length of locking truss k can be expressed as a function of pðkÞ, AðjÞðjakÞ, and y. When the yarn pitch or the in-planeangle between the yarns decreases, the locking truss shortens and provides compressive resistance to further deformation.The undeformed geometric properties of the fabric are denoted by a subscript ‘‘0’’: pðkÞ0 , LðkÞ0 , bðkÞ0 , AðkÞ0 , y0, dðkÞ0 , and aðkÞ0 . A morecomplex unit cell geometry, such as that of the extensible elasticae unit cell of Nadler et al. (2006), might improve thepredictions of the model in some cases, but the simulations of King et al. (2005) and Parsons et al. (2010b) showed thetruss unit cell to be remarkably accurate.

When yarn slip velocities vary across the weave, the amount of yarn within the unit cell can change. We thereforedefine L

ðkÞto be the unloaded yarn lengths of the unit cell (Fig. 3). The unloaded yarn lengths, initially equal to the

undeformed yarn lengths, LðkÞ0 , appear in the constitutive relations for yarn tension in Section 2.5. They also determine thestretches of the yarns, lðkÞ:

lðkÞ ¼‘ðkÞ

LðkÞðk¼ 1;2Þ: ð1Þ

Whether or not yarn slip occurs, the fabric can be described at each cross-over point by yarn pitch vectors, 2pð1Þ and2pð2Þ, connecting adjacent cross-over points (Fig. 4). In the absence of yarn slip, the cross-over points move in an affinemanner with the yarns and are therefore material points. In this case, the pitch vectors are material lines, which can beboth transformed into the deformed configuration by the standard deformation gradient, F¼Grad xðX,tÞ, where X is

3. When yarn slip occurs, the amount of yarn within the unit cell can change. This change is quantified by the unloaded yarn lengths, LðkÞ

. For a fabric

out yarn slip, LðkÞ¼ LðkÞ0 .

Fig. 4. Approach to modeling yarn slip in woven fabric. The continuum represents the cross-over points of the weave, not the yarns. The weave

deformation gradient, Fw, maps the yarn pitch vectors, pð1Þ0 and pð2Þ0 , into the deformed configuration of the fabric.


the position of a material point in the undeformed (reference) configuration and x is the position of the same point inthe deformed configuration. When yarn slip occurs, however, the warp yarns and weft yarns slide relative to each anotherat the cross-over points. The cross-over points are no longer material points, and the pitch vectors are no longermaterial lines.

In contrast to the conventional approach, we use the continuum to represent the position of the cross-over points, notthe position of the yarns. The gradient of the deformation of this continuum with respect to the reference configuration,the weave deformation gradient, Fw, thus defines the in-plane stretch and shear of the analytic unit cell:

pð1Þ ¼ Fwpð1Þ0 and pð2Þ ¼ Fwpð2Þ0 , ð2Þ

from which pð1Þ, pð2Þ, and y are calculated. Without yarn slip, Fw¼ F, and the model of Parsons et al. (2010b) is recovered.

In order to complete the description of the unit cell, the crimp amplitudes of the yarns, AðkÞ, are calculated from the sumof the transverse forces acting on each yarn and the balance of momentum. The transverse forces include both the force inthe cross-over spring and the transverse components of the forces in the yarn trusses. (Further details are provided inAppendix A and Parsons et al.).

2.2. Kinematics of a woven fabric with yarn slip

The motion of the fabric is described with respect to the weave reference configuration, X0, the initial configuration ofthe weave points (Fig. 5). In this configuration, we define unit vectors G1 and G2 to coincide with, respectively, the initialwarp yarn direction and the initial weft yarn direction at each weave point. As the fabric deforms, a weave point at X in thereference configuration moves to position x¼/ðX,tÞ in the deformed configuration, Xt . The yarn directions at this weavepoint are now described by unit vectors fgkgk ¼ 1;2. Although the weave reference configuration is constant, the yarn

material points XðkÞ located at X evolve as yarn slip occurs. We therefore define separate yarn material configurations, XðkÞ,

to describe the initial, undeformed positions, XðkÞ¼wkðX,tÞ, of the yarn material points currently associated with the weave

point at X. X and XðkÞðX,tÞ differ by a linear translation along the initial direction of yarn-k. The undeformed lengths of

differential yarn elements located at the weave point at X can also change with yarn slip according to FðkÞ¼Grad wkðX,tÞ.

The kinematics of the fabric are determined from the motions of both the weave continuum and the material points onthe constituent yarn families. The description of the kinematics is similar to the formulation of Laursen and Simo (1993)for the simulation of general frictional contact problems. For each weave continuum point x, there are corresponding

contacting material points on the two yarn families, xð1Þ ¼/1

�Xð1Þ

,t�

and xð2Þ ¼/2

�Xð2Þ

,t�, both equal to x, which are

located at Xð1Þ¼w1ðX,tÞ and X

ð2Þ¼w2ðX,tÞ, respectively, in the material configurations. When yarn slip occurs, X

ð1Þand X

ð2Þ

are not coincident and may both differ from X. At each weave point,

/1

�Xð1ÞðX,tÞ,t

��/ðX,tÞ ¼ 0 and /2

�Xð2ÞðX,tÞ,t

��/ðX,tÞ ¼ 0: ð3Þ

Fig. 5. Kinematics of a woven fabric with yarn slip. As the fabric deforms, a weave point at X in the reference configuration, X0, moves to position

x¼/ðX,tÞ in the deformed configuration, Xt . The yarn material points currently located at x are given by XðkÞ¼wkðX,tÞ in the yarn material

configurations.


Taking the material time derivative of the preceding equations (holding the weave point X fixed), we obtain the differencesbetween the material velocities of the currently associated points, which we define to be the slip velocities of the yarns:

vslipðkÞðX,tÞ � vðkÞ�XðkÞðX,tÞ

��vðX,tÞ ¼�

@/k

@XðkÞ

@XðkÞ

@tðk¼ 1;2Þ: ð4Þ

Aligned with the current yarn direction, the slip velocity is the instantaneous velocity of yarn-k relative to the velocity ofthe weave point. Differentiating again provides the yarn slip accelerations:

_vslipðkÞðX,tÞ ¼ _vðkÞ

�XðkÞ

,t�� _vðX,tÞ: ð5Þ

2.3. Balance laws of the weave and yarns

The yarn slip velocities and accelerations are used to develop the equations of conservation of linear momentum andconservation of mass for the fabric.

Considering the fabric as a membrane and homogenizing over the continuum, we express, in the material configuration,the global equation of motion of each yarn family asZ

XðkÞrðkÞ0

_vðkÞ XðkÞ

,t� �

dSðkÞ ¼

ZXðkÞ

DivXðkÞPðkÞ X

ðkÞ,t

� �þ

�JðkÞ��1

tðkÞ w�1k X

ðkÞ,t

� �,t

� �( )dSðkÞ, ð6Þ

where rðkÞ0 (k¼1,2) is the initial homogenized areal density of each yarn family, PðkÞðXðkÞ

,tÞ is the first Piola–Kirchhoff stressin the fabric (with units of force per unit length) due to the forces in each yarn family, J

ðkÞ¼ det F

ðkÞis the area ratio of the

material configuration (XðkÞ) to the reference configuration (X0), and tðkÞ is the interaction force in the referenceconfiguration due to contact and friction with the opposite yarn family. (In Eq. (6), we have not included the externalforces acting on the fabric.) Next, we use the Piola transformation to express the divergence of the stress in terms of thereference configuration:

DivXðkÞ PðkÞðX

ðkÞ,tÞ ¼ JðkÞ divx rðkÞðx,tÞ ¼ JðkÞðJw

Þ�1 DivX PðkÞðX,tÞ: ð7Þ


Using also JðkÞ Jw� ��1¼ J

ðkÞ� ��1

and dSðkÞ ¼ JðkÞ

dS0, we write, in the weave reference configuration, the equation of motion ofeach yarn family asZ

X0

rðkÞ0_vðkÞðX

ðkÞðX,tÞ,tÞ J

ðkÞdS0 ¼

ZX0

DivX PðkÞðX,tÞþtðkÞðX,tÞn o

dS0, ð8Þ

or, in strong form,

rðkÞ0 JðkÞ_vðkÞðX

ðkÞðX,tÞ,tÞ ¼DivX PðkÞðX,tÞþtðkÞðX,tÞ: ð9Þ

With Eq. (5) and

tð1ÞðX,tÞ ¼�tð2ÞðX,tÞ, ð10Þ

the equations of motion of the two yarn families can be combined to form a single equation for the motion of the weavecontinuum:

ðrð1Þ0 Jð1Þþrð2Þ0 J

ð2ÞÞ _vðX,tÞ ¼DivX PðX,tÞ�rð1Þ0 J

ð1Þ_vslipð1Þ

ðX,tÞ�rð2Þ0 Jð2Þ_vslipð2Þ

ðX,tÞ, ð11Þ

where the stress contributions from the two yarn families are superposed to define the homogenized weave stress,PðX,tÞ ¼ Pð1ÞðX,tÞþPð2ÞðX,tÞ. The homogenized weave stress is calculated from the forces and moments acting on therepresentative unit cell at X, via the constitutive relations of the yarns, as described in Parsons et al. (2010b). In Eq. (11),we see that the spatial variation of the stress in the fabric accelerates both the weave continuum and the individual yarnsrelative to the weave continuum. In the following derivation and the implementation of Section 2.6, in order to simplifythe equations, we approximate the fabric as initially well-balanced, rð1Þ0 � r

ð2Þ0 �

12r

f0, where rf

0 is the initial areal density ofthe fabric. (Kevlar S706, our model fabric, is well-balanced.)

The instantaneous yarn slip accelerations at the weave point at X can be determined from the relative motion of thetwo yarn families. The equation of motion of each yarn family (Eq. (9)) is first expressed in the non-orthogonal {gk} basisvia the contravariant basis vectors gk. Considering slip in the warp yarn direction (and dispensing with arguments X and t),

rf0

2Jð1Þ_vð1Þ � g1 ¼Div Pð1Þ � g1þtð1ÞT1 and

rf0

2Jð2Þ_vð2Þ � g1 ¼Div Pð2Þ � g1þtð2ÞT1 , ð12Þ

where tðkÞT1 ¼ tðkÞ � g1 is the tangential traction imposed upon yarn-k in the warp direction by the frictional interaction withthe opposite yarn family or, more simply, the inter-yarn friction in the warp direction. Because the position of the weft yarnalong the warp yarn defines the location of the weave point, the acceleration of the weft yarn in the warp direction(Eq. (12b) is the same as the acceleration of the weave point in the warp direction. We thus subtract Eq. (12b) from Eq. (12a)to calculate the yarn acceleration in the warp direction relative to the weave point, _vslipð1Þ

¼ _vslipð1Þg1, which is the yarn slipacceleration in the warp direction:

_vslipð1Þ�

�_vð1Þ� _vð2Þ

�� g1 ¼

rf0

2Jð1Þ

!�1nDiv Pð1Þ � g1þtð1ÞT1

o�

rf0

2Jð2Þ

!�1nDiv Pð2Þ � g1þtð2ÞT1

o: ð13Þ

With tð1ÞT1 ¼�tð2ÞT1 and taking Div Pð2Þ � g1 � 0 because tð2ÞT1 is typically the dominant force accelerating the weft yarn in thewarp direction, we find

_vslipð1Þ¼

rf0

2Jð1Þ

!�1

Div Pð1Þ � g1þ 1þJð1Þ

Jð2Þ

!tð1ÞT1

( )ð14Þ

and, analogously for yarn slip in the weft direction,

_vslipð2Þ¼

rf0

2Jð2Þ

!�1


Jð1Þ

!tð2ÞT2

( ): ð15Þ

These equations express the momentum balance of the yarns in a reference frame anchored at the weave point.Conservation of mass is enforced by evolving the unloaded yarn lengths of the representative unit cell at each weave

point. The rates of change of these yarn lengths, _LðkÞ

, are determined from the gradients of the yarn slip velocity fields.The yarn slip velocities are calculated by integrating the yarn slip accelerations of Eqs. (14) and (15). The rate of change ofeach length of unloaded yarn within the analytic unit cell is then calculated from the yarn slip velocity gradient in thedirection of the yarn:

_LðkÞ¼ �pðkÞ0 Grad

�vslipðkÞ

�� Gk, ð16Þ

where pðkÞ0 is the yarn half-pitch over which the yarn length is defined and vslipðkÞ¼ vslipðkÞ=lðkÞ is the instantaneous slip

velocity of unstretched yarn at the weave point. The right-hand side of Eq. (16) is negative because increasing slip velocityin the yarn direction causes yarn material to be lost from the unit cell. Integration of _L

ðkÞprovides the area ratios

JðkÞ¼ LðkÞ=LðkÞ0 .


2.4. Constitutive relation for inter-yarn friction

A constitutive relation for the inter-yarn friction at each weave point is developed from the results of yarn pull-outexperiments. It is expressed in a form similar to that used by Anand (1993). In the calculation of the inter-yarn friction, werecognize that the relative sliding of two yarns can be represented as the sliding of the contact point in the direction ofeach yarn (Zavarise and Wriggers, 2000).

The inter-yarn friction in each yarn direction, tð1ÞT1 ¼�tð2ÞT1 and tð2ÞT2 ¼�tð1ÞT2 , transfers loads between yarn families and inhibitsthe relative sliding of the yarns. When the inter-yarn friction in the direction of yarn-k equals the slip resistance in thatdirection, yarn slip occurs. The slip resistance, sk, is a positive scalar. In this implementation, it is assumed to be a function onlyof the contact forces between the yarns. The slip function in each yarn direction, f slip

k , determines whether or not slip occurs:

f slip1 ¼ tðkÞT1

�� s1r0 and f slip2 ¼ tðkÞT1

�� s2r0, ð17Þ

where for yarn slip in the warp direction, for example:

if vslipð1Þ ¼ 0, tðkÞT1

�� ¼min 1þJð1Þ

Jð2Þ

!�1

Div Pð1Þ � g1

��,s1

8<:

9=;,

else tðkÞT1

�� ¼ s1:

ð18Þ

In Eq. (18), the first condition results from setting _vslipð1Þ¼ 0 in Eq. (14), which determines the inter-yarn friction when yarn

slip is not occurring. The preceding equations result in the following slip rule in the warp yarn direction:

_vslipð1Þ¼

0 if f slip1 o0,

rf0

2Jð1Þ

!�1


Jð2Þ

!tð1ÞT1

( )if f slip

1 ¼ 0:

8>>><>>>:

ð19Þ

The equations for yarn slip in the weft direction follow analogously. Because inter-yarn friction always opposes yarn slip, thesigns of the inter-yarn friction tractions must obey tð1ÞT1 � vslipð1Þr0 and tð2ÞT2 � vslipð2Þr0.

The slip resistance associated with each weave point is due to a complex distribution of pressure between the slippingyarn and the crossing yarns (Fig. 6a). This pressure distribution is modeled in the analytic unit cell by two contact forces(Fig. 6b). The first contact force is the force in the cross-over spring, FI, representing the transverse force, or normal force,between the yarns due to cross-sectional compaction. The second contact force is the force in the locking trusses,FðkÞL , representing the force due to locking of the weave. Homogenizing over the in-plane area of the unit cell, we simulatethe total in-plane resistance to yarn slip as a linear function of these two forces:

sk ¼1

4pð1Þ0 pð2Þ0

�mNFIþ2mLFðkÞL

�, ð20Þ

where mN and mL are not actual coefficients of friction but constants of proportionality determined from yarn pull-outexperiments. Single yarn pull-out experiments at different cross-loads and pull-out velocities were performed on theKevlar S706 fabric (King, 2006). We fit to these experiments an analytic model based on the unit cell of the fabric anddetermined mN ¼ 0:12 and mL ¼ 0:6.

Although we have expressed the balances of linear momentum and the slip functions in homogenized form suitable forfinite element implementation, it is helpful to think of yarn slip at the scale of the representative unit cell at each weavepoint. If we approximate yarn slip to be driven only by yarn tensions (usually the dominant components of PðkÞ), then yarnslip begins when the gradient of in-plane yarn tension across the unit cell in the yarn direction overcomes the friction forceassociated with the cross-over point. Yarn slip then continues, even without a tension differential sufficient to initiate slip,until the inter-yarn friction can decelerate the yarn. It is at the scale of the cross-over point that yarn slip, by its verydefinition, actually occurs in a woven fabric.

Fig. 6. Simulation of inter-yarn friction: (a) Actual friction force is due to a complex distribution of contact pressure. (b) Contact pressure is simulated by

the forces in both the cross-over spring and the locking trusses in order to calculate the slip resistance, sk ¼ 1=ð4pð1Þ0 pð2Þ0 Þ � ðmNFIþ2mLFðkÞL Þ.


2.5. Constitutive relations of the yarns

The forces and moments acting on and within the analytic unit cell are determined from the constitutive relations ofthe yarns. These constitutive relations were developed by King et al. (2005) from experiments on Kevlar KM2 yarns andKevlar S706 fabric. The forces and moments that the constitutive relations provide are used to calculate the homogenizedweave stress in the fabric, the crimp amplitudes of the yarns, and the slip resistances of the yarns.

Yarn stretch: Single yarn tension tests showed the KM2 yarns to be linear elastic, and the yarn tensions in the unit cellare therefore given by

T ðkÞ ¼ kðkÞ�‘ðkÞ�L

ðkÞ�ðk¼ 1;2Þ, ð21Þ

where kðkÞ is the measured stiffness of the yarns and LðkÞ

is the unloaded yarn length. (LðkÞ¼ LðkÞ0 in the absence of yarn slip.)

The strength of the yarns was taken to be T ðkÞmax ¼ 112 N, the average load per yarn at which Kevlar S706 specimens failed inuniaxial tension at a nominal strain rate of 0.01 s�1. Rate-dependence was not included in this constitutive relationbecause Cheng et al. (2005) showed that the modulus and ultimate strength of KM2 fibers are virtually rate-independentfor strain rates ranging from quasi-static to 2400 s�1.

Changes in crimp amplitude: The changes in crimp amplitude of the yarns, DAðkÞ ¼ AðkÞ�AðkÞ0 , are in part controlled by theresistances to bending and compaction of the yarns. The bending response is approximated as linear elastic, resulting inbending moments,

MðkÞb ¼ kðkÞb

�bðkÞ�bðkÞ0

�ðk¼ 1;2Þ, ð22Þ

in the torsional springs at the pin-joints of the unit cell (Fig. 2a). Crimp changes can occur without compaction of the yarnsby simple crimp interchange, one yarn family straightening at the expense of the other family becoming more crimped.Decreases in crimp amplitude can occur also by compaction of the yarn cross-sections. The resistance to compaction issimulated in the unit cell by the compressive force in the cross-over spring, termed the interference force,

FI ¼ K I

�eaI�1

�, ð23Þ

where I¼�DAð1Þ�DAð2Þ measures the compaction or ‘‘interference’’ of the yarns at the cross-over point and materialparameters K I and a were determined from transverse compression experiments.

In-plane shear: When sheared, the fabric exhibits an initial stiff, elastic response due to rotation of the yarns betweenthe cross-over points, which is represented in the unit cell by the moment in the cross-over spring,

Ms ¼ Ksge: ð24Þ

The elastic deformation is followed by a compliant, rate-dependent, dissipative response due to rotation at the cross-overpoints resisted by friction,

_gf ¼ _g0

Ms

M0

� �b

, ð25Þ

where M0 is the moment at which dissipative rotation occurs at reference dissipative rotation rate _g0 and exponent b

captures the rate sensitivity of the response. The elastic and dissipative rotations result in total relative yarn rotationg¼ geþgf . The parameters Ks, M0, _g0, and b were fit to the results of shear-frame experiments conducted at different rates.

Locking: Locking begins when warp and weft yarns contact each another in the plane of the fabric. Further in-planemotion is resisted by cross-sectional compaction of the yarns. This resistance is provided by the compressive force in thelocking trusses of the unit cell (Fig. 2b):

FðkÞL ¼

0, dðkÞ0 �dðkÞr0

Kd

�dðkÞ0 �dðkÞ

�c, dðkÞ0 �dðkÞ40

8><>: ðk¼ 1;2Þ, ð26Þ

Table 1Initial geometric properties of Kevlar S706.

Warp quarter-wavelength (weft half-pitch), pð1Þ00.374 mm

Weft quarter-wavelength (warp half-pitch), pð2Þ00.374 mm

Warp crimp amplitude, Að1Þ00.060 mm

Weft crimp amplitude, Að2Þ00.090 mm

Warp half-length, Lð1Þ00.378 mma

Weft half-length, Lð2Þ00.384 mma

In-plane angle, y0 901

a Calculated properties.


where Kd was fit to the slope of the dissipative part of the shear-frame experiment response and the exponent c waschosen such that shear-locking occurs at the appropriate yarn rotation angle. The locking trusses have stiffness incompression only.

Further details of the fitting of the constitutive relations are provided in King et al. (2005) and Parsons et al. (2010b).The initial geometry of the analytic unit cell was taken from measurements of Kevlar S706 (Table 1). The KM2 yarns had alinear density of 600 denier and were woven at a thread count of 34 yarns per inch.

2.6. Finite element implementation of woven fabric model with yarn slip

A membrane finite element was developed for explicit integration in order to simulate the dynamic deformation ofwoven fabric. It has degrees of freedom representing the displacements of the weave points, ui (i¼1,3), the changes incrimp amplitude of the yarns, DAðkÞ, and the slip displacements of the yarns, uslipðkÞ. The yarn slip accelerations of Eqs. (14)and (15) are calculated from the gradients of the in-plane yarn tension components of PðkÞ (Eqs. (30b) and (37) of Appendix B.1),and the rates of change of the unloaded yarn lengths of Eq. (16) are calculated from the gradients of the yarn slip velocity fields(Eq. (41) of Appendix B.1). At the boundaries of the finite element mesh, either yarn slip velocities or in-plane yarn tensions arespecified. Failure of the element occurs when the element-averaged tension in either yarn family exceeds the strength of theyarns,

�TðkÞ�e4T ðkÞmax. The finite element was implemented into Abaqus/Explicit as a user element for the simulation of ballistic

impact experiments. Further details of the finite element are provided in Appendix B.

3. Ballistic impact experiments and simulations

Ballistic impact experiments performed on Kevlar S706 with initial projectile velocities vi ¼ 1502550 m=s weresuccessfully simulated with the woven fabric model. At these projectile velocities, significant yarn slip occurs.The patterns of yarn slip in the simulations agree with those of the experiments. Yarn slip is shown to increase theenergy absorbed by the fabric at high projectile velocities. In further simulations, the energy absorbed is shown to be afunction of inter-yarn friction and the boundary conditions imposed on the fabric.

3.1. Procedure for ballistic impact experiments and simulations

The ballistic impact experiments were performed with a closed chamber gas gun (Fig. 7) at the MIT Institute for SoldierNanotechnologies (Parsons et al., 2010b). The specimens were rectangular single layers of Kevlar S706 fabric, clamped attwo ends such that the square gauge section measured 15.2 cm per side. The projectiles were cylindrical polycarbonateslugs weighing 3.6 g each. Each slug was 12.65 mm in diameter and had a hemispherical tip ending at a 5.1 mm diameterflat. The initial velocity of each projectile was measured with a laser sensor, and images of the deforming specimen wererecorded with a high speed CCD camera. The images were used to measure both the transverse displacement (deflection)of the fabric and the residual velocity of the projectile (the exit velocity, if penetration occurs) for comparison with thesimulations. Significant slip of the fabric from the clamps was observed in these experiments.

One-quarter of each fabric specimen was simulated with the woven fabric finite elements. The element edge length wasabout 5 mm, resulting in a mesh of 555 elements. The projectile was modeled with an analytic rigid surface. The grip areasof the specimen were clamped between two rigid surfaces. The clamping pressure was chosen such that the slip of thefabric from the clamps during the simulations matched that observed in the experiments. Contact between the clamps andthe fabric was modeled with Coulombic friction with m¼ 0:3, and contact between the projectile and the fabric wasapproximated as frictionless. A preload of 2 N per yarn, determined from the experiments, was imposed in the warp

Fig. 7. Ballistic impact experiments: (a) Gas gun and closed chamber with viewing ports. (b) Fabric was wrapped around steel bars. (c) Fabric wrapped

bars were clamped to the frame inside the chamber.


direction by applying a distributed load to the edge of the fabric prior to clamping. The time of computation with a single64-bit CPU (2.40 GHz clock speed, 1066 MHz front-side bus speed) was about 15 min.

3.2. Ballistic impact of woven fabric: simulations versus experiments

During ballistic impact of woven fabric, the energy of the projectile is transformed into the motion and stretch of theyarns. The mechanics of this process are well characterized in the literature by Smith et al. (1956), Roylance et al. (1973),Roylance (1980), and Ting et al. (1998), among others. Upon impact, longitudinal strain waves propagate along the contactedor ‘‘primary’’ yarns away from the point of impact. In the wake of these strain waves, the yarns translate inward toward theprojectile. At the cross-over points of the fabric, portions of the strain waves are diverted along the orthogonal ‘‘secondary’’yarns via the frictional interaction between the yarns. In addition to the longitudinal strain waves, transverse displacementwaves develop in the primary yarns and propagate away from the projectile more slowly than the longitudinal waves do. Atthe cross-over points, portions of the transverse displacement waves are also diverted along the orthogonal secondary yarns.

Compared to simulations without yarn slip, simulations with yarn slip provide a significantly better match to theballistic impact experiments at high initial projectile velocities. They predict more accurately the energy absorbed by thefabric than the simulations without yarn slip do. The predictions improve because the model with yarn slip can accuratelyaccount for the reduction in yarn tension associated with yarn slip in the section of the fabric struck by the projectile.

Fig. 8. In simulations of ballistic impact, yarn slip improves the predictions of the residual velocity of the projectile (vr) and the energy absorbed by the

fabric: (a) Residual velocity, as a fraction of initial velocity (vi), versus the initial velocity of the projectile. (b) Energy absorbed by the fabric versus the

initial velocity of the projectile.

Fig. 9. Yarn slip has little effect on transverse displacement at vi � 170 m=s: (a) Prediction of transverse displacement at point of impact for vi ¼ 171 m=s.

(b) Prediction of transverse displacement profiles through point of impact for vi ¼ 164 m=s.


3.2.1. The three regimes of initial projectile velocity

During the ballistic impact experiments, the residual velocity of the projectile, vr, as a function of the initial velocity ofthe projectile, vi, exhibits three distinct regimes: (1) below the ballistic limit, vi � 200 m=s, the projectile does notpenetrate the fabric; (2) at intermediate velocities, vi � 2002275 m=s, the residual velocity is about 50% of the initialvelocity; (3) at high velocities, vi4275 m=s, there is a jump in the residual velocity to approximately 90% of the initialvelocity (Fig. 8a). Plotting the results in terms of energy in Fig. 8b, we see that the energy absorbed by the fabric,E¼ 1

2 mprojðv2i �v2

r Þ, reaches a maximum within the intermediate velocity regime and then, on average, declines through thehigh velocity regime. Shim et al. (1995), Tan et al. (2003), Lim et al. (2003), and Termonia (2004) all observed these sameregimes in their own experiments on the ballistic impact of woven fabric. With simulations of these experiments, Parsonset al. (2010b) showed that, without yarn slip, the woven fabric model is accurate at low and intermediate velocities butunderpredicts the energy absorbed at high velocities.

Yarn slip improves the predictions of the simulations at vi4275 m=s, but it does not have much effect on themacroscopic deformation or failure of the fabric at lower projectile velocities. Below the ballistic limit, significant yarn slipoccurs, particularly at the free edges of the fabric, but this yarn slip hardly changes the deflection of the fabric. Similarresults are seen at intermediate projectile velocities, where the residual velocity and energy absorbed in simulations with

Fig. 10. Yarn slip at vi ¼ 171 m=s. Images of experiment (top) and simulated contours (middle and bottom) of yarn slip displacement, uslipðkÞ , for

vi ¼ 171 m=s at time after impact: (a)–(a00) 30 ms; (b)–(b00) 100 ms; (c)–(c00) 200 ms; (d)–(d00) 340 ms; (e)–(e00) 400 ms. Contour plots are mirrored; yarn slip

toward the projectile is defined as positive in all quadrants.


yarn slip are nearly identical to those in simulations without yarn slip (Fig. 8). At high projectile velocities, however, thetransition to the high velocity regime is predicted to occur at a lower initial projectile velocity with yarn slip than it iswithout yarn slip, providing a better match to the experiments. Furthermore, at high projectile velocities, the simulationswith yarn slip predict the fabric to absorb up to 220% more energy than the simulations without yarn slip do (Fig. 8). Next,we examine the deformation of a fabric specimen at an initial projectile velocity within each regime in order to explore therole of yarn slip during ballistic impact.

3.2.2. Ballistic impact below the ballistic limit

Below the ballistic limit, the deformation extends over the majority of the fabric specimen, and yarn slip ispredominantly manifested by unraveling at the free edges of the fabric. At vi ¼ 171 m=s, just below the ballistic limit,plots of deflection at the point of impact and along the center warp yarn show that the additional compliance provided byyarn slip causes less than 1 mm of additional deflection (Fig. 9). At t¼ 30 ms, shortly after impact, the simulation exhibitspositive slip displacements, uslipðkÞ (Fig. 10). When the projectile first strikes the fabric, it generates longitudinal strainwaves and tension gradients in the primary warp and weft yarns (the yarns contacted by the projectile). These tensiongradients cause the primary warp and weft yarns to slip toward the projectile. (Contour plots are mirrored; yarn sliptoward the projectile is defined as positive in all quadrants.) Initially, the yarns slip most near the point of impact, withyarn material supplied by the stretch of sections of yarn away from the point of impact. Once the longitudinal strain wavesreach the edges of the fabric, however, the tension gradients in the primary yarns fade. Yarn slip in the warp directionvirtually stops. Yarn slip continues in the weft direction, most conspicuously at the free edges of the fabric, where the weftyarn tension is necessarily zero. When the longitudinal strain waves in the weft yarns reach the free edges of the fabric, theinward velocity of the weft yarns increases. Inter-yarn friction at the edges is not sufficient to pull the secondary warpyarns (the warp yarns not contacted by the projectile) toward the projectile, and therefore slip in the weft direction occurstoward the projectile, resulting in unraveling of the weave.

Because the ends of the weft yarns are not clamped, they can be pulled through the weave in a manner similar to that ofyarns in a yarn pull-out experiment. At the free edges of the simulated specimen, at t¼ 340 ms and t¼ 400 ms, we seeextensive areas of fabric with slip displacements in the weft direction of 4–7 mm (Fig. 10). The yarn pitch is 0.75 mm, andthe mean pervasive slip displacement therefore predicts the unraveling of 7–8 warp yarns. (Unraveling is not explicitlysimulated; physically, the simulated specimen can be visualized as having a ‘‘fringe’’ of weft yarn tails extending out fromeach of its free edges.) At t¼ 400 ms, we also see areas of negative yarn slip in the warp direction caused by the reflectionsof the transverse displacement waves.

Two additional aspects of wave propagation in woven fabric are apparent at vi ¼ 171 m=s. First, in both the experimentand the simulation, the transverse displacement waves travel more quickly in the direction of the clamped yarns than theydo in the direction of the unclamped yarns (Fig. 10). The velocities of the transverse displacement waves are proportionalto the tension in the yarns (Smith et al., 1956), and the tension in the clamped, warp yarns is greater than the tension inthe unclamped, weft yarns. Second, the simulation predicts the longitudinal strain waves to propagate more quickly in theyarn family with less crimp (the warp yarns), consistent with the numerical results of Ting et al. (1998) and theexperiments of Figucia et al. (1982), among other results in the literature. In Fig. 10, the extent of propagation of thelongitudinal strain waves is indicated by the occurrence of yarn slip, and yarn slip travels more quickly across the fabric inthe warp direction than it does in the weft direction. The warp yarns have less initial crimp than the weft yarns do, andthey are further decrimped by the initial preload of the fabric. Yarn crimp increases the distance that the strain wave musttravel, lowering the effective (in-plane) longitudinal wave speed.

Fig. 11. Yarn slip in unloaded fabric specimen that was tested at vi ¼ 171 m=s: (a) Unraveling of the weave at the free edges of the specimen caused by

slip in the direction of the weft yarns. (b) Increase in warp yarn pitch near the point of impact also caused by yarn slip in the weft direction.

Fig. 12. Yarn slip in the direction of the weft yarns changes the unloaded weft warn length, Lð2Þ

: simulated contours of Lð2Þ

for vi ¼ 171 m=s at (a) 30 ms,

(b) 100 ms, (c) 200 ms, (d) 340 ms, and (e) 400 ms after impact. Because unraveling is not explicitly modeled, weft yarn tails are effectively being pulled

into the weave at the free edges, increasing Lð2Þ

.


The slip displacements of the yarns in the simulation at vi ¼ 171 m=s are consistent with the yarn slip observed in theunloaded fabric specimen. The frayed sections of the specimen measure about 6 mm, and there are seven unraveled warpyarns at the free edges of the specimen (Fig. 11a). Both experimental measurements agree with the weft yarn slipdisplacements at the edges of the fabric in the simulation (Fig. 10e00). The unraveling of the warp yarns is furthermanifested in the simulation by a large increase in the unloaded weft yarn length, L

ð2Þ. Initially equal to Lð2Þ0 ¼ 0:384 mm,

Lð2Þ

increases nearly twofold at the free edges of the fabric where the imaginary tails of the weft yarns are pulled into theweave (Fig. 12). There is not a large amount of yarn slip in either yarn direction at the interior of the fabric, but, in theunloaded specimen, we see bowing of the warp yarns and an increase in the warp yarn pitch near the point of impact(Fig. 11b). This interior yarn slip is predicted in the simulations by weft yarn slip displacements (Fig. 10a002e00) and anincrease in the unloaded weft yarn length (Fig. 12) near the point of impact. (At small crimp angles, bð2Þ, the unloaded weftyarn length is nearly equal to the warp yarn half-pitch, pð2Þ.)

3.2.3. Ballistic impact at intermediate initial projectile velocities

At intermediate projectile velocities, the deformation still extends over the majority of the fabric specimen, but theprojectile ultimately penetrates the fabric. In the simulations, neither the deflection of the fabric nor the residual velocityof the projectile is significantly affected by the occurrence of yarn slip. At vi ¼ 227 m=s (Fig. 13), for example, the residualvelocity is 84 m/s in both the simulation with yarn slip and the simulation without yarn slip. The simulated contours ofwarp yarn tension in Fig. 13a0 and a00 show that yarn slip initially reduces the yarn tension at the point of impact. Soonthereafter, however, the yarn tensions of the two simulations become similar, and the fabric deforms and fails in nearly thesame manner, with or without yarn slip enabled. Both simulations correctly predict that failure occurs because the warpyarns break. Very few weft yarns break in the experiment.

3.2.4. Ballistic impact at high initial projectile velocities

At high projectile velocities, the fabric fails before the deformation can propagate extensively over the specimen. It failswhile the yarns are still slipping in the warp direction, and, as a result, yarn slip plays a far more important role at highprojectile velocities than it does at low or intermediate projectile velocities. At vi ¼ 362 m=s, for example, the simulationwith yarn slip predicts lower warp yarn tension at the point of impact and greater ultimate deflection of the fabric than thesimulation without yarn slip does (Fig. 14). Yarn slip blunts the tension concentration at the point of impact, enabling thefabric to continue to absorb energy and slow the projectile. The simulation with yarn slip predicts vr ¼ 335 m=s, comparedto vr ¼ 352 m=s for the simulation without yarn slip, a decrease in residual velocity corresponding to an over 160% increasein energy absorbed. Furthermore, the simulation with yarn slip correctly predicts failure to occur by rupture of the warpyarns, but the simulation without yarn slip predicts the weft yarns to break first.

For vi ¼ 362 m=s, the advance of the yarn tension waves at t¼ 15 ms (Fig. 14a0,a00) indicates that the longitudinal strainwaves in the fabric travel significantly faster when yarn slip occurs than they do when yarn slip does not occur. By one-dimensional wave propagation theory, the longitudinal strain wave speed of a single elastic yarn is taken to be cðkÞ ¼

ffiffiffiffiffiffiffiffiffiE=r

p,

where E and r are the Young’s modulus and the volumetric density, respectively, of the yarn. In a balanced fabric withoutyarn slip, the additional mass associated with the crossing yarns slows this speed to cf ¼

ffiffiffiffiffiffiffiffiffiffiffiE=2r

p, according to the theory of

Roylance et al. (1973). However, when yarn slip occurs, the two yarn families are no longer rigidly coupled, and thelongitudinal strain waves therefore travel at speeds exceeding cf . (The in-plane wave speeds differ from these theoreticalvalues due to the crimp of the yarns.)

Fig. 13. At intermediate projectile velocities, yarn slip initially reduces the warp yarn tension, T ð1Þ , at the point of impact, but its effect wanes as the

deformation of the fabric continues and the gradients of yarn tension disappear. Images of experiment (top) and simulated contours of Tð1Þ (middle and

bottom) for vi ¼ 227 m=s at time after impact: (a)–(a00) 25 ms; (b)–(b00) 40 ms; (c)–(c00) 110 ms; (d)–(d00) 250 ms; (e)–(e00) 300 ms.


Just prior to failure with vi ¼ 362 m=s, yarn slip velocities, vslipðkÞ, of up to 132 m/s exist in the simulation (Fig. 15).Combined, these slip velocities form a cross-shaped pull-out zone similar to that reported by Bazhenov (1997) for theballistic impact of an aramid woven fabric. They result in predicted maximum slip displacements of 2.9 mm in the warpdirection and 10 mm in the weft direction. The yarn slip in the warp direction almost entirely reverses itself after theprojectile penetrates the fabric, but evidence of yarn slip in the weft direction is abundant both in the simulation att¼ 200 ms and in the unloaded fabric specimen (Fig. 16). In the simulation, the broad triangular areas of yarn slip at thefree edges of the fabric are composed primarily of 3–7 mm of slip in the weft direction, consistent with the sevenunraveled warp yarns at the free edges of the unloaded specimen. In addition, also in agreement with the unloadedspecimen, the simulated fabric exhibits, near its free edges, significant areas of increased warp yarn pitch, 2pð2Þ (Fig. 16b, b0).

At vi ¼ 362 m=s, the simulation with yarn slip still underpredicts the deflection of the fabric and, hence, the energyabsorbed. The discrepancy may be due in part to the statistical variation inherent to ballistic impact experiments. Anotherpossible cause of the difference between the simulation and the experiment is the failure criterion of the fabric:an element is effectively removed from the calculation when the element-averaged tension in either yarn family exceedsthe strength of the yarns. This failure criterion is likely too severe for several reasons. First, in the experiments, for vi lessthan about 450 m/s, it is primarily the warp yarns that break. Most of the weft yarns do not break, and the projectile

Fig. 14. At high projectile velocities, yarn slip increases the time to failure and the deflection of the fabric by decreasing the warp yarn tension, T ð1Þ , at the

point of impact. Images of experiment (top) and simulated contours of T ð1Þ (middle and bottom) for vi ¼ 362 m=s at time after impact: (a)–(a00) 15 ms;

(b)–(b00) 30 ms; (c)–(c00) 85 ms.


penetrates the fabric by pushing aside the intact weft yarns. During the time that the projectile is separating the intactweft yarns, the fabric should continue to deform and absorb energy. Second, in the experiments, failure is presumablymore gradual than it is in the simulations. Instead of failing in sections as they do in the simulations, yarns in theexperiments can fail individually as tears propagate, potentially increasing the time that it takes the projectile to penetratethe fabric. Finally, although the Kevlar yarns exhibited a sharp load drop at rupture during single yarn tension tests, failuredoes occur by the rupture of individual fibers. A failure model which simulates the progressive failure of the fibers of theyarns, such as that used by Zohdi and Powell (2006) and Xia and Nadler (2011), might also improve the predictions of thesimulations.

3.3. Mechanics of energy absorption with yarn slip

Yarn slip in the warp direction increases the energy absorbed by the fabric at high projectile velocities by slowing theinitial increase in warp yarn tension. Plots of simulated warp yarn tension versus time at the point of impact, usually thelocation of maximum tension in the fabric, exhibit two increases in tension, separated by a sharp drop in tension (Fig. 17a).During the first increase in tension, both the tension and the rate of increase in tension are proportional to the initialvelocity of the projectile. Yarn slip slows this initial increase in tension by dispersing the stretch of the warp yarns awayfrom the point of impact. Once the primary warp yarns become uniformly stretched, however, the tensions at the point ofimpact in the simulations with yarn slip increase to rival those of the simulations without yarn slip, and the benefits ofyarn slip disappear. The drop in tension is caused by slip of the fabric from the clamps, and the second increase in tensioncoincides with the arrival of the transverse displacement waves at the clamps. During both the drop in tension and the

Fig. 15. At high projectile velocities, yarn slip in both directions is still occurring at the time of failure. Simulated contours of yarn slip velocity, vslipðkÞ , for

vi ¼ 362 m=s at time after impact: (a), (a0) 15 ms; (b), (b0) 30 ms; (c), (c0) 40 ms. Contour plots are mirrored; yarn slip toward the projectile is defined as

positive in all quadrants.

Fig. 16. Yarn slip in an unloaded specimen that was tested at vi ¼ 362 m=s (top) and simulation of the test (bottom): (a), (a0) Slip displacement in the

direction of the weft yarns predicts unraveling of the weave at the free edges of the fabric. (b), (b0) Yarn slip in the direction of the weft yarns increases

the warp yarn pitch at the edge of the fabric.


Fig. 18. Predictions of energy absorbed by the deformation of the fabric: (a) Yarn slip does not increase energy absorbed at intermediate projectile

velocities (vi ¼ 227 m=s shown). (b) Yarn slip increases energy absorbed at high projectile velocities primarily by enabling more of the fabric to stretch

and to accelerate (vi ¼ 362 m=s shown). Yarn slip itself does not absorb significant energy prior to failure.

Fig. 17. Compared to simulations without yarn slip, simulations with yarn slip exhibit different histories of yarn tension at the center of the fabric:

(a) Warp yarn tension with yarn slip increases more slowly during the initial increase in tension. (b) Weft yarn tension with yarn slip is lower throughout

the deformation.


subsequent increase in tension, the tension gradients in the warp yarns are generally small, and, consequently, very littleyarn slip occurs in the warp direction. Therefore, it is only when failure occurs during the first increase in tension that yarnslip is able to increase the energy absorbed by the fabric.

Yarn slip in the weft direction reduces the initial projectile velocity at which the transition from the intermediatevelocity regime to the high velocity regime occurs (Fig. 8). Simulations with yarn slip enter the high velocity regime atvi � 270 m=s, matching the experiments. Simulations without yarn slip, on the other hand, do not enter the high velocityregime until vi � 290 m=s and thus significantly overpredict the maximum energy that the fabric can absorb. Forvi � 2702290 m=s, the warp yarn tension at the point of impact during the first increase in tension, while initially lower, isultimately higher with yarn slip than it is without yarn slip (Fig. 17a, vi ¼ 285 m=s for example). In the simulations withyarn slip, at all initial projectile velocities, the weft yarn tensions at the point of impact are reduced by yarn slipthroughout the deformation (Fig. 17b), causing the warp yarns to bear higher loads than they would without yarn slip.The ends of the weft yarns are free, and tension gradients driving yarn slip persist in these yarns throughout thedeformation of the fabric. As a result, for vi � 2702290 m=s, the warp yarns in simulations with yarn slip fail during thefirst increase in tension, while the warp yarns in simulations without yarn slip survive until the second increase in tension,absorbing a considerable amount of energy in the interim.

Yarn slip improves ballistic performance, but it does not itself absorb large amounts of energy. In simulations atvi ¼ 227 m=s, we see that the vast majority of the energy absorbed by the deformation of the fabric is in the form of strainenergy of the yarns and kinetic energy of the weave continuum (Fig. 18a). Both the energy dissipated by inter-yarn frictionat the cross-over points and the kinetic energy due to the motion of the yarns relative to the cross-over points are

Fig. 19. Energy absorbed at vi¼400 m/s as a function of (a) inter-yarn locking friction parameter, mL, and (b) inter-yarn normal friction parameter, mN.

The energy absorbed depends more strongly on the normal parameter than it does on the locking parameter. (Values determined from yarn pull-out

experiments were mL ¼ 0:6 and mN ¼ 0:12.)

Fig. 20. Effects of inter-yarn normal friction parameter, mN, on ballistic performance: (a) Ballistic limit increases with increasing inter-yarn friction.

(b) Energy absorbed decreases at high projectile velocities with increasing inter-yarn friction.


relatively small. At vi ¼ 362 m=s, yarn slip delays failure, and the simulation with yarn slip absorbs over 160% more energythan the simulation without yarn slip does. Here, too, the energy absorbed by the deformation of the fabric is mostly in theform of strain energy of the yarns and kinetic energy of the weave continuum (Fig. 18b). The portion of energy absorbeddirectly by yarn slip prior to failure is less than 18% of the total energy absorbed. In simulations of ballistic impact witheach yarn explicitly modeled, Duan et al. (2005, 2006a) and Zeng et al. (2006), among others, found yarn slip to makesimilar contributions to the total energy absorbed.

3.4. Effects of inter-yarn friction on the ballistic resistance

The effects of inter-yarn friction on the predicted ballistic resistance of the fabric were studied by varying the frictionparameters mN and mL in Eq. (20) at vi ¼ 400 m=s. At this initial projectile velocity, a simulation with mN ¼ 0:12 and mL ¼ 0:6,the values determined from yarn pull-out experiments, predicts the fabric to absorb 220% more energy than a simulationwithout yarn slip does. Varying the locking parameter, mL, with the normal parameter, mN, held constant at 0.12, we seethat the energy absorbed is not a strong function of mL (Fig. 19a). Locking forces are small during ballistic impact becausethe fabric is under tension in both yarn directions and shear angles are not large. Friction due to locking is most importantbelow the ballistic limit, where it prevents unloaded yarn lengths from becoming unrealistically small in sections of thefabric that are unloading. Because the dependence on mL is weak, mL is kept constant in the remainder of this section.Varying mN, with mL held constant at 0.6, we see that the energy absorbed depends more strongly on mN than it does on mL

(Fig. 19b). Increasing mN from its experimentally determined value decreases the energy absorbed considerably atvi ¼ 400 m=s, but decreasing mN does not significantly affect the energy absorbed.

Fig. 21. Effects of boundary conditions on yarn slip and energy absorbed by the fabric: (a) Fixing the fabric at two edges eliminates the peak in energy

absorbed at intermediate projectile velocities, but it has little effect at high projectile velocities. (b) Clamping the fabric (somewhat loosely) at all four

edges reduces the peak in energy absorbed at intermediate projectile velocities, and, in simulations with yarn slip, it also reduces the energy absorbed at

high projectile velocities by increasing the contact force between the yarns, F I .


Decreasing the inter-yarn friction in the simulations reduces the predicted ballistic limit of the fabric. The ballistic limitdecreases from vi ¼ 182 m=s without yarn slip to vi ¼ 169 m=s with mN ¼ 0:05 (Fig. 20a). At projectile velocities at or closeto the ballistic limit, the majority of the fabric specimen deforms before failure occurs. Near the time of failure, the tensiongradients are small in the clamped warp yarns, and therefore yarn slip cannot occur to relieve the warp yarn tensions atthe point of impact. The unclamped weft yarns, on the other hand, are still slipping at this time, causing the portion of theload transmitted to the warp yarns to be higher than it is without yarn slip. Thus, decreasing the inter-yarn friction lowersthe initial projectile velocity at which the warp yarns break because it increases the slip of the weft yarns.

Decreasing the inter-yarn friction in the simulations increases the energy absorbed in the high velocity regime, but thebenefit diminishes at the highest projectile velocities. On average, the energy absorbed in the high velocity regimeincreases as the inter-yarn friction is decreased from the no-slip condition to mN ¼ 0:05 (Fig. 20b). The decrease in inter-yarn friction increases the slip velocities of the yarns. Failure is delayed with decreasing inter-yarn friction because theassociated increasing slip velocities cause increasingly large reductions in warp yarn tension relative to the case withoutyarn slip. In addition, the transition to the high velocity regime is pushed to lower initial projectile velocities as the inter-yarn friction is decreased because slip of the weft yarns also increases with decreasing inter-yarn friction. For vi4475 m=s,however, the energy absorbed is only a weak function of the inter-yarn friction. At these velocities, the simulations predictfailure to occur before the inertia that resists yarn slip can be overcome.

3.5. Effects of boundary conditions on yarn slip and energy absorbed by the fabric

The results of these experiments and simulations are, to some extent, specific to the boundary conditions imposed onthe fabric: two edges clamped with significant slip of the fabric from the clamps. Here, with simulations, we examine theeffects of changing the boundary conditions in order to assist in drawing general conclusions about the role that yarn slipplays during ballistic impact. First, we completely fix the fabric at two edges, and then we clamp all four edges but allowslip of the fabric from the clamps.

Preventing slip of the fabric from the clamps in the simulations reduces the maximum energy absorbed by almost 70%(Fig. 21a). With the fabric fixed at two edges, the warp yarn tension at the point of impact does not drop sharply (as it doesin Fig. 17a); it increases until the fabric stops the projectile or failure occurs. Failure therefore occurs at smaller deflectionsthan it does when slip from the clamps is allowed. The result is a reduction in the ballistic limit to less than 100 m/s andthe elimination of the peak in energy absorbed at intermediate projectile velocities. The large discrepancy in energyabsorbed between the cases with yarn slip and those without yarn slip at vi � 2702290 m=s disappears because there is nolonger a drop in warp yarn tension to be bridged. The fabric without yarn slip does absorb more energy at vi � 2502300 m=sthan the fabric with yarn slip does, but the extra energy absorbed is due to the fabric tearing at the clamps before the projectilecan penetrate the fabric. Yarn slip alleviates concentrations of yarn tension at the clamps, and the fabric with yarn slip failsonly at the point of impact. At high projectile velocities, the energy absorbed in the simulations with the fabric fixed at theclamps is about the same as that in the simulations in which the fabric slips from the clamps. The clamping of the fabric is lessimportant at high velocities than it is at intermediate velocities because the fabric fails before it can slip from the clamps.

Clamping the fabric (somewhat loosely) at all four edges in the simulations reduces the peak energy absorbed by 33%relative to the simulations in which the fabric is clamped only at two edges (Fig. 21b). In these simulations, the fabric is in theshape of a cruciform, and all four grip areas are clamped with a pressure identical to that used in the simulations with two


edges clamped. Failure occurs because the warp yarns break, and the predicted ballistic limit is about the same as that for onlytwo edges clamped. At intermediate projectile velocities, simulations both with yarn slip and without yarn slip predict thefabric to absorb significantly less energy than it does with two edges clamped. Clamping all four edges reduces the slip of thewarp yarns from the clamps, which eliminates most of the concurrent drop in warp yarn tension at the point of impact. Thewarp yarn tension at the point of impact thus increases more rapidly with four edges clamped than it does with two edgesclamped. Throughout the intermediate velocity range, the fabric with yarn slip and the fabric without yarn slip absorb aboutthe same amount of energy. In this velocity range, the tension gradients in both sets of primary yarns disappear by the time offailure, and therefore yarn slip cannot occur to reduce the yarn tensions at the point of impact. At high projectile velocities, thesimulations still predict the fabric with yarn slip to absorb substantially more energy than the fabric without yarn slip does,but clamping all four edges causes the fabric with yarn slip to absorb less energy than it does with only two edges clamped.Clamping the two previously free edges causes the yarn tensions at the point of impact to become nearly equi-biaxial.Compared to the case with only two edges clamped, crimp interchange decreases, and yarn compaction increases, causing theinterference force between the yarns, FI, to increase accordingly. As a result, the slip resistance calculated via Eq. (20) alsoincreases, and yarn slip and energy absorbed diminish when all four edges are clamped at high projectile velocities.

4. Discussion of results

Enabling slip between the yarns in simulations of ballistic impact can increase the energy absorbed by the fabricconsiderably, improving the predictions of the simulations. Yarn slip diffuses the stretch of the yarns away from theprojectile, reducing the tension in the yarns at the point of impact. At high projectile velocities, this reduction in yarntension occurs at a time when failure would otherwise occur. Failure is therefore delayed, and a fabric with yarn slipabsorbs up to 220% more energy than a fabric without yarn slip does. The improvement in energy absorbed at highprojectile velocities is consistent with the results of Zeng et al. (2006) and Duan et al. (2006b), who both observed anincrease in energy absorbed with increasing yarn slip in discrete simulations of aramid fabrics. Yarn slip, however, doesnot increase the energy absorbed at intermediate projectile velocities. At intermediate velocities, the fabric deformsextensively, and the primary yarns no longer sustain gradients of tension at the time of failure. Because the gradients oftension have dissipated, yarn slip can no longer occur to relieve the yarn tension at the point of impact.

Yarn slip reduces the ballistic limit of the fabric by up to 7% when only two edges are clamped. The ballistic limitdecreases with increasing yarn slip because slip of the unclamped primary yarns increases the load on the clampedprimary yarns. A similar reduction in the ballistic limit with increasing yarn slip was observed by Rao et al. (2009) indiscrete simulations of a KM2 fabric.

The effect of yarn slip on the energy absorbed by the fabric is a function of the boundary conditions imposed on thefabric. In the simulations of the experiments (two edges clamped, with slip of the fabric from the clamps), the transitionto the high velocity regime is marked by a sharp drop in the energy absorbed. This transition occurs because the fabricslips from the clamps; it does not occur in simulations during which the fabric is fixed at the clamps. Yarn slip causes thetransition to occur at a lower initial projectile velocity than it does without yarn slip. The change in this transition point,like the reduction in the ballistic limit, is due to yarn slip in the unclamped yarns increasing the load on the clampedyarns. Simulations with all four edges clamped (but with slip of the fabric from the clamps allowed) exhibit a smaller,more gradual transition to the high velocity regime than simulations with only two edges clamped do. With four edgesclamped, the energy absorbed at initial projectile velocities near the transition point does not depend significantly onyarn slip because there are no unclamped yarns that continue to slip throughout the deformation of the fabric. Clampingthe two free edges increases the contact force between the yarns at the point of impact and therefore somewhat reducesyarn slip and the energy absorbed by the fabric at high projectile velocities. This reduction is modest, however, and, athigh projectile velocities, the energy absorbed is generally not a strong function of the boundary conditions because thefabric fails before extensive deformation occurs at its edges.

Our results indicate that yarn slip generally increases or does not affect the energy absorbed by woven fabric. However, Briscoeand Motamedi (1992) and also Termonia (2004), via experiments and discrete simulations, respectively, have reported yarn slip todecrease the energy absorbed by plain-weave fabrics at comparable projectile velocities. The penetrated fabrics in both casesexhibited large amounts of yarn slip and yarn pull-out and few, if any, broken yarns at the point of impact. In these studies, theprojectiles were spherical with a diameter about one-half of that of the blunt-tipped projectiles used in our experiments andsimulations. Furthermore, although precise details are not provided, there are indications that the fabrics tested or simulated bythese investigators were not as tightly woven as Kevlar S706. Thus, yarn slip was detrimental in these cases likely because theprojectile could primarily push its way through the weave when inter-yarn friction was not sufficient to stop the primary yarnsfrom separating. Excessive yarn slip at the point of impact did not occur in the simulations and experiments presented in thispaper because it was prevented by the properties of the fabric and the shape and size of the projectile.

A more complex failure model would improve the predictions of these types of simulation. In future simulations, theelement-based failure criterion will be replaced with an interface approach that will allow each yarn family to breakseparately and tears or slits to propagate. For simulations with smaller or sharper projectiles than those used in this study,failure by yarn separation could also be enabled. Unraveling of the weave can be modeled explicitly by modifying therepresentative unit cell to include only one yarn family once the slip displacement of the yarn at a weave point exceeds thedistance along the yarn to the edge of the fabric.


5. Conclusion

In this paper, we propose a method to simulate yarn slip with a mesostructure-based continuum approach. The key tothis approach is to use the continuum to describe the position of the cross-over points of the weave, not the position of theyarns. The anisotropic continuum properties of the fabric are still determined from a deforming analytic unit cell of thefabric, but additional field variables simulate the slip of each yarn family relative to the cross-over points. The forcesaccelerating yarn slip are determined from the meso-level forces in the analytic unit cell, on both a local and a non-localscale. The resulting slip velocity fields determine the amount of yarn contained within the analytic unit cell.The predictions of simulations are consistent with the results of our own experiments and results in the literature.

The results of the simulations show yarn slip to increase the energy absorbed at initial projectile velocities well abovethe ballistic limit. At these high projectile velocities, gradients of tension exist in the primary yarns when failure occurs.Yarn slip is therefore beneficial because it disperses the stretch of the primary yarns away from the projectile, reducing theyarn tension at the point of impact and enabling the fabric to survive longer than it would without yarn slip. It is analogousto the yielding of metals or polymers in the way that it mitigates the concentration of tension at the point of impact.At projectile velocities closer to the ballistic limit, however, the primary yarns survive to become uniformly stretched, andyarn slip then virtually stops. Thereafter, the yarn tension at the point of impact is about the same in simulations with yarnslip enabled as it is in simulations without yarn slip enabled.

By extension, yarn slip may also be important at initial projectile velocities close to the ballistic limit if the fabric iscomposed of multiple layers. In multi-layer fabric systems, it has been shown that yarn tensions are highest in the firstlayer struck by the projectile (Parsons et al., 2010a). When yarn slip delays the failure of the first few layers of a multi-layersystem, it will increase the energy absorbed by the system even when the initial projectile velocity is close to the multi-layer ballistic limit. Reducing inter-yarn friction therefore has the potential to improve ballistic resistance over a widerange of projectile velocities, provided that the structural integrity of the weave is not compromised by excessive yarn slip.

The simulations also underscore the importance of considering the boundary conditions and the size of the fabricspecimen when predicting the extent and effects of yarn slip. Yarn slip requires the continual existence of gradients of yarntension—either due to yarns with free ends or uneven deformation of the fabric, or both. Yarns with free ends can slipthrough the weave, causing minor reductions in the ballistic limit and increasing the likelihood of failure by yarnseparation. Constraining the fabric specimen at all edges not only prevents yarns from slipping through the weave but alsoincreases the contact force between yarns, thereby further reducing yarn slip. Large specimens will exhibit more yarn slipthan small specimens because, although the velocities and amplitudes of the initial longitudinal strain waves will notchange significantly, the supply of unstretched fabric to recruit will increase with increasing specimen size.

This paper demonstrates that a mesoscale-based continuum approach to modeling woven fabric can be used tosimulate yarn slip. Without modeling every yarn in the fabric, this approach simulates the actual geometry andmechanisms of deformation of the yarns. As a result, it provides not only physically motivated macroscopic predictionsbut also predictions of meso-level quantities, such as the tension, pitch, orientation, and crimp amplitude of the yarns,which are essential for anticipating failure and optimizing the design of woven fabrics. This approach is not limited to thesimulation of ballistic impact; it has the potential to simulate many other types of event during which yarn slip occurs,such as the slit damage of fabrics and the forming of composites.

Acknowledgments

This research was supported by the U.S. Army Research Office through the MIT Institute for Soldier Nanotechnologiesunder contract number W911NF-07-D-0004.

Appendix A. Calculation of yarn crimp amplitudes

The out-of-plane geometric properties of the unit cell, the yarn crimp amplitudes, AðkÞ, are determined from the sum ofthe transverse forces acting on each yarn truss. The compressive force in the cross-over spring, FI, representing theresistance to compaction of the yarns, is opposed by the net transverse reaction force due to the forces in the yarn trusses,FðkÞN . FðkÞN is calculated from the yarn tensions, the forces in the locking trusses, and the moments in the torsional springs atthe pin-joints:

FðkÞN ¼ T ðkÞ sin bðkÞ�FðjÞL sin aðjÞ 1þpðjÞ9 cos y9 cos bðkÞ

‘ðkÞ

!þ

MðkÞb

‘ðkÞcos bðkÞ ðk¼ 1;2Þ, ð27Þ

where j¼ 1;2 ðjakÞ. These transverse forces determine the out-of-plane acceleration of each yarn:

mðkÞ €AðkÞ¼ FI�2FðkÞN , ð28Þ

where mðkÞ is the mass of yarn truss k. Integration of Eq. (28) provides the change in crimp amplitude of each yarn family, DAðkÞ.


Appendix B. Finite element implementation of woven fabric model with yarn slip

A finite element was formulated to simulate the dynamic deformation of woven fabric with yarn slip. Its degrees offreedom are the displacement of the weave points, the changes in crimp amplitude of the yarns, and the slip displacementsof the yarns.

B.1. Formulation of the finite element

The biquadratic finite element was defined in Abaqus/Explicit via the VUEL user subroutine. At each node, the vector ofdegrees of freedom, d, is composed of the three components of the displacement of the weave point, ui, the change in crimpamplitude of each yarn family, DAðkÞ, and the in-plane slip displacement of each yarn family relative to the weave point, uslipðkÞ:

dTh i

¼ u1 u2 u3 DAð1Þ DAð2Þ uslipð1Þ uslipð2Þ� �

: ð29Þ

These degrees of freedom are explicitly integrated forward in time by a Newmark scheme. At tnþ1,

dnþ1¼ dnþDt _d

nþDt2

2€d

n, ð30aÞ

€dnþ1¼M�1 fnþ1

ext þfnþ1int

� �, ð30bÞ

and

_dnþ1¼ _d

nþDt

2€d

nþ €d

nþ1� �

: ð30cÞ

M is the diagonal mass matrix, and fext and fint are the vectors of external and internal nodal forces.The fabric was approximated as a membrane. The majority of the resistance to ballistic impact is due to the in-plane

deformation and out-of-plane inertia of the fabric. Conversely, twist and out-of-plane bending provide very littleresistance. Simulations with membrane and shell elements showed the membrane approximation to be sufficient(Parsons et al., 2010b), and we therefore concluded that the additional complexity and computational cost of shellelements were not necessary.

A four-sided element with nine displacement nodes and two additional sets of ‘‘dummy’’ nodes was used to simulatethe fabric (Fig. 22). The displacement nodes correspond to the weave points of the fabric. In turn, the first set of dummynodes represents the changes in crimp amplitude of the yarns, while the second set of dummy nodes represents the in-plane slip displacements of the yarns relative to the weave points. The initial coordinates of the displacement nodes andthe nodes representing the changes in crimp amplitude are identical, and both sets of nodes are interpolated withquadratic shape functions Na (a¼1,9). To simulate the slip displacements of the yarns, we use the four corner nodes andlinear shape functions Nb (b¼1,4).

The weave deformation gradient at each integration point is calculated from the nodal positions and shape functions ofthe element. The element is isoparametric, and the in-plane position of a weave point in the reference configuration isgiven, as a function of the natural coordinates n¼ ðx1 x2Þ

T, by the shape functions at each node and the initial nodal

Fig. 22. The woven fabric finite element has 22 total nodes: (a) To simulate the displacements of the weave points and the changes in crimp amplitude of

the yarns, we use nine nodes, second order shape functions Na, and 3�3 Gauss quadrature points Q9. (b) To simulate the yarn slip displacements, we use

four nodes, first order shape functions Nb, and 2�2 Gauss quadrature points Q4.


coordinates:

XJðnÞ ¼X9

a ¼ 1

NaðnÞXJa ðJ¼ 1;2Þ: ð31Þ

Correspondingly, the three-dimensional position of a weave point in the deformed configuration is given by

xiðnÞ ¼X9

a ¼ 1

NaðnÞxia ði¼ 1;3Þ: ð32Þ

The relevant components of the weave deformation gradient at a point within the element are therefore calculated as

FwiJ ¼

@xi

@XJ¼X9

a ¼ 1

Na,Jxia: ð33Þ

Provided that the fabric initially lies in the 1–2 plane, Fw determines, at each integration point within the element, theorientation and length of the yarn pitch vectors, 2pð1Þ and 2pð2Þ.

The internal nodal forces accelerating the degrees of freedom are calculated by multiplying the linear momentumequations by appropriate trial functions and integrating over the domain to develop the expressions of the principle ofvirtual power. Multiplying the momentum equation of the weave continuum (Eq. (11)) by trial weave point velocities dvi

and integrating over the element, we calculate the momentum balance in weak form:

Z@Oe

0

dvi

X2

J ¼ 1

@PiJ

@XJ�rf

0

2Jð1Þ_vslipð1Þ

i þ Jð2Þ_vslipð2Þ

i

� ��rf

0

2Jð1Þþ Jð2Þ

� �_vi

( )dS0 ¼ 0, ð34Þ

where i¼1,3 and _vslipðkÞi ¼ vslipðkÞ � ðgkÞi. Using integration by parts and discretizing with vi ¼

P9a ¼ 1

Navia, we compute theinternal nodal force accelerating each weave point degree of freedom as

ðf int uia Þ

e¼�

Z@Oe

0

X2

J ¼ 1

Na,JPiJþNarf

0

2Jð1Þ_vslipð1Þ

i þ Jð2Þ_vslipð2Þ

i

� �( )dS0: ð35Þ

Analogously, for the degrees of freedom representing yarn slip, we multiply the strong forms of the momentum balances ofEqs. (14) and (15) by trial yarn slip velocities dvslipðkÞ and integrate over the element:Z

@Oe0

dvslipðkÞ Div PðkÞ � gkþ 1þJðkÞ

JðjÞ

!tðkÞTk�

rf0

2JðkÞ_vslipðkÞ

( )dS0 ¼ 0, ð36Þ

where ðk¼ 1;2; jak). During ballistic impact by a projectile, yarn tensions are high and shear angles are typically small.Therefore, in calculating the yarn slip accelerations, we approximate yarn tension and inter-yarn friction as the only forcesacting on the yarns. Shear and locking forces, usually negligible until shear angles become substantial, are omitted from

PðkÞ in order to simplify the calculation of the non-local forces driving yarn slip. With vslipðkÞ ¼P4

b ¼ 1

NbvslipðkÞb , the

contribution to the internal nodal force vector from each element can thus be written as

f int slipkb

� �e¼

Z@Oe

0

Nb GradTðkÞcosbðkÞ

2pðjÞ0

!� Gkþ 1þ

JðkÞ

JðjÞ

!tðkÞTk

( )dS0 ðk¼ 1;2; jakÞ, ð37Þ

where T ðkÞ cos bðkÞ is the in-plane yarn tension. A similar procedure is followed to compute the internal nodal force

corresponding to each crimp amplitude degree of freedom DAðkÞ (Parsons et al., 2010b). From the weak form of thetransverse momentum balance, we compute the contribution to the internal nodal force vector from each element as

f intDAka

� �e¼

Z@Oe

0

NaFI�2FðkÞN

4 pð1Þ0 pð2Þ0

dS0 ðk¼ 1;2Þ, ð38Þ

where, in the analytic unit cell, FI is the interference force between the yarns and FðkÞN is the net transverse component of

the force in each yarn. All the preceding expressions for the internal nodal forces are integrated numerically over the finiteelement.

The mass of the fabric is lumped at the nodes in order to construct a diagonal mass matrix. The initial total mass of eachelement is given by

Me0 ¼ r

0f

X9

q ¼ 1

JqxWq, ð39Þ

where Jqx ¼ detð@x=@nÞ and Wq are the Jacobian determinant and weight, respectively, at integration point q. Within each

element, corner nodes, most often shared by three other elements, are allocated proportionally less of the mass of the


element than mid-side and center nodes are. For a weave point degree of freedom with initial mass mia, balance of linearmomentum shows that the initial mass of a crimp amplitude degree of freedom at the same initial nodal coordinates canbe approximated as mka ¼mia=4. For the yarn slip degrees of freedom, the mass of the element is apportioned equally tothe four corners of the element. Because each yarn slip degree of freedom corresponds to only one yarn family, the mass ateach node is mkb ¼ 1=2�Me

0=4.It is apparent in the inertial terms of Eqs. (34) and (36) that the mass of an element may change as yarn slip occurs.

The mass matrix is therefore updated at every increment according to the current area ratios JðkÞ

at each integration point.Element failure occurs when the tension in either yarn family, averaged over all integration points of the element,

exceeds the strength of the yarns,�T ðkÞ�e4TðkÞmax. With this failure criterion, we have approximated that, once one yarn

family breaks, the projectile will then easily push the other yarn family aside. When an element fails, its contribution tothe internal nodal force vector is eliminated.

B.2. Gradient calculations for determining the yarn slip accelerations and the unloaded yarn lengths

This model is non-local because the yarn slip accelerations and the rates of change of the unloaded yarn lengths arefunctions of, respectively, gradients of in-plane yarn tension and gradients of yarn slip velocity. The calculation of thesegradients at an integration point requires information from other points in the element, from adjoining elements, or fromboth. In this implementation, the gradients are calculated from the shape functions of the element.

For the calculation of the yarn slip accelerations, continuous fields of in-plane yarn tension, TðkÞ cos bðkÞ, are constructedat the nodes of the elements from the values at the integration points. When constructing these fields, we evaluateT ðkÞ cos bðkÞ at the first order integration points because, like stresses, yarn tensions are more accurate there than at thesecond order integration points (Barlow, 1989). The values at the integration points are then linearly extrapolated to theelements’ corner nodes, b¼1,4. For a node shared by two or more elements, the in-plane tensions in each adjoiningelement, weighted by the areas of the integration points, contribute to the in-plane tensions at the shared node. The resultis continuous fields of in-plane tension at the nodes,

�T ðkÞ cos bðkÞ

�b, from which the required tension gradients at each

integration point can be calculated:

GradðT ðkÞ cos bðkÞÞ � Gk ¼X4

b ¼ 1

X2

J ¼ 1

Nb,JðnÞðTðkÞ cos bðkÞÞbðGkÞJ , ð40Þ

where ðGkÞJ is the component of Gk in the J-direction.The rates of change of the unloaded yarn lengths at each integration point are calculated from the gradients of the

unstretched yarn slip velocities, vslipðkÞ:

_LðkÞ¼ �pðkÞ0 GradðvslipðkÞ

Þ � Gk ¼�pðkÞ0

X4

b ¼ 1

X2

J ¼ 1

Nb,JðnÞvslipðkÞb ðGkÞJ : ð41Þ

The unstretched yarn slip velocities at the nodes, vslipðkÞb , are computed from the vslipðkÞ

b nodal degrees of freedom and thestretch of each yarn family at the nodes, lðkÞb :

vslipðkÞb ¼

vslipðkÞb

lðkÞb

, ð42Þ

where lðkÞb is calculated by a procedure similar to that used for the in-plane yarn tensions at the nodes.

B.3. Numerical integration procedure

The nodal degrees of freedom are integrated forward in time by the Newmark algorithm (Eq. (30)). The contribution ofeach element to the internal nodal force vector must be provided.

The internal nodal forces corresponding to the weave point displacements and changes in crimp amplitude are

integrated numerically with the 3�3 Gauss points Q9 (Fig. 22a). At time tnþ1, Eq. (30) provides the displacements,

velocities, and accelerations of the nodal degrees of freedom. By interpolation of the weave point displacements, ðuiaÞnþ1,

and the changes in crimp amplitude, ðDAðkÞa Þnþ1, the weave deformation gradient and crimp amplitudes, respectively, are

calculated at each integration point. These quantities determine the deformed geometry of the analytic unit cell. Thecorresponding unloaded geometry of the unit cell at each integration point is determined by the unloaded yarn lengths,

ðLðkÞÞnþ1. The unloaded yarn lengths are integrated forward in time via ð _L

ðkÞÞnþ1 from Eq. (41):

ðLðkÞÞnþ1¼ ðL

ðkÞÞnþDt

2ð_LðkÞÞnþð

_LðkÞÞnþ1

�, ð43Þ

where ðLðkÞÞ0¼ LðkÞ0 and ðL

ðkÞÞn and ð _L

ðkÞÞn are state variables from the previous increment. From ðL

ðkÞÞnþ1, the deformed

geometry of the unit cell, and additional state variables, we determine the yarn forces and moments at each integration


point via the constitutive relations. The yarn forces and moments are then used to compute the homogenized stress in the

fabric and the internal nodal forces ðfint uÞnþ1 and ðfint DA

Þnþ1.

The internal nodal forces accelerating yarn slip are integrated numerically with the 2�2 Gauss points Q4 (Fig. 22b),according to the value of the slip function ðf slip

k Þnþ1 (Eq. (17)) at each integration point. In order for yarn slip to occur, the

absolute value of the inter-yarn friction in the direction of yarn-k must equal the slip resistance in that direction, ðskÞnþ1.

The slip resistances are calculated via Eq. (20) from the yarn forces ðFIÞnþ1 and ðFLkÞ

nþ1. The gradient of in-plane yarntension at tnþ1 is approximated from ðT ðkÞ cos bðkÞÞnb , the continuous nodal field of in-plane yarn tensions from the previoustime increment. If ðf slip

k Þnþ1¼ 0, then the inter-yarn friction and tension gradient in the direction of yarn-k in Eq. (37) are

set to zero at that integration point.

B.4. Yarn slip at the boundaries of the fabric

Boundary conditions must be defined to control yarn slip at the edges of the finite element mesh. At a fixed edge, theyarn slip velocities vslipðkÞ

b of both yarn families are set to zero. At a free edge, the in-plane yarn tensions�

T ðkÞ cos bðkÞ�

bof

unclamped yarns are set to zero.The yarn slip displacements near the edges of the mesh determine when yarns unravel from the weave. If the slip

displacement of a yarn at a weave point exceeds the distance along the yarn to the edge of the mesh, then the weave pointhas unraveled. Unraveling could be simulated by modifying the analytic unit cell to represent only one yarn family.

References

Abbott, N.J., Skelton, J., 1972. Crack propagation in woven fabrics. J. Coated Fibrous Mater. 1, 234–252.Anand, L., 1993. A constitutive model for interface friction. Comput. Mech. 12, 197–213.Assidi, M., Boubaker, B.B., Ganghoffer, J.F., 2011. Equivalent properties of monolayer fabric from mesoscopic modelling strategies. Int. J. Solids Struct. 48,

2920–2930.Barlow, J., 1989. More on optimal stress points—reduced integration, element distortions and error estimation. Int. J. Numer. Methods Eng. 28,

1487–1504.Bazhenov, S., 1997. Dissipation of energy by bulletproof aramid fabric. J. Mater. Sci. 32, 4167–4173.Boisse, P., Borr, M., Buet, K., Cherouat, A., 1997. Finite element simulations of textile composite forming including the biaxial fabric behaviour. Composites

B 28, 453–464.Boisse, P., Gasser, A., Hivet, G., 2001. Analyses of fabric tensile behaviour: determination of the biaxial tension-strain surfaces and their use in forming

simulations. Composites A 32, 1395–1414.Boubaker, B.B., Haussy, B., Ganghoffer, J., 2007. Discrete woven structure model: yarn-on-yarn friction. C. R. Mec. 335, 150–158.Briscoe, B.J., Motamedi, F., 1992. The ballistic impact characteristics of aramid fabrics: the influence of interface friction. Wear 158, 229–247.Cheng, M., Chen, W.M., Weerasooriya, T., 2005. Mechanical properties of Kevlars KM2 single fiber. J. Eng. Mater. Technol. 127 (2), 197–203.Chocron, S., Kirchdoerfer, T., King, N., Freitas, C.J., 2011. Modeling of fabric impact with high speed imaging and nickel-chromium wires validation. J. Appl.

Mech. 78. 051007: 1–13.Duan, Y., Keefe, M., Bogetti, T.A., Cheeseman, B.A., 2005. Modeling friction effects on the ballistic impact behavior of a single-ply high-strength fabric. Int.

J. Impact Eng. 31 (8), 996–1012.Duan, Y., Keefe, M., Bogetti, T.A., Cheeseman, B.A., Powers, B., 2006a. A numerical investigation of the influence of friction on energy absorption by a high-

strength fabric subjected to ballistic impact. Int. J. Impact Eng. 32 (8), 1299–1312.Duan, Y., Keefe, M., Bogetti, T.A., Powers, B., 2006b. Finite element modeling of transverse impact on a ballistic fabric. Int. J. Mech. Sci. 48, 33–43.Figucia, F., Williams, C., Kirkwood, B., Koza, W., 1982. Mechanisms of Improved Ballistic Fabric Performance. Technical Report, U.S. Army Natick Research

& Development Center.Godfrey, T.A., Rossettos, J.N., 1998. Damage growth in prestressed plain weave fabrics. Text. Res. J. 68, 359–370.Godfrey, T.A., Rossettos, J.N., 1999. The onset of tear propagation at slits in stressed uncoated plain weave fabrics. Trans. ASME 66, 926–933.Hamila, N., Boisse, P., Sabourin, F., Brunet, M., 2009. A semi-discrete shell finite element for textile composite reinforcement forming simulation. Int. J.

Numer. Methods Eng. 79, 1443–1466.Ivanov, I., Tabiei, A., 2004. Loosely woven fabric model with viscoelastic crimped fibres for ballistic impact simulations. Int. J. Numer. Methods Eng. 61

(10), 1565–1583.King, M.J., 2006. A Continuum Constitutive Model for the Mechanical Behavior of Woven Fabrics Including Slip and Failure. Ph.D. Thesis, Massachusetts

Institute of Technology, Cambridge, MA USA.King, M.J., Jearanaisilawong, P., Socrate, S., 2005. A continuum constitutive model for the mechanical behavior of woven fabrics. Int. J. Solids Struct. 42,

3867–3896.Lastnik, A.L., Karageorgis, C., 1982. The Effect of Resin Concentration and Laminating Pressures on Kevlars Fabric Bonded with a Modified Phenolic Resin.

Technical Report, U.S. Army Natick Research & Development Center.Laursen, T.A., Simo, J.C., 1993. A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact

problems. Int. J. Numer. Methods Eng. 36, 3451–3485.Lim, C.T., Shim, V.P.W., Ng, Y.H., 2003. Finite element modeling of the ballistic impact of fabric armor. Int. J. Impact Eng. 28, 13–31.Lomov, S.V., Huysmans, G., Luo, Y., Parnas, R.S., Prodromou, A., Verpoest, I., Phelan, F.R., 2001. Textile composites: modelling strategies. Composites A 32,

1379–1394.Nadler, B., 2009. A theory of the mechanics of two coupled surfaces. Math. Mech. Solids 14, 456–473.Nadler, B., Papadopoulos, P., Steigmann, D.J., 2006. Multiscale constitutive modeling and numerical simulation of fabric material. Int. J. Solids Struct. 43

(2), 206–221.Nadler, B., Steigmann, D.J., 2003. A model for frictional slip in woven fabrics. C. R. Mec. 331, 797–804.Parsons, E.M., Socrate, S., Weerasooriya, T., 2010a. Projectile impact of multi-layer woven fabrics: experiments and continuum-level simulations. In:

Proceedings of the 27th Army Science Conference.Parsons, E.M., Weerasooriya, T., Sarva, S., Socrate, S., 2010b. Impact of woven fabric: experiments and mesostructure-based continuum-level simulations.

J. Mech. Phys. Solids 58, 1995–2021.Popova, M.B., Iliev, V.D., 1993. Simulation of the tearing behaviour of anisotropic geomembrane composites. Geotext. Geomembr. 12, 725–738.Rao, M.P., Duan, Y., Keefe, M., Powers, B., Bogetti, T.A., 2009. Modeling the effects of yarn material properties and friction on the ballistic impact of a plain-

weave fabric. Compos. Struct. 89, 556–566.


Roylance, D., 1980. Stress wave propagation in fibres: effect of crossovers. Fibre Sci. Technol. 13, 385–395.Roylance, D., Chammas, P., Ting, J., Chi, H., Scott, B., 1995. Numerical modeling of fabric impact. In: Proceedings of the National Meeting of the ASME.Roylance, D., Wilde, A., Tocci, G., 1973. Ballistic impact of textile structures. Text. Res. J. 43, 34–41.Shahkarami, A., Vaziri, R., 2007. A continuum shell finite element model for impact simulation of woven fabrics. Int. J. Impact Eng. 34, 104–119.Shim, V.P.W., Tan, V.B.C., Tay, T.E., 1995. Modelling deformation and damage characteristics of woven fabric under small projectile impact. Int. J. Impact

Eng. 16, 585–605.Shockey, D.A., Erlich, D.C., Simons, J.W., 1999. Lightweight fragment barriers for commercial aircraft. In: 18th International Symposium on Ballistics, San

Antonio, TX, pp. 1192–1199.Smith, J.C., McCrackin, F.L., Schiefer, H.F., Stone, W.K., Towne, K.M., 1956. Stress–strain relationships in yarns subjected to rapid impact loading: 4.

Transverse impact tests. J. Res. Natl. Bur. Stand. 57, 83–89.Tan, V.B.C., Lim, C.T., Cheong, C.H., 2003. Perforation of high-strength fabric by projectiles of different geometry. Int. J. Impact Eng. 28, 207–222.Tanov, R.R., Brueggert, M., 2003. Finite element modeling of non-orthogonal loosely woven fabrics in advanced occupant restraint systems. Finite Elem.

Anal. Des. 39, 357–367.Termonia, Y., 2004. Impact resistance of woven fabrics. Text. Res. J. 74 (8), 723–729.Ting, C., Ting, J., Cunniff, P., Roylance, D., 1998. Numerical characterization of the effects of transverse yarn interaction on textile ballistic response. In:

International SAMPE Technical Conference Series, pp. 57–67.Xia, W., Adeeb, S.M., Nadler, B., 2011. Ballistic performance study of fabric armor based on numerical simulations with multiscale material model. Int. J.

Numer. Methods Eng. 87, 1007–1024.Xia, W., Nadler, B., 2011. Three-scale modeling and numerical simulations of fabric materials. Int. J. Eng. Sci. 49, 229–239.Zavarise, G., Wriggers, P., 2000. Contact with friction between beams in 3-D space. Int. J. Numer. Methods Eng. 49, 977–1006.Zeng, X.S., Tan, V.B.C., Shim, V.P.W., 2006. Modelling inter-yarn friction in woven fabric armour. Int. J. Numer. Methods Eng. 66 (8), 1309–1330.Zhang, G.M., Batra, R.C., Zheng, J., 2008. Effect of frame size, frame type, and clamping pressure on the ballistic performance of soft body armor.

Composites B 39 (3), 476–489.Zhu, B., Yu, T.X., Tao, X.M., 2007. Large deformation and slippage mechanism of plain woven composite in bias extension. Composites A 38, 1821–1828.Zohdi, T.I., Powell, D., 2006. Multiscale construction and large-scale simulation of structural fabric undergoing ballistic impact. Comput. Methods Appl.

Mech. Eng. 195, 94–109.

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Modeling yarn slip in woven fabric at the continuum level: Simulations of ballistic impact

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