penetration mechanics of textile …vant to the penetration mechanics of textile structures intended...
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TECHNICAL REPORT
Id""'II
PENETRATION MECHANICS
OF TEXTILE STRUCTURES
o .yDAVID ROYLANCE
, SU-SU WANG
I
CONTRACT NO. DAAG 17-76-C-0013
Approved for public release; JUNE 1979disf,'ibution unlimited.
Clothing, Equipment and Materials Engineering Laboratory
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P-NETRATIONA-EHANICS OF TEXTILEESTRCTUS 4 6. PE•FORMING ORG. REPORT NUMBERIE.rRA.... .A 'IS -O ...... •.......... CEMEL #218 -
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TEXTILES BODY ARMOR TEXTILE STRUCTURESFIBERS BALLISTIC PROTECTION PENETRATION
BALLISTICS STRESSKEVLAR STRAIN
20. ABSTRACT C(°olnb -•t reverse sd H mnocey ••d Identify by block number)
This report reviews those aspects of wave propagation and dynamic fracture rele-vant to the penetration mechanics of textile structures intended for use in per-
: ~ sonnel ballistic protection, and then describes the development and implementa-tion of numerical analyses for use in instances for which closed-form analysesare intractable. These numerical treatments are used to assess the manner inwhich fiber material properties influence ballistic resistance, and this is doneby performing simulations of missile impact on four fabrics of actual interest:ballistic nylon, Kevlar 295, Kevlar 49*, and graphite. Following this parametri
Dr 1473 EDITION OF I NOV\ýS IS OBSOLETE -UNClASSIFIED
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materials study, the numerical treat'ment is extended to include the effect oflinear and non-linear viscoelastic relaxation on fabric response to impact.Finally, a special purpose computer code is described which was developed tostudy stress wave effects occurring at fiber crossovers.I *Trademark of E.I. du Pont Co. for its aramid fiber material. Use of thistrademark does not imply government endorsement of a commercial product.
- 1
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UNCIASSIFI1DSECURITY CLASSIFICTION OF T--S PAGEf-..n D.ta Et.td)
II.7,
FOREORD
This work was carried out for the U.S. Army Natick
Research and Development Command DAAG-17-76-C-0013,
Swith Dr. R.C. Laible acting as technical monitor.
The authors gratefully acknowledge the considerable
assistance of Dr. Laible, as well as that of Dr. W.D.[ •Claus, Dr. G.C. DeSantis, and Ms. M.A. Wall.
Accession For
1XTZS G&;a&IDDC TABUnannounced yJustification
By__
D Dist r!ihuti n .L
Aviihnbilit. Ccde s
D~st I s3.2-ce-I
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TABLE OF CONTENTS
Page Number
Foreword ........... .............. 1
List of Figures 5I. Ballistics of Transversely Impacted 9
Fibers
Introduction ...... ............... 9
Longitudinal Wave Propagation ........ 11
Transverse Impact of Fibers .... ..... 15
Use of Rate-Independent Theory inPreliminary Design ... .......... ... 19
Selection of a Failure Criterion . . . 25
II. Numerical Analysis of Impact on Woven
Panels
Method of Analysis .... ........... ... 31
Mathematical Formulaticn. ......... 32
Solution Stability, Convergence, andAccuracy ...... . .............. 39
Parametric Materials Study. . ........ 47
III. Effect of Viscoelastic Materials Response
Viscoelastic Constitutive Relations . 62
Results for Single Fibers. .......... .. 65
Results for Woven Panels ............ 71
Nonlinear Viscoelastic Response. . . .. 75
I ___
X m771MJ.
TABLE OF CONTENTS (concinued)
Page Number
IV. Numerical Analysis of Wave Propagation
in Two Crossed fibers
Introduction ...... .............. .. 2
Method of Solution ................ 83
Results and Discussion ........... ... 97
Conclusions ...................... .. 106
References .. . . .. 109
Appendix A - The FABRIC Code .... .... . . . . . 11Appendix B - The XOVER Code .. .. ... . .. 136
tA
I4 A'
LIST OF FIGU•,S
FigureNumber
1 Wave Propagation in a transversely impacted fiber.
2 Predicted impact strain for linear rate-independentfibers.
3 Predicted impact tension for linear rate-independent fibers.
4 Effect of fiber stiffness on ballistic response;10 = 10 g/den for tension, 10% for strain, 0.03g/den for strain energy, and 900 gm/den sec forenergy absorption rate.
5 Prediction of optimum stiffness for nylon fibers.
6 Variation of breaking tenacity with loading rate-Zhurkov model.
7 Variation in transverse critical velocity due tofracture rate effects.
8 Idealization of impacted fabric panel as anassemblage of pin-jointed tension members.
9 Free-body diagram of forces acting at a fabriccrossover point, showing the influence of the fourfiber elements meeting there and the elastic
10 Propagation scheme for the iterative wave propaga-
tion algorithm.
1 Stability of the numerical scheme as indicated bya minimum in the discrepancy between energy lostby the projectile and energy absorbed by the fabric.These data were obtained from a simulation of a400 m/sec impact on Kevlar 29 fabric at times afterimpact as shown, and for various values of thestability ratio oL defined by Equation 36.
12 Illustration that the numerical scheme convergesto accurate values with time, as indicated by theenergy discrepancy ratio. Note that nonoptimumvalues of the stability ratio (- in this figure)lead to divercence at longer times.
13 Illustration that the numerical scheme predictsvalues of final projectile velocity after penetra-tion in agreement with experimental observation.
14 Computed and experimentally observed values ofcone deformation cone size at time of projectilepenetration. The V5 0 is that value of impactvelocity at which penetration occurs nearlyinstantaneously.
15 Distribution of strain along orthogonal fibers* pessing through the impact point. Curves are
16 drawn for various fabric types, at various times• after a 400 m./sec impact.
16 Effect of initial projectile velocity on thedevelopment of strain at the point of impactfor nylon fabric.
17 Relative ability of the various fabric types toslow the projectile during impact. Ordinal valuesrepresent the ratio of current to initial project-ile velocity.
18 Energy absorbed by a Kevlar 29 panel after a 400rm/sec impact, illustrating the partition of impactenergy into kinetic and strain energy in the panel.
19 Illustration of the relative ability of the fourfabric types to absorb impact energy. The curvesare terminated at the right by projectile penetra-tion, as indicated by a maximum-breaking strainfailure criterion.
20 Development of strain at the point of impact in thevarious fabric types after a 400 m/sec impact.
6
21 "Master" curve for impact-induced strain at thepoint of impact. Ordinal values representstrain normalized on the basis of the strain whichwould be generated in a single fiber by impact atthe same velocity, while abscissal values areadjusted by a factor equal to the fourth root ofthe fiber modulus.
22 Wiechert spring-dashpot model for linear visco-elastic fiber response.
23 Normalized strain plotted against Lagrangian fibercoordinate for various times after impact.
24 Normalized tension distribution along fiber.
25 Numerical values for tension distribution for t41.08 microsec after impact.
26 Stress distributions along orthogonal fibersrunning through impact point for linear elasticand viscoelastic fabrics (t = 30.4 microsec).
27 Distribution of strain along orthogonal fibersrunning through impact point.
28 Stress histories at impact point for linearelastic and viscoelastic materials.
29 Stress relaxation at impact point for variousimpact velocities.
30 Stress distributions along orthogonal fibersrunning through impact point for linear elasticand nonlinear viscoelastic fabrics.
31 Comparison of stress relaxation in linear and non-linear viscoelastic fabrics.
32 Stress histories at impact point for linearelastic, linear viscoelastic, and nonlinear visco-elastic fabrics.
33 Schematic of model for numerical analysis of twocrossed fibers.
4. 7
. - .- ,• j5
34 Discrete element of fiber.
35 Strain distributions in two crossed fibers ofKevlar 29, 28.7 microsec after impact at 400 m/sec.
36 Influence of the fiber modulus on the fraction ofstress wave intensity which is transmitted througha fiber crossover, in the absence of fiber-fibersliding.
37 A comparison of the reflection-only bounce modelfor wave propagation in an impacted fabric, in Icomparison with the fabric model of this report.
38 The influence of fiber-fiber sliding on thefraction of stress wave intensity which is re-flected at fiber crossovers, as indicated bycomputer experiments on Kevlar 29 fibers.
39 The influence of fiber-fiber sliding on the extentto which a portion of the propagating stress waveis diverted from the primary fiber to begin pro-pagating along the transverse secondary fiber.
40. The influence of sliding on the extent of stresswave intensity propagated beyond fiber crossovers.
I
4
¢,I
PENE-TRATION MECHANICS OF TEXTILE STRUCTURES
I. BALLISTICS OF TRANSVERSELY IMPACTED FIBERS*
Introduction
Although impact of single fibers or fiber assem-
blies is an important subject in its own right, being
relevant to climbing ropes, aircraft carrier arrest
cables, high-speed weaving, etc., the principal develop-
ments in this area have been made by workers whose
major interests have been in the impact resistance of
woven or non-woven textile structures. The most notable
of these structures have been the lightweight armor vests
used by police and military personnel, but among other
important applications can be listed aircraft engine
containment shrouds, flak blankets, and vehicle seat
belts. Ballistic nylon has been used successfully for
these vests since the Second World War, although
current developments have emphasized the du Pont aramid
fiber marketed as Kevlar**. Although, as will be shown
below, excellent single-fiber ballistic response does
not necessarily guarantee a superior vest, any under-
standing of textile structure ballistics must be pre-
* From Reference 1 (see Page 109). Used by permission
of Textile Research Journal-
"*Trademark of E.I. du Pont de Nemours & Co., Inc.
ii9
ceeded by an understanding of single-fiber response.
A strong motive for discussing fibers is that
single fiber tests are often used as screening tests
for ballistic protection materials. As an example,
one often encounters tabulations of "transverse
critical velocity", that ballistic velocity at which
a transversely impacted yarn experiences nearly
instantaneous failure. Typical data is shown below.
Transverse critical velocities of textile
fibers. [2]
V r m/sec
Nylon 616Polyester 472
Nomex 442
Fiberglass 274
Kevlar 29 570
Such tests are often indicative of relative
ballistic resistance, but perfect correlations cannot
be guaranteed. In the above tabulation Kevlar 29 proves
to be the best ballistic material when put into a panel,
in spite of its having a lower transverse critical
• Numbers in brackets refer to references listed on
pages 109-110.
10
veloc'ity than nylon.
Longitudinal Wave Propagation
Wave propagation phenomena in fibers and thin
rods are considerably less complicated than in a
general medium, since the possibility of unrestrained
transverse contraction in fibers eliminates (to a
good approximation) the simultaneous propagation
of independent dilatational and distortional waves
which are present in general. The equation of motion
for fibers or rods is simply [3]:
E. )27 (1)
where u is the longitudinal particle displacement,
is the material density, E is the longitudinal Young's
Smodulus, and x and t are the space and time coordinates.
This is the well-known wave equation, whose solution
represents a disturbance traveling at a velocity
IF (2)
3.3"
z'4
Conventional textile units employing stiffness per unit
linear density are very convenient in wave propagation
analyses, since the factor P is included implicitly
in the modulus. For modulus expressed in grams per A
denier and wavespeed in meters per second, Equation 3
becomes: A
(3)
where k = 88,260 is the necessary units-conversion
factor. In these equations, as well as those to
follow, the modulus is taken to be the "dynamic"
stiffness relevant to the high strain rates corres-
ponding to wave propagation tests. The development- of
such dynamic constitutive relations from experimentalfiber-impact data has been described elsewhere (4,5].
Coiisider a fiber fixed at one end whose free end
is suddenly subjected to a constant outward velocity
V in the longitudinal (fiber) direction. After a time
it ,the strain wave twill have propagated into the fiber
a distance ct, while the free end will have displaced
outward an amount Vt.. The strain resulting from the
impact is then the displacement Vt divided by the
3.2Ii 12 - . ...
affected length ct:
S"- -- = .. __ _-(4) , l
cL C v;A-a
The corresponding stress is Z
The above relations have assumed a linear
S~elastic material whose stiffness E is independent of
-the strain. In this case the wavefront will propagate
as a sharp discontinuity (a shock wave) at which the
strain rises instantaneously from zero to the value
given by Equation 4. Many ballistic fibers are
nonlinear, however, and the effect of material non-
linearity leads to some complication of the above
description. A nonlinear fiber can be characterized
as having a strain-dependent modulus E = E( r ),so
that Equation 3 becomes:
2C_ cc=, CG' (6)
13I
The shape of the wavefront is now dependent on the
shape of the dynamic stress-strain curve. If the curve
is concave toward the strain axis, so that the modulus
decreases monotonically with strain, each suceeding
increment of strain in the propagating wave travels
more slowly than the previous increment. The wave is
then dispersive, and broadens as it travels. If on
the other hand portions of the stress-strain curve
are away from the strain axis, then portions of the
strain wave will overtake more slowly propagating
increments of lesser strain, and the wave will contain
shock components. In general, a wave may contain both
dispersive and shock components.
In the region behind the wave, material flows in
the direction of the imposed velocity with a "particle
velocity" w. This motion is fed by the strain
developed in the propagating wave, and the particle
velocity is related to the wave to the wave speed by:
C2
ccr=) J (6ýJr. (7)
14?
11W
where I0 is the ultimate value of strain generated
by the impact. Since the particle velocity must match
the imposed velocity, we have
wA
The strain C developed by longitudinal impact is
found by solving Equation 8, perhaps numerically.
Transverse Impact of Fibers
As the transverse impact of fibers seems intuitive-
ly germane to impact of woven textile panels, the
technical community interested in lightweight ballistic
protection has devoted intensive effort to this problem A
since World War II. Following the pioneering works of
Taylor [6] and von Karman [7] during the war, valuable
contributions have been made by Peterson et al. [8],
Shultz et al. [9], Wilde et al. [5], among others, but
by far the most prolific of these efforts has been
that of Jack C. Smith and his colleages at the National
Bureau of Standards. Reference [10] provides a review
of most of this work, which contains a wealth of
fi5!
experimental and theoretical contributions ranging over
a period of approximately ten years in the fifties
and sixties.
V V V
U 0= U=W IAIA-
WAVE- SPES ELO=OltS
Figure 1. Wave Propagation in a transversely impacted fiber.
The rate-independent theory of transverse fiber
, impact as developed by Smith can be stated with refer-
ence to Figure 1. This illustrates a fiber, originally
straight in the horizontal direction, which has been
impacted by a projectile traveling vertically upward.
Upon impact, longitudinal waves of the type described
in the previous section are propagated outward from the
point of impact. Behind these waves material flows
inward toward the point of impact at a constant
velocity w, strain Co, and stress In
165
{:.! " +.6
addition to the longitudinal waves, transverse "kink"13 waves are also propagated outward from the impact point.
At the transverse wavefront the inward material flow
ceases abruptly and is replaced by a transverse
particle velocity equal in magnitude and direction to
that of the projectile. The strain and tension are un-
changed across the transverse wavefront, but both the
Jlongituidinal and transverse particle velocities ex-
perience discontinuities there; in this sense the trans-
verse wave is a geometrical shock. The apprently un-
balanced tensions on either side of the transverse
wavefront are compensated by the change in particle
momentum as the wave propagates. Behind the trans-
verse wavefront all particle velbcities are equal in
magnitude and direction to the projectile velocity,
and the fiber configuration is a straight line at a
constant inclination 0 from the longitudinal direction.
The inward particle velocity is found, as in the
I longitudinal case, as
~ 3 ~CC~~6 ~~\ V~c&~~46 (9)
The final strain 0 is unknown as yet, but E( E )
is known as the slope of the dynamic stress-strain
L17
curve. The outward velocity U of the transverse kink
wave, measured relative to a Lagrangian frame attached
to and extending with the fiber, is:
4I 6- (10)
To a fixed observer the transverse wave appears to
propagate in a "laboratory" frame of reference at
Vb(
Finally, the above variables are related to the impact
velocity V through the relation:
V0+. z -~
Equations 9-12 constitute four relations between, V,
w, e 0' O-0, U, and U. The material dynamic stress-
strain curve relates and £0' so that once one
of the parameters (say V) is specified, the other four
independent parameters (w, e 0' U, U) can be found.
For nornlinear stress-strain curves, numerical solu-
tions will likely be more convenient.
Certain limitations to the Smith analysis
described above must be mentioned. First, it is
18M i
rate-independent. Most polymeric fibers exhibit
strong rate dependencies, and these effects are beyond
the capacity of this analysis to describe. Perhaps
a more severe limitation is that the Smith analysis is
not applicable to late-time effects in the wave propa-
gation process. In real situations the outgoing
longitudinal wave soon collides with an obstacle: a
clamp, in the case of single-fiber tests, or a fiber
crossover, in the case of impact in woven textile
panels. Upon such a collision a reflected wave is
propagated from the collision point in the direction
opposite that of the original wave. This reflected
wave in turn soon collides with the outward-traveling
transverse wave, and this collision generates another
two waves which travel away from the collision point. IN
These waves in turn eventually collide with the clamps, :q
or the projectile, or other waves. The result of
-1hese wave reflections and interactions is a situation
which becomes intractable by closed-form mathematical
methods, and this late-time intractability is a
principal reason for the development of numerical
computer solutions.
Use of the Rate-Independent Theory in Preliminary Design
In spite of the limitations of the Smith theory
outlined above, the rate-independent analysis provides
19
a highly useful means of assessing approximate relations
between fiber material properties and ballistic
response. These relations are of considerable -value
in performing preliminary design steps in development
of textile ballistic-protection devices.
Assuming the material to be linear in stress-
strain response (E = constant), the Smith analysis can
be cast in the simple form:
V ec)~ 'cj+6)-E~ (13)
which provides a relation for the strain 4E developed
by impact at a velocity V in terms of the fiber modulus.
The relation can be solved numerically if one wishes to
compute E 0 for a given V, or it can be used directly
to plot CO versus V for the purpose of developing
design curves (see Fig. 2). Once C 0 is known, then
or U, U, and w can be found from either the stress-
strain relation or Equations 9 - 12. Figure 3 shows
such a plot of tension sr0o versus Vwith modulus as a
parameter.
20
- 7 .
* - -I E
30
E 20 gpd 60 UpdUoy
11112 gpdS 20
0 1000 1500
impact Velocity, mlsec
SFigure 2. Predicted impact strain for linear rate-independent fibers.
8 - 1100
,/90•, ; ,•_80
S, 7 0 ICU
,.•,, 60 E
0~0
c 30
E -- 220
4f4-
010
0 100 20304050 5 0 0
Impact Velocity (M/sec)
Figure 3. Predicted impact tension for linear rate-independent fibers.
8 100
Since the above curves rise monotonically with
velocity, one can observe the influence of modulus more
easily by plotting ballistic response at a constant
velocity, and Figure A shows such a plot at V = 400
m/sec. Here are plotted the strain and tension from
the above methods, along with the strain energy
S"= 6 •o developed behind the wave and2 00
the rate of energy absorption (c of the fiber.
(The term, 9c,is shown in mixed units, but it could
be converted to joules/sec once the density and denier
of the fiber are specified.) The rate of energy
absorption at the wavefront must equal the rate at
which the fiber extracts kinetic energy from the
projectile, and it is a reasonable measure of ballistic
efficiency. Note that this energy absorption rate
rises monotonically with fiber modulus, although with
less dramatic improvements after approximately 500 g/den.
8
24
• 1 22
SIV 400 m/seclo8 : \ ~ ~~Tension *....-'"A•
Z- " Energy Absorption Rate0
a. .Af Strain Energy Behind Wave
O0 p
0 I00 200 300 400 500 600FIBER MODULUS, gmn/den
Figure 4. Effect of fiber stiffness on ballistic response ; 10 = 10g/den for tension,. 10% for strain, 0.03 g den for strain
energy, and 900 g/den sec for energy absorption rate.
" ~Of course, one c~annot improve ballistic efficiency
indefinitely by continuing to seek stiffer fibers. In
general, increases in stiffeness are accompanied by de-
creases in breaking Strain, and a point may well be
S~beneficial reduction in impact-generated strain shown in
Figure 2. This effect m~ay be quantified by means of
Equation 13, where one may calculate the critical trans- Iverse velocity by determining the velocity which just
j generates the dynamic breaking strain on impact. If one
23
knows the variation of breaking strain with stiffness,
these calculations may be used to select approximately
an optimum fiber stiffness for ballistic efficiency.
This process is carried through for illustrative pur-
poses in Figure 5.
4010004
•Y • • - 3 0• 800 3_.0
S N 60020
400-
C:u
... 200 ,
0 40 80 120
• Modulus, gpd
SFigure 5. Prediction of optimum stiffness for nylon fibers.
,•; The dashed line in this Figure is the relation•: i between dynamic stiffness and breaking strain as
4- 24
L. dtrmndfomfbrimattst naseisoS•nyonyrn tathd ee ubecedt vriu
• 0)2
drawing treatments by the manufacturer [5]. The solid
line is the the calculated transverse critical
velocity, considering the effect of stiffness on both
impact-induced strain and on breaking strain. An
optimum is observed near 60 g/den, which is in fair
agreement with experimental observation. All this is
a quantification of the often-quoted guideline in armor
design that one seeks the highest possible modulus in
order to spread the impact over a wide area via in-
creased wavespeed, but that the process must not be
carried so far as to induce excessive brittleness. In
hard armor this reasoning has led to the use of
ceramic faceplates to give high wavespeed, backed by
a fiberglass laminate to provide the needed toughness.
Selection of a Failure CriterionI4
The use of simple ultimate breaking strain as a
failure criterion in the above example is overly
simplistic, since it does not incorporate the strong
temperature and rate dependencies that are known to
exist in polymeric material. A versatile fracture model
4 which does incorporate these dependencies and is still
computationally convenient is that due to Zhurkov f1i],
which states that the lifetime t of a solid subjected
to a uniaxial stress s is of the form
25
T.sa- fc tor unis4o
where 0 is a pre-exponential factor with units of
time, U* is an apparent activation energy for the
fracture process, • is a factor with utnits of
volume, R is the gas constant (8.314 J/mole 0 K), and
T is the absolute temperature. For constant tempera-
ture, Equation 14 reduces to
6L~ (15)
where
iW,~ (a*ZT%)(16)
When stress and temperature vary during the loading
process, one can assume linear superposition and write
the Zhurkov criterion in the form:
•i In a constant-stress-rate experiment at constant
•! temperature, for instance,
62
x (18)
To illustrate the order of rate dependency provided by
Zhurkov's model, Figure 6 shows a plot of Equation 18
for the case of drawn nylon fibers. In this figure2.20 x 1019 sec, and 5.13 (g/den)-I
these are the values obtained by Zhurkov [11] by
fitting Equation 15 to creep-rupture data. Such
a plot can be used to depict the time-to-break for a
11< fiber, and the tenacity-at-break, as a function of
the loading rate.
27
4 Drawn Nylon Fibers* Constant Stress RateV T 2300 °K •'
N0E 1001
02- !
COO
a: 10
-4
hrkv- mode infbrbalsistesrespe
S-6-
6 7 8 9 10 11
• ~TENACITY, gm/den
Figur Variation of breaking tenacity with loading rate - Khurkov
model.
As a more direct example of the utilization of
Zhurkov's model in fiber ballistics, the stress pre-
dicted by the Smith theory for a given impact velocity
and fiber modulus can be used in Equation 14 to pre-
dict the time after impact at which the fiber will
rupture. This analysis predicts t:hat there is no
unique critical transverse velocity, but rather a
range of velocities over which the fiber will fail in
experimentally observable times. Figure 7 shows the
4 predicted results for drawn nylon fibers, using an
assumed dynamic modulus of 80 g/den with the same
values of 6L and used in Figure 6. This figura
28
shows that at velocities above approximately 775 m/sec,
rupture occurs in less than fifty microseconds and
would be counted in most high-speed photographic
records as having occurred instantaneously upon impact.
The times-to-break become exponentially longer at
lower velocities, and failure will occur at the clamp
due to wave reflection at times dependent on the
wavespeed and fiber length. This variation in what
may be termed critical velocity for impact may make up
a large part afthe scatter observed experimentally in
determining critical transverse velocities.
100
80 Astrain = 0. 132tt
VIo 60E
,,, 40
"Nylon 6 fiberE E --80 gpd
strain =0. 13520
Sstrain--0. 140
750 775 800 825
I mpact Velocity, tsecr
Ficave 7. Variation in transverse critical velocity due to fractlre rateeffects.
29 '
An important advantage to Zhurkov's model is that
it is derivable in terms of basic reaction-rate
fracture analysis. As such, it provides a means
whereby the materials scientist can predict materials
and processing modifications so as to manipulate the
fracture parameters and improve ballistic performance.
A recent review [12] describes the basic implications
of reaction rate models such as Zhurkov's, as well as
their limitations and experimental corroboration.
The development of these models is somewhat controver-
sial, with several quite divergent approaches having
strong advocates. Zhurkov's model in particular is
often criticized as being simplistic, but is convenient
for use in impact by virtue of its computational con-
venience and its ability to model a wide range of
materials behavior, if only phenomenologically. Finally,
it should be cautioned that the experiments Zhurkov used
in corroborating his model were no faster than the milli-
second time scale, some three orders of magnitude
slower than ballistic impact fractures. Such an extra-
polation is cleariy dangerous and should be verified by
additional experimentation. The plot given in Figure
7 is in reasonable but not excellent agreement with
experimental data given in Smith's papers, indicating
that the approach is promising but needing of further
corroboration.
30
-,j-
II. NUMERICAL ANALYSIS OF IMPACT ON WOVEN PANELS
I Method of Analysis
The method of analysis used in this study is a3
direct numerical approach which attacks the governing
dynamic equations of the problem through a computer-
aided iterative scheme. It may be considered as a
hybrid of the finite element method in selecting
control volumes and the finite difference in establish- Al
ing recurrence formulas.
Sy
V (t)
S~L12
z0(t)N• _.X 11.•......__ ------ x
r C (t)(x# yO z10
Figure 8. Idealization of impacted fabric panel as an essemblage ofpin-jointed tension members.
31
The fabric of dimension L by L shown in Figure 8
is modeled as a network of interconnected fiber
elements impacted at zero obliquity by a rigid missile
of mass.M with an initial striking velocity V . The
network model rather than a continuous membrane is
employed here, for it is not only more consistent
with the discrete fabric structure but also leads to
better agreement with the physical deformation con-
figuration (pyramidal cone rather than hemisphere) as
observed by Wilde (131 in high speed photographs.
The elastic continuum supporting the fabric deformation
is generally very flexible in transverse direction:
bending effects are neglected. The constituent fibers
are considered to have a slender and uniform cross
section that only plane waves propagate uniaxially.
Furthermore, crossovers are modeled as hinged connec-
tions; then, slippage is assumed to be negligible.
Mathematical Formulation
Considering a typical crossover in the panel,
as shown in Figure 9, the impulse-momentum balance
32
kb Az(j+I, k+I1 t)
T1O (j+lk; 0J
. j+I" k+l)
Tol (j,k+l;t)Z Tio(j+l,k+ I;t)
z r Nxly, Z; t) To, (j+l, k+l4t)
0
Figure 9. Free-body diagram of forces acting at a fabric crossoverpoint, showing the influence of the fcur fiber elementsmeeting there and the elastic resistive force provided bySthe fabric backing. I
d.uring time dt may be written as
k-wA r 1 . F t(19) T
where dv is an incremental velocity, T is the resultant
tensile force, and Am is the lumped mass of a
33
fabric element. This relation provides a means of
calculating current velocity from field variables in
previous time increment. For instance, at node (j+l,
k+l), the velocity at time tm+1 may be expressed in a
finite difference form as
y., +. T (20)" ~~~~3+ik~t ".,A9
where is the linear fiber density, the
length of an orthogonal fiber element, , , a numer-
ical factor associated with the crimping and wave of
the fabric structure, and the tensile force T is given
as
-TI.
T . . ,.,o,(21)
in which kb is a backup spring constant and u is the
displacement vector. T and T1 are tensile vectors10 01
in the deformed orthogonal fibers running through the
crossover as shown in Figure 9. The Lagrangian* The .underline is used to denote a vector quantity.
3463,."
coordinates of the node are then evaluated by
U +A-. (22)
AAThe up-to-date strain defined as e(j+l, k+l; m+l) =
Sel0, e011 may now be determined from a continuity
condition;
-1a. (23)
and
e (24)
35
= 4,-
iW
Tensile stresses for the fibers at the crossover
may be computed from material's dynamic constitutive
relationship. For the case of a simple elastic fabric,
T may be calculated by
L A'v
T E. e k(25)
Current missile velocity V (t) may be obtained byp
F 00 (26)
rr
where the fabric inclination 9 at the impact/' P
point may be evaluated from
el.FF .\n (27)
The total kinetic energy loss of the missile
36
during penetration can be computed from
AE~/ (V -V1, (28)
where Vr is the residual velocity of the missile upon
penetration. The fabric energy absorption and parti-
4 !ition are of great interest, since they provide a means
of evaluating the performance of the material. The
kkinetic and strain energy, AE (t) and &ES(t),f
obtained by the panel during impact may be calculated
by
* I:4E C \\&w l '17 (29)
-V
Where A is the area of the fabric. The above formula-
tions have been coded in FORTRAN, and the computation
algorithm proceeds from one node to the next along a
wave front propagating through the , Due to
geometric symmetry, only half of one quadrant is con-
37
sidered. At each time increment, the code begins at
the origin (the impact point) and works outward until
the wavefront is reached. The space progression con-
sists of a series of passes along lines diagonal to
the orthogonal fibers, indicated by dashes in Figure
10. The algorithm begins at the x-axis (a subroutine
is employed to handle the slightly different condi-
tions along this symmetry boundary), and progresses
along the diagonal until reaching the end of the
octant. For reasons of stability, the rate at which
the code progresses outward from the impact point is
related to the fabric wavespeed; these stability con-
siderations will be discussed in a later section.
y
I I I I /4-3_ _ - _-__ L_ -_• -I_
I I I. I / /
I ' / , -
i ~ ~ ' -T i I .T
S(j+l, W~) •
% S,, olution Front"I
Impact A
Point
Figure 10. Propagation scheme for the iterative wave propagationalgorithm. 38
The numerical method requires appropriate initial
and boundary conditions in order to proceed with the
computation. The initial condition is that all nodal
points are at rest except that the initial projectile
velocity is imposed at the center of the panel, i.e.
A.Y* vý (31)
The boundaries of the fabric are assumed to be
rigidly clamped during impact, thus
"•-"•s L/. (32)
Solution Stability, Convergence and Accuracy
Solution stability and convergence are directly
related to the theory of characteristics for the hyper-
bolic system. The Von Neumann stability criterion [141
for the probem may be written as
where cr is the wave velocity in the fabric. Hence
the selection of &t and &X cannot be arbitrary.
r3
The value of cr is not known prior to the analysis, but
a preliminary study by Roylance [15] indicates that it
is a fixed fraction of the wave velocity ir a single
Ifiber, cf, i.e.
C - (34)
where a is a numerical factor. It generally haE
value greater than unity, which may be attributed Lo
the effective increase on lineal density caused by
fiber crossovers. in a square-woven fabric, the lineal
density of a fiber along which a wave is propagating
is effectively doubled. This retards the wave velocity
according to Equation 2 by a factor of 4C = -F2. The
stability condition is then of the form
(35)
In the current analysis, • is obtained by physical
considerations; therefore &t is constrained by Equation
(35). The parameter & has its optimum value obtained
from stable solutions and is defined as
(36) 0
40
Co .73 !U
In the absence of a complete theory for finding
the exact value of this parameter, it is a common
practice in the use of numerical methods to have the
computer determine it. The optimum R may be obtained
by changing its value continuously in test cases until
a known solution is matched and a variation of the
quantity will not yield any appreciable difference in
the results. Unfortunately, there is no existing solu-
tion for this problem and one must resort to numerical
tests of smoothness or of conservation laws for this
purpose. A convenient measurement of stability and
convergence is to study the rate of energy conserva-
tion of the system. In this study, an energy discrep-
ancy parameter Y is introduced for the purpose, and is
defined as
11 (37)
where Ef is the total energy absorption by the fabric,
and E is the projectile energy loss defined previously.p
Figures 11 and 12 give illustrations of the dependence
of solution stability and convergence on the values of
parameter. An optimum value =f-2 is obtained
from these results, in agreement with physical reasoning. I-•
41
1;. 10N •
8 I si
SO. I0
Stabilit Rati
10~
•i in ~~~~th dicrpac betwe n enrg los 40 he poetl n
I AP
• energyt absorbe byth fbi.These dt eeobandfo
14f00
6c
2o'
412
0.11
Stability Ratio
Figure U. Stability of the numerical scheme as indicated by a minimumin the discrepancy between energy lost by the projectile andenergy absorbed by the fabric. These data were obtained froma siimilation of a 400 rn/sec impact on Keviar 29 fabric at timesafter impact as show~n, and for various values of the stability
ratio d..defined by Equation 36.
42
(
S ,10-f=/
L~=
•00000
0.
0; K0 20 30 40 1S ~TIME, j•sec
•!Figure 12. Illustration that the numerical scheme converges toaccurate value•s with time, as indicated by the energy
S, discrepancy ratio. Note that nonoptimum values of-• the stability ratio (Yj in this figure) lead to diver-• gence at longer times.
• • Assessment of accuracy of the numerical analysis
• is somewhat problematical, as no closed-form mathematic-
•. al analyses are available against which to check the
g[• code results. Certain experimental observations are
•.•,available, however, one of which is shown in Figure 13.
iF ci
&ý• This figure is a plot of residual projectile velocity
M S
0.53
Figure 13. Illustration that the numerical scheme predicts values offinal projectile velocity after penetration in agreementwith experimental observation.
KEVLAR 29600 ONE LAYER "
o .0 //"
~400-
-J/
>-0 -- EXPERIMENTAL -
X / - - FABRIC CODE200 /
1 7' /
0/0 200 400 600 800 1000
MISSILE IMPACT VELOCITY, M/SEC
after penetration of a Kevlar panel, as a function of
initial velocity. The good agreement of the predicted
and observed results is important, since it provides
some assurance that both the transient response and
the final fracture processes are being modeled reason-
:• ably. It might also be mentioned that this particular
plot is one which plays an important role in the design
process, so that the ability to generate it numerically
without prior ballistic data or any idealizing assump-
tions is of considerable practical importance.
44-
Another result of the numerical calculations which
may be checked against experiment has to do with the
shape of the transverse deformation cone, since the
cone may be followed by high-speed photography during
the impact event. It should be mentioned that this
photgraphic evidence provided the initial impetus to
the development of the present pin-jointed fiber
model, as opposed to various membrane approaches which
have been attempted in the past. The photographs
clearly show a pyramidal defomnation cone which re-
flects the orthogonal nature of the woven structure, as
opposed to the circular cones which would be predicted
by axy-symmetric membrane analyses. The present
numerical treatment predicts this pyramidal shape
correctly. A convenient indicator of deformation is
the size of the cone at the time of projectile
penetration, as this parame- !r also reflects both tran-
sient and fracture properties of the panel. Figure 14
shows the predicted and observed cone size at penetra-
tion for a Kevlar panel, and again it is seen that
agreement is satisfactory.
1< 45
One Layer, Kevlar 29
E 12i- I2.. -- - ExperimentalMt
-Computed
1 10
E
0 300 600
F,'v• I mpact Velocity, m/sec
Figure 14. Computed and experimentally observed values of conedeformation cone size at time of projectile penetra-tion. The V is that value of impact velocity atwhich penetr•ion occurs nearly instantaneously.
M
46
Parametric Materials Study
The utility of the numerical model will be
illustrated by means of a number of computer experiments
in which the influence of fabric materials properties
* ion ballistic resistance is assessed. These results help
validate the reliability of the model in that it can
be shown to generate data in agreement with experimental
observations. It also provides a means of illustrating
certain phenomena, such as transient wave propagation,
which are not generally observable experimentally; in
this regard one's intuitive understanding of the impact
even is improved considerably.
Numerical results have been obtained for a
series of four simulated orthogonally-woven square
panels 203 mm on a side, impacted at zero obliquity by
a 0.22-caliber projectile weighing 1.10 gram. Such a
projectile is commonly used in experimental work to
simulate the effect of fragment impact. The edges of
the panels were assumed to'be clamped, although penetra-
tion generally occurred before the arrival of stress
waves at the clamps; the nature of the edge boundary
conditions is therefore relatively unimportant. Rather
than perform straightforward parametric tests in which
one variable, such as fiber modulus, is varied while
others are held constant, it was decided to simulate
147
I ns"2fý
a series of actual fabrics for which input data was
available either from the weaver or from laboratory
measurement. The computer results can thus be compared
directly with laboratory ballistic tests, although in
general, more than one variable is changed in each
simulation. In particular, the fabric panel weight
varies slightly for each material type, although this
effect was expected to be small relative to the large
change resulting from the markedly different fiber
moduli.
For the purpose of these parametric tests, only
very simple constitutive and fracture models were
employed. Although more realistic models are available
as described elsewhere in this report, the numerical
data necessary for input into these models are generally
not available. For this reason the fiber stiffness was
set to a constant value obtained from handbook quasi-
static stress-strain data, and the failure criterion
was a simple maximum-breaking-strain check, where the
maximum allowed was also taken from quasistatic tensile
test results. In spite of these questionable choices,
the results of the computer simulations are of con-
siderable interest.
The data for the four fabric types are shown
below:
•_ , ,,- . 9-
GEOMETRIC AND MATERIALS PROPERTIES USED IN
FABRIC STUDIESTM TM
Fiber Nylon Kevlar 29 Kevlar 49 Graphite
Tensilemodulus,gpd 80 550 990 2650
FractureStrain, % 14.0 4.0 2.2 L.i
FabricMass, gm 19.53 17.38 25.75 27.09
Yarn denier1050 1167 1485 1500
Ycrn/cm 17 16 16 16
Figure 15 shows typical computer predictions of
strain wave profiles obtained at various times after a
400 m/sec impact on the various fabrics. Unlike impact
on a single fiber, in which a constant level of strain
is propagated outward from the impact point, the
array of fiber crossover junctions around the impact
point in a fabric serves to reflect a portion of the
outward-propalating wave back toward the impact point.
As a result, the strain is always greatest at the point
of impact, and grows continuously with time (unless
the projectile is slowed appreciably by the panel).
Both the level of strain and the rate of propagation
49
-... .• .w -o
are governed by the fiber modulus and density. As
shown in the figure, graphite fibers have the highest
modulus of the four materials, and thus propagate the
lowest level of strain at the highest rate. As the
modulus is decreased, the strain level is increased and
the wavespeed is decreased.
Single Layer, 8"x 8"1V p 400 m/secp
12-
,2 • 8 Nylon, 11. 59 microseconds
L._
4.Kevlar 29, 5. 80 microseconds
S9.02 microsecondsI 12.25 microseconds
Graphite, 5. 87
00 2 4 6
Distance from Impact, cm"Figure 15. Distribution of strain along orthogonal fibers passing
through the impact point. Curves are drawn for variousfabric types, at various times after a 400 m/sec impact.
50
The strain history in the fabric is directly
related to missile striking velocity as indicated in
Figure 16. When the velocity is greater than a
critical impact velocity Vcr, strain at the impact point
continuously rises until penetration, duo; to the- con-
tinual arrival of wavelets reflected from crossovers
and boundaries. In contrast, if the -elocity is
smaller than Vr, the impact strain develops to a
level below the breaking strain and remains relatively
constant for the rest of the dynamic process. Here
the effect of unloading due to projectile deceleration
is able to balance the increase of strain due to wave
reflection. The wavy strain history in the figure may
be caused by the dispersion and interaction of the
traveling and reflected wavelets, and perhaps, some
numerical fluctuation.
16
ge16Ef12fVp 400 m/seoc
S' • !1•Vp 300 m/sec
TIME AFTER IMPACT, -oec
Figure 16. Effect of initial projectie velocity on the developmentof strain at the point of impactt for nylon fabric.
I LI -
41
Penetration dynamics can also be illustrated by
the missile deceleration as shown in Figure 17, where
reductions o'- missile velocity by various fabric
materials are given. Note that the ability of the
various fabrics to decelerate the projectile increases
monotonically with the fiber modulus.
1.00 S •it NYLON
S0.9 6KEVLAR 49W KEVLAR 29 0
S0.94-0
Vp =400 M/SEC
zA0A
- 0 2 4 6 8 10 12TIME AFTER IMPACI; MICROSECONDS
Figure 17. Relative ability of the various fabric types of slow theprojectile during impact. Ordinal values represent theratio of current to initial projectile velocity.
The energy extracted from the projectile is
partitioned into strain and kinetic energy in the panel.
This energy partition is easily computed, and Figure 18
indicates that approximately half the total fabric
energy absorption is stored in the form of strain
energy. The kinetic energy associated with transverse
velocity is approximately equal to that associated with
in-plane velocity components. Energy absorption is a
52
S0.6
/JE
"" Etota= 0.4
1._.
Q 0.2,-"xnetic, z
0 1E• . strain
• •- kinetic, x-y
0 4 8 12
Time After Impact, microseconds
Figure 18. Energy absorbed by a Kevlar 29 panel after a 400 m/secimpact, illustrating the partition of impact energyinto kinetic and strain energy in the panel.
53
- d
Q 6
E Kevlar 29 &A
S0.4-0
Toim Kevlar 49 (i
F=re 19Nylutrton ofterltv bliyo h orfbi c
p of Graph ite c
I-0
o" 4 8 12•• Time After Impact (microseconds)
, Figure 19. Iflustration of the relative ability of the four fabricS~types of absorb impact energy. The curves are termin-
ated at the right by projectile penetration, as indicatedby a maximum-breaking strain failure criterion.
544
convenient indicator of panel ballistic performance,
and Figure 19 illustrates the relative energy absorp-
tion capabilities of the four panel materials studied.
It is seen that the high fiber modulus of the
graphite panel leads to a rapid rate of energy absorp-
tion, but that fracture occurs before the panel has
been able to extract as much of the projectile's
impact energy as the lower-modulus fabrics. Conversely,
nylon requires a long time to penetration, but the
energy absorption rate is too slow to lead to a large
total energy absorption. The Kevlar 29 panel exhibits
the best combination of energy absorption rate and
long time to penetration, and is thus predicted to be
the superior ballistic material of the four types
studied.
E.Y
55
_____ I
It should be mentioned that the accuracy of these
results is limited by the questionable assumptions
which had to be made due to the present lack of know-
ledge as to dynamic fiber properties which could be used
as input data for the code. For these rather stiff
fibers, the use of a linear elastic constitutive law
is probably not a serious error; however, the use of
maximum-breaking-strain failure critrion is almost
certainly to blame for some inconsistencies in the
results. In particular, Kevlar 49 is known to be
essentially as good as, if not superior to Kevlar 29
as a ballistic material. The authors feel the shape
of the energy absorption curves in Figure 19 is
accurate, but that the location of the failure point
is poor in the case of Kevlar 49. This points out the
need for more complete dynamic fracture data on these
fibers, so that more realistic models such as that
described by Equation 17 may be employed.
It is natural to seek some simple relationship
between fiber material properties and fabric ballistic
resistance. The preceeding results lead one to expect
that the most important parameter governing the stress
history in the fabric before fracture isthe fiber
modulus. The modulus controls wavespeed through the
relation c= /-E, and thus the distance the impact dis-
turbance will have traveled in a given time. The
56
modulus also controls the level of strain which will be
generated by impact at a given velocity. The relation
is not known explicitly for fabrics, but can be deter-
mined by performing computer experiments using the
numerical code.
Figure 20 depicts the computed strain history at A
the point of impact for 400 m/sec impacts upon the four
model fabrics. It is clearly seen that an increase in
fiber modulus decreases the strain for a given time, in
correlation with the same result for single fibers.
The fabric impact is considerably more complex than
single-fiber impact, however; the point of impact feels
iot only the continuing influence of the projectile,
but is also continually bombarded by wavelets reflected
and diverted from adjacent fiber crossovers. The
situation is too complex to permit simple generaliza- A,
tions, but the nonlinear form of the strain histories
for the various fabrics can be taken to reflect the
influence of wave interactions occurring in a region
whose size increases quadratically with •ime. Note
also that the shape of the strain histories varies
consistently with fiber modulus: the time for arrival
of the first peak, for instance, decreases monotonically
with modulus.
57
SNylon
rr
S 10
g.4
CLE
4-.
.~ 5S~~Kevlar 29 I
jo Kevlar 49
j G raph ite
(AA
0--
0 4 8 12Time After Impact, microseconds
Figure 20. Development of strain at the point of impact in thevarious fabric types after a 400 m/sec impact.
If one normalizes the magnitudes of the ordinal
values in Figure 20 by the value of strain which would
be developed in a single fiber by transverse impact at
the same projectile velocity, the strain magnitudes of
the four curves achieve comparable values. This pro-
cedure essentially compensates the curves for the
effect of the fiber modulus on the impact-induced strain.
58
The shift of the curves along the abscissa, however,
is less clear. The rate at which the strain increases
at the impact point is governed by the complex inter-
actions of waves traveling about within the constantly
expanding region of influence, and is beyond simple
visualization. On average, the time necessary for a
wave to reflect and return eventually to the impact
point should decrease inversely with the wavespeed,
i.e. inversely with the root of the modulus. However,
the size of the region in which stress waves are
traveling at any given time also depends on the wave-
speed, and one would expect that a larger region of
influence would decrease the rate at which reflected
and diverted wavelets are able to return to the impact
point.
It is found that the time after impact at which
the first peak in the strain occurs varies linearly,
with good correlation, with the fourth root of the
fiber modulus (or the square root of the wavespeed).
Using this observation, which is likely related to the
geometry of the region of influence, one can compensate
the abscissal values of Figure 20 by the factor E0 .25
The result of the ordinal and abtcissal normalization
is shown in Figure 21, where a curve valid for all four
fabrics is developed.
59
3
03
0 0 K
O00 0 00050A60
ipc -n a- Nylon
/-03- Kevlar 29
0
t b- - Graphite
00 L0 10 20 30 40 50 60
t (ILSoc) [ E (gpd)] 0 2
Figure 21. "Master" curve for impact-induced strain at the point ofimpact. Ordinal values represent strain normalized on
2'the basis of the strain which 'would be generated in asingle fiber by impact at the same velocity, while abscissalvalues are adjusted by a factor equal to the fourth root ofthe fiber modulus.
This master curve represents an improved means
of performing preliminary armor design. Since the
normalizing factors are known once the dynamic modulus
of the fiber is specified, one can generate a strain
vs. time curve from Figure 21 applicable for a particu-
lar fabric and impact velocity. The time for rupture
is the time at which the impact-induced strain exceeds
60
.pp m . . . "
the fiber's dynamic breaking strain. As in the fiber
case, we then see that ballistic resistance is a
balance between high fiber modulus leading to high
wavespeeds and lower strains, and fiber breaking strain.
This approach is approximate in severa± respects,
however, and is thus limited to preliminary design.
First, it is seen in Figure 21 that perfect correlation
among all four test fabrics is not attained, the nylon
showing a deviation at high strain. Similar deviations
in other fabrics might be observed as well. Second,
the curve of Figure 21 was generated from computer
experiments at relatively high velocity, so that pro-
jectile slowdown was not an appreciable factor. At
low impact velocities, the fabric is able to decelerate
the projectile and even bring it to rest. The effect of
projectile slowdown is to generate unloading waves in
the fabric which travel simultaneously with those
previously described. This unloading would have a
strong influence on the curves such as that in Figure
21, causing the curve to pass through a maximum and
decrease thereafter in those cases in which the fabric
is able to defeat the projectile. For these cases,
complete treatment using the numerical code would be
necessary.
V4
A~
III. EFFECT OF VISCOELASTIC MATERIALS RESPONSE
Viscoelastic Constitutive Relations
In the course of the iterative calculations
described earlier, a constitutive material law must
be evoked at each element in order to compute the
element tension from its strain (or strain history).
One would expect that a model incorporating viscoelastic
effects would be necessary for proper simulation of
polymeric materials, and in fact there is considerable
direct evidence [16] that relaxation does indeed occur
in the ballistic time frame. This is also to be
expected in light of the dynamic mechanical spectrum of
nylon, for instance, in which a beta relaxation is
observed having an apparent activation energy of
approximately 60 kJ/mole [17]; this relaxation iscalculated to occur in approximately five microseconds
at room temperature.
62
I (t)Sk, k2• kj
S..... ....IMA
t'• I
Figure 22. Wiechert spring-dashpot model for linear visco-elasticfiber response.
A general viscoelastic model well suited for
computing tensions from prescribed strains is the
Wiechert model, depicted schematically in Figure 22.
This model takes the polymer response to be analogous
to that of an array of Newtonian dashpots and Hookean I
springs. The differential tension-strain law for the
Sjith arm of the model is
k -. +- . (38)
63
where the dots indicate time differentiation, o- is the
tensile stress and e is the strain. Casting this
equation in finite difference form relative to a
discrete time increment At and solving:
3 ________(39) 1
where the superscripts t and t-1 indicate values at the
current and previous timesrespectivelypand 't =
/ is a relaxation time for the jth arm. The totalt
tension at time t is the sum of all the plus the
tension in the equilibrium spring ke
C5= (40) -
e
g+
This tension-strain calculation is performed at each
element node. In addition to storing all the kj and
It j, the computer must also store the previous strain
and tension values at each node.
The choice of the k. and 'j should be such as
to model the polymer viscoelastic response in a time
64
_scale comparable to the ballistic event, which takes
place on a microsecond time scale. It is of course
difficult to conduct such conventional tests as creep
or stress relaxation on this time scale, but guidance
as to proper model parameter selection can be obtained
from dynamic mechanicai spectra, using the activation
energies of the appropriate low-temperature relaxations
to effect a temperature-rate conversion. For nylon
fibers, for instance, one would fit the Wiechert mo&el
to the beta relaxation, ignoring the alpha and gamma
relaxations as not being appropriate to the ballistic
time scale at room temperature.
Results for Single Fibers
As a means of developing a proper context for
the study of viscoelastic response of a woven textile
panel, some results obtained in an earlier study [18]
which used a direct numerical simulation of visco-elastic relaxation in a transversely impacted single
fiber will be reviewed briefly. The numerical approach
for this study was identical to that described for
fabric structures, except that it considered a single
fiber discretized as a series-connected assemblage of
pin-jointed finite elements. As in the fabric case,
65
773=7
this treatment produced numerical values for the
position, veloci y, strain, and tension of each finite
element of fiber as a function of time after impact.
A variation of this treatment will be described in some
detail in Chapter IV of this report.
Figures 23 and 24 show the distribution of
nondimensionalized strain and tension along the fiber
at various times after impact, plotted against the
Lagrangian fiber coordinate. These distributions were
obtained from the Wiechert model using only a singles sring-dashpot arm in parallel with the equilibrium
si;h •hree-element model is commonly known as
the "standard linear solid", or the "Zener solid" The
distributions for these two figures are for a choice
of model parameters ke = 80 gm/den, k1 = 20 gm/den,,
and c1 50 /4sec. The values of the ordinates
have been normalized by the strain or tension which
the rate-independent Smith theory predicts for a linear
elastic material at the same impact velocity.
664
- -
g,ýr; Z,
1.2-
0.6 I0.2-7- 20.54-' 30.81--4 10--U.LSEC /I SEC 11 SEC 11LSE C
0.4-
0.2
Figure 23. Normalized strain plotted against Lagrangian fiber coordiniate
for vriou tims afer ipact
67
0.0
10.27 20.54---0 30.81 41.08---'
To
0.4
I 0.2-
0 2 4 6 8 10 12 14
X (cm)
Figure 24. Normalized tension distribution along fiber.
The distributions in figures 23 and 24 demonstrate
several features typical Of viscoelastic wave propaga-
tion:the magnitude of the wavefront attenuates as it
propagates along the fiber, the strain ata given
position increases with time from its original value,
and the tension decays with time. Smith [19] used the
method of characteristics to show that the wavefront
attenuation is given by:
68
A= E.~~4 e0 L (42)
where k kl/(k1 + ke) is the relative strength of
the viscoelastic relaxation. The wavefront tension
magnitude predicted by Equation 41 is shown in Figure 4,
25, which also serves to illustrate the numerical
accuracy of the direct analysis. (Here a 0.15 m
•?• •fiber was divided into 200 finite elements). Some
numerical overshoot is evident at the discontinuous
wavefront, but the distribution extrapolates to the
analytically-predicted value.
By means of Laplace transforms, Smith [201 also
obtained approximate expressions for the strain and
tension distributions in a longitudinally impacted
fiber. These expressions predict that the tension andstrain at the point of impact will approach the
limiting values
cc~e) C7.~I~ 494_c 43
69
Ii - - •
(44)
Where the numerical coefficients are for = 0.2.
At x =0, the distributions in Figures 25 and 23
approach limiting values greater than Equation 43 for
tension and greater than Equation 44 for strain. Thus
stress relaxation is slightly less and creep slightly
greater for transverse impact than for longitudinal
impact; Smith [19] reached this same conclusion in his
work on transverse impact.
1.010 00
0
0.8
0.6 Method of Characteristic Solution
T/T°
0.4 0
0.2
00 2 4 6 8 10 1 14
X (cm)
Figure 25. Numerical values for tension distribution for t - 41.08
microsec after impact.
70
Results for Woven Panels
The three-element Wiechert model (the standard
linear solid) used in the previous section was also
employed to examine the influence of viscoelasticrelaxation during ballistic impact of woven panels.
The model parameters were chosen to simulate ballistic
nylon: kI =20 gm/den, ke =80 gm/den, It 5 Asec.
Results have been obtained for a simulated 0.2 m x 0.2--m
panel weighing 19.5 gm, impacted with a 0.22-caliber
fragment simulating projectile weighing 1.10 gms at
various impact velocities.
14 -1.4
12 -. 2
10 -1.0
i'0
rLL0 I 4
lii pc o4 o iea lsi n icelsi arc t•
IAJ
6- -0.6
4- - 0.4-
2 0.2
DISTANCE FROM IMPACT POINT, cm
Figure 26. Stress distributions along orthogonal fibers running throughimpact point for linear elastic and viscoelastic fabriozs (t
30.4 microsec).71
Figure 26 shows the distribution of stress along
the orthogonal fibers running through the point of
impact at t = 30.4 a-sec after an impact at Vp = 300
m/sec. The results for the viscoelastic fabzic are
compared with those of an ideally elastic fabric
having a stiffness equal to that of the unrelaxed
viscoelastic material (100 gm/den). The nonuniform
distributions along the fiber are due as described
earlier to the continual reflection of wave components
from fiber crossovers, resulting in a maximum in stress
at the impact point. An appreciable difference in
stress levels between the linear elastic and Visco-
elastic cases is observed, especially near the wave-
front. Relaxation of the stress due to the rate-
dependent material behavior may be expressed by the
ratio of the viscoelastic and the elastic and the
elastic stress, T/TO, as shown in the figure. It is0
found that a large amount of relaxation occurs near
the wave front while an equlibrium state of relaxation
is reached in the region away from the disturbance front.
72
",'•
1.2-
U'_W 0.6-
04 t:10.36 psec t=-20.04 psec t:=30.41/ IL
0.0-
S0 1
0i' 2 3 4 5 6 7
DISTANCE FROM IMPACT POINT, cm
Figure 27. Distribution of strain along orthogonal fibers runningthrough impact point.
The wave attenuation during this 300 m/sec impact
is also demonstrated in Figure 27, where relative strain
Sdistributions F- /&t are given for various times
-4 after impact. As illustrated in this figure, the
magnitude of the Wavefront attenuates significantly as
it propagates along the orthogonal fibers, and the
strain at a given position increases with time after
impact from its original values.
73S1& -w' . .. ..., ., ,1
1I.0 1
0.8.o -12 30
- 0.6 - 8 •~08Al
P=9IxI i
(00 5 10 15 20 25 30
TIME AFTER IMPACT, p6sec
Figure 28. Stress histories at impact point for linear elastic andviscoelastic materials.
The stress and strain histories at the point of
impact shown in Figure 28 give another indication of
viscoelastic dissipation in the response of the panel.They increase continuously with time due to the re-
flection of wavelets from crossovers; however, stress
relaxation and strain creep of the viscoelastic material
occur simultaneously with this general increase. The
viscoelastic stress at a given time is smaller than the
elastic case as shown in the figure. The relative
relaxation at the point of impact, denoted by the ratio
of the viscoelastic and elastic case, develops gradually
and reaches a steady state at long times. The relaxa-
tion histories for different missile striking velocities
74
are given in Figure 29. The similarity of their magni-
tudes is a manifestation of the linear material res-
ponse. Again, these stresses approach an asymptotic
equilibrium state at times longer than the clh.aracteris-
tic material relaxation time.
(L . 1.0
V•= 400 m/sec V Vp .00 -n/mc 01 - 250m/secP o
-J8
S< 0.6-'• .j
w
0 5 10 15 20 25 30TIME AFTER IMPACT, j.sec
Figure 29. Stress relaxation at impact point for various impactvelocities.
Nonlinear Viscoelastic ResponseF •The use of the Wiechert model as described in
the previous sections is sufficient to illustrate the
most important features of rate-dependent ballistic
materials. Howevermost materials do not meet the
75
rigid requirements of linearity necessary for a truly
rigorous application of concepts of linear viscoelasti-
city. For more detailed simulations of fabric and fiber
response, one must turn to general constitutive rela-
tionships which more accurately model the behavior of
these materials. Unfortunately, there does not exist a
general concensus of the most realistic means of
achieving this goal. At present, the subject of non-
linear viscoelasticity is being pursued actively by
several groups, several of which are employing highly
divergent approaches to this problem. It is not
possible here to select any single approach as having
significantly greater merit than certain others.
However, the ease with which various constitutive
laws may be incorporated into the direct analysis scheme
makes it possible to assess relatively easily the in-
fluence of various assumed material models. As an
illustration of this capability, some results using
Eyring's model of thermally-activated nonlinear
viscoelasticity will be presented.
The computationally-convenient Wiechert model
can be extended to include the effect of material non-
linearity by rendering the springs and/or the dashpots
nonlinear. If,for instance ,one uses a power-law spring
and a nonlinear Eyring dashpot [21], defined as
76
0- -OLCy- (45)
then the finite-difference equation relating tensions
and strains in the jth arm of the model is:
)6)65 V-4(46)
4- A. )
A relation such as this requires an iterative numerical
solution for at each element and at each time3
step; the computer effort is increased but the princi-
ples of the impact algorithm are straightforward. The
principal obstacle to the use of nonlinear models in
the direct analysis is not the incorporation of the
models into the computational scheme, but rather the
determination of the material parameters (the b's,
k's,A's,and &.'s in Equation 46) applicable to the
microsecond time scale of polymer relaxations.
To illustrate the effect of nonlinear constitutive
models on panel ballistic response, a series of computer
experiments was performed on three different simulations
of nylon fabric: one using only linear elastic response
77
(only the equlibrium spring in the Wiechert model), one
using the standard linear solid model for linear visco-
elastic response (the equilibrium spring plus one spring-
dashpot arm), and the last being a standard linear
solid but with the dashpot made a nonlinear Eyring
element. The intial modulus was taken as 100 gm/den,
the relaxed modulus as 80 gm/den, and the relaxation
time for the standard linear solid as five Xsec (the
same as in the previous section). The concept of rela-
xation time (the time to complete 63.2% of the total
relaxation) has no meaning for the nonlinear element,
since the rate of relaxation changes nonlinearly with
the stress. Lacking any experimental data in this time
scale, the A and 04 were arbitrarily chosen so as to
cause relaxation in approximately the same time scale
as the standard linear solid. A and Ot were set at
e t3 -wsec c and 0.7 den/gm,respectively. The nonlinear
constitutive equation was solved at each element using
Muller'smethod [22t, which increased the computationtime relative to that of the standard linear solid by
roughly one-third.
78
10/ .1.0
8 -0.8. near Elastic Model.To \ T To
E 6\ -- 0.64- .. Ta
Tow 4- -0.4
0)
2 Nonlinear Model, Ta'• \ 0.2
0 I I t-.- .-- 0• _0 1.0 2.0 3.0 4.0 5.0
DISTANCE FROM IMPACT POINT, cm
Figure 30. Stress distributions along orthogonal fibers running
through i ,topact point for linear elastic and noninuearS~viscoelastic fabrics.
," One means of comparing the various constitutive
models is in terms of the distributions of strain along
P• the orthogonal fibers running through the impact point,
÷:• at various times after impact. Figure 30 shows the
Sdistributions for the elastic and nonlinear viscoelasticK
materials 20 ,Msec after a 300-rn/sec impact. Also
shown is the ratio between the nonlinear and the
', elastic cases. This ratio is a measure of the stress
i relaxation in the fabric; it is greates: at the wave-
S~front, as the large gradient of strain there produces a
S~similarly large rate of relaxation.
55
IN
0. A79
1.0
- Standard Linear Solid
0.
U w Nonlinear ModeiW
>_0.4-
, 0.2 --
010 1.0 2.0 3.0 4.0 5.0
DISTANCE FROM IMPACT POINT, cm
Figure 31. Comparisor. if stress relaxation in linear and nonlinearviscoelastic fabrics.
The nonlinear stress relaxation is compared in
Figures 31 to that produced by the standard linear
solid. Although the equilibrium value far from the
wavefront is approximately the samae in both cases,
strong differences are evident near the wavefront. These
are due to the relatively more rapid response to higher
strain gradients in the nonl'n-ar material. Another
indicator of viscoelastic fabric rcE:ponse is the stress
at the point of impact. The stress and strain at the
impact point increase with time due to the continual
arrival theze of wavelets reflected from fiber cross-
overs, but in viscoeLastic materials both stress relax-
80
ation and creep strain are superimposed on this over-
all increase. In Figure 32 the point-of-impact stress
histories for the three materials are plotted, as well
as the stress relaxation ratio defined as before as
the ratio between the viscoelastic and elastic stress.
As in the earlier two figures, the linear and non-
linear viscoelastic models approach essentially identical
equlibirum values at long times, but are markedly
different near the wavefront due to the more rapid
response of the nonlinear material to large gradients.
l.0
16 -- 0.9
-0.8E Linear Elastic Model I0
I, 12 Standard Linear Solid•I- 12----Nonlinear Mndel c" "z
Q -
rj
0 5 1o 15 20 25 30TIME AFTER IMPACT, u.Lsec
Figure 32. Stres, hiistories at impact point for linear elastic,
-inear viscoelastic, and nonlinear viscoelastic fabrics.
IJ
IV. NUMERICAL ANALYSIS OF WAVE PROPAGATION
IN TWO CROSSED FIBERS
Introduction
This chapter describes study of the dynamics of a
special but highly important physical system: that of
two fibers, one having been transversely impacted at
zero obliquity by a high-speed projectile, and the
other crossing the first perpendicularly at some
distance from the impact point. This system is germane
to the understanding of impact and wave propagation
phenomena in woven textile panels used for ballistic
protection. The wave propagation phenomena occuiring at
the fiber crossover have a strong influence on the
response of a woven panel to impact, since these panels
typically have on the order of forty crossovers per
inch. The nature of these crossovei interactions may
be one of the factors causing what appears to be an
excellent fiber in single-fiber ballistic tests to
exhibit less ballistic protection when woven into a
textile panel structure than a nominally inferior fiber.
As mentioned earlier, this situation obtains in the
case of the Kevlar ballistic protectio:. vests now
being used by military and police personnel: the Kevlar
vest outperforms the older nylon vest, in spite of
82 K
nylon's having a higher transverse critical velocity.
The inability to predict vest performance from single-
fiber test data is a matter of considerable concern tothe armordesign community, and this study of fiber
crossover dynamics was begun to clarify this situation.
Method of SolutionA
II VP
V WON
x
Figure 33. Schematic of model for numerical analysis of two crossed
83
S / [
System Idealization. The system of two crossed
fibers is modeled as in Figure 33, where the origin of
coordinates is placed at the midpoint of the clamped
primary fiber, which extends along the x-axis. The
projectile moves along the y-axis only, and impacts the
primary fiber at the origin. From symmetry, only half
the primary fiber need be considered. The secondaryprimary fiberatsm a - ryditnefoth
fiber extends along the z-axis and intersects theprimary fiber at some a'-_-zrary distance from the
origin. At the crossover point, the secondary fiber is
assumed to follow the motion of the primary fiber in
the direction perpendicular to the primary fiber (in
the x-y plane), but is allowed some measure of slip in
the direction parallel to the primary fiber. Motion of
the primary fiber is assumed to occur in the x-y plane
only, while the secondary fiber may move in all three
directions.
i+1
•X
Figure 34. Discrete elemnt of fiber.
4 84
To proceed, the fibers are discretized as a series
of n pin-jointed finite elements of equal length as
shown in Figure 34. The masses of the elements are
taken to be lumped at the nodal end points of these
elements, and at these nodes are defined vector co-
ordinates ,i' velocity Zi' and tension T.. The
scalar strain 6: at each element will be computed
from the coordinates of the nodes at either end of
the element. The tension T. has the same direction
as the element itself (approximating the element's
assumed inability to support a bending moment), whileA
Vi is not constrained in direction. These elements are
now described as in the fabric analysis by simple
governing equations: impulse-momentum balance, strain-
displacement relation, constitutive relation, etc.
These relations are cast as a recursive algorithm for
proceeding from one element to the next along the
fiber length, and then repeating the process at a new
increment of time. The computer solution thus is re-
ferenced to a Lagrangian frame of reference attached
to and extending with the fiber, which effectively
reduces the problem to one dimension.
Momentum Balance. A consideration of impulse-
momentum balance at the i + 1st node provides a means
of computing the current velocity at that node in terms
85
J• m i" .• • - " -
of its velocity in the previous time increment and thetensions acting on it during that time increment. (Inthe follce ing, subscripts on a variable refer to thenode at which it is defined ,while superscripts t and
t-l refer to values at the current and previous timesrespectively). The impulse-momentum balance can bewritten in finite difference form as
S/( 4 7 )
A7- - krk (48)
Letting A = &m/ 2t, a fixed parameter, equation 48
may be solved for vti+l :
This vector expression can be written in scalar formby reference to the inclination angles 1 and &i i+l
of the T. and T. vectors respectively:
44.86
1.4Z. 1+i
Then the x and y components of velocity are:
- . +A (Tk4' .*o. (52)
A
-T.
+A( T (53)
where =T 1T is the tension magnitude. The boundarySconditions are easily incorporated into- the impulse-
momentum balance: at the first mode, the velocity isSt = t
set equal to the current projectile velocity (v )
tand at the clamp the velocity is set to zero (vn 0).
Strain-displacement Relation. Having computed
the velocities at the ith and i+lst nodes, the strain in
the element between these nodes is computed-as
87
L3
(54)
I
where L. is the element length. Continuing:1
LIi}• - • '•. - I(55)
where
(56)
and
< t
SConstitutive Relation. Knowing the strain E t),
the tension magnitude Tt is computed from the material'sd
dynamic stress-strain law. These relations are as
described earlier, and currently available constitutive
models include linear elastic, nonlinear elastic (cubic
88
polynomial and exponential strain hardening), linear
viscoelasticity (Wiechezc model), and nonlinear visco-
elasticity (Eyring model).
Computation of New Projectile Velocity. The
algorithm described above proceeds from one element to
the next along the length of fiber, and i-" started by
imposing the initial projectile velocity on the first
node. At the end of the first time increment, a strain
will have developed in the first element due to the
velocity difference between the first and second nodes.
(Initially, all velocities, tensions,and strains are
set to zero.) This strain produces a tension as cal-
culated from the constitutive relation, and this tension
produces a velocity in the second node beginning at
the next time increment.
After each time increment, at the completion of
the lengthwise recursive calculations, a new projectile
tvelocity v can be computed by means of a momentum
balance using the tension at the first node:
89
S • •-• •-----, .-- "-Mg! - -i
where M is the projectile mass, T is the component ofp y
fiber tension in the projectile travel direction, and
the factor 2 accounts for the other half of the primary
fiber extending in the -x direction. T is:-y
TF.,T cO49s~ cco'T.s
Crossover Fiber Calculations. Computation of
field variables along the secondary fiber proceeds in a
lengthvise manner exactly as described above, although
the vector resolutions become slightly more complicated
due to the motion in three rather than two dimensions.
The secondary algorithm is started by imposing on the
first node of the crossover fiber the velocity imparted
to it by the primary fiber. As stated earlier, the
secondary fiber is allowed a measure of slip along the
primary fiber but is constrained to follow it in the
direction normal to the primary fiber. Denoting the
node on the primary fiber nearest the crossover point as
ix, the velocit" of this node resolved in directions
parallel and perpendicular to the primary fiber there
I0 90
are:
A&T - A - (60)
krs. +AY. F>.SO&E 61
47 ~ 3~ os,& -~~t. ~J~.e.(61)
In equation 62, the notation of the form yl or y2
indicates field variables for the primary and secondary
fibersrespectively. The velocity imposed on the first
(crossover) node of the secondary fiber is:
AS 4C A3-1. 5~1.A 15 ,T E)"- .IM., ( 63 )
AID +At (64ti.• ---(,.,I,• ".LA+ •9 (64)
ý- •(65)
91.
__-J-------
where •s is a slide factor which permits no slidingS
when set to 1 and unrestrained sliding when set to
zero.
The crossover node ix will change with time if the
tangential slip along the primary fiber is sufficient.
After each time increp..nt, a new position of the first
node on the secondary fiber is computed, and ix is
assigned to the nearest node on the primary fiber.
tThe momentum-balance calculation of in the
primary fiber must be modified when the i+1 node is
also the crossover node ix, since the secondary fiber
applies its own tension to that node. Denote the direc-
tion angles of the primary and secondary fibers at that
node as 41 i tI4' and 0Z W 4>Z• ,respect-
ively,( a = 0). Then the components of tension
applied by the secondary fiber, resolved along directions
parallel and perpendic'ilar to the primary fiber, are:
+ (66)TZ Cs
> v It
TZJ W7.T Cos 4>Z Cos C.os Zco..c~ilj (67)
92
where the direction cosines are computed from the
current nodal coordinates. The primary fiber is allowed
to feel the impulse of the [,.rpendicular component
fully, but the parallel component is reduced by the
slide variable a The usual computation of vts "i+l
is the adjusted as
4:4where (68)
A*1 A. +z A [TZý_ (69)
+ X'sA CS 04A4where the - symbol indicates a computer replacement
operation; i.e. the additional impulse from the
secondary fiber is added to that already computed from
equations 63 and 64.
Stability, Accuracy,and Efficiency. Criteria fcr
stability and accuracy of the above method are related
as in the fabric case to the theory of charactristics A
for hyperb(.lic systems of partial differential equationi A_
and are similar to those for finite-difference solutions
of wave propagation problems [14]. Given a wave
equation of the form
93
which is to be solved by approximating ýt and Zx
by finite differences At and &x, a stability
ratio d can be defined as
(71)
The finite difference scheme is stable and accurate for
1= , stable but increasingly inaccurate for dc< 1,
and unstable for X(> 1. The choices for Ax and
St are thus not independent, but are related by the
wavespeed for the choice of OL= 1.
In the direct analysis of the fibers described
above, this stability criterion is equivalent to adjust-
ing the rate of march of the computer solution along
the fiber to match the rate of propagation of the
strain wave. Conceptually, this requirement is related
to the necessity of programming the finite governing
equations so as to model the actual continuous dynamic
process as accurately as possible. If a major disturb-
94
ance - such as the passage of a strain wave with its
accompanying energy input - takes place in a finite
element which is not considered explicitly -in the
computational scheme, divergent numerical results are
very likely.
Once a stable computational scheme has been a
developed one usually attempts to increase its accuracy
to whatever limit is desired by decreasing the size of
the elements; i.e. Dy increasing the number of nodes. JSince for 6C = 1 a decrease in &x requires a corres-
ponding decrease in &t, the computation time - and
therefore the expense - required for analysis of a
given impact event increases as the square of the
number of nodes. The element size is therefore chosen
so as to balance the conflicting requirements of
economy and accuracy. As an example of computation
time, the CPU requirement for the IBM 370/168 system
was 0.168 minutes for a problem in which the strain wave
propagated 0.2 m along the primary fiber and 0.1 along
the secondary fiber, with a length increment of 2.0 m m.
As a means of improving code efficiency, the program
employs logical flags which terminate the length loop
computation when the computer passes the point along the
fiber length corresponding to the wavefront.
95
Accuracy assessment for the case of two crossed
SI fibers is difficult, since no experimental or closed-
form mathematical analysis of this problem is available,
but some assurance of accuracy is derived from computer
runs in which the secondary fiber is placed at the
origin (the impact point). In this case, response of
both the primary and secondary fibers is found to be
that predicted by independent analyses. Data such as
that previously presented in Figure 23 is obtained along
the primary and secondary fibers. In certain cases to
be discussed below, the numerical overshoot and
oscillation observed near wavefrcnts cause problems in
interpretation of results. Thessoscillations are a
result of the inability of the discrete difference
equations to model discontinuities accurately. Although
the method is conditionally stable and the oscillations
are damped out away from the discontinuity, problems
of interpretation remain near the discontinuity. The
oscillation at the wavefront is diminished by the
material viscosity, and in some cases an "artificial"
viscosity may be included solely for the purpose of
smoothing the numerical results.
96
Results and Discussion
Primary Fiber i1.6
41.45 %
. Crossover PositionQ 4
~~0.2•-
0 j Secondary FiberS0 ,
0 5 10 15 20
X, cm
Figure 35. Strain distributions in two crossed fibers of Kevlar 29,28.7 microsec after impact at 400 m/sec.
Figure 35 shows typical results obtained from the
above described computer treatment, in this case for
two crossed fibers of Kevlar 29, the crossover point "l
being 10 cm along the primary fiber from the impact
point. The fibers were assumed to respond elastically,
and no sliding was permitted at the crossover ( sl
97
The figure shows the distribution of strain in each
fiber 28.7 /?"Sec after impact at 400 m/sec, where the
abscissa measures the distance along the secondary
fiber from the crossover point. The dotted line at
strain 1.45% depicts the level of strain which would
be generated in a single fiber atthis impact velocity.
In this example no viscosity has been included, and the
large overshoot at the wavefront causes problems in
interpretation of results. In spite of this oscillatory
behavior, however, an increase in strain in the primary
fiber behind the crossover due to the wavelet reflected
from the crossover is evident, as is a reduction in the A
strain intensity in the region of the primary fiber
beyond the crossover. More easily measured is the level
of strain intensity propagated along the secondary fiber.
SComputer experiments were conducted on thecrossover system for a range of fiber moduli and slide
factors, and graphical output similar to Figure 35 used
to determine coefficients of wave reflection, trans-
mittance, and diversion. These coefficients are defined
as that fraction of the outward-propagating strain wave
which is reflected backwards by the crossover, thefraction which passes through the crossover and continues
its outward propagation, and the fraction which is
diverted and begins propagating along the fiber passing
98
-a~
trasve~el thouh the crossover. As a means of
~btainiflg these coefficients in spite of the uncertainl
ties caused by the numerical fluctuations near the
wavefronts, the computer was se odtrfil h
average strain level over apotooftef.erenh
away from the oscIllation region. In order to guaranltee
conservation of energy, the sum of the squares
of tche
aboe tre coffcients should equal unity; this was
in fact obtained and of fers some assurance
as to the
4accuracy of the numerical values.
OL 3a 99
0
0.-
40-
0-0
00
a 971
0 1000 2000 3000
modJulus, gpd
Figure 36. Influence of the fiber modulus on the fraction of stress
wave Intensity wbich is transmitted through a fiber
crossover, in the absence of fiber-fiber sliding.
99 -..
I A......
___ _ ---t;--. .,-
The variation in the transmission and diversion
coefficients with fiber modulus is shown in Figure 36.
The coefficient of reflection was near 1% over this
range of moduli, but showed considerable scatter, It is-
seen that the diversion coefficient is of a much larger
magznitude than the reflection coefficient, and that it varies
more strongly with the fiber modulus. The major
portion of the crossover influence on wave propagation
is thus ascribed to diversion rather than reflection.
This observation is of significance, since an
approximate treatment of fabric impact by Freeston and
Claus [231 sought to predict the increase of strain at
the impact point by considering wave propagation along
a single fiber which reflects a certain portion of the
outward-propagating wave at a series of discrete points
along its length. The analysis is then reduced to a
bookkeeping procedure in which one keeps track of
inward and outward-propagating waves in each of the
elements between these reflection points. This scheme
leads to a very simple computer code and one would hope
it could provide at least approximate guidance indesign.
100
•g.++•;~~ý Y .. ,. •
4
3
I-
0 0 2TIME, MICROSECONDS
Figure 37. A comparison of the reflection-only bounce model for wavepropagation in an impacted fabric, in comparison with thefabric model of this report.
Unfortunately, it appears that the reflection-only
model predicts much too high a strain level, except at
times very early in the ballistic event. Figure 37
shows the impact-point strain history as predicted by
the reflection-only model for a 400 m/sec impact on a
Kevlar 29 panel with 1575 yarns/m (the curve developed
by the "BOUNCE" code). This prediction is compared
101
with that of the more general code described earlier
(the curve labeled "FABRIC"). At very short times, the
BOUNCE code results are likely superior, as they give
strains equal to that developed in a single transversely-
impacted fiber; the Fabric code shows a numerical lag
in the development of strain. The two codes achieve
similar values at near 1.5 microseconds, but after this
the bounce code increases rapidly to unreasonably high
values of strain and thus predicts penetration too
early. It is interesting, however, that the value of
the reflection coefficient chosen by Freeston and Claus
in order to bring their model into line with experiment
was very nearly that found explicitly in the crossover
study (0.01).
As the slide factor - decreases from-wniity toward
zero, representing less fiber-fiber friction at cross-
over points, one would expect that the reflection and
diversion coefficients would approach zero and that
the transmission coefficient would approach unity. At
&L.S = 0, there is no coupling between the two fibers
(until the arrival of the transverse kink wave, which
generally occurs later than the arrival of the longi-
tudinal wave). As seen in Figures 38, 39 and 40,
respectivelyfor Kevlar 29 fibers, this trend is
quantified by the results of the crossover computations.
102
0.010
SModulus a 5 5 0 gpd /
50.005-
z2
S~0W.-
C)
LLJ
0 , -_ I a
0 0.2 0.4 0.6 0.8 1.0
SLIDE FACTOR
Figure 38. The influence of fiber-fiber sliding on the fractionof stress wave intensity which is reflected at fibercrossovers, as indicated by computer experiments onKevlar 29 fibers.
K
103
ON.
0.ID r
Modulus • 550 gpd
I2 0.10-IL
i oi
/0/ 00
z I0 0.050
00
0 0.2 0.4 0.6 0.8 1.0SLIDE FACTOR
Figure 39. The influence of fiber-fiber sliding on the extentto which a portion of the propagating stress waveis diverted from the primary fiber to begin pro-pagating along the transverse secondary fiber.
104
1.00
-44
X I odulusu55O,0 I
•4%,L%
0 4 0'
-:- Z 0.990
o 00
z
SI i p
0.980 0.2 0.4 0.6 OB 1.0
SLIDE FACTOR
Figure 40. The influence of sliding on the extent of stress waveintensity propagated beyond fiber crossovers.
In principle, it would be possible to include the
effects of fiber-fiber slippage in the two-dimensionalfabric code by incorporating the formulae of this
present chapter into the general code. Such an in-
corporation, however, would likely render the fabric
L code so much slower as to be uneconomic. In addition,
one has at present no real means of assessing the slide
parameter prneeded for the computation. It might be
possible, h.iwever, to adjust the viscosity of the
105
T2material model artificially in order to produce results
similar to those caused by fiber slippage. This method
would be similar to that employed during wave propaga- Ition calculations in geometrically dispersive composites
[24). If this approach is pursued in the future, the
data 6f Figures 38-40 will be useful in providingguidance as to the desired effect. As a final comment
on this work, it may be stated that the FABRIC impact
code provides a simulation of impact on textile fabrics
which is already of sufficient accuracy that inclusion
of the fiber crossover effects would not be considered
necessary, at least in the case of tightly woven panels
which do not exhibit extensive fiber slippage during
impact.
CONCLUSIONS
The numerical analyses described in this report
offer a means whereby the designer of personnel armor
may perform computer-aided design and analysis of what
up until now has been an impossibly intractable
problem. Perhaps as useful as the ability to perform
such analyses, however, is the extent to which the
armor designer's intuition of the mechanics of penetra-
tion is enhanced by this tool. These numerical codes
106
are easily implemented on any modern computer system,
and run very economically. They are also extremely
amenable to user modification in order to permit easy
implementation of various constitutive laws, fracture
models, etc.
Certain areas still exist, however, for significant
improvement in this treatment. First, the codes are
presently limited to zero-obliquity impact by a
projectile whose lateral dimensions are small compared
with the region of influence during impact. Relativelyminor code modifications would be necessary to include
oblique impacts by large and arbitrarily shaped
projectiles. In this manner the influence of projectile
geometry could be modeled. Second, and more important,
is the necessity to incorporate more acccurate models
of material response. As was demonstrated within this
report, rather sophisticated constitutive and
fracture algorithms can be implemented within the
codes with no serious difficulties. More work is
needed, however, to determine the extent to which these
or other models are applicable to fabric response in
the ballistic time frame, and to determine the numerical
parameters to be used in the models.
Two examples of this latter problem may be repeated
here. First, recall that the treatment of nonlinear
107
viscoelastic effects was limited not by the code
i capability, but by the present lack of understanding
as to which of several possible nonlinear constitutive
laws would most accurately model ballistic response.
Regardless of the model selected, work is required to
obtain experimentally the numerical data required as
input parameters.
Second, the otherwise very successful materiais
parametric study described in the report is deficient
in that it predicts that Kevlar 29 should far outperform
Si Kevlar 49. In fact, the two aramid fabrics perform
almost equally well, and some evidence suggests that
Kevlar 49 is actually superior. The authors have no
doubt that this discrepancy lies not in the wave-
propagation aspects of the code predictions, but in the
dynamic failure criterion used. Micrographs of
ballistically-fractured fibers show extensive
fibrillation, and evidence exists which suggest that
Kevlar 29 and 49 differ primarily in their extent of
fibrillation during fracture. Experimental work aimed•i at elucidating the nature of the fracture mechanism
is needed; incorporation of the resulting information
into the penetration codes should follow without
difficulty.
This document reports research undertaken incooperation with the US Army Natick Re-search and Development Command under 18Contract No. DAAG 17-76-._001 3 and hasbeen assigned No. NATICK/TR /0921in the series of reports approved for publica-tion.
REFERENCES
1. D.K. Roylance, Textile Res. J., 47 (1977) 679.
2. R.J. Coskren, N.J. Abbott, and J.H. Ross, AIAAPaper No. 75-1360, Nov. 1975.
3. H. Kolsky, Stress Waves in Solids (Dover Publica-tions, New York, -- 63.
4. D.K. Roylance, MIT Tech. Rept. R77-1, Dept. ofMaterials Sci. and Eng., Jan. 1977.
5. A.F. Wilde, J.J. Ricca, D.K. Roylance and G.C.Tocci, Tech. Rept. AMMRC TR72-12, Army Materialsand Mechanics Research Center, Watertown, Mass.(1972).
6. G.I. Taylor, Scientific Papers of G.I. Taylor,(Cambridge University Press,1958), paper no. 32.
7. T. von Karman and P. Duwez, J. Appl. Phys., 21(1950) 987.
8. D.R. Petterson, G.M. Steward, F.A. Odell, and R.C.Maheux, Textile Res. J., 30 (1960) 411.
9. A.B. Schultz, P.A. Tuschak and A.A. Vicario, J.Appl. Mech. (1967) 392. N70. C.A. Fenstermaker and J.C. Smith, Appl. Polymer•i• ! Syrup., 1 (1965) 125.
11. S.N. Zhurkov, Intl. J. Fracture Mech., 1 (1965) 311. :Z
A12. K.L. DeVries and D.K. Roylance, Progress in Solidi ~State Chem% 8 1.973)'283.
13. A.F. Wilde, Textile Res. J., 44 (1974) 772.
14-. S.J. Crandall, Engineering Analysis, (McGraw-Hill,SNew York, 1956) 396.
15 D.K Roylance, A.F. Wilde and G.C. Tocci, Textile71Res. J., 43 (1973) 34.
1Q9i 9
16. J.c. Smith, C.A. Fenstermaker, and P.J. Shouse,Textile Res. J., 35 (1965) 743.
17. N.G. McCruin, L.E. Read and G. Williams, Anelasticand Dielectric Effects in Polymeric Solids(Wiley, London, 1967).
18. D.K. Roylance, J. Appi. Mech., (1973) 143.
19. J.C. Smith and J.T. Fong, J. -Res. Natl. Bur. Stan., ~
20. J.C. Smith, J. Appl. Phys.- 37 (1966) 1697.
21. H. Eyring and G. Halsey, Textile Res. J., 15 (1945)295.
22. B. Carnahan, Applied Numerical Methods, (Wiley,New Yr, 1969) 201
23. W.D. Freeston and W.D. Claus, Textile Res. J., 43(1973) 348.
24. W.L. Bade, Tech. Dept. AFWL-TR-72-8, Air ForceIWeapons Laboratory, Albuquerque (1972).
-10
APPENDIX A -The FABRIC Code
II General. FABRIC is a FORTRAN coding of the
numerical analysis of fabric impact which was
described in Chapter II. In its present form, the
code is restricted to zero-obliquity impact on an
orthogonal fabric mesh consisting of only one fiber
type. These constraints could be relaxed through
suitable code modifications. The code was developed
and implemented on MIT's IBM 370/168 computer system,
but was later implemented without difficulty on the
NARADCOM computer. The code was run at MIT in a batch
mode, but could easily be modified for interactivei• ~ terminal operation: this would likely consist primarily !
of adding terminal queuing for data input and graphical
display for output data.
Code input and output. The input data needed by
FABRIC is detailed in a series of comment lines at the
beginning of the code listing. Briefly, these include
specification of the impact velocity and projectile
mass, the fabric idealization (principally the number
of fibers per unit length), the constitutive and frac-
ture properties of the fibers, and such run paramatersas maximum alloted time and printing increment.
A typical data input set, for a 300-m/sec impact
on a nylon panel is given below:
LINEAR VISCOELASTIC FABRIC (BALLISTIC NYLON)
1.1 300. 19.533 10.16 5280
0.14 0.0 2. 1.4142 1.
3
100. 0.2 5.
36 1 1
Code output consists first of a series of values
relative to the initial conditions which were read in
and which the computer requires in order to begin the
recursive calculations.
After each time increment (or less often, depending
on the value used for the print skip increment INC), the
code prints-the current values of the field variables
at each node in the fabric octant. Currently, these
are simply dumped in order of the calculation scheme as
defined by Figure 10. This presentation of data has
been sufficient for the research studies discussed in
this report, but for production design work, graphical
or some other high-order output would likely be prefer-
112
able. The output for the first time increment of the
above 300 i/sec nylon impact is given below for
illustration. TI0 and T01 are the tensions in the two
orthogonal fibers passing through a node as shown in
Figure 9, EPS10 and EPS01 are the corresponding strains,
VZ is the transverse velocity and XCD and ZCD are the
x- and z-coordinates of the i,j node. This print also
presents the current time after impact, the current
projectile velocity, the energy lost by the projectile
and the partition of impact energy into a strain and
kinetic components within the fabric. Clearly, a great
deal of data is made available by the code, and the user
should modify the output format so as to provide the
most convenient display of results for his needs.
NAN
113
11 -- __
EXECUTItON BEGINS***ILINEAR VISCOELASTIC FABRIC (BALLISTIC NYLON)
INPUT PARAMETERS:oYARN DENIERP DENYRiN (DEN)= O.528OOE+O4INITIAL PROJECTILE VEL.OCITYYVPROJ (M/SEC)= O.30000E+03PROJECTILE MASSY PMASS (CM)::- 0.11000E+O1FABRIC PANEL LENGTH XL (CM)= 0+i0i6OE--iO2ELEMENT LENGTH DXLY (CM)= 0o29029E+OOMAXIMUM IMPACT DURATION TMAX (MICROSEC)= O,20000E+OI.STABILITY COEFFICIENT, CDTM= Oo14142E+O0.INCREMENTAL. TIME ['TM (MICROSEC)= O.69116E+O()NUMBERS OF LAYERSY CLYR= O.10000E+O1
STRAIN WAVE VELOCITY CWAVE(M/SEC)= 0+29698E+04INITIAL MODULUS OF YARN, EYRN(GR/DEN)= 0,10000E+03BACK UP ELASTIC SPRING CONSTANT XK (OR/CM/CM)= 0+0
NUMBER OF NODES AL-ONG MODEL PANELY jTr= 36NUMBER OF TIME INCREMENT STEPS, NTINC= 2PRINTING FREQUENCYYINC= I
ACTUAL FABRIC MASSYFMASSA (GM)= Oo19533E+02MODEL FABRIC MASSY FMASSM (OjM)= O.16689E 02CRIMF'=FMASSA/FMASSM= 0. 11704E"f-01UNIT ELEMENT MASS *9,E05 (LJNITM)= 0.35877E+04HALF UNIT ELEMENT MASS *9.E 05 (HUNITM):= 0+1L7938E+04
INITIAL PROJECTILE KINETIC ENERGYP XKE (JOULE/GM)= O.25342E+O1
MATERIALS PROPERTIES#OPTION--MATERIAL MODELY IPT= 3INITIAL MODULUSY EYRN(GR/DEN):= O.1OOOOE+03
VICELATCMODl-- 1TNADLNA OI PARAMETER
MODL GASS MDULS (F*)::; 010 0,SOOOE+0
MODEL VSCOUS RACTIO= 140000E+OO
114
r,,. Ný,,l I,•z.x IW W
1 1) a. w0000
00JC W6 N J NJW0000
• Z0000 0 " 000 0000
0 0 000 14,400- 0000 0000
7,z 0000
0 wwww
0000 N vN000 0000
. .. . .4.4 .4 ..00000 0000 0000 0000
0000 0000 00I0lW 0000 0000
In 0 0 0 0 0
0000
.0 N m 0. W lJw*
z• •0000 000 0000zo
SL•J•" O •..4 .4.4 .
0000
,'• ,?.,1 a) +,-'4-+
Wi. . 0000 0000 .... 0000 0000
0c .'I~ . .**...* w*ý 0000 0000IT0 ,.~0W 0000 0000
'-44 Itr10 01
0000 14'CD 0 0
CL U0000 0000 0000
C-- 4+4J 4- ILUla
U 0000
-C4 00000 00000 L 00000 00000
. 000000000 00000
Q0000 VI 000 0 0000
00 0 CDinDo oo 0 00 00000
S" 0 0
D.l0 00004 0O
4 0 00000 00000 0T0 00000 00000
W ")zJ %0c
'Ol C It 0 'CJ00P-%0cc0 00W '-0?0 -000....................0 0 0 0 0 0
0 -r 0 0000 0000 0000 0C, 0 0.4 i40.
I- 0000
X't 0000 X 0O00,.•*0 1- tf) Lf M 0 00000 00t 000
SvO0000,,O00000•. 0o0 000vo. v• o.,0 0 0 0 0 0
-1 15
U. w
Program Requirements. In its present form, the
FABRIC code is self-contained, needing no external
software support. All its subroutines, for instance,
are contained within the listing given below. One
qualification to this statement however, is the inclu-
sion of a call to Subroutine UERTST within the nonlinear
equation-solving subroutine ZNOLNR. UERTST is an error-
handling routine available through the proprietary
"International Mathematics and Statistics Library"
(IMSL). If IMSL is not available at the user's location,
the call to UERTST could be removed with little risk:
no error-handling capability was needed during any of
the computer experiments described in the body of this
report.
FABRIC needs no tape or disk support; all compu-
tations are performed directly within core. Core
requirements and run times are dependent both on problem
specifications and on the computer system, but by way
. of illustration some parameters observed during a typi-
cal run on the IBM 370/168 system will be mentioned.
During an impact simulation of a Kevlar 29 single layer
at 300 m/sec, the computed time-to-penetration was
116
J4-
4 -- '. -3
42.87 /Lsec. The panel was idealized as having a A
fiber spacing of 0.3175 cm, and the time step for
optimum code stability was 0.32234 /h•sec. A total oi
133 time increments was therefore necessary for the
run, with each time increment involving an additional
node relative to the previous step. The total run
time for this job was 1.089 minutes ($12.92 at weekend
rates), and a total of 182 kilobytes of CPU core was
required. When this same problem was run at NARADCOM,
3 min 50 sec of CPU time was required, which illustrates
the system dependency of such job parameters.
R
A7
FLOW DIAGRAM AND LISTING,
e O # layers
[bO'kI max'
ipt (.e 1,2,3 or 4)
read Material Properties
pt= 1: 0, E1 E2, 3
=2 :E,b
3 g,
4= Eo,E12 ,A3,
Compute Auxilliary Parameters:
t = ( AX/C)/ d s
Print Out-put Data
2
Ž1
-ý111'1~ --ýrIFP-
Initialize Arrays:
Xj,k = (j-l)dx
Yj,k (k-l)dx j,k 1,40
set velocity, tension
strain energy at
node j,k to zero 'AEBegin Time Loop
'V Vz(l,l) =V
3P
loop until
Space Loop t. ge. tmax
or
New Projectile Velocity
Eq. 26,,27
=Print Field Variables _
en d
la,9
IN
___"v ro_________- __ -- ,
i IA,
SSpace Loop T(In subroutine BNDRY if i j)
fcompute new velocities
Eq. 20-21 -
compute new coordinates
Eq. '?
compute element strain
Eq. 23-24 1
compute element tension
(subroutine TENSN)
Eq. 25,40,46
compute kinetic and
strain energies
Eq. 29,30
4
120
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APPENDIX B - The XOVER Code
General. XOVER is the FORTRAN program written to
carry out the calculations described in Chapter IV
of this report. It is identical in concept to the
FABRIC code, differing primarily in the dimensionality
of the problem. XOVER is essentially a treatmei,. of
transverse ballistic impact on a single fiber as
described earlier by Roylance [18], but it has been
modified to include a second fiber, transverse to the
first, which receives its loading from the motion of
the first (primary) fiber. The code is made more
complicated than the single-fiber case by the necessity
of allowing motion in all b-hree directions for the
secondary fiber, and in computing the initial values of
the secondary fiber in terms of the primary fiber
motion. As described in the text, allowance is made
for slippage of the secondary fiber along the primary.
Code requirements. As was true for the FABRIC
code, XOVER r:equires little hardware or software
support. At present, the code includes a call to an
MIT library subroutine PRTPLT for the purpose of
obtaining a rough plot of the nodal variable values
on the system line printer. This subroutine call could
136I
be removed without affecting the performance of other
parts of the code. For a typical run, in which two
crossed Kevlar fibers were impacted at 300 m/sec, the
job run time was 0.305 minutes and 170 kilobytes of
core was reserved.
Definition of program variables. XOVER is
conveniently described by means of a listing of the
principal program variables: (* denotes input data)
"A 8.826 x 106 (dt/dl)
CROSS* Initial crossover position (fraction of XLI)
CZ Wavespeed (m/sec)
DEN* Fiber denier
DL Length increment (cm)
DT Time increment (sec)
t El (J), E2(J) Strain at jth node in primary and secondaryfiber
SEIOLD(J), Strain at previous time incrementE2OLD(J)
EMAX* Maximum strain permitted
EEMAX Maximum strain in fibers
G* Instantaneous modulus (gm/den)
IPLOT* .EQ. 0 if no printer plot is desired
11,12,13* Node numbers at which plot desired (13 onsecondary)
137
JX Crossover node on primary fiber
JJX Initial crossover node
KMAX Maximum number of time increments
LPSKIP* Length print skip
LTSKIP* Time print skip
NLINC* Number of length increments on primary"fiber
PMASS* Projectile mass (gm)
PVELOC* Projectile velocity (m/sec)
SLIDE* Slip factor (1-no slip, 0-no friction)
TAU* Viscous relaxation time (sec)
TITLE* Alphanumeric title (80 characters max.)
STI (J), Tension in primary and secondary fibers! T2%'J)
J(gm/den)I T1OLD(J),T2)LD(J) Previous tension (gm/den)
TMAX* Maximum time (sec)
,Ul(J), X-component of velocity in primary andU2(J) secondary fibers (m/sec)
Vl (J), Y-component of velocity
V2 (J)
VLAMDA* Viscous fraction (0-elastic, 1-purelyi' Viscous)
138
i
Wl (J), Z-component of velocityW2 (J)
XLl, XL2 Half-length of primary and secondaryfibers (cm)
Xl(J),Note thX-coordinate of jth node on primary and
secondary fibers
require b Y-coordinate~YV(J)
Zl (J), Z-coordinateZ2 (J)
Note that in its present form, XOVER is written
: explicitly for viscoelastic material response of the
S~standard linear solid type. For elastic fibers, the
:• user should set VLAMDA to zero and G to the Young's
• • modulus; TAU could be any nonzero-.value. The variables
required by the code are indicated as input information
and are specified in the above list by an asterisk; an
example of a typical input data set is given below.
139 3
EXAMPLE:
VELOCITY STUDY, V = 300 m/sec KEVLAR 29
1.O0E+06 300 20. 10. 0.5 1.0
1500 550 0.14-0.4 0.
100 5 5 287E 0.05
1 25 75 25
1) 204A TITLE
2) 6E10.3 PMASS, PVEL0C, XL1, XL2, CROSS, SLIDE
3) 4E10. DEN, G. TAU, VLAMPA
4) 31 10, 2E 10.3 NLINC, LPSKIP, LTSKIP, TMAX, EMAX
5) 41 10 IPLOT, Ii, 12, 13
Typical Output. Typical output from the XOVER code
for impact on two crossed Kevlar 29 fibers at 300
m/sec is given below:
140
- - -
11 C- 0.
F. .C4 .c- V
ZrU0 20 c-aI-~ C-~iL~ c, ~ Va 0, It U
c Ln C0 cr wC. &I I a in
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- . C a. s-i A P r C -. tr c 0 C- or .oxz -
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OLtlr ;wwt I-
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0 C, 0 C' 4n 0 0' 0 *
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c Lf. l- vc . Cc l. C . I (W * -C-,.'O C .
r% P- . f, CC,0C. C,00C.c C, C, -C. ,C ,CC n0CC.,C0 ' _, CIý 40 C.
. 1.* I *~~r- c- C.U . C ' 0 N ~ 0 - F -- I . - 0U 0'C O C
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6-~~ c -z LCC0CC~c.CCCC
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c ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = c -j 1c ,L r% vr - ra r0 -CC t caa
m t. 1 L! r C 1:W A r Cý W jý : ý C a Qý . el r c cC14 %Cw C c~ Z m ý . C Z
w- N se.Ncmmt. nj, z j 4 n fkr C D rC% a Ma , . .(rC
143
all
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V. 3c 4..,4 . U
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t" - .. .C4g N ~ 0, -.14's PK. >L .ccl - F
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8- M 15- C. W H *; -1`2 Eý r-. ~ f- IX.- 1-14 ~ CacO PCk1, E liWE-3 - k be H1-4ý* C. C C. 6 k ' C) a U. E - 4I$ ll C. r *.)r rMlk r I CL a. . xt ILo -I e ,IL U' tI * 0 E- U C or 6~-4 .. U CL 1 . U ). - tMH3 c -a -iv Ot*Iuu 1. 1 -
3r ta - . * 1. .
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149
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C -Ccuo - N* I-C' -C +- fl. ,-a0a.t 4 c0r4PI E-. 11 It xj WL C 11 .4 + 1 f .L. 't -- :E
ý- - r -- l - > P-. ft >0Ef 0 1 c el X C 1 i -e.F,-f - I - C- f*- - E, E*- J
A~~~ ~ ~ IL CZ-( l ,O I= , .eI C * al'. -C -r E- 1ý E,..C4 31 *KE - - .3c37j r0 c l0 D t 14 Ic P. t~ Lof* 11 ;c t, .'m.. E-4 C3 cr r.mK.a-E-
41 r : t : * C , w- U C': .m MP~ C~.w~ 1 .c ly r.,--4 2 C.~ r c ~ c --1-4 I ' ir LL II I- v C U i o . * P s L) oc r- Q C M b .4 U' rs- In-'- w) - -
* . N'~ Z2 %~D *f is. 3L~ t. C.CC-'f, C
90 . CA C*m- ci z C''ý CZ~- en M. _r'~~ E-
(Ii U " c m L L C C, fr -2 t- %C r.~ CC 4L v- issfs C s-
Ii I% ww vW sqv-c C% fý r ,rt -r r"I r- r, fC r as C C mCC cc cca- 0' v o* c* C. e I- - ý - V.
C )CCCCCC C4 .C 9 LCCC -C C:. t, C C C.C C, c
0150
ýKM Rp
kInI-
at' I l ~ In$- - i .t '
Insi.. s an In 63 r- C
f, N .IL 0 SK Ck c
C.. % = IA * IM inE-
3r 9EE IA U) H- L)~ WAH- 'ge 0- N H~'a *r t , t - n% L C)V
Ný U. .t . N,- It b'. a. t
C U)I IN w ~ . r, 1a- AC4- H InU ult tý 0,-41N Lr- I.
II *IZ .N . 1 sIW )z S-i Cl-K 1 . (J1.:a: M~ P- cr cal : at M lH- 96.i.
cr~- r- L 4 N, it. ul U I .it i w- - b-d i
z 43 sr. Ix 9. M. L- a 6i 's tqc %:& =- o- N, b S . LJIf. 2r.
b-t; ,-,i. IS * , CL- .3e- -I C * g ( -
c * c a. % nI . -- $-I U.. W~ IC L It. % C. Ow b. KbIC ' C-- U'. u,- Ui .it *1 1-4 W " ;- C ; l p;C r' en
V ý ~ W U. N E-HO IbI= Z Z SEer -E 'D- r- f. r.raH
3c ~ ~ ~ ~ C.~ i cm! m -4 U V~. A.NK w U) )U ,c l L xUP r:cr -~~~~~~~ Iii. C.' ..- b .. E . H l* s
Eir V., C6 SII In gb i.. 0- L N, P.CK'm* VC" Ix-z is Cdf .IlC ci). -0r-C ,W :.cc' H- "N cj r, r.o b. :1 1 31
f-i- V,~t" .00 0 HL 1x 3;1:>IC G x . I-cr 'I * .* 9 ' m
JX O'. I% cr. K rm W. K Ix -ý i
Ca~~.a r3bi-t Cu' . c - C', CN~ ~ I c -C N
tL [a. C, r.CL 9L a L
C% r j . I-- Ma. i- M, cr IN- M.Cb :1 jr %D
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C'~~ I*0 C - N ('LC
C C, CC C- C CC. c C. C.)
151