Quasi-Static Uniaxial Tension Characteristics of 

Plain-Woven Kevlar KM2 Fabric 

by Martin N. Raftenberg and Thomas J. Mulkern 

ARL-TR-2891 December 2002 

Approved for public release; distribution is unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position unless 

so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the 

use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2891 December 2002 

Quasi-Static Uniaxial Tension Characteristics of 

Plain-Woven Kevlar KM2 Fabric 

Martin N. Raftenberg and Thomas J. Mulkern 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
i

Contents 

List of Figures ii

List of Tables iv

1. Introduction 1

2. Description of the Experiments 3

3. Results 3

3.1 Force-Displacement Curves ............................................................................................3 

3.2 Initial Effective Ply Thickness ........................................................................................3 

3.3 First Piola-Kirchhoff Stress.............................................................................................7 

3.4 Fabric Work per Cross-Sectional Area ...........................................................................8 

3.5 Fabric Strength and Displacement and Work at Failure .................................................9 

4. Analytical Fits to the Fabric Stress-Crosshead Displacement Curves 11

4.1 Bilinear Fit.....................................................................................................................11 

4.2 Exponential Fit ..............................................................................................................14 

4.3 Quartic Fit......................................................................................................................21 

5. Discussion 26

5.1 Force-Displacement Curves ..........................................................................................26 

5.2 Force-to-Stress Conversion ...........................................................................................28 

5.3 Fabric Strength ..............................................................................................................28 

5.4 Failure Conditions for Warp vs. Fill Directions............................................................29 

5.5 Bilinear vs. Exponential vs. Quartic Fits to Stress-Displacement Curves ....................29 

6. Concluding Remarks 31

6.1 Summary of Results ......................................................................................................31 

6.2 Recommendations for Future Work..............................................................................31 

7. References 32
ii

List of Symbols 33

Report Documentation Page 35 

List of Figures 

Figure 1. The PPTA monomer that comprises Kevlar....................................................................2 

Figure 2. A plain-weave structure consisting of mutually orthogonal families of warp and fill 

yarns...........................................................................................................................................2 

Figure 3. Instron 4505 loading frame. ............................................................................................4 

Figure 4. An Instru-Met capstan webbing grip used in the tests. ...................................................5 

Figure 5. A specimen clamped in the Instron 4505 apparatus........................................................5 

Figure 6. Applied force-crosshead displacement data from six specimens pulled in uniaxial 

tension along the warp direction................................................................................................6 

Figure 7. Applied force-crosshead displacement data from six specimens pulled in uniaxial 

tension along the fill direction. ..................................................................................................6 

Figure 8. Fabric stress-crosshead displacement results from six specimens pulled in uniaxial 

tension along the warp direction................................................................................................7 

Figure 9. Fabric stress-crosshead displacement results from six specimens pulled in uniaxial 

tension along the fill direction. ..................................................................................................7 

Figure 10. Work per cross-sectional area-crosshead displacement results from six specimens 

pulled in uniaxial tension along the warp direction...................................................................8 

Figure 11. Work per cross-sectional area-crosshead displacement results from six specimens 

pulled in uniaxial tension along the fill direction. .....................................................................8 

Figure 12. Fabric strength (first Piola-Kirchhoff stress at failure) for the 12 tests.........................9 

Figure 13. Crosshead displacements at failure for the 12 tests.....................................................10 

Figure 14. Work at failure per cross-sectional area for the 12 tests. ............................................10 

Figure 15. Sketch of a typical force-displacement curve for a fabric loaded in uniaxial 

tension......................................................................................................................................12 

Figure 16. The bilinear fabric stress-crosshead displacement curve of equation 3. .....................13 

Figure 17. Least-squares error vs. locking strain for test W1; the value of ∆xlock that 

minimizes ε was selected.........................................................................................................14 

Figure 18. The small-displacement slope of the bilinear fit to the fabric stress-crosshead 

displacement results from each of the 12 tests.........................................................................15 

Figure 19. The large-displacement slope of the bilinear fit to the fabric stress-crosshead 

displacement results from each of the 12 tests.........................................................................15 
iii

Figure 20. The locking displacement of the bilinear fit to the fabric stress-crosshead 

displacement results from each of the 12 tests.........................................................................16 

Figure 21. Least-square errors for bilinear fits to the fabric stress-crosshead displacement 

results from the 12 tests. ..........................................................................................................16 

Figure 22. Bilinear fit to the fabric stress-crosshead displacement data from test W1. ...............18 

Figure 23. Bilinear fit to the fabric stress-crosshead displacement data from test F1..................18 

Figure 24. Characteristic-stress parameter of the exponential fit to the fabric stress-crosshead 

displacement results from each of the 12 tests.........................................................................19 

Figure 25. Characteristic-length parameter of the exponential fit to the fabric stresscrosshead displacement results from each of the 12 tests........................................................20 

Figure 26. Least-square errors for exponential fits to the fabric stress-crosshead 

displacement results from the 12 tests. ....................................................................................20 

Figure 27. Exponential fit to the fabric stress-crosshead displacement data from test W1. .........22 

Figure 28. Exponential fit to the fabric stress-crosshead displacement data from test F1............22 

Figure 29. The linear-term coefficient for the quartic fits to the fabric stress-crosshead 

displacement data from the 12 tests.........................................................................................23 

Figure 30. The quadratic-term coefficient for the quartic fits to the fabric stress-crosshead 

displacement data from the 12 tests.........................................................................................24 

Figure 31. The cubic-term coefficient for the quartic fits to the fabric stress-crosshead 

displacement data from the 12 tests.........................................................................................24 

Figure 32. The quartic-term coefficient for the quartic fits to the fabric stress-crosshead 

displacement data from the 12 tests.........................................................................................25 

Figure 33. Least-square errors for quartic fits to the fabric stress-crosshead displacement 

results from the 12 tests. ..........................................................................................................25 

Figure 34. Quartic fit to the fabric stress-crosshead displacement data from test W1. ................27 

Figure 35. Quartic fit to the fabric stress-crosshead displacement data from test F1...................27 

Figure 36. Least-squares errors for the bilinear, exponential, and quartic fits to the warpdirection fabric stress-crosshead displacement curves. ...........................................................30 

Figure 37. Least-squares errors for the bilinear, exponential, and quartic fits to the filldirection fabric stress-crosshead displacement curves. ...........................................................30 
iv

List of Tables 

Table 1. Tensile properties of Kevlar yarns....................................................................................1 

Table 2. Properties of the specimens and tests. ..............................................................................6

Table 3. Failure conditions for the six warp tests. ........................................................................11 

Table 4. Failure conditions for the six fill tests. ...........................................................................11 

Table 5. Parameters for bilinear fits to stress-displacement curves from the six warp 

tests. .........................................................................................................................................17 

Table 6. Parameters for bilinear fits to stress-displacement curves from the six fill tests............17 

Table 7. Parameters for exponential fits to stress-displacement data from the six warp 

tests. .........................................................................................................................................21 

Table 8. Parameters for exponential fits to stress-displacement data from the six fill 

tests. .........................................................................................................................................21 

Table 9. Parameters for quartic fits to stress-displacement data from the six warp tests. ............26 

Table 10. Parameters for quartic fits to stress-displacement data from the six fill tests...............26 
1

1. Introduction 

Plies of plain-woven Kevlar* aramid fabrics are the usual components of soft body armor (types I 

through III-A, as defined in National Institute of Justice [1987]). According to E. I. du Pont 

de Nemours and Company (DuPont) (undated), “Kevlar KM2 is a new, very high strength, high 

toughness (form of Kevlar) designed for improved ballistic fragmentation resistance and energy 

absorption capacity.” Table 1 compares single-yarn mechanical properties of KM2 to those of 

other forms of Kevlar. The KM2 properties in this table were obtained from publications by 

H. H. Yang of DuPont. Kevlar KM2, 29, 49, etc., are all composed of the same paraphenylene 

terephthalamide (PPTA) monomer shown in Figure 1. These different forms of Kevlar differ in 

the degree of crystallinity, which reflects the degree of molecular alignment and hydrogen 

bonding between neighboring molecules (Scott 2001). In Table 1, Ey is the yarn stiffness, or 

slope of a quasi-static, tensile stress-strain curve; σy,fail is the yarn strength, or maximum stress 

attained on a tensile stress-strain curve; and ey,fail is the strain corresponding to maximum stress. 

Note that KM2 is distinguished by relatively large values for both σy,fail and ey,fail, which 

translates into a large value for toughness, or work per unit volume at failure. 

Table 1. Tensile properties of Kevlar yarns. 

Yarn Type Ey 

(GPa) 

σy,fail

(GPa)

ey,fail

Kevlar 29a 70 2.9 0.036 

Kevlar 49a 135 2.9 0.028 

Kevlar 68a 99 3.1 0.033 

Kevlar 119a 55 3.1 0.044 

Kevlar 129a 99 3.4 0.033 

Kevlar 149a 143 2.3 0.015 

KM2b 63 3.3 0.040 a Yang (1993), p. 26. 

b Yang (2000), p. 219. 

A single ply of plain-woven fabric consists of two mutually orthogonal families of yarns: 

one called “warp” and the other called “fill” (Figure 2). In the present study, single plies of 

plain-woven 600-denier Kevlar KM2 (style 706) were tested under quasi-static uniaxial tension. 

Twelve specimens were tested; six were loaded along the warp yarn direction and six along the 

fill yarn direction. 

Section 2 describes the experimental procedure. The immediate data were applied force and 

crosshead displacement of the Instron machine. In section 3, these data are presented and 

 *

 Kevlar is a registered trademark of E. I. du Pont de Nemours and Company. 
2

Figure 1. The PPTA monomer that comprises Kevlar. 

Figure 2. A plain-weave structure consisting of mutually orthogonal 

families of warp and fill yarns. 

processed to obtain fabric stress-crosshead displacement curves. In the process, warp and fill 

values for fabric strength, σfail, are obtained. Least-squared-error fits to the stress-displacement 

curves are presented in section 4. Fits are obtained in three forms: (1) three-parameter bilinear 

curves, (2) two-parameter exponentials, and (3) four-parameter quartics. Sections 5 and 6 follow 

with discussions and conclusions, respectively. 

Reproduced from Warner (1995), p. 14. 

Reproduced from Warner (1995), p. 271. 
3

2. Description of the Experiments 

The uniaxial tension tests were performed under the guidance of American Society for Testing 

and Materials (ASTM) standard D5035-95 (ASTM 1995). An Instron 4505 load frame, such as 

that shown in Figure 3, was used (Instron Corporation 1988). The Instron load cell was rated at 

100-kN maximum load. Instru-Met capstan webbing grips (Figure 4) were employed in place of 

the “wedge action grips” in Figure 3. Figure 5 shows a specimen clamped in preparation for a 

test. 

All specimens were cut from the same sheet of style 706 fabric obtained from Hexcel Schwebel. 

Each specimen had an initial length of 1.219 m and an initial width, wo, of 50.8 mm (Table 2). 

At each end of the specimen, a length of 50 mm was clamped into the capstan grip. The 

specimen was then wrapped twice around each capstan grip. During each test, the load frame 

crosshead speed was constant at 2.12 mm/s. This quantity is denoted dx/dt, where x is the 

distance between the moving and stationary crossheads at time t. The applied force was sampled 

25 times/s. 

Twelve specimens were tested. Six were elongated along the warp direction and had the fill 

direction associated with the width. These are tests W1–W6. The other six were elongated 

along the fill direction and had the warp direction associated with the width. These are denoted 

tests F1–F6. 

3. Results 

3.1 Force-Displacement Curves 

The quantities directly measured were the crosshead displacement, ∆x, and the corresponding 

force, F, both as functions of time t. Data for ∆x and F are plotted in Figures 6 and 7. 

3.2 Initial Effective Ply Thickness 

Calculation of first Piola-Kirchhoff stress, σ, from force F requires an evaluation of the ply’s 

initial thickness, do. This quantity does not admit a direct measurement because of the ply’s 

woven geometry. Instead, it was estimated based on three facts: (1) A ply of style 706 KM2 

contains 34 yarns/in (Scott 2001), or 1.339 yarns/mm width; (2) A yarn of 600-denier KM2 

contains 400 filaments (Scott 2001), hence, a ply of style 706 600-denier KM2 contains 

535.4 filaments/mm width; and (3) A single filament has a circular cross section and a nominal 

diameter of 12 μm (Yang 1993, pp. 28 and 30), which corresponds to a filament cross-sectional 

area of 1.13 H 10!4

 (mm)2. Hence, a single ply of style 706 600-denier KM2 has a crosssectional area of 0.0606 (mm)2

/mm width. This constitutes an effective thickness, do, of 

60.6 μm, which is added to Table 2. 
4

Figure 3. Instron 4505 loading frame. 

Reproduced from Instron Corporation (1988), pp. 1-4 and 1-5. 
5

Figure 4. An Instru-Met capstan webbing grip used in the tests. 

Figure 5. A specimen clamped in the Instron 4505 

apparatus. 
6

Table 2. Properties of the specimens and tests. 

wo 

(mm)

dx/dt 

(mm/s) 

do 

(μm)

50.8 2.12 60.6 

Figure 6. Applied force-crosshead displacement data from six specimens 

pulled in uniaxial tension along the warp direction. 

Figure 7. Applied force-crosshead displacement data from six specimens 

pulled in uniaxial tension along the fill direction. 
7

3.3 First Piola-Kirchhoff Stress 

Engineering stress, σ, can now be evaluated with the relationship 

σ = F

w do o

. (1)

Engineering stress is a component of the first Piola-Kirchoff stress tensor (Malvern 1969, p. 222). 

The values for wo and do in Table 2 were used to produce the fabric stress-crosshead 

displacement curves in Figures 8 and 9. 

Figure 8. Fabric stress-crosshead displacement results from six 

specimens pulled in uniaxial tension along the warp direction. 

Figure 9. Fabric stress-crosshead displacement results from six 

specimens pulled in uniaxial tension along the fill direction. 
8

3.4 Fabric Work per Cross-Sectional Area 

Let W denote the fabric work per unit undeformed cross-sectional area, cumulative up to a given 

level of crosshead displacement ∆x, or 

. (2)

Numerical integration was performed by the trapezoid rule. Results are presented in Figures 10 

and 11. 

Figure 10. Work per cross-sectional area-crosshead displacement 

results from six specimens pulled in uniaxial tension along 

the warp direction. 

Figure 11. Work per cross-sectional area-crosshead displacement 

results from six specimens pulled in uniaxial tension along 

the fill direction. 
9

3.5 Fabric Strength and Displacement and Work at Failure 

For a given test, the strength, or maximum value attained by the first Piola-Kirchhoff stress, is 

denoted by σfail. Visible degradation to the specimen, in the form of slack and pulled-out yarns, 

became apparent at about the time at which maximum stress was reached, hence, the use of the 

subscript “fail.” The corresponding values of crosshead displacement and work per area are 

denoted by ∆xfail and Wfail, respectively. Note that Wfail is proportional to toughness, or the work 

at failure per unit volume. The quantities σfail, ∆xfail, and Wfail are plotted in Figures 12–14, 

respectively, and listed in Tables 3 and 4. Mean values, averaged over six tests, for the strength 

were 2.23 GPa for the warp direction and 2.67 GPa for the fill direction. 

(W|F) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6

Figure 12. Fabric strength (first Piola-Kirchhoff stress at failure) for 

the 12 tests. 

The specimens loaded along the warp direction consistently exhibited smaller strength (failure 

stresses) and larger failure displacements than those specimens loaded along the fill direction. 

The work per area at failure showed percentage-wise less systematic difference between warp 

and fill, although five fill specimens required more work to fail than did any of the six warp 

specimens. 
10

(W|F) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 13. Crosshead displacements at failure for the 12 tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 14. Work at failure per cross-sectional area for the 12 tests. 
11

Table 3. Failure conditions for the six warp tests. 

Test σfaila

(GPa) 

∆xfailb

(mm) 

Wfailc

(J/mm2) 

W1 2.33 55.1 33.5 

W2 2.07 54.4 28.9 

W3 2.18 49.8 29.9 

W4 2.23 47.7 30.2 

W5 2.31 50.4 31.9 

W6 2.28 52.1 31.0 

Mean 2.23 51.6 30.9 

Standard deviation 0.10 2.8 1.6 

a

Plotted in Figure 12. 

b

Plotted in Figure 13. 

c

Plotted in Figure 14. 

Table 4. Failure conditions for the six fill tests. 

Test σfaila

(GPa) 

∆xfailb

(mm) 

Wfailc

(J/mm2

) 

F1 2.77 43.6 38.6 

F2 2.69 43.9 35.7 

F3 2.47 42.2 31.8 

F4 2.75 43.9 37.3 

F5 2.61 40.5 34.6 

F6 2.72 40.5 36.3 

Mean 2.67 42.4 35.7 

Standard deviation 0.11 1.6 2.4 

a

Plotted in Figure 12. 

b

Plotted in Figure 13. 

c

Plotted in Figure 14. 

4. Analytical Fits to the Fabric Stress-Crosshead Displacement Curves 

4.1 Bilinear Fit 

According to Grosberg (1969), Figure 15 sketches a force-displacement curve obtained for a 

generic fabric. The curve has three regions: (1) an initial nonlinear (negative-curvature) region 

primarily governed by “inter-fiber friction,” (2) an intermediate nonlinear (positive-curvature) 

region corresponding to the phenomenon of fiber uncrimping, and (3) a final linear region that 

reflects the stiffness associated with the elongation of uncrimped yarns. The curve abruptly 

terminates at the end of this third region, corresponding to failure of the fabric. 
12

(3) 

Figure 15. Sketch of a typical force-displacement curve for a fabric 

loaded in uniaxial tension. 

In Figures 6 and 7, the warp and fill stress-displacement curves from a ply of 600-denier KM2 do 

not exhibit the initial negative-curvature region in Figure 15. Because of Kevlar’s high strength, 

the initial negative-curvature region may be present but applicable to only a negligible fraction of 

the total load excursion. 

The two-part stress-displacement curve sketched in Figure 16 consists of two linear portions 

joined at the locking displacement, ∆xlock. According to Jinlian and Newton (1993), this bilinear 

fit of Figure 16 (albeit applied to stress-strain rather than to stress-displacement) may have been 

first proposed by Alsawaf (1985). The slope of the small-displacement portion is denoted β1, 

and that of the large-displacement portion is denoted β2. The displacement corresponding to the 

maximum stress, at which failure is assumed to occur, is denoted ∆xfail. This bilinear stressdisplacement curve is described by 

Reproduced from Grosberg (1969), p. 339.
13

∆x 

σ

0 ∆ ∆xfail xlock

β1 ∆xlock

β1 ∆xlock + 

β2(∆xfai l - ∆xlock)

1

β1

β2

1

Figure 16. The bilinear fabric stress-crosshead displacement curve of 

equation 3. 

Parameter ∆xfail is determined directly from the data (see Tables 3 and 4 and Figure 13). A leastsquares-error procedure is used to obtain ∆xlock, β1, and β2 for each of the 12 tests. 

For a given test, all data at strains larger than ∆xfail are discarded. This leaves N points of 

measured data: (∆xi, σi), i = 1, 2, ..., N, where 

∆x x N fail = ∆ . (4) 

The procedure is to pick a ∆xlock that satisfies the condition 

 0 ≤ ∆x x lock fail ≤ ∆ . (5) 

Define Nl by the conditions 

∆x x N lock ∆ l ≤ ; (6a) 

∆x x N lock ∆ l +1 > . (6b) 

The total squared error of the fit is given by 

. (7)
14

The dependency of ε 2 on ∆xlock is partly embedded in the limits of summation. For a given 

∆xlock, minimization of ε 2 with respect to β1 and β2 is imposed. 

. (8)

∆xlock is then varied, covering the range of equation 5 in small increments. Minimization of ε

with respect to ∆xlock is imposed graphically (Figure 17). 

Figure 17. Least-squares error vs. locking strain for test W1; the value 

of ∆xlock that minimizes ε was selected. 

The results for β1, β2, ∆xlock, and ε for the 12 tests are given in Figures 18–21, respectively, and 

in Tables 5 and 6. For illustration, the bilinear fits obtained for tests W1 and F1 are displayed in 

Figures 22 and 23, respectively. 

Values for ∆xlock are consistently larger for warp than for fill. Values for β2 are consistently 

smaller for warp than for fill. Values for β1 are mixed with regard to warp and fill. Values for ε 

are consistently larger for warp than for fill, indicating that the bilinear form is a better 

description for the fill than for the warp response. 

4.2 Exponential Fit 

Consider the exponential fit to the entire fabric stress-crosshead displacement curve (0 ≤ ∆x 

≤ ∆xfail): 

. (9)
15

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 18. The small-displacement slope of the bilinear fit to the fabric 

stress-crosshead displacement results from each of the 12 

tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 19. The large-displacement slope of the bilinear fit to the fabric 

stress-crosshead displacement results from each of the 12 

tests. 
16

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 20. The locking displacement of the bilinear fit to the fabric 

stress-crosshead displacement results from each of the 12 

tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 21. Least-square errors for bilinear fits to the fabric stresscrosshead displacement results from the 12 tests. 
17

Table 5. Parameters for bilinear fits to stress-displacement curves from the six warp 

tests. 

Test β 1

a

(MPa/mm) 

β 2

b

(MPa/mm) 

∆xlockc

 

(mm) 

εd

(GPa) 

W1 9.455 81.02 31.9 1.024 

W2 7.329 75.80 31.2 0.7887 

W3 11.59 80.86 28.3 0.8737 

W4 12.67 83.21 26.2 0.8089 

W5 10.80 80.93 27.4 0.8076 

W6 9.125 83.50 29.6 0.8273 

Mean 10.16 80.89 29.1 0.855 

Standard deviation 1.92 2.76 2.2 0.088 

a

Plotted in Figure 18. 

b

Plotted in Figure 19. 

c

Plotted in Figure 20. 

d

Plotted in Figure 21. 

Table 6. Parameters for bilinear fits to stress-displacement curves from the six fill 

tests. 

Test β 1

a

(MPa/mm) 

β 2

b

(MPa/mm) 

∆xlockc

 

(mm) 

εd

(GPa) 

F1 15.85 101.1 20.0 0.7780 

F2 12.90 99.02 20.5 0.7036 

F3 11.91 97.41 19.7 0.6102 

F4 12.51 102.1 19.9 0.7045 

F5 15.91 100.4 17.9 0.6371 

F6 16.78 102.0 17.5 0.6483 

Mean 14.31 100.3 19.3 0.6803 

Standard deviation 2.10 1.8 1.2 0.0608 

a

Plotted in Figure 18. 

b

Plotted in Figure 19. 

c

Plotted in Figure 20. 

d

Plotted in Figure 21. 

This functional form seems to have been first proposed by Jinlian and Newton (1993). The two 

constants, ξ and η, remain to be determined by a least-squares-error procedure. 

For a given test, the displacement corresponding to the maximum measured stress is denoted 

∆xfail. All data at displacements larger than ∆xfail are then discarded. We are left with N points of 

measured data: (∆xi, σi), i = 1, 2, ..., N, where 

∆x x N fail = ∆ . (10) 
18

Figure 22. Bilinear fit to the fabric stress-crosshead displacement data 

from test W1. 

Figure 23. Bilinear fit to the fabric stress-crosshead displacement data 

from test F1. 

The total squared error is 

. (11)
19

For each test, the particular set of values for ξ and η is found that minimizes the error ε. The 

minimization conditions 

, (12)

lead to two coupled nonlinear equations for ξ and η that can only be solved numerically. 

Instead, a “brute force” approach was applied, in which ξ and η were varied in small increments 

and the minimum in ε was sought. The results for ξ, η, and ε are given in Figures 24–26, 

respectively, and in Tables 7 and 8. The exponential fits to tests W1 and F1 are displayed in 

Figures 27 and 28, respectively. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 24. Characteristic-stress parameter of the exponential fit to the 

fabric stress-crosshead displacement results from each of the 

12 tests. 

Values for ξ are consistently smaller for warp than for fill. Values for η are also generally 

smaller for warp than for fill, although there is some overlap. Both of these observations reflect 

the generally smaller stress levels for warp than for fill for a given crosshead displacement. 

Values for ε are consistently smaller for warp than for fill, indicating the exponential form to be 

more applicable to warp than to fill data. Recall from section 4.1 that the reverse was the case 

for the bilinear form. 
20

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 25. Characteristic-length parameter of the exponential fit to the 

fabric stress-crosshead displacement results from each of the 

12 tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 26. Least-square errors for exponential fits to the fabric stresscrosshead displacement results from the 12 tests. 
21

Table 7. Parameters for exponential fits to stress-displacement 

data from the six warp tests. 

Test ξ a

 

(MPa) 

η b 

(mm)

ε c

(GPa) 

W1 64.89 15.04 0.5656 

W2 52.33 14.39 0.8494 

W3 84.82 14.92 0.4731 

W4 104.3 15.07 0.5847 

W5 94.07 15.35 0.7299 

W6 66.09 14.35 0.7630 

Mean 77.8 14.85 0.6610 

Standard deviation 19.8 0.40 0.1420 

a

Plotted in Figure 24. 

b

Plotted in Figure 25. 

c

Plotted in Figure 26. 

Table 8. Parameters for exponential fits to stress-displacement 

data from the six fill tests. 

Test ξ a

 

(MPa) 

η b 

(mm)

ε c

(GPa) 

F1 223.7 16.34 1.170 

F2 175.4 15.33 1.222 

F3 159.5 14.58 1.229 

F4 190.5 15.52 1.376 

F5 237.0 15.80 1.167 

F6 267.4 16.32 1.179 

Mean 208.9 15.65 1.224 

Standard deviation 40.8 0.67 0.079 

a

Plotted in Figure 24. 

b

Plotted in Figure 25. 

c

Plotted in Figure 26. 

4.3 Quartic Fit 

Consider the quartic fit to the entire fabric stress-crosshead displacement curve (0 ≤ ∆x ≤ ∆x fail),

σα β γ δ =+ + + ∆∆ ∆ ∆ xx x x () () () 234 . (13) 
22

Figure 27. Exponential fit to the fabric stress-crosshead displacement 

data from test W1. 

Figure 28. Exponential fit to the fabric stress-crosshead displacement 

data from test F1. 

The authors know of no previous attempts to apply this quartic functional form to woven fabric 

data. The four constants, α, β, γ, and δ, remain to be determined by a least-squares-error 

procedure. 
23

For a given test, the displacement corresponding to the maximum measured stress is denoted 

∆xfail. All data at displacements larger than ∆xfail are then discarded. We are left with N points of 

measured data: (∆xi, σi), i = 1, 2, ..., N, where 

∆x x N fail = ∆ . (14) 

The total squared error is 

. (15)

The four minimization conditions, 

, (16)

lead to four linear algebraic equations for α, β, γ, and δ. The results for α, β, γ, δ, and ε are 

given in Figures 29–33, respectively, and in Tables 9 and 10. The quartic fits obtained for tests 

W1 and F1 are displayed in Figures 34 and 35, respectively. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 29. The linear-term coefficient for the quartic fits to the fabric 

stress-crosshead displacement data from the 12 tests. 
24

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 30. The quadratic-term coefficient for the quartic fits to the 

fabric stress-crosshead displacement data from the 12 tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 31. The cubic-term coefficient for the quartic fits to the fabric 

stress-crosshead displacement data from the 12 tests. 
25

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 32. The quartic-term coefficient for the quartic fits to the fabric 

stress-crosshead displacement data from the 12 tests. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 33. Least-square errors for quartic fits to the fabric stresscrosshead displacement results from the 12 tests. 
26

Table 9. Parameters for quartic fits to stress-displacement data from the six warp tests. 

Test α a

 

(kPa/mm) 

β b 

(kPa/mm2) 

γ c 

(kPa/mm3) 

δ

 d

 

(kPa/mm4) 

ε e

(GPa) 

W1 10200 !727.0 30.47 !0.1186 0.1906 

W2 14490 !1451 55.49 !0.3798 0.2853 

W3 11820 !820.2 40.86 !0.2257 0.1492 

W4 14000 !1122 59.99 !0.4589 0.1885 

W5 14270 !1298 62.52 !0.4877 0.2008 

W6 15050 !1450 60.14 !0.4160 0.1736 

Mean 13300 !1145 51.58 !0.3478 0.1980 

Standard deviation 1880 313 12.96 0.1449 0.0464 

a

Plotted in Figure 29. 

b

Plotted in Figure 30. 

c

Plotted in Figure 31. 

d

Plotted in Figure 32. 

e

Plotted in Figure 33. 

Table 10. Parameters for quartic fits to stress-displacement data from the six fill tests. 

Test α a

 

(kPa/mm) 

β b 

(kPa/mm2) 

γ c 

(kPa/mm3) 

δ d 

(kPa/mm4) 

ε e

(GPa) 

F1 9546 !877.6 101.1 !1.200 0.1386 

F2 10360 !1346 115.7 !1.341 0.1675 

F3 12480 !1885 146.6 !1.800 0.1179 

F4 9936 !1422 126.3 !1.513 0.1750 

F5 8712 !884.5 124.0 !1.686 0.1306 

F6 7796 !569.2 113.9 !1.578 0.1493 

Mean 9810 !1164 121.3 !1.520 0.1465 

Standard deviation 1600 476 15.3 0.221 0.0219 

a

Plotted in Figure 29. 

b

Plotted in Figure 30. 

c

Plotted in Figure 31. 

d

Plotted in Figure 32. 

e

Plotted in Figure 33. 

Values for γ are consistently larger in the fill tests than the warp tests. The reverse is true for δ. 

For α and β, no systematic relative orderings of warp and fill values are apparent. Values for ε 

are generally smaller for fill than for warp data fits, indicating that the quartic fit works 

somewhat better for fill data. 

5. Discussion 

5.1 Force-Displacement Curves 

The force-crosshead displacement curves, discussed in section 3.1, from the six warp-direction 

tests (Figure 6) exhibit greater intertest variation than those from the six fill-direction tests 

(Figure 7). Such intertest variations for either warp or fill may reflect initial differences between 
27

Figure 34. Quartic fit to the fabric stress-crosshead displacement data 

from test W1. 

Figure 35. Quartic fit to the fabric stress-crosshead displacement data 

from test F1. 

the six specimens. All 12 specimens were cut from a single sheet of KM2 obtained from Hexcel 

Schwebel, so that there is no reason to expect substantial interspecimen variations. Other 

possible sources for the differences between results from the six warp-direction tests are intertest 

differences in the degree of specimen slippage, yarn rotation, and/or nonuniform load 

distribution among the 68 yarns across the specimen’s 50.8-mm width. 
28

The force-displacement curve for each warp-direction test displays one or more jogs (Figure 6). 

Note that no such jogs are evident in the fill-direction tests (Figure 7). The most plausible 

explanation for these warp-direction jogs is sudden specimen slippage. One possible site for 

slippage was the clamps at either end. The two 50-mm-long clamped ends were not remeasured 

following the test, so slippage at the ends cannot be ruled out. A more likely type of slippage 

would have involved the sudden overcoming of frictional restraint somewhere along the two 

wrappings around either capstan. 

Yarn rotation could have resulted from initial misalignment of the specimen, whereby the 

loading direction would not have coincided exactly with the direction of yarn orientation. The 

load’s off-axis component would then have caused yarn rotation (“scissoring”) with relatively 

little fabric resistance. 

There is no apparent reason why either specimen slippage, yarn rotation, or nonuniform loading 

would have been more likely in the warp-direction tests than in the fill-direction tests. In fact, 

slippage, in particular, seems more likely in the fill direction because of greater strength, smaller 

elongation, and greater toughness relative to the warp direction (see section 5.4). 

5.2 Force-to-Stress Conversion 

Equation 1 was used to relate the applied force to the stress within the fabric ply. If the applied 

force were not distributed uniformly across the 50.8-mm width, then the 68 yarns would not have 

been equally loaded. In this case, equation 1 would yield an average stress across the specimen 

width and not the actual stress in each yarn. Similarly, in a given yarn all 400 filaments might 

not have been equally loaded. 

Equation 1 makes use of an effective ply thickness, do. In evaluating do in section 3.2, the crosssectional area of each filament had to be estimated based on a nominal filament diameter of 

12 μm and the assumption of a perfectly circular filament cross section. 

5.3 Fabric Strength 

For a single ply of Kevlar KM2 style 706, Tables 3 and 4 and Figure 12 show that σfail varied 

between 2.08 and 2.34 GPa with a mean of 2.23 GPa in the six warp-direction tests and between 

2.62 and 2.78 GPa with a mean of 2.67 GPa in the six fill-direction tests. These results can be 

compared with values in the literature. 

Johnson et al. (1999) reported a strength of 3.34 GPa for a single ply of plain-woven 850-denier 

KM2 (style 705). They do not specify whether their specimen was loaded in the warp or fill 

direction, nor do they specify any details of the experimental procedure. (The focus of their 

paper is on the development of a fabric constitutive model and not on the experimentally 

obtained input to the model.) This value of 3.34 GPa is substantially larger than both the warp 
29

and fill values in Tables 3 and 4, respectively, for a ply of plain-woven 600-denier KM2 (style 

706). Whether this discrepancy originates in real differences between the two batches of 

specimens or in experimental error cannot be determined. Note that 850- and 600-denier KM2 

differ only in the number of filaments per yarn. Since the number of filaments per yarn was 

taken into account in the normalization procedure of obtaining stress from force, the distinction 

between 850 and 600 denier cannot account for the discrepancy. 

The authors know of no other stiffness value reported in the literature for woven KM2 fabric. 

However, Yang (2000) reports a yarn strength of 3.3 GPa for KM2. This exceeds the 2.23-GPa 

mean warp-direction value (Table 3) by 34% and the 2.67-GPa mean fill-direction value (Table 

4) by 19%. Substantial degradation in yarn strength is known to occur in the weaving and fixing 

processes (Scott 2001), so that these substantial differences between fabric strength and 

prewoven yarn strength are plausible. Another consideration is that the single-yarn specimen 

studied in Yang (2000) may have been helically twisted prior to testing, whereas yarns in woven 

fabric are generally untwisted. 

5.4 Failure Conditions for Warp vs. Fill Directions 

Tables 3 and 4 and Figures 12–14 show smaller fabric strength, σfail, greater displacement at 

failure, ∆xfail, and smaller work at failure (toughness), Wfail, in the warp-direction tests relative to 

the fill-direction tests. Since, in the weaving process, adjacent warp yarns are respectively raised 

and lowered to accommodate a fill yarn, warp yarns initially have more slack, or crimp, than fill 

yarns. Hence, one would expect a greater ∆xfail under tensile loading in the warp direction than 

in the fill direction because more ∆x goes into uncrimping yarns in the former case. The 

observation that σfail and Wfail were smaller in the warp direction than in the fill direction can 

most likely be attributed to a greater mechanical degradation incurred by the warp yarns during 

the weaving and/or finishing processes. 

5.5 Bilinear vs. Exponential vs. Quartic Fits to Stress-Displacement Curves 

Figure 36 collects ε results from bilinear, exponential, and quartic fits to warp data, and the 

values are listed in Tables 5, 7, and 9. Figure 37 does the same for fits to fill data and the values 

are listed in Tables 6, 8, and 10. 

The exponential form, with only two parameters, is generally a better fit to warp data than is the 

three-parameter bilinear form (although there is some overlap). On the other hand, the bilinear 

form generally provides a better fit to fill data than the exponential form. In the case of stressstrain curves for woven cotton fabrics, Jinlian and Newton (1993) found the eponential fit 

superior to the bilinear fit in both the warp and the fill directions. 
30

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 36. Least-squares errors for the bilinear, exponential, and 

quartic fits to the warp-direction fabric stress-crosshead 

displacement curves. 

(WIF) 1 (W|F) 2 (W|F) 3 (W|F) 4 (W|F) 5 (W|F) 6 

Figure 37. Least-squares errors for the bilinear, exponential, and 

quartic fits to the fill-direction fabric stress-crosshead 

displacement curves. 
31

Visual comparison of the four-parameter quartic fits in Figures 34 and 35 with the bilinear fits in 

Figures 22 and 23 and the exponential fits in Figures 27 and 28 reveals the quartic fit to be the 

closest, at least for tests W1 and F1. The observation that the quartic fit is the closest for the 

other tests as well is borne out by Figures 36 and 37. Here, all quartic values for ε are smaller 

than any bilinear or exponential value. The quartic fit has apparently not been proposed 

previously in the fabric literature. 

This observation of a close fit obtainable with a quartic can most likely be transferred from fabric 

stress-crosshead displacement curves to fabric stress-fabric strain curves. However, this 

transferability might not be the case if the effective gauge length varies during the course of a 

test since this would alter the shape of the stress-strain curve relative to the stress-displacement 

curve. 

Computationally, the quartic fit should not be substantially more expensive to evaluate than the 

exponential fit. These should both be less expensive than the bilinear fit, which requires a check 

on the current strain level relative to ∆xlock. 

6. Concluding Remarks 

6.1 Summary of Results 

Quasi-static, uniaxial tension tests were conducted on 50.8-mm-wide single-ply specimens of 

plain-woven Kevlar KM2 fabric (style 706.) Six specimens were pulled along the warp 

direction, and six were pulled along the fill direction. Fabric stress-crosshead displacement 

results were presented from each test (Figures 6 and 7). The strength (maximum stress achieved) 

was found to range from 2.07 to 2.33 GPa with a mean of 2.23 GPa in the warp direction and 

from 2.47 to 2.77 GPa with a mean of 2.67 GPa in the fill direction (Tables 3 and 4 and 

Figure 12). 

Three least-square-error fits were obtained to the fabric stress-crosshead displacement curve 

from each test: (1) bilinear, (2) exponential, and (3) quartic. For both warp and fill data, the 

quartic fits were generally the closest. 

6.2 Recommendations for Future Work 

In the present study, it was not possible to determine fabric strain from crosshead displacement 

because the effective gauge length was undetermined. The experiments should be repeated using 

an electromechanical extensometer or an optical technique to measure local fabric deformation 

over a known initial gauge length. A challenge with the former approach is the need to grip the 

fabric specimen without inflicting damage. 
32

The tests were performed at a single small crosshead displacement rate, producing quasi-static 

results. In ballistic impact much larger loading rates are achieved. An Instron or MTS Systems 

Corp. machine would allow for examination of strain rate dependence over a modest range. A 

Kolsky bar approach (Shim et al. 2001) would allow still larger strain rates to be achieved. 

7. References 

American Society for Testing and Materials (ASTM). “Standard Test Method for Breaking 

Force and Elongation of Textile Fabrics (Strip Method).” ASTM D5035-95, West 

Conshohocken, PA, July 1995. 

Alsawaf, F. B. Ph.D. Thesis. University of Manchester, Manchester, UK, 1985. 

E. I. du Pont de Nemours and Company. “Kevlar KM2, Preliminary Information Bulletin.” 

Wilmington, DE, undated. 

Grosberg, P. “The Tensile Properties of Woven Fabrics.” Structural Mechanics of Fibers, 

Yarns, and Fabrics. Volume 1, edited by J. W. S. Hearle, P. Grosberg, and S. Backer, New 

York: Wiley-Interscience, pp. 339–354, 1969. 

Instron Corporation. “The Series 4500 Floor Model Load Frames.” Manual no. M10-4500-51, 

issue B, Canton, MA, June 1988. 

Jinlian, H., and A. Newton. “Modeling of Textile Stress-Strain Curve of Woven Fabrics.” 

Journal of China Textile University (English Edition), vol. 10, no. 4, 1993. 

Johnson, G. R., S. R. Beissel, and P. M. Cunniff. “A Computational Model for Fabrics 

Subjected to Ballistic Impact.” 18th International Symposium on Ballistics, vol. 2, edited by 

W. G. Reinecke, Lancaster, PA: Technomic Publishing, pp. 962–969, 1999. 

Malvern, L. E. Introduction to the Mechanics of a Continuous Medium. Englewood Cliffs, NJ: 

Prentice-Hall, 1969. 

National Institute of Justice. Ballistic Resistance of Police Body Armor. NIJ Standard 0101.03, 

U.S. Department of Justice, Washington, DC, April 1987. 

Scott, B. R. Personal communication with M. Raftenberg. E. I. du Pont de Nemours and 

Company, Wilmington, DE, 2001. 

Shim, V. P. W., C. T. Lim, and K. J. Foo. “Dynamic Mechanical Properties of Fabric Armour.” 

International Journal of Impact Engineering, vol. 25, pp. 1–15, 2001. 

Warner, S. B. Fiber Science. Englewood Cliffs, NJ: Prentice-Hall, 1995. 

Yang, H. H. Kevlar Aramid Fiber. Chichester, UK: John Wiley, 1993. 
33

Yang, H. H. “Aramid Fibers.” Comprehensive Composite Materials, Volume 1, Fiber 

Reinforcements and General Theory of Composites, Oxford, UK: Elsevier Science, pp. 199–

229, 2000. 

List of Symbols 

Ey single-yarn stiffness 

F force exerted on the specimen 

N the value of i corresponding to σfail

W work per unit undeformed cross-sectional area 

Wfail work at failure per unit cross-sectional area 

do initial effective thickness of the fabric specimen 

ey,fail engineering strain at failure in a single-yarn specimen 

t time 

i index identifying a particular data point (∆xi, σi)

wo initial width of the fabric specimen 

x distance between the loading frame’s moving and stationary crossheads 

∆L elongation of the fabric specimen from its initial length 

∆x displacement of loading-frame crosshead 

∆xN Nth measured crosshead displacement 

∆xfail displacement of loading-frame crosshead at fabric failure 

∆xi the ith measured crosshead displacement 

∆xlock locking displacement for the bilinear fit to a stress-displacement curve 

α linear-term coefficient in the quartic fit to a stress-displacement curve 

β quadratic-term coefficient in the quartic fit to a stress-displacement curve 

β1 small-displacement slope for the bilinear fit to a stress-displacement curve 

β2 large-displacement slope for the bilinear fit to a stress-displacement curve 
34

γ cubic-term coefficient in the quartic fit to a stress-displacement curve 

δ quartic-term coefficient in the quartic fit to a stress-displacement curve 

ε least-squares error associated with a fit to a stress-displacement curve 

ξ, η exponential-fit parameters 

σ First Piola-Kirchhoff axial stress (engineering stress)

σfail strength of a fabric specimen

σi stress corresponding to displacement ∆xi
35

Kevlar, KM2, personnel protection, body armor, mechanical properties, constitutive 40

properties, fabric

Quasi-static uniaxial tension tests were conducted on 50.8-mm-wide single-ply specimens of plain-woven DuPont

Kevlar KM2 aramid fabric (style 706). Six specimens were pulled along the warp direction, and six were pulled along

the fill direction. The strength was found to range from 2.07 to 2.33 GPa, with a mean of 2.23 GPa in the warp direction

and from 2.47 to 2.77 GPa, with a mean of 2.67 GPa in the fill direction. A bilinear, an exponential, and a quartic fit

were obtained to the fabric stress-crosshead displacement curve from each test, with the quartic fit generally found to be

the closest.

ARL-TR-2891

U.S. Army Research Laboratory

ATTN: AMSRL-WM-TD

Aberdeen Proving Ground, MD 21005-5066

Martin N. Raftenberg and Thomas J. Mulkern

Quasi-Static Uniaxial Tension Characteristics of Plain-Woven Kevlar KM2 Fabric 1L162618AH80

December 2002 Final, June 2001–July 2002

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