COMPOSITES
SCIENCE AND
TECHNOLOGY
Composites Science and Technology 65 (2005) 183–189
www.elsevier.com/locate/compscitech
A shear-lag approach to the tensile strength of paper
Leif A. Carlsson
a
a,*
, Tom Lindstrom
b
Department of Mechanical Engineering, Florida Atlantic University, P.O. Box 3091, 777 Glades Road, Boca Raton, FL 33431-0991, USA
b
Department of Fiber and Polymer Technology, The Royal Institute of Technology, 10044 Stockholm, Sweden
Received 12 December 2003; received in revised form 21 June 2004; accepted 22 June 2004
Available online 14 October 2004
Abstract
A shear-lag approach to the prediction of the tensile strength of paper is outlined and examined. It is demonstrated that transition of fiber strength to paper strength requires long fibers for sheets with weak fiber–fiber bonds, or low relative bonded area.
Fiber pull-out is encountered even for highly bonded sheets. Predictions of tensile strength for papers of the same fiber length,
but with different beating degrees, and for papers with different fiber lengths and beating degrees, are in quantitative agreement with
previously published data.
2004 Elsevier Ltd. All rights reserved.
Keywords: A. Fibres; Tensile strength
1. Introduction
Models for prediction of the tensile strength of paper
have been developed by several authors, starting with
the pioneering contributions of Kallmes et al. [1,2] and
Page [3]. The subject has been reviewed by e.g. van
den Akker [4], Waterhouse [5] and de Ruvo et al. [6].
Such models provide a more or less accurate tool for
the prediction of the tensile strength of paper from
accessible structural parameters of the sheet, the tensile
strength of the fibers, and the shear strength of the fiber–
fiber bonds. An often cited strength theory is the one
developed by Page [3]. He envisaged a failure process,
where bonds are progressively broken in the failure
zone, and the still bonded fibers take more and more
of the load until they become overloaded and break.
This hypothesis leads to the premise that the sheet
strength directly scales with the fraction of fibers that
are broken across the rupture zone (‘‘scan line’’).
*
Corresponding author. Tel.: +1 561 297 3421; fax: +1 561 297
2825.
E-mail address: [email protected] (L.A. Carlsson).
0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compscitech.2004.06.012
Through dimensional arguments and guidance from
some previously carefully conducted experiments, Page
derived a simple expression for the sheet strength in
terms of zero-span strength (which he considered as a
measure of fiber strength), and the shear strength of
the fiber–fiber bonds. At the time (1969) most of the
governing parameters in the theory were inaccessible
by experiments. Page, however, presented comprehensive experimental verification of the theory based on literature data where one parameter was varied while
other parameters that were not measured (due to lack
of test methods) were assumed to be constant.
Kallmes et al. [1,2] were first to develop a theory for
prediction of the tensile strength of paper. They assumed that local rupture of a fiber or a bond promoted
further such failures, rapidly leading to total failure of
the sheet. Papers with weak bonds were considered to
fail due to bond failures rather than rupture of the fibers. Bond failures were treated by removing the contributions from debonded fibers from the integration of
the sheet load. Kallmes et al. [1,2] hypothesized that
poorly bonded papers failed at a (small) strain corresponding to a maximum of the theoretically predicted
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L.A. Carlsson, T. Lindstrom / Composites Science and Technology 65 (2005) 183–189
load-elongation curve and, based on these grounds, derived an expression for the strength of weakly bonded
papers. The theory was correlated with experimental
data with variable success in Part 4 of their report [2].
Recent strength prediction approaches have taken a
step into more complex analysis of the failure process.
Karenlampi [7] was probably the first to consider a statistical distribution of fiber properties and developed an
approach considering both bond and fiber failures. The
results led to an integral equation for the tensile
strength. The theory developed in [7] was used in a later
paper by Karenlampi [8] to perform simulations for
some hypothetical sheets using simplified assumptions,
but did not compare the strength predictions to experimental data or predictions using previously published
strength models. Feldman et al. [9] developed computer
simulations based on the Monte-Carlo method using
random numbers to arbitrary position up to 10,000 fibers across a scan line. Shear-lag expressions based on
CoxÕs [10] and Page and SethÕs [11] works were used to
obtain the distribution of fiber stress along its length.
The shear stress at the bond sites was determined from
equilibrium. The simulation can accommodate fibers
with different length and strength and varying bond
strength.
On the experimental side, the determination of several structural parameters such as relative bonded area
(RBA) and fiber–fiber bond strength, fiber strength,
and fiber pull-out forces are still far from routine and
standardization. It has also become evident that the
zero-span tensile test for estimation of fiber tensile
strength is in violent contradiction to Saint–VenantÕs
principle and is severely influenced by several uncontrolled test variables such as unavoidable slippage in
the jaws [12,13]. This test method thus cannot be used
for accurate measurement of parameters governing paper strength.
The approaches carried out by Karenlampi [7,8] and
Feldman et al. [9] have substantially improved the
understanding of the failure process of paper, but the
complexity of the integral equations and extensive computer simulations possess severe obstacles for day-today use of paper analysts and designers. The approaches
developed by Kallmes et al. [1,2] and Page [3] are simple
and useful but are not built on a satisfactory representation of the failure process. We will outline here the simple shear-lag model for strength prediction of paper.
The model was originally developed for strength and
fracture work analysis of unidirectional short fiber composites. The similarity between a short-fiber composite
and a sheet of paper was first recognized by de Ruvo
et al. [6] who also suggested that the Kelly–Tyson
[14,15] model could be directly used for the strength prediction of paper. de Ruvo et al., however, did not incorporate the very important bonding parameter, i.e.,
RBA, in the analysis, and they did not examine predic-
tions of the strength of paper using the shear-lag model
in a quantitative manner. We will here examine the
shear-lag approach in some detail. Parametric studies
will be presented, and the ability of the model to predict
strength of paper is examined and discussed.
2. Shear-lag model
Davison [16] conducted single fiber pull-out experiments on various papers and found that fibers may be
extracted intact even from well-bonded sheets, and suggested the weak link in paper strength is the shear
strength of the fiber/fiber bond. Fiber pull-out in the
context of tear strength was later studied by Yan and
Kortschot [17] who found that the energy consumed in
the pull-out process is dominant, but that the work involved in fiber fracture also contributes to the tear
strength. The mechanism of simultaneous pull-out and
fracture of fibers fits into the framework of the shearlag theory developed by Kelly and Tyson [14,15] who
examined stress condition and energy dissipation factors
associated with fiber and fiber/matrix interfacial failures
in aligned (unidirectional) short fiber composites. The
energy dissipation mechanisms they examined are fiber
pull-out and fiber/matrix interfacial debonding, where
pull-out requires supply of frictional work due to pulling
out the fibers from the matrix, while debonding refers to
fiber/matrix bond breakage. Further research into the
contributions to the fracture work of paper due to fiber
pull-out and breakage has been presented by Karenlampi and Yu [18]. The shear-lag approach is a good
candidate for analysis of the fracture energy of paper
loaded under in-plane tension, although in this paper
we will restrict attention to the tensile strength.
A central concept of the shear-lag theory is the transfer of load into the fiber from the surrounding matrix.
Fig. 1 shows a differential element of a circular cross-section fiber where a change in axial fiber stress, dr, is
introduced by shear stresses, s, acting at the fiber–matrix
interface. Kelly and Tyson [14,15] assumed that the
shear stress is constant along the fiber length at the instant of fracture, which corresponds to ideal plastic
yield, and a linear axial stress build-up in the fiber, see
Fig. 2. If the fibers, of uniform length (l), are short,
the axial stress in the fiber will not reach its ultimate va-
σ
σ
+ dσ
τ
Fig. 1. Element (free-body) of a circular cross-section fiber showing
axial stress build-up through shear at the fiber/matrix interface.
L. A. Carlsson, T. Lindstrom / Composites Science and Technology 65 (2005) 183–189
185
located symmetrically with respect to the fracture plane,
Fig. 3, will fracture rather than being pulled out,
lc ¼
rf d f
;
2sb
ð1Þ
where df is the fiber diameter, and sb is the shear strength
of the fiber–matrix interface.
The tensile strength of a composite reinforced with
aligned fibers of length l < lc (neglecting any contribution from the tensile strength of the matrix) is [14,15],
(a)
Xt ¼
(b)
Fig. 2. Shear and axial stress diagrams for short fiber.
lue, rf, before the fibers are pulled out of the matrix, as
illustrated in Fig. 3. Kelly and Tyson [14,15] defined a
critical fiber length, lc, as the length beyond which a fiber
V f sb l
;
df
l < lc ;
ð2Þ
where Vf is the volume fraction of fibers. Notice that
when such a composite fails, all fibers bridging the failure plane will pull out from the matrix as shown in
Fig. 3. For a composite with fibers longer than the critical length, however, only fibers with the ends within a
distance of lc/2 from the fracture plane will pull out from
the matrix while fibers with the ends further away will
break. Statistically, if the fibers are distributed uniformly
throughout the volume, the fraction of fibers that will
pull out is lc/l, while the remaining fraction will break.
The tensile strength of the composite becomes [14,15]
lc
ð3Þ
X t ¼ V f rf 1 ; l > lc :
2l
Turning the attention to the fracture process of a
sheet of paper, the shear-lag approach needs modification to accommodate the discrete nature of the fiber network and the random in-plane orientation of the fibers.
Consider first a strip of paper of cross-sectional dimensions b and t (width and thickness), consisting of a number, Nf, of unidirectional long fibers that all break
simultaneously along the ‘‘scan line’’. The volume fraction fibers in the sheet composed of fibers and voids is
given by
Vf ¼
qs
;
qc
ð4Þ
where qs is the apparent sheet density, and qc is the cellulose density. If interactions between failing fibers is neglected, the tensile strength (in N/m2) of such a paper
becomes
Xs ¼
Fig. 3. Pull-out of fibers (l < lc) during fracture of a short fiber
composite.
N f F f qs
¼ rf :
bt
qc
ð5Þ
For an actual sheet of paper consisting of short fibers
loaded in tension, however, some fibers will be pulled
out while some fibers will break during the fracture
process. The critical fiber length, lc, corresponds to the
length when shear stresses at the fiber–fiber bond sites
transfer sufficient load into the fiber to make it fail in
tension. The total shear force transmitted to a fiber of
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L.A. Carlsson, T. Lindstrom / Composites Science and Technology 65 (2005) 183–189
length l (<lc), through the fiber–fiber bonds, is calculated
following the Kelly–Tyson approach and the loading
diagram in Fig. 2
The theoretical maximum tensile strength of a random sheet, T 1
s , where failure occurs due to fiber failures
only (l ! 1), is, according to Eq. (12b)
l
V ¼ sp RBA;
2
T1
s ¼
ð6Þ
where s is the (uniform) shear stress at the fiber/fiber
bond site, p is the perimeter of the fiber cross-section,
and RBA is the relative bonded area, i.e., fraction of
the fiber surface area occupied by bonds [3]. Critical
conditions, defining the critical fiber length, lc, occur
when the shear stress equals the fiber–fiber bond shear
strength, sb, and the shear force V, equals the tensile
load at failure of the fiber, Ff. Eq. (6) yields,
2F f
:
lc ¼
sb pðRBAÞ
ð7Þ
During fracture of the short fiber (l < lc) paper
(Fig. 3), each fiber will be pulled out over a length
between 0 and l/2, in average l/4. The average load
per fiber associated with fiber pull-out is (Eq. (6)),
V ¼ sb plðRBAÞ=4:
ð8Þ
The failure stress of the sheet corresponding to fiber
pull-out is obtained from Eq. (5) by replacing rf with
V/Ac, where Ac is the cross-section area of the fiber
Xs ¼
qS sb plðRBAÞ
:
4qc Ac
ð9Þ
For random in-plane orientation of the fibers, the
unidirectional strength given by Eqs. (2) and (3) must
be reduced by a factor. Page [3], Davison [16] and van
den Akker et al. [19], used a factor of 3/8, while de Ruvo
et al. [6] and Feldman et al. [9] used 1/3. Although the
difference is significant, experimental data on paper
strength are not precise enough to pinpoint the exact
factor to use. We used a factor of 3/8 here unless stated
differently. Furthermore, due to the voidy structure of
paper, the strength is commonly expressed as ‘‘tensile index’’ or ‘‘specific strength’’, which is the ultimate stress
divided by the apparent sheet density, qs
Ts ¼
Xs
:
qs
ð10Þ
With the incorporation of an isotropy correction factor
(3/8) and conversion to tensile index, the strength is given by
3sb plðRBAÞ
; l < lc ;
32qc Ac
3rf
lc
Ts ¼
1
; l > lc ;
8qc
2l
Ts ¼
ð11aÞ
ð11bÞ
where the critical length, Eq. (7) is
lc ¼
2rf Ac
:
sb pðRBAÞ
ð12aÞ
3rf
:
8qc
ð12bÞ
At l = lc, T s ¼ T 1
s =2 (Eq. (12b)) with no strength discontinuity at l = lc. This may, perhaps, be counter intuitive, but follows from the definition of the critical fiber
length, Eqs. (12).
3. Parametric study
Here, we will briefly examine the influence of some of
the key parameters governing the sheet strength, viz.
RBA, fiber–fiber bond shear strength (sb), and fiber
length (l). We will keep the other parameters constant
at values given by various literature sources. In particular, we will assume that the cellulose density, qc, is fixed
at 1.55 · 103 kg/m3 [16], the fiber cross-section, Ac is
2.54 · 1010 m2 [16], and the fiber perimeter, p, is
9.0 · 105 m [3]. The fiber–fiber bond shear strength,
sb, has been measured for chemical pulp by Schniewind
et al. [20], and for thermomechanical pulp by Thorpe
et al. [21] as also reported also by Uesaka et al. [22].
Both types of pulp fibers showed large variability in
sb; Schniewind et al. [20] found an average shear
strength of 2.52 MPa with a standard deviation of
84% of the mean, while Thorpe et al. [21] found shear
strength values between 3.20 and 13.4 MPa, with an
average of 8.12 MPa.
Fig. 4 shows the paper strength, Ts (normalized by
T1
s ), calculated using Eqs. (11) plotted vs RBA at two
values of the bond strength (2.5 and 5 MPa). Fig. 4(a)
shows the influence of fiber length on the tensile strength
at a bond shear strength, sb, of 2.5 MPa. The open circles at T s =T 1
s ¼ 0:5 represent the transition from failure
entirely controlled by fiber pull-out to failure involving
fiber fractures. It is noted that the strength increases
more rapidly with RBA, and approaches higher levels
when the fiber length increases. However, the ideal
strength ðT 1
s Þ is not approached even for ‘‘fully
bonded’’ sheets (RBA = 1) for the range of fiber lengths
shown (2–3 mm). This is apparently due to the low bond
strength. Thus, to better utilize the strength potential of
the fiber, sheets with weak fiber/fiber bonds require long
fiber lengths. Fig. 4(b) shows the strength vs RBA for a
sheet with 2 mm long fibers with stronger fiber–fiber
bonds (sb = 5 MPa). Comparing the results in Figs.
4(a) and (b) reveals that the strength of the more
strongly bonded sheet develops much more rapidly
(the slope of the curve is increased by a factor of two),
and reaches higher values, close to 90% of the ideal
strength, at RBAs close to unity. Consequently, the
L. A. Carlsson, T. Lindstrom / Composites Science and Technology 65 (2005) 183–189
(a)
(b)
Fig. 4. Strength predicted from shear-lag model: (a) sb = 2.5 MPa;
(b) sb = 5 MPa.
187
beaten in a ‘‘pebble mill’’ to various beating degrees in
an effort to preserve the fiber length. They conducted a
painstaking effort using light scattering techniques to
determine the RBA for the sheets. Unfortunately, Ingmanson and Thode did not measure the fiber length,
bond strength and fiber strength. The zero-span tensile
strength data could possibly be used to estimate the fiber
strength, but recent analysis on this test [13] advice
against such procedure. The bond strength should be
fairly constant, and a fiber length of 2 mm should be
reasonable [3]. We will further assume that the cellulose
density (qc) is 1.55 · 103 kg/m3 [16], fiber cross-section:
(Ac) is 2.54 · 1010 m2 [16], and fiber perimeter: (p) is
9.0 · 105 m [3]. The bond strength and fiber strength
are left open as fitting parameters.
Fig. 5 shows tensile strength data for classified pulp
obtained from Ingmanson and ThodeÕs [23] study plotted vs RBA, along with a ‘‘best visual fit’’ theoretical
curve based on Eqs. (11) with rf = 360 MPa and
sb = 3.4 MPa. The fiber strength (rf = 360 MPa) is within the range measured by Davison [16] (267–668 MPa
for 170 fibers), and the bond strength (sb = 3.4 MPa)
is well within the range found by Schniedwind et al.
[20] (2.52 ± 2.12 MPa). Based on these parameters the
shear-lag model provides quite an excellent description
of measured strength.
Watson and Dadswell [24] determined a unique set of
experimental data on the on the influence of fiber length
and degree of beating on the tensile strength of chemical
pulp papers. A set of different pulps were prepared from
chips cut at different lengths prior to delignification so as
to obtain a set of papers with narrow fiber length averages. Watson and Dadswell prepared papers from the
pulps produced at various fiber lengths refined (beaten)
for various amounts of time in a Lampen mill (0, 18, 36
and 72 min). Fig. 6 shows a graph of tensile strength vs
strength of the fiber bond is an important parameter
also for highly bonded sheets.
4. Comparison to strength determined experimentally and
by simulation
In this section we will examine how predictions of
strength using the shear-lag model compare to previously published experimental data, and simulation results. The prediction of tensile strength using Eqs. (11)
requires the fiber strength (rf), shear strength of the fiber–fiber bonds (sb), density of cellulose (qc), fiber
cross-sectional area (Ac), fiber length (l), perimeter, p,
and RBA, all in all seven parameters, most that are difficult to measure.
A rare set of consistent tensile strength and RBA data
for bleached chemical papers was presented by Ingmanson and Thode [23]. Their bleached sulphite pulp was
Fig. 5. Experimental and theoretical strength data for bleached
sulphite pulp paper. The experimental data are obtained from [23].
188
L.A. Carlsson, T. Lindstrom / Composites Science and Technology 65 (2005) 183–189
10
RBA=0.90
8
RBA=0.50
RBA=0.30
Ts 6
km
4
RBA=0.16
2
Data from Watson and Dadswell, 1961
0
0
2
4
6
Fiber Length, mm
Beating Time
:
:
:
:
72 min
36 min
18 min
0 min
of high relative bonded area, unless the fiber/fiber bonds
are very strong. Comparison to experimental data was
not possible in a rigorous manner because some of the
important parameters remain undetermined. The comparisons presented within this ramification, however,
show very reasonable fits to published strength data.
The simplicity of form and physically sound foundation
of the model make this approach very appealing as a
candidate not only for strength analysis but also for
modeling of the fracture work of paper.
8
Fig. 6. Experimental and theoretical strength data for chemical
papers prepared from araucaria klinkii chips. The experimental data
are from [24].
fiber length. The points are the experimental data and
the curves are obtained from Eqs. (11) by changing only
the RBA. The data used in the predictions were the same
as for Fig. 5, except for the fiber strength, rf (400 MPa),
and bond strength, sb (2.5 MPa). It should be pointed
out that refining (beating) may possibly increase both
bond strength and RBA. The RBA and bond strength
appear as a group (product) in Eqs. (11) and (12). Hence
each curve shown in Fig. 6 could be represented by a
change in bond strength.
A final comparison is made to the simulation results
of Feldman et al. [9]. They considered a hypothetical
isotropic unbleached pulp sheet consisting of 2.235 mm
long rectangular cross-section fibers of 35.6 lm width
and 3.5 lm thickness, with a tensile strength of 200
MPa. The fiber–fiber bond strength was 6 MPa and
RBA was 0.306. Further, Feldman et al. used an
isotropic reduction factor of 1/3, rather than 3/8. With
a reduction factor of 1/3, and the above data Eqs. (11)
yield a tensile strength of 37.9 MPa with 83% of the fiber
broken and 17% being pulled out. Feldman et al., using
their computer simulation analysis, obtained a strength
of 33.4 MPa and 23% of the fibers pulled out. We cannot
judge which result is more accurate since no experimental data are provided. The results are surprisingly close,
considering the simplicity of the shear-lag model put
forward here with a constant shear stress and neglecting
interactions between fibers.
5. Concluding remarks
The applicability of the shear-lag approach to the
tensile strength of paper has been examined. The analysis clearly indicates the importance of fiber length for
weakly bonded sheets, and also the role of relative
bonded area as a mean for utilizing the strength potential of the fibers. It is also demonstrated that a significant extent of fiber pull-out is expected also for sheets
Acknowledgements
The financial support for this work from the Foundation for Strategic Research (SSF) through the Forest
Products Industry Research College, Sweden, is greatly
acknowledged. Several persons assisted with various
things throughout this study, i.e., Dr. Doug Coffin (Miami University), Dr. Nicholas Shevchenko (University of
Delaware), and Mr. Francis Aviles (FAU). We thank
Ms. Trudy Jefffries (FAU) and Ms. Melissa Morris for
typing, and Ms. Shawn Pennell (FAU) for the artwork.
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