Presented at 1999 Paper Physics Conference San Diego Determination of Load Determination of Load - - Bearing Bearing Element Length in Paper using Element Length in Paper using Zero/Short Span Tensile Testing Zero/Short Span Tensile Testing Warren Batchelor Australian Pulp and Paper Institute Dept of Chemical Engineering Monash University
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Presented at 1999 Paper Physics Conference San Diego
Determination of LoadDetermination of Load--Bearing Bearing Element Length in Paper using Element Length in Paper using Zero/Short Span Tensile TestingZero/Short Span Tensile Testing
Warren BatchelorAustralian Pulp and Paper Institute
Dept of Chemical EngineeringMonash University
Presented at 1999 Paper Physics Conference San Diego
IntroductionIntroduction
Paper strengthEffect of drying treatmentDefects in fibresLoad-bearing element
Presented at 1999 Paper Physics Conference San Diego
LoadLoad--bearing elementbearing element
NOT a fibreFibres can be made up of many elementsJoined by kinks etc
Properties:Length, lCross sectional area, CYoung’s modulus, E
Presented at 1999 Paper Physics Conference San Diego
Test and Sample DimensionsTest and Sample Dimensions
Lsheet
Wsheet
WJ
G
Jaw
1
Jaw
2θ
A=LsheetWsheet
Presented at 1999 Paper Physics Conference San Diego
ProbabilitiesProbabilities
Single load-bearing element in sheet, length, l, angle: P1: probability that element is gripped by Jaw 1P2: probability of element gripped by Jaw 2 if also gripped by Jaw 1
Assumption: Wsheet, Lsheet, Wj much greater than l.
θ
sheet
j
sheet WW
LlP θcos
1 =
lGP / cos12 θ−=
Presented at 1999 Paper Physics Conference San Diego
Short span testShort span test
G
l
G
GG ∆+
GG∆
=ε :strain Overall
Presented at 1999 Paper Physics Conference San Diego
Load on single fibreLoad on single fibre
curl plane ofout todue load bearingnot fibres offraction is where
coscos2)1(
or is fibre oriented located,randomly by on contributi Average
cos
isdirection loadingin component and
cos
is jawsboth spanning fibreon force , strain,At
)/(cos
0 34
21
3
2
1
c
lGjcav
f
f
f
f
dlG
AlW
ECfF
PPF
ECF
ECF
θθθπ
ε
θε
θε
ε
∫−
−−=
=
=
Presented at 1999 Paper Physics Conference San Diego
bonding fibre-fibre Non orientatio Random 7.0sAssumption
(2) 932
83)1(
:obtainexpansion seriesTaylor Use
(1) 1247arccos
83
11212)1(
get gIntegratin
2
23
lG
GlA
WECfF
lG
lG
lG
lG
lG
AlW
ECfF
jcav
jcav
<
−−=
−−+
−
−−=
πε
πε
Presented at 1999 Paper Physics Conference San Diego
Comparison between equations 1 and 2Comparison between equations 1 and 2
Approximate limit for accuracy of Taylor’s series approximation
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1G/l
Forc
e (a
rbitr
ary
units
) Full function (Eq 1)
Taylor seriesexpans ion (Eq2)
Presented at 1999 Paper Physics Conference San Diego
( )
−
−=
GGCGE
GlGCGEA
WfGIGF j
c
)()(932
)()()(83 )1)((,
π
εε
Total forceTotal force
For gap, G, if have I(G) load bearing elements with
then the total force, F, is
where is the average of the I(G) elements
π9/32Gl >
)()()( GlGCGE
Presented at 1999 Paper Physics Conference San Diego
Average length of elementsAverage length of elements
Assumptions:)0( )0()0()0()0()0( lCElCE ≈
lGGGlGCGE << if oft independen )( ,)()(
0
),(
)0,(932)0(
=
−=
G
frac
frac
dGGdF
Fl
ε
ε
π
Presented at 1999 Paper Physics Conference San Diego
Comparison of loadComparison of load--bearingbearingelement length (mm) with element length (mm) with
arithmetic and lengtharithmetic and lengthweighted fibre lengths (mm)weighted fibre lengths (mm)
Presented at 1999 Paper Physics Conference San Diego
ConclusionsConclusions
Load-bearing element length approximately the same as arithmetic fibre length for these pulpsDrying treatment- no effect on average load-bearing element length
Presented at 1999 Paper Physics Conference San Diego
AcknowledgementsAcknowledgements
Daniel Ouellet, Paprican and University of British Columbia Pulp and Paper CentreRussell Allan, Amcor Research and Technology CentreGlenda Spencer, Michael Briggs and Geoff Ennis for assistance with the experimentsFunding from the Commonwealth Government of Australia through the Cooperative Research Centre for Hardwood Fibre and Paper Science