Seo is with International Paper Co., Long Meadow Rd., Tuxedo Park, NY 10987. .... does not change the location of the .... increased to l"2E,uand 0.88r, the loca-.
Paper Physics
An optimization approach for the determination of i*ptane elastic constants of paper Yung B. Seo, B. Castagnede, and Richard E. Mark
ABSTRACT An amffate mahodfor the dztermination ofthe inplane shear modulu G, ofmachine-madr paper is giuen here. This work is the extension ofan earlier papn where it wds s/town that G* and the in-plnne Poisson ratio n * Are recouerablr fom measuremenis of ,ht Young moduhu E. and thb shear modufu G" conduaed at uarious oientation angbs. Howeuer: reliablz ualues for G" are ueryt dfficuh to obtain. Hne we show that G - can be accurately extraaed with onty afrw datafor E uersus orirnt/tion anglz plus oi, good ualueforr . Tbe kttn may be measured by a lner speckb photograpby technique. The influence ofthe number ofdata, the precision ofthese data, and the angular spacing benaeen data points are fuscribed by numerical simulntions based on large perturbation analysis. Also, a simple riterion is establishedfor the optimum angulttr region (the so-callzd minimum enor boundnryl where the needzd measurements should be performed to achieue the uery best accuraqt in the determinatio, of G;
I{EYWORDS: Anisotropy, e ltuticiqt, e latic snength,
equations,
optimiztztion, paper Poisson ratio, quantitatiue arnlysis, shear snength, simuLttion.
T I
tisu'ellknoulthatthedetermin-
ation of some of the in-plane elastic constants, specificalll, G-. and 1.,. of
papel' matelials is intricate. For instance, G... can be measut'ed by the cirlinder method, the torsional pendu-
lum method, the off-axis tension method, and so on. All these methods present difficulties (1, 2) and are sometimes ver] inaccurate in their results due to systematic elTors in thickness measurements that can amnlifv the
en'or of the shear modulus measurement(3,.D. Soundlybased procedures for accurate detetmination ofthese elastic con-
stants of paper materials are highly desirable. G and other elastic constants have rblevance for the solution of engineering design problems where paper is used stmcturally. Also, such parameters are valuable for comparing theoretical predictions for paper mechanical behavior rnith exper{mental results. We start u.ith an assumption of orthotropic synmetry for paper
(1, 5). Laser speckle photography (LSP) (g 6) pror,-ides precise estimates of the in-plane Poisson ratios (better lhanl07o in absolute precision). Hov,ever, such precision is delusir,e for inplane shear moduli, as rve have noted.
In an earlier work, in'e suggested that G.. could be deterrnined indirectly
0).
Tliis paper reports on progress made in pursuit of this idea.
Optimization of shear modulus Experimental values for elaslic constants are usually scattered around the "true" elastic moduli because of metrological errors and specimen fluctuations. We use a least-square algo-
rithm to estimate the shear modulus that is best fitted to our erperimental data. This algorithm is given in the appendlx. Equation 1 is the basic equation that describes the relationship be-
Seo is with International Paper Co., Long Meadow Rd., Tuxedo Park, NY 10987. Castagnede is with the Laboratoire D'Acoustique de I'Universit6 due Maine, Ave Olivier Messioem, B.P. 535,72017 Le Mans Cedex, France. Mark is senior research associate with the Empire State Paper Research Institute, State University of New York, College of Environmental Science and Forestry, Syracuse, NY 13210. Seo and Castagnede were postdoctoral associates at ESPBI where this work was done.
tween elastic constants
in
tu,'o-dimensional orthotropic case (\ee equation I
Reprinted from Tappi Journal, Vol. 75, No. 11, November 1992. Copyright 1992 by TAPPI, and reprinted by permission ofthe copyright owner.
)
the ).
(7
Paper Plrysics w'here
Equations:
Er_, = Young modulus
measured in
uniaxial tension test ofoff-axis stlip (angle O from the macl'rine clirec-
tion)
E,n,r
:
Young modulus measnred
I cos*O sinoO F= -F--+ L^LE.
cos'O sintO
wnd:n
irr
-+
uniaxiai tension test of machine directional strip (x direction)
Young rnodulus measured in uniaxial tension test of cross machine directionai strip (y direction)
t""l (l)
Lq-TJ
,or2o rinta
G.r'=
E,,t :
f'
1[ ,or4o
trL
@ 'u^, t**T-q ,in4
G,, = In-plane shear modulus u.,. : In-plane Poisson ratio measured
.
.orr"
,rrtol
(2)
I
in unia.xial tension test.
Ifone has a given data set for E,,Enu, {., and u,,, then Eq. 1 can be solved for G_r,i
(see equation 2)
and the differences were piotted
The best strateg* to obtain a reliable shear modulus G_. is to choose a procedure such that th8 coeffrcients ofvariation (CO\D of the experimental values of
E (includingE.o and -8.,,) and
u_u
do
not have too much effect on the COV of
the estimated shear moduius. If the O sin: O (which is associu", in Eqs. 1 and 2) approaches zero, the effects of the COVof the Poissonratio on the COVof estimated shear modulus uill decrease. The term cos2 O sin2 O has its maximum at 45o from the machine direction ani decreases as O goes to either the MD (0') or CD (90"). Horvever', it is not so simple to minimize the effects of the COVofYoung's moduli on the COV of the estimated shear modulus, and il is necessaty to find a s1'stematic rvay'to clo it. Ifrve use three Young's moduli--E,nt,8,,1, E oand one Poisson ralio, the shear modnvalue of
cos2
ateduith Poisson ratio
lus can be uniquely deternined either trq.
1
b1'
or2. To see the influence of
the experimental en'or. of 8.,-, on that estimation of shear moduius, 1.1 r Euand 0.9 x E'..lr-ere used in the equation, r'r-here the given constantvalues of E,u,r, 8",r, fl and u.,. r'rithout pertr"rrba", tion u'ei'e inselted, as O changes fi'om zero lo 90'. The shear modulus estimated in this \1'a)'was compared uilh the gir.en one,
210
No".^rb"r 1992'Iappi Journal
as
percentages in Fig. 1 and aiso in Fig.2 (dotted lines). where.9,., 5.52 GPa,
: : 2.03 GPa, and u,,
G,,
E
: ",,:2.34GPa, 0.5 rvere assumed. in Fig. 1, u'here
was used to caiculate the shear modulus, one may obsen'e hr.o spikes, centered around g' atid 78u, that appear u,'hen error esti1.1 x -8"
rnations of shear nodulus are perl fonned. These anornalous spikes coltrastlrith a generally smooth, rather flat curve
of low
error. The dotteri iines
in lrig. 2 proricle a magnified r.ier,v of this en'oi (1.1 x ,E,.'nvas uscd for lhe E._, for +,he lor,r,er cuite). inn-hich the resultsforthe r.ange 15' to 70' are plotted-tiris range is
upper cunre and 0.9 x
appropriate for experiniental.,r.ork and will avoici the i'egions of the anomalous spikes.
What is the i'eler,ance of lite arionialous spikes? It can be leadily ciernon-
strated that spike localions
at'e
fitnctions of sigrrificant (large) vai'.res of A,O". Fiquatir-rn 2 can be
reuritten inthe
form:
If one is simulating experimental errors forE^, it is clear that the above fraction will become infinite as the value of ll E the r.alue "approaches of cr , and is undefined tf E = 11o" . positive.
^
tuns out that u,hen O : 9", the relation shoun in Eqs. 2 and 3 will bioiv up Frorn nunrerical substitution, it
(make a spike) rvhen E is measu,r'ed at ol' nea-r' 9" i .ith an elror of the magnitude assurned in this case ( + 10% ). At t-ri' near O: 78". one obtains the same trend for that enor condition. Spikes do not occut' betu'een 0'and 90" rvhen LU n. = 0, but do occur if A-U" is not infinitesimal. The presengs 6f -spikes does not change the location of the
minimum en'ol boundary' (the minimum dislance betueen the upper and lolr,er curr,es), i.vhich for the case just de-qcribed (and shorln b"l'the dotted iines) js located at approximately33"in Fig. 2. F'oi' the least en'or in estirnating G. under these error constraints, $re should use 'rhe measurecl Young's r:roduii in the of 38'. The au".icinity' lhols have t'ounrl the minimum error' bounriarn concept quite usefui and have
,
n-rlo ui,
o
-=-- c{ I
(3)
(J
used the simulation procedure extensiveil, to veriff it.
The angle of the rninimum er-ror boundar-r' is dependent upon the elastic
In Eq.3, it can be showl thai 1, 0. is aiwa"vs positive, and l,r,Er- u is alu-a., s
constants of the material. Poisson ratio does not affect tiris angle. The anisot-
lopl' of shear moduli, u'hich desclibes
1. The shear modulus (G,,) estimation error in percent when the value 1.1 x E is substitutedfor F"in Eq. 2. Theanisotropyof Young's moduli
2. The shear modulus estimation error In percent when 1.1 x E substituted for E"(concave upward curves) and when 0.9 x E"
is 2.36.
substituted (convex upward curves). The anisotropy of Young's moduli is 2.36 for the dotted lines and 5.0 for the solid line curves, respectively.
z
z o
9 F
F
Eo Fo
=^.
0.75
tr'd ouJx
oF ITJ
is is
X
qs
(/,>g
-^
DH
?6
OF vE =ul tr
Eff
uJ
UJ
tr
I a
I
at,
-0
1s
30
45
75
60
10
90
/10
30
50
60
70
ANGLE FROM MACHINE DIRECTION, degrees
ANGLE FROM MACHINE DIRECTION, degrees
3. Angle of minimum error boundaryvs. anisot-
20
l. The COV of shear modulus estimated at different angle centers with five Young's moduli
ropy of Young's moduli /:\-
Angle
E.^
ffH
qz
=-Y 2d
sg
ratio
Eb) fg
6= =5 14o do z
Jo
2
ANISOTROPY OF YOUNG'S MODULI
Mean
Poisson perturbation
Mean
5% 10"/" 207o 307"
2.105
28.5
2.044
2.156
32.1
2.158 2.017
34.9 24.4
1.969 2.030
Poisson
COV,
ratio
Mean
cov,
2.075
2.027
2.138 2.008 2.04
8.88 8.28 1 1.3
12.4
2.094 2.038 2.093 2.080
7.77 7.57 9.60 12.0
COV,
5.92 7.16 8.98 10.34
@=60o
Mean
Mean
COV,
ha/'t trhalinn
5o/" 10% 207" 30%
Mean
COV,
o/ /o
O=45'
Angle
o=38'
o=30"
1Fo
2.044 2.236 2.030 2.048
o/ /o
COV, %
-z.5JZ
19.6
0.322
23.3 17.5
:
26.5
*:
too much scattering, not worthwhile to report. E^. = 5.52, E^^ = 2.34. G.. = 2.03, u. = 0.5 were used. Y6Lng s modiilus was p6iturbed to 6q" in cov. Poisson ratio was perturbed lo 5'k, 1O"k, 2O"k, and 30% in COV.
the variation of in-plane shear moduli with orientation in the sheet (6), does affect the location of the point. Horvever, for paper materials, the anisotropy of the shearmoduliis restricted to values from 0.992 to 1.169 experimentally. A range of anisotropy of shear moduli from 0.9 to 1.2 u.as therefore assumed 16'. 8). The result of this exer-
cise $'as that the location of the minimum error boundary was affected by less than 1o, u'hich is quite inconsequential. Thus, for paper materials, it can be safell'said thatthe anisolropy of shear moduli does not actually affect the location of the point of minimum en'or boundarl' signifi cantll'.
Hou'ever, the anisotropy of Young's
moduli. r.vhich is defined as the ratio of E u,,tto E,,r (6, 8), has some influence on
the location of the point of minimum
en'or'.
ln Fig. 3. the anisotlopl' ol
Young's moduli is 2.36 for the case sho\ 'r') by dotted lines ancl 5.00 for the solid lines. The angle fol the uinimum en'or boundarf is ^- 38'for the formei' and - 34'for the latter. ir'ovember 1992 Tappi
Jot."d 2l I
Paper Pbysics A master curue that includes the minimum er:ror boundary values for the above two conditions is shoun in Fig. 3. This illustration sho',r's that the minimum er"ror boundary moves to a Ior,ver angle as the E
't' (91 data pontt)
lnteNai anglg
Mean
Poisson ratio
CCV,
peftufuation
the angle of the minimum eror boundary if the anisotropy ofYoung's moduli has been obtained by experiment. To
obtain a shear modulus ll'ith a smail COV, one can measure the Young's moduli around this angle and calculate the shear modulus w-ith the equations in the Appendh. If one performs the same exercise
5%
2.037
1OT"
2.135 2.234 2.119
15% 20Yo
lnterval
Poissan ratic perturbation
E./e., = 5.0 and assumed
er-ror increased to l"2E,uand 0.88r, the location of the minimum errorboundaryis nol changecl, but the magnitude of er. ror is more than doubied.
Simulation approach
angle
15"/'"
20%
f
(19 data points)
Mean
LVV.
%
anisotrop-v
^/E"o increases. With this curve, one canfind
-with
ll. The COV cf shear modulus estimated fi.om Young's moduli raken at sei rnter.vats
_
1A (10 data
11.29 14.24 21 .26
2.128
19.77
2.020
COV, %
2.033 2 072 2.039 z.az
9.493 16.79
20.49 16,70 15.70 17.08
z too 2-026
points)
Mean
I
%
__
ltr
17 data pcints)
COV
2.146
1
i0
|
2.0
17.75 17.65
t.88 4.ZJ
2.125
14.87
2.039
14.52
E-o = 5.52. E,.- 2.3a. G,. = 2.O3. r,,, = 0.5 were used. Young's modulus was perturbed to 59. in COV. Poisson ratio was perturbed to 5%, 10%, 15%, and 20c./. in COV.
The simulation procedure is nowintroduced to show the effectiveness ofthe minimum en"or boundary. The proce-
dure generates perturbed Young's
moduli and a Poisson ratio compulationally as foilows:
.
1.
For a set of initial given param-
eters
(i.e.,,O,n,1,
t".,
G.,., u,. ),
the_8"'s
ale calculated.
.
2. These -E.o's and u.,. are
perturbed
b1 the use of a normal distribution and a certain COV, e.g., 10%.
.
With the optimization algor"ithm clescrdbed in the appendlr, G,, is
3.
calculated and compa.ed ro the value
before perturbation.
The perturbations used are large (on the order of 107o or more in COV). It is important to under.stand the char-
acteristic changes (spikes) thatwill oc-
cui' in G . u-hen noninfinitesimal pelturbations of .Uo and u., ar.e assurned: the lirocerlure .rields lelilrble foi'G,,. The t.e-qults ar.e compatible ri-ith those obtained u.hen infinitesitnal pettulbations are considet.ed. but the lattel conditions are generallr. r-alLres
the ltractical capabilities of mea.ruling elastic constants exper.ibe1'oncl
mentzrlll'. To use the perturbation pro-
212
No.=r,b.r 1992 lappiJournal
cedure, four elastic constantS, E..1, E"n, G,., and u\. \\'ere set up for a tlpical paper. These values are shourr in the
labies and are derived from erperiments. Second, i-._^, f,", ancl i',_,*. u,ere calculated from the gir-en elastic coitstants by using Eq. 1. The angle O was r arierl flom 15'to 75". Thirrli.r'. ihc fir'c
Young's moduli, 8.n1, Eo;, Eo, 8.,--, and -&",,, and the one given Poisson
ratio rvere perturbed, t'espectivehr. within the nonnai clistribution -co that the COV of the Young's moclnli u-as 5%.The COV of the Poisson latio rl as perturbed to 5% , l\c/c , l.1cr'r .20% , ancl 30r/.. With these perturbecl Ycrmg's moduli ancl one pertut'becl Poissorr latio, the shear modulus u as calculated. This procedure'uvas replicateci 20 tinres for one COV of Poisson ra.tio fcr ever;; angle O. Aftei' that, the COV of the
and 45', lhe COV of shear moclnlus is sirrallel than the othei's. For O : 75o, the shear ntodulu-q is estirnated poor'11',
because the erroi' bonndarr. ai.ouncl 75"is too high to be r-i,qecl in its e-qtimatioti. (See Fig. I fcr high spike at 7E'.) The COV of the estirnatecl sheai' moclr-rlus at O : 38'ls seen to be the "qmallest for ail czrses. in accoldance u'ith erpec-
tations.
It
is u'orthu'hile 'lo consider
tr
hat
happens if ail the Young's moduii, at a
certain constant inten. a1 from lhe machine clirect"iiin to lhe closs-machine clirection, are u..ed. I'ol that case. the (-lOV of the eslimated sliear modulus tvas gleatll, increaseci, as sholr.i in Table II. Thi-s occurs because per.trirbcrl value.rr of Yor-rng's motluli neai'
the machine dilection ot' the crossmachine clilecrion inllate the COY of
esrimatecl shear moduhrs u'as obtaineci. The lesuli,s of this sirnuiation ai'e snr"r-t-
rhe estimatecl shcar'mr.idtrlus on ing ilr Lhe extren:r:i1- large et'i'ol lio,.iirdaries
tnarized in Table I. The anisott'op1.cf this mater{alis 2.36 and the angle oftire minirnnm en'ot'hounclarf is i3B"in Fig. 2 (clotted lire erample). AtO : ii0",.j8',
at tho-qe a,ngle-.. We alsc finrl that the angular spacing of rneas'-rlellents i't'oni the ntini-
mlin] errol'
l-',c''undat'1.'
aff'ects t he CO\r
lll. The COV of shear modulus with five Youno's moduii (YM) centered on the minimum error boundanr of 38'
Angles for YM
aeterminaticn Interval of
(E
o,
angle
E"o)
Eu,Eu,E,,
(E_o,E"",
3"
Eu,Fr.,E"o) 5.
(E^d,
E2,,,
Eu,E*r,F".) 10.
Poisson ratio perturbation
Mean COV, %
Mean COV, %
Mean
5"k
2.026 5.23 2.056 7.93 2.063 8.1 I 2.005 .9
2.094 2.038 2.093 2.080
2"064 2.A49 2.080 2.160
107o
20% 300k
1 1
Angles for YM determination E^o E"", E"", Er",5". Interval of angle
E*
5.92 7.16 8.96 10.3
COV, %
6.25 6.93 12.4 12.6
E,u. Eu, E u, qo
tc-
Poisson ratio pefturbation
10% 20o/"
30%
Mean CAV, %
Mean
2.071 2.017 2.037 2.088
2.027 2.053 2.023 2.921
5.94 9.42 10.4 13.3
COV, %
10.6 12.1
14.3
12.3
E-" = 5.52. E."= 2.34. G.,= 2.o3. u., = 0.5 were used. Young s modulus was periurbed to 5.o in COV. Poisson ratio was perturbed to 5%, 10%, zo"k, and 30o/. in COV.
of shearmodulus estimation, as sho.,ul in Table III. The interval of 3" ma3' be too small to be practical. and makes no difference in COV of sheal moduilts
turbed 57o, the COV chailges fi'om 5.497r (seven values) to 5.92% ffive vaiues). The COV of the shear modulus does not change enough to justif,v the
estirration cornpared with an in[en'a] of 5". But, if the interr.al is toc lat'ge, there is a danger that at a certain
ef'fort of obtaining m.oddi.
angle, aYoung's modulusr,alue iiral'be
Summary
located in rire legion \\tel'e a large
elrol
It
hlo more
Yr-rung's
mine in-plane shear modrilus wilh high degree of coufirlence b3' using
lesult is a i'elativelv high COV of the
snali number of
e-stirnatecl sl-lear' modullls. After eramininst hi.l,ehn ior'. it i. c,rnchrdeci tlrat
utth one exllerimental r'alue of
5'is
practical and is
small enongh not to include point,r in he l;-r|ge ei'r'nI l,orrnda|r' r'egion. The authcrls alsovai{ed the uulxbei' ofpoints taken at'onnd the angle ofthe minimum ei'i'ol bontrdarl- and found that the cliil'elence bet'.r.een the COV's of sheiu'mor'iulus det'it'ecl froin seven t
Your-rg's nrot-luli,
e.
g., t,,,.,, 8
..r, 8,,,,,,
8,,r,
E ,,,,,E ,r,E ,,,or-ril those cierived fi'orrr fir-e
mocliili (8,,,,i,8.,;, Err.E r, d..) i.s small. I'r,r' 9r11p1111p. .,iith Poissorr l'lrtio lrer.
COV of estimated shear rnocluhn is relatively srnallrn-hen itis compared k-r the effect of the COV ofYound-. moduli. In practice, one maJl- selecl a value for
boundar'.v erists. When the intei. valis20'. one oftheYoung'smodulilies cio-qe to that lai'ge er'r'oi' region, and the
the inteir-al of
and -9.,i and a Poisson ratio, the shear modulus can be calcul.aled with great confidence. The effect of ihe COV of Poisson ralio on the magnitude of the
has been shor,rn thal one ca,n delera a
mea-qured Young'-s
rncduli in sevei"ai clirections, together Pois-
son ratio. Surprisingi-v, it is ncL alu'avs heiplhl to use a ialge nui-nber ofYoung's moilLrli clata oriented at r-arious angle.c
tc estimate the shear modulus.
Ii
is
bettel to select certain clirer:tions and use thos€i ciii'ecLrcnal Yi-'ung's nroduli
to get a reliable -*hear' mtiilulus. B\. calc:uiating the anisr-rli'ogr,v cf Young's
tn,rrlr;li i)ie :ing:le t) n1 ;1 r'nii1;ryiurn er! i'o1'boL1n(iarj, ca,n be cleteirnined. With live Your.rg's rrrorlu-ii I,,,_,, li,,, _-., Il,.r. E,_, . ..,
Poisscn ratio lrom the literature (5J appropr"iate to tiie paper furnish ancl in-plane anisotrolry. and use it in the estimation of in-plane shear moduius under the assumption that the difference fronr the true Poisson ratiouill be less than 20%. In such a case, lri'r"h the additional assumption of a5% errorin measuring the Young's mocluli, the inplane shear modr-rlus u'il1 be confi dentlrdetenninecl liith less than 10% eitoi'. It is aiso possible to estimate the en"ol involveri in the cleter"iiina'iion of i-]rc sheal murlulils hr rr.ing our sitlrrlaticn pi'ocecir"u:e if the erper-imerrtal erroi's ir.;v0l."'ed iir measunng Young's nrc,ciuii and P,:isson ratio nre knorr-.:r. ff
November l?92 Tappi
J"".""i 213
Paper Physics Literature cited 1.
Castagnede, B. and Seo, Y.
2.
70(9): 1 13(1987). Castagnede, B. and Mark, R. E., Study of an inhomogeneous strainfield for a translu-
B., Tappi J.
cent material b1'laser speckle photograph1', C. R. Acad. Sci. Paris,305 (Series 221(198?).
3.
4. 5. 6. 7.
8.
Ii):
Ta1'lor, D. H. and Craver, J. K., "Anisotropic elasticity of paper from sonic velocity
measurements," in Consolidation of the PaperWeb, F. Bolam (ed.), Tech. Sect. B.P. & B.M.A., London, 1966, p.852. Jones. A. R., Tappi 51(5):203(1968). Mann, R. W., Baum, G. A. and Habeger, C. C., Tappi 63(2): 163 (1980). Castagnede, 8., Mark, R. E., and Seo, Y. B., J. Pulp Paper Sci. 15(5): J178(1989). Jones, R. M., Mechanics of composite materials, Scripta Book Co., WashinpJton, DC, 1975, p. 5.1. C-astagnede, B.,
Mark, R. E., and
Seo, Y. J.
Pulp Paper Sci. 15(6): J201(1989).
glatefulll acknotvledge Empire State Papel Research Associates. Inc. (ESPRA) lbr ploliding a majol portion ofthe financial support The authors
lbt'this lesealch. Receir,ecl
fol revierv Aprii 24,
1989.
Accepted Feb. 15, 1992.
214 Not.-b.r
1992 Tappi Journal
Appendix With the objective of optimizing the in-plane shear modulus, G*, we proceed to utilize the terrns of Eq. 1:
Let
Xi : X2, :
,&'*d
i
-
The partial differential of the 2cos'zOsin2o.'-/
+ sin4oi/''"d*llg@i
= 1,2,3,...,n-2
where
n. = number
of Youngrs moduli used
foroptimizingS:1/GThe Euclidean norm, defined by the
expression:
is minimized for z determinations of Young moduli in order to opti-
mizeS.
coszOisinzOi
cos4@i/r'*,r
F(n,,Xzt)-L(SX +X2,):
above exTression with respect to S is: 2LX\(SXlr +X2,) = 0
and the calculated in-plane shear modulus G*- is: G.r. =
1/.s
=
-r
(xr)'D(xrX2,)
