NBSIR 79-1735 /t

Characterizing the Interfiber

Bonding of Paper Pulps:

Tensile Behavior of Some

Mathematical Models of Paper

Networks.

Jack C. Smith

Polymer Science and Standards Division

National Measurement Laboratory

National Bureau of Standards

September 1. 1978

Final Report Covering the Period

April 1, 1977 through September 30. 1977

Prepared for U.S. Department of Energy

Washington, D.C. 20234

NBSIR 79-1735

CHARACTERIZING THE INTERFIBER

BONDING OF PAPER PULPS:

TENSILE BEHAVIOR OF SOME

MATHEMATICAL MODELS OF PAPER

NETWORKS.

Jack C. Smith

Polymer Science and Standards Division

National Measurement Laboratory

National Bureau of Standards

September 1, 1978

Final Report Covering the Period

April 1, 1977 through September 30, 1977

Prepared for U.S. Department of Energy

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

SUMMARY AND CONCLUSIONS . , , » , . , . o , , 1

INTRODUCTION . o o » . » o o . . » o . . . 1

THE PARALLEL- SPRING MODEL = . o

.

' o . , » . . 2

THE SQUARE-NETWORK MODEL , , o , , = « » . . . 6

SQUARE- NETWORK MODEL CALCULATIONS . o . o o , . . 12

DISCUSSION OF RESULTS » o o .,o 15

REFERENCES o , o o « o . o o . . » . . 26

1

SUMMARY AND CONCLUSIONS

The behavior of a thin web-like paper network during

elongation has been approximately simulated by two simple

mathematical models. One of these is a parallel-spring model

consisting of a series of segments formed by connecting springs

in parallel between two rigid bars. The other is a squarenetwork model consisting of a network with square meshes form.ed

from springs of equal length. The square-network model

provides the better simulation of an actual network^ but a

complicated computer program is required to calculate behaviors.

The parallel-spring model is mathematically simple, and can

be used to suggest general behaviors which can be verified

by calculations for other more suitable models.

In these studies emphasis is placed on the effect of

breakage of a central bond when the model is stretched. The

resulting distorted mesh configurations, the drop in tensile

force, and the energy lost by the network are calculated.

An attempt is made to interpret the behavior of an actual

paper network in the light of these results.

There is an energy parameter E obtained experimentally

by averaging the energy losses incurred in a series of bond

breaks when a web-like paper network is elongated. This energy

parameter has been used to characterize the adhesive force

between fibers constituting the network. The model studies

suggest that the values of E are approximately independent

of the size and shape of the test specimens, provided the

mesh sizes of the specimen networks are the same.

The model studies indicate that a force drop parameter F,

analogous to the energy parameter, would be feasible for

characterizing adhesion. This parameter, which has not been

tested experimentally, would be obtained as the average of

force drops resulting from bond breaks in a specimen as it
ii

is elongatedo According to the model studies the values of F

are approximately inversely proportional to the initial lengths

of the specimens, and approximately independent of the v/idths,

provided the mesh sizes of the specimen networks are the same.

The two parameters E and F are sensitive to mesh size.

The model studies suggest that if two test specimens have the

same initial lengths but different mesh sizes, the values of

E and F obtained will be approximately proportional to

lengths characterizing the mesh size of each specimen network.

Results of calculations with a square-network model show

that the energy loss incurred by any one bond break is not a

linear function of the local force at the bond. It possibly

cc is approximately proportional to f , where f is the local

force, and a is a constant between 1 and 2. For a squarenetwork model a is close to I .5 • Calculations with a square-network model shov; that the

force drop incurred by any one bond break is not a linear

function of the force at the bond. It possibly is proportional

to f^, v^here p is a constant between 0 and 1. For a squarenetwork model P is close to O.5 • One concludes from the model studies that the parameters

E and F can characterize adhesive force between paper pulp

fibers. The parameters, however, have some objectionable

features. They are sensitive to mesh sizes and distribution

of mesh sizes in the specimen network, and they are not

linear functions of the average of the bond breaking forces.

The value obtained for one of these parameters is a complicated

function of mesh size and bond adhesive force. For instance

a formulation for the parameter E based on the square-network

model would be

n

1=1

V7here n is the number of bond breaks. K is a constant of the
network j which may change slightly from break to breaks

f is a local bond breaking force, and a a constant, g is

quasi-linear function of a characteristic mesh length x. A formulation for the parameter F would be similar, but

different constants and functions would be used.

1

i:?TRO~ip:Tic:i

in previous repor-os .--pi a neinoi nas ceen i-s

•Q ^ --0 3 3” ^

^^v-i '*

for.Tt of a lovr-d.ensit7 open vrec frcn pulp fleers so c uated. If a specimen from “his hanisheet is elonzaoe

break in a sensitive pensile tester,,

caused by the breakage of a bond beo'veen fibers cons

the handsheet network. By integrating under this 2ur

vicinity' J of the Jag it is possible to determine a ch

CL’np,-

_i- o — o :::: broken bona. This energy is use

adhesion between two fibers

The selection of th'

applicability noust be Judged on the basis of e:c

evidence. Vihiat the parameter ac well understood. Sotie ouestion

- —

j

bond break? How is the energy loss related to

stress at the bond Just before break? That is

piesh size? Vfnat is the cctitiupi soeciiten size :

1 -p, c; ' in tnis report fwo cruie na

network are described, and their

calculated. The concepts developed and the results o add to an intuitive unders tandin^ of the enerzv para
2

THE PARALLEL- SPRING MODEL

The parallel-spring model is depicted schematically

in figure 1. It consists of a number of springs in a seriesparallel arrangement o The dots indicate v/here the springs

are bonded to each other and to a series of rigid bars. There

are 2M-1 junction points on each bar, and there are 2N-1 bars.

In the unstrained state each spring has a length £, called

the mesh length. The spring constant is given by k/2c The

model is elongated so that each spring is extended an amount

where S is the strain^ The bond B then breaks, and the

tvjo springs previously joined at that point are inactivated.

The initial force sustained by the model before break is = {2y[-l)vJ ( 1 . 1 ) The initial elongation of the model is

AL = 2(N-l)i/ (1.2)

The initial energy stored in the model is

Eb = |(2M-l)k6'2(N-l)i(5

= 2(N-1) (2M-1)£' |k(5^ (1.3)

After the break the elongation of the model is maintained

constant, and the force drops to a value F^^. The horizontal

bars to which the springs are attached remain horizontal.

In the tvvo segments of the model on either side of the broken

bond the force F^ is sustained by 2M-2 parallel springs,

so that

Fg = (2M-2)k-y-i

or

F 9 (AL) = (l.A)

(2M-2)k

v:here (AL)-, is the elongation of the segment.
3

The elongation in the other 2(N-2) segments is given by

F X

(AL)„ = —^—— (1.5)

(2M-l)k

where (AL )2 is the elongation of the segment.

The total elongation of all the segments is

2(N-l)iif = 2(N-2)Fy 2F,/

+

(2M-l)k (2M-2)k

Thus

(2M-1)ic<5' = Fj, =

N-2

+

2M-1 F

N-1 2(M-1) (N-1)

a

or

^b = 1 +

2M-1

i 2(M-1)(N-1) N-lj = |l +

^

a

[ 2(M-l)(N-l)^ a

Hence

Fa = TF^

where

T =

2(M-1) (N-1)

( 1 . 6 )

(1-7)

2(M-1) (H-1) + 1

T is the fraction of the force remaining after a bond break.

The force drop resulting from the break is given by

AP = F^(l-T) (1.8)

The energy in the two segments of the model adjoining

the broken bond is

1

^ ^ a 2 (2M-2)k

and the energy in the other 2(N-2) segments is

i(AL)

F

1 ^a ^

2 'a (2M-l)k
Thus ohe total energy is

E, = 2(K-2)- §

_I s. r. -i

(2M-l)k

2-iCL (2M-2)k

= cL<^N-2 - L2M-: 2M-2

1 a ^

k

^ ,2.(.H-l)[lhi ^ . . , .2.M-.I l^.j2 (2M-i)2k2^2

2M-1 ' N-1 2(M-1) (N-l)j 2k

= 2(N-1) (2M-l)i. ^k^^

T

= 2(N-1) (2M-l)iT

( 1 . 9 ) In order to describe the way strain energy is distributed in a nodel it is useful to associate energy v.'ith each

of the junction points in such a v:ay that the sum of the

associated energies is equal to the total energy stored.

At any given junction point the associated energy is equal

to one-half the stun of the energies in the springs joined

at that point. The factor of one-half is introduced because

v.’hen the associated energies are summed over all the junctio

points, the energy of each spring is counted tv:ice. At

junction points on the top and bo'^tom boundaries there is

only one spring, but the factor of one-half is still

necessary.

A dimensionless average associated energy E can be

defined by means of the relation.

E^ = (2K-1) (2N-l)E'|k^^l ( 1.10

The factor (2M-l)(2N-l) gives the total

in the netvrork. The factor is the

number of points

energy stored in
9

( 1 . 11 )

one spring before a bond break, and is introduced into

expression to make Z dimensionless. 2y equating relati-;

(1.9) and (1.10) an alternative relation,

_ 2N-2 p, ^ - 21TT ^

is obtained.

The energy loss resulting from a bond break is,

fromi eq (1.9)^

AE = E. -E = (I-T)E. (1.12)

An alternative formi is obtained from eqs (1.9) and (l.lO).

±S = 2(N-l)(2M-l)i;i - p^Z).iki2

(1.13)
6

THE SQUARE-IJETV/ORK MODEL

The square-network model is depicted schematically in

figure 2. It consists of springs forming a square netv/orko

There are 2M-1 columns of junction points and 2N-1 rows of

junction points. The depicted network has 7 columns (M=4)

and 7 rows (N=^) . The unstrained length of each spring is i,

and the spring constant is given by k/i . The model is

elongated by an amount 2(N-1 )£^^o The bond at point B thenbreaks, and the four springs previously joined at that point

are inactivated. The resulting configuration is calculated by

a computer program described belov;.

In order to discuss the model it is convenient to identify

junction points by the coordinates i = o, 1, « . .m; j = o, i, ... n. The corresponding numbers 1=1, 2, ... M; J = 1, 2, ... N, hov/ever, are retained for use in the formulas for force

and energy, as they are more convenient for counting purposes.

Because of symmetry it is only necessary to calculate the

configuration of the upper right-hand quadrant; i.e., the

junction points corresponding to positive values of i and j.

When the model is strained each of the junction points

moves from its original position by an amount the

horizontal direction and t). ./in the vertical direction. Thus

^ J v;hen the model is first extended and no bond break has occurred.

ij = 0

Ti. . = iSj ( 2 . 1 ) After the bond break the values of f • -j 'H* • readjust

except for certain constraints at the boundary and the

1=0, j = 0 axes of symmetry. These are

= C. m

"'^in ^iO

= n-^ = 0

'il ( 1,-1 =

^iii

j

.- 1,-1

( 2 . 2 )

(2.5)

(2.4)
(2.5) "Oj -

i^-lj =

I

’1-lJ =^IJ J

( 2 . 6 )

The values of J, rj attained after bond break are found

by solving a S3''stem of equations in which the horizontal and

vertical forces at each junction point are balanced out, plus

the boundary conditions (2.2 - 2.6). To formulate the equations

applicable at the point i, j , let

Ra = ^ + A^t]

j

Rg = + (T+^t]) (

p ^

' (2-7)

R^ =]jA^€^ + (l+A^q)^

where A^ is a difference operator involving the indices (i, i-l)^

Aq the indices (j, j-l), the indices (j+1, j) and A^ the

indices (i+1, i) . Thus, for instance, A^ means . The quantities Ri are the distances from a point ij to an

adjacent point. The horizontal force balance equation is then

k(Rp-l)(H-Ap?) k(Rp-l)Ag5

_ k(R3-l)A3? k(R^-l)(H-Aj^g)

or

(l+Ay )(1-1/Rp) + Apf(l-1/R(,)

- A^f(l-1/Rg) - (1+Ap5)(l-1/Rj3) = 0. (2.8)

Similarly for the vertical components

Apn(l-1/Rp) + (l+AQTi)(l-l/Rg)

- (l+AgTi)(l-l/Rg) - A^ti(1-1/R,^) = 0. (2.9)
8

Zquarions (2.S) and (2.9) £-PPly a'c the general junction

point ij, out vrhen i = 0, mor j = 0, n they must be modified

lO conform vrith boundary or s^.Tnmetry conditions.

Ic should be noted that the horizontal forces acting

at a point i,j are dependent mostly upon the values of

-• , r_. and ... Therefore eq (2.3) wh'ch has the

I orm

'• _

> ^ ""l-i _ 1 . ^ j. j

^

j.

' — i \j —.Jc -L-r ^ 3 XJ — ‘ f- d-* .S’- •» -.I.1 ) ~ ( 2 . 10 )

v:as expressed as a Taylor's series expansion in terms of

only these three most important variables to become

l(k) , ,(k) (k) _ 11 ar

.

— — u>

•>o

(k)

/^ 5 . = 0.

.-S..5 .

^ ^ i f

- ~ \ / - \ - / ^

^ ^ ^ (-^-^l) ^ (i^) 2.no As,. _. ana

ivnere tne lis. .. means 51.,.'^ ~5.- -• --Ij J

( 2 . 11 )

-o -kd -<j

. have similar meanins:s (k) pf ’ ,(k)

;c . are

. ^(k) (k) . -(k+1) evaauatea as zunctions oi ' , . Tne values oi ^ ' are obtained b^' solving the system of equations (2.11)

corresponding to different values of jc

k set of five of these equations, for instance, would

1^"1

L s- —

C,X,

-- _

C^.x1---.i: ;::dw u r'educDior-.

II

X-^ -i- 5. X

e -r 4- C .X , = H

t t

II

-L S easily solved by a
ihe ccr-Ti^rj-r’

orner ror ir.; (£.5)

ei'Gcanded

‘or 5, r) cculd

;icns ' —- j ~3

I *

le otner

‘11 10 variacles.

ecuauioas aa Mi uracacvms ;u_a ce ccaaane;

not jvere.

The leorK /..Cl 11 etc: oeovreen

-C3 ; -•er J ICO or 20C

Ahlch zhe inirial s y

’e ohan 200 iO:

one ;one TlS

va_ues

L
10

case to rewrite eqs (2.8) and (2.9) ^ expressing terms of the

form (1-1/r) as a truncated series, although this was not

done for the calculations reported here.

After sufficiently accurate values of ^ and r\ were

obrained, the tensile force at the top of each column

of springs v;as calculated using the formiula

Tin = k(l+ni-n^^„_p(l-l/P^)

B'

( 2 . 12 )

R., =A/f.. i . + {l+n5-r|, ) IB Y'^ i, n-1 ' ''i,n-l'

V/hen i = 0, the value of Tq^ simplifies to

T.

’On

The force afrer break F then becomes

a n

F = +2 T

.

a On in

(2.15)

(2.14)

The fraction of the force remaining after a bond break T is

given by the ratio

The average associated energy E is found from its

definition, eq (1.10), after calculating the associated energy

for each junction point and summing over all the junction

points to obtain the stored energy after break E . The value of E can be obtained by a different calculation,

if F^ is knovm as a function of S from a series of calculations

a on a given model. The energy after break E^ is found by

Ci. integrating under the curve of F^ versus initial elongation

AL = 2(N-l)i^. Thus,

•2(N-l)i^

F^-d[2(N-l)fi‘]

= 2(N-1) (2M-l)k^ / Tc^d6

•^0 (2.15)
11

From the definition eq (l.lO) E is then given by

- 2N-2 ^ ~ 2N-1 (2.16)

In the square-netv/ork model T is not a constant

independent of 8 as it is in the parallel-spring .model

,

If T were constant, eq (2.16) would reduce to the relation

eq (loll) given for the parallel-spring m.odel.

It should be noted that the behavior of the squarenetwork model is the same as that of the parallel- spring

model before the central bond breaks. Therefore eqs (l.l),

(1.2) and (1-3) stre the same for both models. The quantity

T is defined by eq (1.6) and the quantity S by eq. (l.lO)

in both models. Therefore eqs (1.8) and (I.I3) are valid for

both models. Relation (2,l6) is a general definition valid

for both models, but relations (1.7) and (1,11) apply to the

parallel-spring model only. Therefore eqs (I.9) and (1.12)

also are valid only for the parallel-spring model.
12

SQUARE- NETWORK MODEL CALCULATIONS

A number of calculations were carried out for different

square-network models to determine the number of iterations

required for accurate values of T and E. The model for v;hich

M=ll, N=21, i=l was studied the most extensively. Results of

calculations for this model are given in table 1. These

results indicate that for values of S greater than O.O5 , 100

or 200 iterations are required to obtain values of T and E

accurate to more than four significant digits. When S is O.O5

or less, more than 200 iterations are desirable. Check calculations were also carried out on the other models studied,

and 200 iterations were determined as sufficient.

Calculations were carried out on five different models.

The iniuial strain S for these models was 0.20 . A mesh length

of 1=1 v;as assumed for the calculations. Values of M and N

were varied in order to study how T and E depended on the

shape and on the number of junctions in the model. Results of

vhe calculations for these models are given in table 2.

The configuration of the mxOdel for which M=ll, N=21 is

plotted in figure p. Only the upper right-hand quadrant of the

model is shov/n, as the complete model is symmetrical with

respect to this quadrant. The distortion due to the bond break

is seen oo be concentrated along the central column of springs

( 1=1)0 Buckling occurs (R<1) in the regions where the springs

are indicated by dotted lines. Although not easily discernable

in the figure, the central vfidth (J=l) has been reduced to 0.95^

times the original width, as a result of bond break.

An energy map of the model for v^hich M=ll, N=21 is shown

as an isomevric plot in figure 4. As before only the upper

right-hand quadrant of the model is shown. The dimensionless

value plouted at each junction point is the ratio of the

associated energy after bond break to the associated energy

before bond break. According to this map the associated energy

remains constant at most of the junction points, it increases

at points immediately to the right and left of the broken bond
13

and decreases to a lov;er level at junction points above and

below the broken bond.

Figure 5 shows the configuration after bond break, and

figure 6 shov/s the energy map of the model for which M=21,

N=41 . This model, of the same shape as the previous model,

has approximately four times as many junction points. The

corresponding figures for the two models are similar.

Table 3 presents values of T and E as functions of initial

strain S in the case of the model for which M=ll, N=21. A force

elongation curve for this model after one central bond has been

broken can be plotted using data from table 3* The force has

the value (2M-l)Tk<^, and elongation the value 2(N-l)i^^. The

essential characteristics of the curve are shown by figure

which is a plot of Ti5^ vs for this model. Unfortunately the

values of T are so close to unity that this curve lies very

close to the force-elongation curve, shown as a dashed line,

of the model before the central bond was broken, and the nonlinear character of the curve is not readily apparent.

The non-linear character of the force-elongation curve

can be demonstrated by plotting the difference in force before

and after break as a function of S. This is shown in figure 8,

in which (l-T)^^' is plotted vs cf . As a matter of interest

(1-T) for the parallel-spring model (M=ll, N=21) was calculated

Its value was 0,002^938 . The linear function rj, 0024938^ is

shown as a dashed line on the plot. The value of T for the

parallel-spring model exceeds the values of T calculated for

the square-network model. Evidently the force drop incurred

by a bond break in the square-network model is greater than

the force drop incurred in the parallel-spring model.

The area under the tS curve in figure 7 is equal to

, and according to eq (2.l6) the average associated

2N-2 2

E is equal to ^ times this quantity. Thus values of

2N-1
14

S can be calculated from T and S according to eq (2.l6) and

the result compared with the values of E in table 3y which

were calculated from the associated energies present after

break. This should provide a check on the accuracy of the

computer calculations. The results of this calculation are

given in the last column of table The values of E obtained

by the two methods are seen to be in fair agreement.
X'

DISCUSSION OF RESULTS

1. Configurarional Distortions

Associated Energies

It has been assume

break results only in local distorticns, and most of t

“tions and tne R.edis tribution G _

- the Hesul t of a Eono. Break.

w> ^ Ct lov:- density papei’ o :o rk

.iT.enrally [ — ^ 1 y — —wJw a bond

j.1. j loss is also concentrated locally. It uould be desirable to

demonstrate this ivith a mathematical model, but the squarenetwork m.odel is not suitable for this purpose. The distortions

and energy losses are concentrated along the ccl'mmn of springs

directly in line with the broken bond. Possibly a better model

A-ould be one in vrhich the meshes 'were of a different shape,

such as a hexagon, and so arranged that load bearing fibers

could not be easily aligned in the direction of stretch. Such

a model vvould require less stretching force, have a nonlinear

force-elongation curve, ana probably have a greater tendency

to buckle laterally. It would tetter simulate the behavior

of a thin paper network, but at the price of greater mathemiatical complexity.

In the energy m.aps, figures - and £, there is an increase

in associated energy at the junctions to either side of the

broken bond. These bonds therefore should break immediately

alter oreakase Oiiv centrax ;nd, anc should break successively in a tearing action that ^ ^ ^ Q

across the model nefwork. This tearing action is probably

common to many miodels, and actually occurs in real paper networks. In the paper netvrorks hov:ever it seldom, goes to ccm.- pletion, because the density of fibers is not sufficiently

uniform.

2. The Effect of Mesh length

In experim.ental work specimens of standard di.m.ensions

(usually 2x1 cm.) are elongated, and the applied force recc:

as a function of the elongation. V.lienever a bond breaks

L
Id

force decreases sharply, and the energy loss resulting from

the bond break can be determined by integrating under the

force-elongation curve in the vicinioy of the force drop.

Certain of these energy losses are averaged to obtain a

parameter characterizing bond strength.

If it is assumed that all of the breaks occur at approximately the same local force level, and the specimens have

uniform mesh size, this energy parameter might provide a

reliable indication of the relative bond strength. For

instance bonding could be studied in thin handsheets of the

same mass per unit area made from a pulp that had been

subjected to various beating treatments. However if one

attempts to compare bonding in standard handsheets made from

different pulps, a difficulty arises. One pulp may be coarser

than the other. Thus a handsheet of a standard mass per unit

area made from a coarse pulp ivould have fewer bonded junctions

and larger meshes on the average than would a handsheet made

from a finer pulp. In order to compare energy parameters for

these different pulps it is necessary to know what is the

effect of mesh size.

To study this effect in models, consider two parallelspring models: model Aj M=ll, N=21, i =1 and model Bj M=21,

N=4l, ^=0.5 • These models have the same external dimensions,

but the mesh length of model B is half chat of model A. Let

the two models be expended the same amount, so that 6 and

therefore the local force kcT is the same in each. From

eqs ( 1 . 1 ), ( 1 . 7 ) and (I. 8 ) the force drop in model A is

21ki^/A01 0.05ki, and in model B is Alk^/loOl 0.026k^.

From eqs (l.p), (1.7) and (1.12) the energy loss in model A

is

^

2.1* ^k^^, and in model B is

^

1.02*^k(S^.

Ev'idently model B with half xhe mesh size of model A also has

approximately half the force drop and half the energy loss.
17

In any parallel-spring models from eqs (l.l), (l*7) ana

( 1 . 8 ) AF =

(2M-l)k<^ 2(M-1)

2 (M-1 ) (N-1 ) + 1 “ 2(M-i) (N-1) + 1^

which for large M becomes

AF N-1•k<^ Substituting the initial length L=2(N-1)X inro this expression

gives

AF k^ (4.1)

From eqs (1.7) and (1.12)

2(N-1) (2M-l)i l,.c:2

2(N-l)(2H-2) + 2(N-1) , 1, f2 = 2{iyl-i)tM-i) + 1

' which for large M becomes

ZiE ~ (4.2)

According to eqs (4.1) and (4.2), if any two parallel-spring

models for which M is large are extended so that the local

force kS is the same in each model, the ratio of the energy

losses zXE-i/AEg la the two models will be the same as the

ratios of the respective mesh lengths H^/

2

’ 1^ la addition

the initial lengths of the two models are the same: i.e.,

2 (N^-1)^^-2 (N

2 ~l)i 2

^ ratios of the force drops will be

equal to i • Table 4 gives the results of calculations for a number

of parallel-spring models, all of which have the same initial

length. According to these results the force drops and energy

losses are almost linearly proportional to the mesh length 2,

despite the relatively low value of M in some of these models.

It is interesting to observe that values of total force before

break F^ and the total energy before break do not influence

these results significantly. The values of F^ and E^ are

proportional to 2M-1 however, so this is just a manifestation
16

of ohe near independence of ohe values of AF and AE on the

values of K, for K sufficiently large (e.g., M=ll)

.

Tacle 3 gives results of calculations for some squarenetvrork models having the same initial length. The M, N and

t parameters for these models are the same as those in table 4.

The force drops and energy losses for these models decrease

as the mesh length i decreases, but are not linearly proportional to the mesh length. For instance the three models for

v.'hich K=ll have I ratios of 1.0:0.30:0.23, AF' ratios of

1=0:0.37:0.31 and AZ ratios of 1.0:0.64:0.37* The two models

for which M=21 have h ratios of 1.0:0.30, Z\F ratios of

1.0: 0 . 3- and AF ratios of 1.1:0.33. Apparently the dependence

of AF and AZ upon x- becomes more linear as M increases.

F.esults for the parallel-spring and square-network models

suggest that the force drops and energy losses observed in

a paper fiber network have an appro >'J.mately linear dependence

on a characteristic mesh length. The actual dependence,

ho’wever, must be determined experimentally.

The effect of mesh length can also be deduced by a different argument as follovvs: Select a model (M,N) for which

calculations have been made so that T and Z are known for a given c ; i.e., T-T(3), Z=Z(£5). Select a mesh length Z . The

initial length of the model then is 2(N-l)^. For each mesh

length selected the model is elongated an amount Al-2 (N-l)iJ'.

This assures that the local tension at the bond break is

alvrays the same (k§) . From eqs (l.l), (1.5) and (I.I3 ) = {2V-1) (i-n-c!? ( 4 . 3 )

.•Is T and T are functions of S vrhich is kept constant, AF

.dependent of mesh length and AZ is directly

the mesh length. This conclusion is valid

arallel-spring or square-network t 2*pe of model,

ion just obtained seems to conflict vrith that

lysis presented above. Hovrever in that analysis
19

a number of models of dLfferenz mesh length buo having "Ch^

same initial length v/ere elongated, vrhereas in the presern

analysis the initial length of the model depends on

length.

This latter deduction may be useful in experimental vrork,

as illustrated in the following situation: Suppose that tvro

handsheets have been manufactured from the same culc stock.

jet the first handsheet have twice the area! .3i.oy as tne

second, so that its characteristic mesh length is one-half

as large. According to the analysis above, bond breaks in booh

0 0 1 0 0 1 ^ but a specimen

nded tvvice as much as

i" V-. -? c ^ ^ ^ . Thus the

the second nanosneeo

s suggests zhat energy

one from the first in order to attain

average energy loss per bond break in one secon

parameters found by tests on handsheets of different areal

density can be scaled to find the value corresponding to a

standard mesh length.

p. The Effect of Samiple Size and Shape

If several models, each with the same miesh length I, are

elongated to the same tension, the force drops and energy

losses are given by eqs (^.3) and (A.^). For square-network

models it is necessary to know ohe appropriaoe values of r

and E before the effects of sample size can be predicted.

For parallel-spring miodels hov/ever, simple e:q)ressions for

T and E are known, so that for large values of M eqs (-.1)

and (9.2) apply. Equation (9.1) predicts thao the product of

the force drop by the length of the specim.en, LAF, is a cons'cant for all models having "che same mesh lengoh 2, and

eq (9o2) then states that the energy losses are constant.

Table 6 gives values of the force drop and energy loss

for some parallel-spring models. Note that the quantities

LAF/k52 and AE/0.pic5^>^ have the same value in this oable
20

because of the choice of units. It is apparent that AF and

AS for parallel-spring models are closely predicted by

eqs (^.1) and (4.2). The three models for v/hich M=ll

(and width W=2 (M-1)£ =20i) have values for LZIF and AS of

approximately 2.095^ and the two models for which M=21 (W=40i)

have values of approximately 2.0^8. Evidently LAF and AS

approach a value of 2.0 with increasing M, as they should

according to eqs (•4-.1) and (4o2) . Table 7 gives values of the force drop and energy loss

for some square -network m.odels. The values of LAF and ZiE for

these models are roughly constant, but there seems to be

some dependence upon both M and N. For instance the three

models for which M=ll (W=20i) have lengths that increase in

the ratio 20:40:80, The LAF values for these models are

10.27 : 11 o 79 ’•12.80 . The two models for which M=21 (W=40Z)

have length ratios of 40:80 and LAF values of 11.90:12.27 . Similar results are obtained for the zlE values.

The results for the square -network models suggest that

various values of LAF and AE obtained by tests on specimens of

the same characteris-cic mesh length but of different size and

shape, are roughly comparable. For most accurate results

however all test specimens should have the same standardized

dimensions

,

For the parallel-spring models, from eqs (l,l), (1.9)^

(1,8) and (1.12)

This equation states that for the parallel-spring models the

ratio Z1E/z\F is proportional to the average elongation at

which bond breaks occur, regardless of the value of M,

Experimentally it is desirable to have significantly large

force drops for a given energy loss, so the ratio AE/AF

should be kept small by making L as small as feasible.

AE _ ^b

AF “ F^

2(N-l)i(^ l5

2 2 (^» 5 )
For the square-netv;ork models, from eqs and (4,-4-)

# = [4 - |^e)/(i-t)] 1^" (4.6)

Table 8 gives values for the ratio zXE/z\F for some squarenetwork models. Parallel-spring models have values of AE/aF

that are proportional to the initial length L in accordance

with eq (4.5) j but for the square -network models the dependenc

of zXE/aF upon L is not quite linear, and there also seems to

be a small dependence upon M. This is best shown by comparing

2 /AE '' values of the quantity v;hich for parallel-spring

models is equal to unity but for the square-network models

in table 8 varies between 1.2 and 1,5 depending upon the model

Of most importance, however, is the confirmation that

the quantity AE/AF can be kept low by choosing a small value

of L. This situation is limited som.ewhat in experim.ental tests

A specimen length at least twice the specimen width is

preferred, in order to avoid excessive stress distortion near

the clamps.

Although the results of tables 7 and 8 indicate that

the values of LZ\F, zXE and Z\E/LAF are insensitive to the value

of M, there are circumstances where it is advantageous to

vary the width of the specimen, as in the following

experimental situation; Suppose that tests are being conducted

on a given specimen, but the force drops observed are Simall,

so that they cannot be measured with much certainty. The

calculation of energy loss requires that the force drop be

known as accurately as possible. Thus the experimental length

should be small in order that the force drop be large. Further

improvement is then achieved by decreasing the specimen width.

This decreases the value of force at break F^ without significantly affecting the value of the force drop. The amplification of the recording instrument can then be increased and

the force drop better resolved.
22

To Dependence of Energy Loss Upon the Local Breaking Force.

According to eq (-^-eT) the energy loss AE is proportional

to the product of the quantity (l - ty the square of

the local breal-cing force k6. For parallel-spring models ^

and therefore the quantity (l - -^

;-~

-a

-E ) is independent of the

value of this force. Thus for these models energy loss is

proportional to the square of the local force at break. For

square -network models however the value of E depends upon

the value of S, so this simple quadratic relationship no

longer applies.

Figure 9 is a plot of the quantity* (1 - as a

function of S. The ordinate in this case is proportional to

the energy loss in a square -network model for vrhich M=ll and

N=21. This relationship, calculated from the data of table

is shovm by the solid line. A relationship having the same

value for ^=0.10, but for vrhich the ordinate is of the form

-2

K3 , whore K is a constant, is shown by the dashed line.

A. corresponding relationship of the form is given by

the dotted line. This latter relationship provides a better

fit to the data than does the quadratic relationship.

The relationship between energy loss and local breaking

force in an experimental test specimen is not knovm, and it

does not seem feasible to determine it experimientally « It is

possible that further studies with other more realistic

mathematical models may provide more information on this

relationship

.

3 . Interpretation of the Energy Parameter.

In the introduction an energ^^ parameter E for characterizing the adhesion betvreen pulp fibers was described. This

energy parameter is obtained experimentally by elongating thin

'web-like specimens in a sensitive tensile tester, and measuring

the energy loss incurred each time a bond brealis, as denoted

by a sharp drop in the force-elongation curve. The energy losses

for a selected number of bond breaks are then averaged.
The parameuer is supposea lo prcviie a rr:ea3ure

adhesive force ceoiveen fibers, cue the previous discussicn

shows that E depends on the struceure f the scecinen neeucr>

meshes in a specim.en net'wc rk range over a variety

. the energy loss incurred by a break depends upon

he hole that opens up . If all the bonds in t he

were of the same strength the parameter ‘would

.re that strer.gth, but could stxj.j. taxe on 2. range of values depending on the size distribution of the

meshes in the soecimen networks

The force of adhesion between fibers in the net'A'crk varie;

from bond to bond= It has Juse been shown thae the energy loss

incurred as the result of a bond.d break is not 1

>-) 0 p “I -j- related

the bond. Ther average

es not reflect c- S UH'C J- 0 average

of the local breaking forces.

Results of calculations 'with so.uare- net'work models

suggest that the energy less has only an approximate linear

relationship to the mesh length. This introduces a slight

additional complication in the interpretation of 'what the Z

parameter actually measures.

The interpretation of the E param.eter may perhaps be

better expressed by the following formulation: let the energy

losses be measured for each of a series of n brealcs, and let

the energy loss AE^. for the break be given by

AE . = K_. f . *^g ( X . ) 1 Vi. 1 1'

is a constant of the net'work that may vary from break to

break if the network is significantly altered by the breaks,

f^ is the local force on the bond at breal'i, and a is a

constant having a value probably bet'ween 1 and 2, Eor a

square -network model ct has a value close to 1.5 • represents the functional dependence of AE^. upon a characteristic mesh ler^gth associated with the bond i. This
9

rela~icr.shlp is prctatly alnioso a linear one.

Under these assiinptions the characteristic energy E

is given by

(^*•7)

This fcrniila gives an approxiitate idea of the nature of the

parameter Z. Hovrever one should renenber that this formula is

based on the results of souare-netvrork model calculations.

le souare-netvrork mode, is only a cr-aoe representation of the paper networks that are tested experimentally.

It is likely that vrcrk vrith more appropriate models will

provide an improved understanding of the nature of E.

6. An Alternative Force Drop Parameter.

Adhesion between pulp fibers could be characterized

’eazis wnen a test specimen is eiongatea.

parameter could be cbtained by averaging force drops incurred

in a series of bond

The parameter F has not been tested e e riment ally, but its

use as a parameter to characterize adhesion seems feasible.

In order to calculate an energy parameter E it is

necessary to knovr the slope of the force-elongation curve in

the vicinity of a force drop. The force drop parameter F could

be calculated more simply, as only the force drops are used.

In order that the force drop parameters be comparable,

the test specimens must all have the same initial length. In

other respects the force drop parameter F is similar to the

energy parameter E. Both parameters are sensitive to mesh size

and mesh size distribution in the test specimen network.

Neither parameter is linearly related to the average of the

local bond breaking force. However the functional deoendence

00 r v«; .. ne average oon ireaxins lorce is iioierent Irom

‘ ^ - V- ' (e.y) the local force drop resulting from
tL
26

REFERENCES

1. J. Co Smith and E» Lc Graminski, Characterizing the

Interfiber Bond Strengths of Paper Pulps in Terms of a

Breaking Energy, Progress report covering the period

Jan. 1, 1976 to June 30, 1976, NBSIR 76 - 1148. Available

fromi National Technical Information Service, PB 264,689.

2. J. C. Smith and E. L. Graminski, Characterizing the

Interfiber Bond Strength of Paper Pulps in Terms of a

Breaking Energy: Effect of Beating, Progress report

covering the period July 1, 1976 through Sept. 30, 1976,

NBSIR 77 - 12860 Available from National Technical

Information Service, PB 276,47b.

3. J. C. Smith and E. L. Graminski, Characterizing the

Interfiber Bonding of Paper Pulps: Effect of Preparation

Pressure on Tensilo Test Specimens, Progress report

covering the period Oct. 1, 1976 through March J>lj 1977

j

NBSIR 78 - 1459. Available from National Technical

Informiation Service, PB 280,279*
27

Table 1.

Values of T and E as Functions of Number

s

of Iterations for

M=ll,

No . 0 f

a Square-Metvrork

N=21, 1 =1

Model

Iterations T

0.01 100 0.95227 0.92927

200 0.95254 0.92927

0.07 100 0.95779 0.97112

200 0.95714 0.97100

0.05 20 0.96778 0.97606

50 0.96679 0.97587

100 0.96614 0.95567

200 0 . 96564 0.97561

0.10 20 0.97806 0 . 9"+6^7

50 0.97829 0.94678

100 0.97827 0.94677

0.15 20 0.98716 0.95270

50 0.98770 0.95267

100 0.98772 0.95262

0.20 10 0.97787 0.97979

20 0.98771 0.95679

50 0.98594 0.95644

100 0.98595 0.95641

200 0.98597 0.95640
28

Table 2 .

Values of T and E for

^^=0.20, X =1,

Some Square-Net'/rark Models

200 iterations

M N T E

11 11 0.97554 0.92506

11 21 0.98597 0.95640

21 21 0.99511 0.96659

11 41 0.99258 0.97654

21 41 0.99626 0.98221
29

Table

Va.lues of T and E as Functions of Initial

Strain for a Square -Network Model

M=ll, N=21, H =1

5 T e' E

eq (2.16)

0.01 0.95254 0.92925 0.92951

0.05 0.95714 0.95100 0.95250

0.05 0 , 98584 0,95561 0,95658

0.08 0.97477 0.94270 0.94528

OolO 0.97827 0.94657 0.94685

0.15 0.98175 0.95052 0.95078

0.15 0.98552 0,95262 0,95275

0.18 0.98507 0.95510 0.95505

0.20 0.98597 0.95640 0,95626

Note; T for the parallel-spring model is 0,99751

E for the parallel-spring model is 0.97518

Note: Values forc$=0.01^ 0.05^ O.O5 and 0,20 were obtained

after 200 iterations. All other values were obtained

after 100 iterations.
30

cv OJ

CD rH

03

CQ I—

aao

bD

05

2h03 CmW<m0m o CM r^! o CJ\ rH o

in C7^ CM CM I—‘P K^

r

-=r 1—

CM

p

O o O LOl

m

p

C7^ OJ 1— CO LA >

<]

OJ 1—

1 1 O o

CO 1—

>

<]

C\J 1—CO tp

OJ

CQOhoQ0o

ChO

Eh I—i I—I I—I I—

OJ CJ ^ I 1 I 1 E 1 I 1 rH rH OJ rH 0

T5o P in -=r OvJ CM M3 G [p M3 LA o ^A M CM 1 1 LO LA O rA CM o lA P o UA LA CM CM p: 1—

03 00 M3 LA

CO t—

OOOO P \ LA CM CM 1 1 1—

1 E>m 0 p • • • • • 1—

CD <] o O

O

OO s1 < OO o OO 0M0

E3O' CO

OJ 0I—

03 Eh

M

<3

Uj0

02

m 02 o • • • o O P P 1—

1 1 ! 1 P P P OJ CM CM JZ^ P

bO bO M G 0 0 G G H P

G3

/P P

Xh LA lA

G

G

O uA LA OJ CM

G P P O O O O P ^ I—I I—I I—I I—I I—

^ CM OJ ^ OJ OJ

02Op0oop I—I I—I I—! I—I I—

rH OJ OJ -=r ^

OJ 1—i I—

rH CM I—

CM

CQ 0 0 P3 1 1 s p s P 0 o LA • o LA • T3 CO • O O o o o ’I/ _ • O o o o o O O CM pi CM CM CM O CM CM CM CM OJ S G \ P- -=t CO -=:j" 00 rH \ P- jrj- 00 03 O A • o A 1—

P P LA P P 1—

0

ShOPo

CM

II I—

I S

CM LA LA LA LA OJ CM II O O O O P

-P

faO G0I—

0PPGH
M

11

11

21

11

21

M

11

11

21

11

21

31

Table 6.

Values of Force Drop AF and

for Some Parallel-Spri

Energy loss

ng Models.

AE

N isF/kd LzXF/kbL AE/0 . 5kf"^.c

11 20 0.1045 2.090 2.090

21 40 0.0524 2.095 2.093

21 40 O0O512 2.047 2.047

41 80 0.0262 2.097 2.097

41 80 0.0256 2.049 2.049

Table J.

Values of Force Drop zlP and Energy loss zXE for Some Square -Network Models.

N L/i AF/k6 LAF/kt^. An/O . z)k.

11 20 0.5137 16.27 12.93

21 40 0.2946 11.79 16 . 54

21 40 0.2825 11.30 15.50

41 80 0 . 1600 12.80 19.25

41 80 0.1533 12.27 18.08

Table 8 .

Values of AE/AF for Some Square -Network Models

M N

1 AE

1 dF

2 ZiE

rj AF

11 11 20 12.59 1-259

11 21 40 28.07 1.405

21 21 40 27-43 1-372

11 41 80 60.14 1.505

21 41 80 58 » 98 1.474
Figure 1

.

Schematic representation of parallel-spring model. Model has

2M-1 parallel columns of springs attached to 2N-1 rigid

transverse bars.
3

4

i=0 I 2

1=1 2 3

Figure 2 o Schematic representation of square -network models. Model has

2M-1 parallel columns of springs. Each column has 2N-1

junction points which, excepting the end points, are attached

to transverse springs.
Figure

Configuration of the

positive quadrant of a

square-network model for

which M=ll, N=21j after

a central bond break.

The broken bond is designated by the open circle

in the lower left corner.

Springs in which buckling

occurs are designated by

dashed lines.
r

associated

energy

at

a

Junction

point

after

bond

break,

divided

by

the

associated

energy

before

breaks
Figure 5»

Configuration of the

positive quadrant of a

square -network model for

which M=21, after

a central bond break.

The broken bond is designated by the open circle

in the lower left corner.

Springs in which buckling

occurs are designated by

dashed lines.
Q-i c: •H fn o cd 1 ft

;d 0 & a o' fn 0

<9 cd Cd ?H o

VD

a 0 O > ft cd 0 •H 0 U bO P 1—

U u •H Cd 0 0

hO 0 0 Td ft

•H

a

O ft o ft

plH

w ft o 0

Td0

ft0

•Ho

ft o a M

•H CO O Cd ft 0

f:o ft

•H

ftO

p S p0n

Cd •H>

ft •H 9 cd 0

>3 •% 0

bO

U

cd P 0 0 0 0 P Po

Td P ft 0 cd 0

ft o p

cd •HOO

D2

COcd

cu -p

cd

bO0

f:: cu
FORCE

,

ARBITRARY

UNITS

ELONGATION

Figure 7

Force-elongation curve after central bond break in a squarenetwork model for which M=ll, N=21, T<^ is plotted vs oT, where

S is the elongation per unit length. Dashed line shows forceelongation curve before bond break.
FORCE

DROP,

ARBITRARY

UNITS

.003

ELONGATION

Figure 8.

Curve of force drop due to bond break vs elongation of a

square-network model for which M=ll, N=21<, (l-T)^ is plotted

vs S, where S is the elongation per unit length. Dashed line

shows force drop for equivalent parallel-spring model.
ENERGY

LOSS

.

ARBITRARY

UNITS

0 .05 .10 .15 .20

LOCAL FORCE . ARBITRARY UNITS

Figure 9

Curve of energy loss due to bond break vs bond breaking force

for a square -network model with M=ll, N=21. (l - is

plotted vs elongation per unit length Sj which is proportional

to bond breaking force. Dashed line is plot of kS'^ vs S, and

dotted line a plot of vs S. The constants K and K' are

adjusted so that ordinates of the plots are equal at (^=0 . 10 .
NBS-n4A (REV. 7-73)

!I

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 79-1735

2. Gov't Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Characterizing the Interfiber Bonding of

Paper Pulps: Tensile Behavior of Some Mathematical

M(n(ip1p nf ppppr> Npt.wnr>l<-p ,

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

Jack Co Smith

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

U.S. Department of Energy

Washington, D. C. 202^4

13. Type of Report & Period

4/1/77 - 9/30/77

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The tensile behavior of a thin web-like paper network was simulated

by two simple mathematical models. The mesh distortion, drop in tensile

force and energy loss resulting from breakage of a network junction

were calculated. These results were used to formulate two parameters

for characterizing interfiber adhesion: a parameter averaging the

network losses incurred in a series of bond breaks when the network is

elongated, and a parameter averaging the force drops. The effect of

mesh size, local bond adhesive force, and size and shape of the

' specimen network were calculated. These results based on model studies

I

I were used to interpret behavior observed in an actual paper network.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Mathematical modeling, network; network, tensile properties; paper.

interfiber bonding; paper, low-density handsheets; paper, pulp properties

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