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Unformatted text preview: qgjncgmrmg 3 (“MW WthmM Reprinted from ENCYCLOPEDIA
OF MATERIALS SCIENCE
AND ENGINEERING Editoa‘wianhief
Michele} B. Braver Massacfmsefrs insriruze 0f ch‘r’moiogy PERGAMON PRESS
Oxfm'd  New York  TDYDBLD ~ Sydﬁey  Frankfurt Fiber Bundles: Strength Statistics US Steel Corporation 1985 The Making, Shaping, and
Treating of Steel, 10th edn. Association of Iron and Steel
Engineers, Warrenviile, Pennsylvania Walton C i7 1981 Iron Castings Handbook. iron Castings
Society, Des Plaines, lllinois G. R. Speich Fertile Blanket Materials _ Breeding in nuclear systems refers to the process by
which an isotope with no fuel value is converted by
neutron capture to a form useful as reactor fuel.
For ﬁssion reactions, the principal opportunities for
iertile~to—ﬁssile conversion involve 23“wa 239Pu and
232Thw51> 233U. Fuel for fusion reactions can be
obtained by 7Lil—3 4He + 3T. The most common
applications for materials containing fertile isotopes
are for breeder blankets in thermal and fast spectrum
ﬁssion reactors, and for tokamak or similar fusion reactors. l. Breeder Materials for Fission Fuels Breeder blankets may be incorporated in a reactor
around the core periphery and/or homogeneously
mixed with fuel rods or assemblies within the core.
Metal alloys, oxides and carbides containing 238U
or 232Th are used as blanket materials. 2381302, in
essentially the same metalclad cylindrical form as the
reactor fuel elements, is the most common blanket
material. in contrast to fuels, blanket materials gen
erate the least amount of heat at the beginning of
their reactor life. As conversion progresses, the
blanket composition shifts from U02 to a U02~Pu02
solid solution. During irradiation physical and chemi
cal properties change with both the actinide metal
composition and deposited ﬁssion products. Fission
rates and blanket temperatures increase throughout
irradiation of the blanket, in proportion to the con
centration of the bredin ﬁssile isotope. Th0; and U02*Th02 solid solutions are utilized
for breeding 233'U. Their irradiation behavior is anal—
ogous to UOZWPUOZ. Uranium and thorium carbide
blankets offer higher metal density and improved
breeding efﬁciency compared with oxides. Breeder
materials for l1igh~temperature gasucooled reactors
are prepared as particles of (U,Th)02 or (U,"l"h)C2,
coated with impervious pyrolitic carbon and silicon
carbide, and dispersed in a graphite structure. 2. Breeder Materials for Tritium Lithium irradiated as ceramics or metal alloys is
"the primary source of'iritium for fusion devices and
experiments. Lithiumwaluminum alloys are used in
ﬁssion reactors for breeding where low temperatures
(~375 K) are maintained. Supplies of tritium to tuel large fusion reactors are based on continuous tritium
recovery from a blanket region surrounding the
fusion chamber.. Liquid lithium metal, Li7Pb2 and
lithium ceramics are candidate fusion reactor blanket
materials. Flowing liquid lithium functions as a cool
ant while solid materials require external cooling and
ﬂowing helium for tritium removal. The efﬁciency of
tritium production is principally a function of lithium
atom density. Candidate materials aligned in order
of decreasing lithium atom density are: LiZO (0.93
g cm‘3), Li4SiO4 (0.54g cm’3), Li (0.51 g cm”),
LiTPbZ (0.49g cm’S), Li28i03 (0.36g era—3), and
LiZZrO3 and LizTiO3 (both 0.33 g emut). Li7Pb2 and LiZO provide excellent breeding, but are quite reactive with moisture and air. The three
component oxides have refractory properties and
are chemically stable and readily fabricated. Tritium
diffusion out of these solids is optimized by a micro—
structure with small grain size and open porosity.
Equilibrium of LiEOwLiOH with water vapor affects
the rate of tritium release as T20 and inﬂuences
concentration of corrosive LiOH, which in turn
affects compatibility with structural materials. Near
total release of tritium from L102 and LiA102 occurs
in the 925—975 K temperature range, with near total
retention below 575 K. See also: Fission Reactor Fuels and Materials; Fusion Reac—
tor Materials; Nuclear Materials: An Overview Bibliography Evans T W 1964 The Technology of Thoria and Thoriu—
Urania Compositions, HWw84106. US Atomic Energy
Commission—Hanford Works, Richiand, Washington Roberts 5 "F A 1981 Structural Materials for Nuclear Power
Systems. Plenum, New York Smith D L, Clemmer R G, Davis J W 1979 Assessment of
solid breeding blanket options for commercial Tokamak
reactor. In: McGregor C K, Batzer T H (eds) 1979 Proc.
8th Symp. Engineering Problems of Fusion Research, Pt.
1, IEEE Publication No. 79CH144l5 NPS. Institute of 'Electrical and Electronics Engineers, New York,
pp.433w38 Waltar A 33, Reynolds A B 198i Fast Breeder Reactors.
Pergamon, New York E. T. Weber Fiber Bundles: Strength Statistics The tensile failure of a bundle of brittle ﬁbers in a
ﬂexible matrix is a complex process involving the
failure of. ﬁbers at scattered flaw sites, the overloading
of neighboring ﬁbers at these sites and the growth of
sequences of adjacent ﬁber breaks to some critical
size. current statistical'lheory describes a 'Chaino'f '
microbundles model for this process. An approxi—
mate Weibuil distribution arises for the strength of
the bundle. 1707 Fiber Bundles: Strength Statistics 1. Description of the Failure Process in .5: Bundle On application of a moderate tensile load x (to each
ﬁber in a bundle), ﬁbers fail at random flaws which
have strength below )5, and these failures are typically
quite far apart spatially. At these failure sites, the
load in the broken ﬁber decreases to zero, but the
load on each immediate neighbor increases to say
le, where K, > 1 and is called a load concentration
factor. Moreover, the length of the region, say 6,
over which this overload takes place is typically of.
the order of a few ﬁber diameters. K, and (5 depend
on the mechanical properties of the ﬁber and
matrix. At these failure sites, a few overloaded ﬁbers may
fail, and typically some pairs of adjacent brealts will
result. At these pairs, the ﬁankin g neighbors are then
subjected to the more severe overload 19):, where
K2 > K! > 1, and a few sequences of three adjacent
breaks may occur. Again, the length of the overload
region is of the order of (3. This process continues
until all failure sequences become stable, or one or
more sequences reaches critical size, say K", and a
catastrophic crack propagates through the bundle. in
the latter case, the overload Kgx exceeds a charac—
teristic strength x5 of :1 flanking ﬁber, and failure of
this ﬁber becomes almost certain. 2. The Weibull Distribution and the Strength of
Single Fibers
When tensiontesting single ﬁbers, one typically ﬁnds that the ﬁber strengths follow a Weibull distribution
function of the form: Fibs) m 1 — eXp[ml(x/.ro)9], x 2 0 (1) z where p is the shape parameter, x0 is the scale para—
meter for unit length and 1 is the (dimensionless)
gauge length; the Weibuil scale parameter being x; ""—“
xul “1/0. Roughly speaking, this means that if each of
the ﬁbers in a very large population were submitted
to a load it, then the fraction of ﬁbers that would fail 'is Mr). The mean for the above distribution is Filpxﬂﬂ + Up), the ,standard deviation is l"'1/Px0 ><
[Ill + 2/,0) w F(l + 1/p)2]‘/2, and the coefﬁcient of
variation (standard deviation divided by the mean)
is approximately 1.2/ p. Typically, 4 g ,0 S 1.5 is
measured experimentally, so that the mean ﬁber
strength diminishes quite rapidly with increasing
length I. For example, if p = 6, the ﬁber strength is
reduced to half when the length is increased 64
times. in the simple bundle model considered here, we
will need the distribution function FAX) for ﬁber
elements of length (5, where 6 is of the order of a
few ﬁber diameters. Note that I in typical tension test
is at least a few millimeters, whereas 6 is of the order
of 100 um. Because (3 is so small, extrapolation is 1788 necessary to estimate F5(x)_ The obvious choice is
the Weibull distribution, FAX) I 1 ‘ €XPl‘lx/xalpla x 3 0 {2) where by extrapolation x5 2 x,(r‘3/l)“"‘D = xgd’l/P. Unfortunately, the accuracy of Eqn. (2) in model
ling the true F5(x) can only be demonstrated for the
lower tail of F505), that is, for x < x5, since strengths
in the middle and upper tall are not typically observed
at gauge length l. Fortunately, knowledge of the
upper tail of F505) is not important. To demonstrate
this, results can be obtained under the power dis
tribution function F§(x)=(x/xd)pr Oe‘ixgxa (3)
which has a very different upper tail to Eqn. (2). 3. The Model and Basic Assumptions The bundle is viewed as a planar structure of n
parallel ﬁbers, which are partitioned into a series of
m short sections called microbundles, each with n
ﬁber elements (Fig. 1); each microbundle has length
5 called the ineffective length. Statistically and struc»
totally the microbundles are independent, and the
strength of the structure is that of the weakest micro
buudle (measured on a loadperﬁber basis). The
strengths ot the run ﬁber elements are independent
and identically distributed random variables, with a
common distribution function Fa(x), for rat) as
discussed earlier. fiber break Figure 1
Bundle of ﬁbers in a matrix in the form of a planar tape;
failure is localized within microbundles Each microbundle is a planar arrangement of par
allel elements (Fig. l), which share the load according
to the following simple local loadsharing rule: if the
bundle load is 2: (per ﬁber), a surviving ﬁber element
carries a load er, where K, is called a load con
centration factor and r is the number of consecutive
failed ﬁber elements immediately adjacent to the
surviving element. Furthermore, a failed element m W90wgc lg
ie
at
n
ve int carries no load. In our numerical calculations, we
assume the simplest case K, 5 1+ r/2 (where r =
1,2,3, . . .). In the above description, the (total) load on a
microbundie will fall slightly short of nx if a few
elements have failed at the edge of a microbundle.
However, these boundary effects turn out to be unim
portant, and are neglected for simplicity. Also, load
increases on fiber elements not adjacent to broken
elements are neglected. The consequences of this are
also unimportant. To describe the weakest link, consider a single
microbundle, and let 64):), x 2 (l he the distribution
function for its strength. In other words, G,,{x) is
the probability of failure of an arbitrarily selected
microbundle under the nominal. ﬁber load x (total
load ax). Next, let Hm!,,(x), x a G be the distribution
function for the strength of the bundle of ﬁbers in a
matrix. Since the bundle survives if and only if each
of its in microbundles survives, the probability of
survival of the bundle is [1 m Gﬂ(x)]m under ﬁber
stress x. Thus Hm'n(x), which is also the probability
of failure of the bundle under the ﬁber load x, is lel.li(x) = 1 _ _ Gn(x)imi x P" U Equation (4) amounts to a statement of the weakest
link rule. 4. Analysis of the Model Rosen @964) was the ﬁrst to consider a chainoi
microbundles model of this type; however, he
assumed equal load sharing among nonfailed
elements in each microbundle, an assumption which
is not in keeping with those here and which leads to
very different results. (Smith and Phoenix (1981)
discuss such models.) However, Zweben and 'Rosen
(1970) and Argon (1972) obtain interesting approxi
mate results for versions of the model here. Harlow and Phoenix (1978a,b) were the ﬁrst to
obtain exact results for the present modei, but only
for small 11. Current computer capacity limits exact
calculations to n 3:. M with little hope for signiﬁcant
extension to higher :1. Later, Harlow and Phoenix (1981a,b) developed a
powerful recursion analysis for the treatment of
larger bundles. A major result was that the dis
tribution for bundle strength Hn,,n(x) is extremely
accurately approximated by the weakest linlt form Hm{x):1w[l ~ wane", x 2 e (5) where W(x) is a characteristic distribution function,
depending on F605) and the K, of the local load
sharing rule. Unfortunately, W(x) cannot be "expressed in terms of simple functions, an‘d'the pro« cedure for its numerical calculation is cumbersome. However, W(x} can be approximated (see Sect. 6).
Smith (1979, 1980) has developed two distinct Fiber Bundles: strength Statistics asymptotic analyses. His results are quite easy to use,
and are accurate for the range of p of practical
importance. In what follows, we discuss the most
useful of Smith’s approximations. 5. Exact Results and Approximations for (1 Bundle
with Three Fibers For a bundle with three ﬁbers, the key task is to
compute 030:) for a microbundle whose three
eiements have strengths, say X1, X2 and X3. To do
this we must enumerate all the distinct ways the
bundle can {all under ﬁber toad x, calculate the
probabilities for these, and sum the probabilities to
obtain (33$). One Way is by the event (X: S x,
X2 é x, X3 :3 x), that is, all elements fail under their
initial load x, and this has probability F5(x)3. A
second way is when exactly two elements fail under
load at, and the remaining element fails under the
induced overload sz; one such event is (X 1 Sr,
x < X; 5—: Kgx, X3 3 x). There are three such events
for this way of failure, so that the total probability is
3F5{x)3[F§{K2x) — F1500]. A third way for failure to
occur is for an end element to fail under load I, the
middle element to survive x but fail under 16;, and
the other end element to survive x but fail under K 2):.
There are two such events, one of which is (XI gs x,
x < X2 E Kix, x < X3 S sz), so that the total
probability is 2F5(x){F(K1x) — F5(x}][F.5{K2x) M
F 5 (x)]. A fourth possibility is for the middle element
to fail under load x, and each of the end eiements to
survive x, but fail under their overloads Kjx; this has
probability F5(x}[F5(K,x) M 135.00]? Finally, we
may have a sequence of failures starting with the
middle element, such as the event {)5 <X, 5:. sz,
X2 S x, KIx < X3 $3 Kgx). There are two such events,
and the resulting probability term is 2F5{x){Fa(K1x)
w F§(x)][F5(K2x) * E;(K1x)]. Summing all the above
probabilities yields G305) 3 4Fa(x)Fé(Kix)Fa(K2X)
““ F6(X)F6(K1x)2 _ FaUllFanax)
M 2175002 F5(K1x)+1i‘a(x)3, x 2 0 (6) In the analysis of bundle failure, a useful distribution
function is Gilda), x U, which we define as the
distribution function for the ﬁber load )5, at which k
or more adiacent breaks occur in the microbundie of
n elements. Note that Glg‘lOc) : Gn(x). To compute
G‘EJL‘C), we note that 1 — 02210:) is the probability
that at most only isolated broken elements occur in
the bundle. One possibility is for none of the elements
to fail, and this has probability [1 w F5(x')]3. Another
way is for exactly one element to fail and thisyields
the probability ZFﬁngl'l m F5(K1r)]l1 — F§(x)] +
F5(x){1 — F,;(K,x)'§‘ , where the ﬁrst term cor
responds to an end failure and the second term to a 1709 Fiber Bundles: Strength Statistics middle failure. Finally, exactly two elements may
fail, but these must be the ﬁrst and last elements with
the middle element as a survivor; this yields the
probability F5(x)2[1 W F5(K2x)]. Summing these
probabilities, and subtracting them from unity gives 6530:) = 4Fa(x)Fa(K1x) we 2F5{x)2
" Fa (X)F6(Kix)2 “ ZFalxlea (le)
+ F5002 F5(K2x) + F5003, x 2 0 (7) Lastly, we obtain GglMx) = 1 ~ [1 w 1750013, since
1 — Smut) is the probability of having no breaks in
the microbundle. Figure 2 shows the various distribution functions
above, under both Eqn. (2) and Eqn. (3) with p =
15. The scaling is that of Weibull probability paper,
so that a Weibull distribution plots as a straight line.
(Weibull probability paper has coordinates which are
linear in ln{ln(1 — G)] versus ln(x), where G is the
cumulative probability and x is the load.) Also shown
are the values of the distribution function H,,,,,,(x) for
the strength of a bundle of ﬁbers with m = 1000 and one
0 so
on)
2
'0 0.99
0.90
3
,3: “3 w 0.5o A
a
g lose olo v;
e f
._
“c 30'5 It)"2
>\
.t
5 l0'5 40.3
I)
,0
O
a lO'?
lO'a
10‘9
0.2 cs 0.4 0.5 0.6 or on on LG
Dimensionless load x/xa
Figure2 Distribution functions associated with the failure of a
bundle of three ﬁbers displayed under the Weibull and
power distributions Notice that it makes little difference whether one
uses the Wcibuil distribution [Eqn (2)}, or the power
distribution [EQI‘L (3)] for the ﬁber elements; clearly
the lower tail of F§(x) matters most, {When p is
substantially less than 5, the difference is more
marked; however, such low values of p occur
infrequently for commercial ﬁbers.) Also, the 1710 expected branch points in the distributions at the
critical loads x5/K, and xa/Kg should be noted. For
loads x such that le < x5 < sz, a pair of adjacent
breaks is critical for the bundle, because of the posi~
tion of its rnedian strength along the load axis.
Figure 2 indicates that the bundle strength approxi—
mately follows a Weibull distribution with shape
parameter kp = 30. To see why, and to calculate the
scale parameter, the lower tails of the GE,” (x) are
investigated. Since F50?) ~ (DC/Kg)” (where ~ indi»
cates that the ratio of the two sides tends to one as x —> 0), we can show by direct substitution into Eqn.
(6) and Eqn. (7) that for k x 1,2,3, Gite) ~ (3 w k + ode/me (8) where G§3lEG3, d2: 1, a, = 2m m 1 and d, =
4K“; Kg a K219  Kg  2105+ 1. Thus we have the
Weibull approximation: Hate) e 1  expl~mtn — I“: + lldtlx/xalf‘pl,
x a a (9) where ,0 =15, m =1000, n = 3,412 = 2 and d; =2Kf
— 1. (Beware that the critical It changes if. p or m
are changed signiﬁcantly.) The two stumbling blocks to using the above
approximations as it increases are as follows: ﬁrst it
is determined graphically, and this requires prior
knowledge of Giﬂx); secondly, Glf‘Kx) and dk
become very difﬁcult to calculate as k increases.
Smith (1980) has resolved these issues with his asymp
totic analysis as we now see. 6. Exact and Approximate Results for Large
Bundles For k = 1,2,3, . . . we let (to) where KO E 1. We also let It be the value of k that
Solves at) = more m g 111th Ylk ~1><1ntmnvp<nk> <11) From Smith (1980) we have the Weibull approxi mation: Ham 2 1 w expi~mn cite/min. a
x20 on;
where ' sﬁgkeugwyuangp, k=1.2,...(13) Here, It is interpreted as the critical failure sequence Size. This result is consistent with those of the previous section for it 2 3. Here, we use it in place of n  k + 1, because It is large. Also, we use (1,, as an}?
accurate approximation to dk; the error is very smallﬁ for p 2 5, which is typical. E Fiber Bundles: Strength Statistics To evaluate the accuracy of the above Weibuil
approximation, we recall the Harlow~Phoenix
approximation {Eqn (5)], which for all practical ' purposes represents H,,,,,,(x) exactly. Figure 3 shows 5) C6 115 a plot of the characteristic distribution function W(x)
for Weibull ﬁber elements with p a 5. (This low value
of p yields a true test of accuracy since the differences
are most exaggerated.) Also shown is the distribution
function Hm’nﬁc) for a bundle with ma 2 105 elements
in total. The coordinates are again Weibull co
ordinates.
For k: 1,2, . . . we let gill“) = 1  explri Entrants]. x 2 0 {14) On the scale for W(x), the straight lines on Fig. 3
happen to be the distribution functions 975%), but
on the scale for Hm),,(x) they are Simon. Clearly one
of the (x) for some k is an accurate approximation
of Hm,n(x), and it can be seen from Fig. 3 that k z 5
is the correct choice. This value of k is also the critical
it which satisﬁes Eqn. (11), and thus (x) is simply
the Weibull approximation [Eqn (12)]. Characteristic distribution function thl O 2 0.3 0.4 0,5 0.6 0.?
Dimensionless load X/Ig
Figure 3 Characteristic distribution function for bundle strength
and associated Weibull approximations An accurate estimate of W05) is the inner envelope
formed by the Weibull distributions g; 1(x), x a 0,
that is, We) e min[95ill(x),97:21(x),%l3l(x),. . .1, x20 (15) of The aboveideas are discussedin more detailin
ailPhoenix and Smith (1983), and Smith et al. (1983) at} extend the results to bundles with ﬁbers arranged in. a hexagonal array. 7. Conclusions We have found that the strength of a bundle of
ﬁbers in a matrix approximately follows a Weibull
distribution. The shape parameter for this Weibull
distribution is kp, where k is the critical failure
sequence size and ,o is the Weibul] shape parameter
for the ﬁber. In the example of Fig. 3, we have lip =
5 X 5 = 25, and the scale parameter for this Weibull
distribution is x5(mnd;,)’l/kp, which is clearly much
smaller than 26,5. This is typically observed in
experiments.  Unlike the model itself, the ﬁnal results are sym
metric in the number of ﬁbers :1 and the number of
microbundles m; that is, the total bundle volume
mod is what matters. See also: Fracture: Statistical Theories; Strength of
Composites Bibliography Argon A S 1972 Fracture of composites. In: Herman H
(ed) Treatise on Materials Science and Technology, Vol.
1. Academic Press, New York, pp. 7941 Harlow D G, Phoenix S L 1978a The chainedbundles
probability model for the strength of ﬁbrous materials.
1: Analysis and conjectures. J. Campos. Mater. i2: 195—
214 Harlow D G, Phoenix S L 1978b The chainof—bundles
probability model for the strength of ﬁbrous materials.
II: A numerical study of convergence, J. Campos. Mater.
12: 314—34 Harlow D G, Phoenix S L 1981a Probability distributions
for the strength of composite materials. I: Twolevel
bounds. Int. J. Fracr. 17: 347w72 Harlow D G, Phoenix 5 L 1981b Probability distributions
for the strength of composite materials. ll: A convergent
sequence of tight bounds. Int. J. Freer. 17: 601730 Phoenix S L, Smith R L 1983 A. comparison of probabilistic
techniques for the strength of ﬁbrous materials under
local ioad~sharing among ﬁbers Int. J. Solids Struct. '19:
479—96 Rosco B W l964 Tensile failure of ﬁbrous composites.
AIAA J. 2: 198541 Smith R L 1979 Limit theorems for the reliability of series
parallel loadsharing systems. Ph.D thesis, Cornell Uni—
versity, Ithaca, New York Smith R L 1988 A probability model for ﬁbrous composites
with local loadsharing. Proc. R. Soc. London, Ser. A
372: 539—53 Smith R L, Phoenix S L 1981 Asymptotic distributions
for the failure of ﬁbrous materials under series—parallel
structure and equal loadsharing. J. Appl. Meek. 48: 75—
82 Smith R L, Phoenix S L, Greenﬁeld M R, Henstenburg
R B, Pitt R E 1983 Lowertail approximations for the
probability of failure of three—dimensional ﬁbrous com
posites with hexagonal geometry Proc. R. Soc. London,
Ser. A 388: 35391 Zweben C, Rosen B W 1970 A statistical theory of material
strength with application to composite materials. J.
Mech. Phys. Solids 18: 189406 S. L. Phoenix 1711 Fiber Networks Fiber Networks: Models for Predicting
Mechanical Behavior of Paper The mechanical behavior of a ﬁber network, such
as a paper sheet composed oi pulped wood ﬁbers,
depends on the ﬁber properties and the physical or
geometrical structure of the bonded ﬁbrous network.
The principal objective of a mathematical model is
to predict the mechanical properties of the ﬁber
network in terms of the properties of the ﬁbers and
the structure of the ﬁber network. In paper, the ﬁbers
are often collapsed, ribbonlike structures which are
bonded together, primarily by hydrogen bonds
formed when the sheet is pressed and subsequently
dried (see Hydrogen Bonding in Paper: Theory). The ﬁrst attempt to develop a mathematical model
for predicting the mechanical behavior of a ﬁber
network such as paper was made by Cox (1952),
who demonstrated how ﬁber orientation distribution
inﬂuences the in~plane elastic behavior of the system.
The analysis predicted all the in—plane elastic con
stants of a ﬁber mat in terms of the ﬁber orientation
distribution expressed as a Fourier series. Although Cox’s analysis for iiiplane elastic con
stants was based on an. ideal mat in which the ﬁbers
lie in a plane and extend from one edge of the sheet
to the other, he also discussed the effect of short
ﬁbers in a resinbonded system. Since Cox’s pioneer
ing work, a considerable amount of study has been
done. The basic premises of Cox’s work are generally
accepted as being valid, but it has been subiected to
reﬁnements and extensions The reﬁnements incor—
porate the effects of ﬁber network structure, straining
during the formation of ﬁbermﬁber bonding and dif
ferent ﬁber elastic moduli for tensile and compressive
straining. Extension of the analysis allows predictEOn
of the ﬁber network strength. Work in this area has been reviewed by Algar
(i966), Dodson (1973), Van den Akker (i970, 1972)
and Kallmes (1972a, 1972b). The most recent work
was reported by Perkins (1980, 1982), Perkins and
Maris {1981), and Seth and Page (1981). in this article, the basic assumptions and results of
the current mathematical models for predicting the
inplane mechanical behavior oi ﬁbrous networks are
presented. I. Basic Elements of the Mathematical Model A typical ﬁber shown in Fig. 1 consists of several
straight segments £51, 152, 1,3, . . . . The sum of the
segment lengths equals the total ﬁber length AT. it is
assumed that the ﬁbers are so flexible in bending that
no appreciable load can be transmitted from one
straight segment to the next. If a ﬁber has microcompressions or other damaged
regions along its length, the locations of these regions
determine the ends of the load~bearing segments,
even if the ﬁber is perfectly straight. The segment 1712 Figure 1'
Schematic of ﬁber having three distinct segments and
numerous crossing ﬁbers in contact with it elements are coupled to the network by means of the
crossing ﬁbers. The strains in the sheet are presumed
to be transmitted to the segment elements by bending
and shearing deformation of the crossing ﬁbers and
by shearing deformation of the ﬁber—fiber bonds.
Thus the axial strain in the segment elements is not
uniform but varies from the segment ends, where it
is zero, to the middle, where it has its maximum
value. If the segment is long enough and if the coup
ling of the crossing ﬁbers is strong, the axial strain in 
the middle of a segment is the same as the normal 3 '
component of strain of the sheet for the direction of
the segment. On the other hand, if the segment is
short and / or the coupling is weak, the segment strain
is less than that associated with the sheet. The model is iurther illustrated by Fig. 2, whic
shows a portion of a segment that is coupled by two T
crossing ﬁbers to the remainder of the network. The ,
boundary between the element and the network is depicted by a dashed line. This boundary is assumed _
to be located a distance i from the centerline "
the segment, where i represents the centerwcenter
distance between bonds along a typical fiber; lb ‘
bond length along the ﬁber and w and if are the widt
and thickness of the ﬁber, respectively. If I is small!
in comparison with wf, as would be expected in: ‘
moderately dense paper, the coupling is primarily} attributable to the shearing deformations of the ﬁber“;1 ﬁber bonds. in a very low—density system such a tissue paper, the bending and shearing deformatio
of the crossing ﬁbers may be substantial. On the assumption that the ﬁber is linearly elasti
the stress of at any point along the ﬁber is related to i
the ﬁber strain of at that point by = ‘ 0r = EarEr (ll where E g is the effective axial modulus of the ﬁbe
The effective axial modulus depends not only on the inherent properties of the ﬁber but also on how the ‘
ﬁber is bonded to other ﬁbers in the network. Thus:
there is a stiffening effect due to the reinforcement
of other bonded ﬁbers. Eaf can be considered as at: inherent property of the paper system, or its valui1 ;
pt
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 Fall '10
 PetruPetrina
 Composite Materials, Probability distribution, Distribution function, Weibull distribution, strength statistics

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