Fiber Bundle Strength Statistics

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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 metal-clad 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 bred-in 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 emu-t). 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. 79CH144l-5 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 'Chain-o'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 tension-testing 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 load-per-fiber 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 load-sharing 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 chain-oi- 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 this-yields 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 Ggl-Mx) = 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 above-ideas are discussed-in more detail-in ail-Phoenix 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 chained-bundles 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 chain-of—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: Two-level 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 load-sharing systems. Ph.D thesis, Cornell Uni— versity, Ithaca, New York Smith R L 1988 A probability model for fibrous composites with local load-sharing. 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 load-sharing. J. Appl. Meek. 48: 75— 82 Smith R L, Phoenix S L, Greenfield M R, Henstenburg R B, Pitt R E 1983 Lower-tail approximations for the probability of failure of three—dimensional fibrous com- posites with hexagonal geometry Proc. R. Soc. London, Ser. A 388: 353-91 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, ribbon-like 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 iii-plane 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 resin-bonded 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 in-plane 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 5" ...
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