《复合材料 Composites》课程教学资源（学习资料）第二章 增强体_glass fiber-19 The Single Fiber Composite Test：A Comparison of E-Glass Fiber Fragmentation Data with Statistical Theories

The Single Fiber Composite Test: A Comparison of E-Glass Fiber Fragmentation Data with Statistical Theories Gale A. Holmes, Jae Hyun Kim, Stefan Leigh, Walter McDonough Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8541 Received 19 November 2007; accepted 25 October 2008 DOI10.1002/app.31002 Publishedonline22March2010inWileyInterscience(www.interscience.wileycom) ABSTRACT: The exact theories advanced by Curtin length of the fiber specimens, with the uniformity appa- and Hui et al. to describe the fiber break evolution pro- rently being independent of interfacial shear strength, fiber cess in single fiber composites are found to be incorrect type, matrix type, and fiber-fiber interactions. The theory when compared with experimental data. In contrast to the- of uniform spacings gives an explicit distribution function oretical predictions where the matrix is assumed to be for the ordered fragment lengths. 0 2010 wiley Periodicals, lastic perfectly plastic, experimental data indicate that the Inc. J Appl Polym Sci 117: 509-516, 2010 sizes of the fragment lengths that survive to saturation decrease as the strain is increased. It is also shown that Key words: glass fibers; polymer-matrix composites the break locations at saturation are uniform along the(PMCS); fragmentation; interface; statistics INTRODUCTION tions evolve to a uniform distribution as saturation The interest in the single fiber fragmentation test is approached. This result implies that the ordered (SFFT)methodology lies in the use of the test out- spacings (ie fragment lengths) at saturation are puts to quantify the interfacial shear strength(IFSS which is a fundamental property used to character CDF) described by (1)ve distribution function ize the level of adhesion between the fiber and the matrix in composites. This property is thought to be Pr(Dm-n)sx) obtainable from micromechanical tests such as the SFFT. Currently, the SFFT methodology provides )-(+s9x1(1) only a relative means of quantifying the perform ance of formulations designed to promote adhesion between the fiber and matrix. The outputs from this where 0<x<l and a,= max(a, 0)(i.e, fiber length behavior. Due to the importance of this test, there is length. ngth,i e total number of breaks over the test methodology have been used to quantity om- U(o, 1 length, Du) denotes the (n-i)'h fra an extensive literature on this subject and the These results contrast with the experimental embedded fiber fragmentation test(EFFT) methodol- results obtained by Gulino and Phoenix from three- fiber hybrid microcomposites where a 5.5 um graph- ogies, including single fiber and multifiber array ite fiber was sandwiched between two 13-Hm SK decade to become potential tools for quantifying the glass fibers with an interfiber distance of (31)um impact of fiber-fiber interaction and its impact on fragment length distribution and the evolution of the In a recent publication, Kim et al. analyzed fiber break density with increasing stress conformed of E-glass fibers embedded in single fiber composite expected the distribution of breaking stresses to con- (SFC) specimens using the SFFT methodology the primary result being that the fiber break loca- stand the basis for agreement over such a wide range of stress. The results of Kim et al. indicate that the wide agreement observed in the Gulino and Correspondence to: G. A. Holmes (gale. holmes@nist. gov) Phoenix data may in fact not be universal. This result is important since only the Gulino and Journal of Applied Polymer Science, Vol. 117, 509-516(2010) Phoenix experimental data provide support for the o 2010 Wiley Periodicals, Inc. theories", that have been advanced to quantify the

The Single Fiber Composite Test: A Comparison of E-Glass Fiber Fragmentation Data with Statistical Theories Gale A. Holmes, Jae Hyun Kim, Stefan Leigh, Walter McDonough Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8541 Received 19 November 2007; accepted 25 October 2008 DOI 10.1002/app.31002 Published online 22 March 2010 in Wiley InterScience (www.interscience.wiley.com). ABSTRACT: The exact theories advanced by Curtin6 and Hui et al.7 to describe the fiber break evolution pro￾cess in single fiber composites are found to be incorrect when compared with experimental data. In contrast to the￾oretical predictions where the matrix is assumed to be elastic perfectly plastic, experimental data indicate that the sizes of the fragment lengths that survive to saturation decrease as the strain is increased. It is also shown that the break locations at saturation are uniform along the length of the fiber specimens, with the uniformity appa￾rently being independent of interfacial shear strength, fiber type, matrix type, and fiber–fiber interactions. The theory of uniform spacings gives an explicit distribution function for the ordered fragment lengths. VC 2010 Wiley Periodicals, Inc. J Appl Polym Sci 117: 509–516, 2010 Key words: glass fibers; polymer-matrix composites (PMCS); fragmentation; interface; statistics INTRODUCTION The interest in the single fiber fragmentation test (SFFT) methodology lies in the use of the test out￾puts to quantify the interfacial shear strength (IFSS), which is a fundamental property used to character￾ize the level of adhesion between the fiber and the matrix in composites. This property is thought to be obtainable from micromechanical tests such as the SFFT. Currently, the SFFT methodology provides only a relative means of quantifying the perform￾ance of formulations designed to promote adhesion between the fiber and matrix. The outputs from this test methodology have been used to quantify com￾posite performance and model composite failure behavior. Due to the importance of this test, there is an extensive literature on this subject and the embedded fiber fragmentation test (EFFT) methodol￾ogies, including single fiber and multifiber array configurations, which have evolved in the last decade to become potential tools for quantifying the impact of fiber–fiber interaction and its impact on critical flaw nucleation in composites.1 In a recent publication, Kim et al.2 analyzed experimental data from the sequential fragmentation of E-glass fibers embedded in single fiber composite (SFC) specimens using the SFFT methodology, with the primary result being that the fiber break loca￾tions evolve to a uniform distribution as saturation is approached. This result implies that the ordered spacings (i.e., fragment lengths) at saturation are modeled by a cumulative distribution function (CDF) described by (1).3,4 PrðDðn
jÞ xÞ ¼ X j r¼0 n r
Xn
r s¼0 ð
1Þ s n
r s
½1
ðr þ sÞx n
1 þ ð1Þ where 0 < x < 1 and aþ ¼ max(a,0) (i.e., fiber length is 1), n denotes the total number of breaks over the U[0,1] length, D(n
j) denotes the (n
j) th fragment length. These results contrast with the experimental results obtained by Gulino and Phoenix5 from three￾fiber hybrid microcomposites where a 5.5 lm graph￾ite fiber was sandwiched between two 13-lm SK glass fibers with an interfiber distance of (3  1) lm along the specimen length. From their data, the final fragment length distribution and the evolution of the fiber break density with increasing stress conformed to Weibull distributions. Curtin6 noted that he expected the distribution of breaking stresses to con￾form to a Weibull at low stresses but did not under￾stand the basis for agreement over such a wide range of stress. The results of Kim et al. indicate that the wide agreement observed in the Gulino and Phoenix data may in fact not be universal. This result is important since only the Gulino and Phoenix experimental data provide support for the theories6,7 that have been advanced to quantify the Correspondence to: G. A. Holmes (gale.holmes@nist.gov). Journal of Applied Polymer Science, Vol. 117, 509–516 (2010) VC 2010 Wiley Periodicals, Inc.

HOLMES ET AI physics of the sequential fragmentation process that fragmentation of E-glass/DGEBA/m-PDA specimens occurs in the sfFt and carbon fiber/DGEBA/m-PDA specimens, where It is important to note that the theories and the DGEBa denotes the diglycidyl ether of bisphenol-A supporting Monte Carlo simulations assume that the and m-PDA denotes meta-phenylenediamine matrix is elastic-perfectly plastic (EPP). Although this assumption has been repeatedly shown to be incorrect for most polymer matrices, the EPP EXPERIMENTAL assumption is generally considered to be a reasona- Unsized E-glass fibers, w 15 um in diameter, were ble approximation for capturing the key features of obtained from Owens Corning. The fibers were the sequential fragmentation process in the SFFT either used as received(bare E-glass fibers)or methodology The EPP assumption leads to the con- treated with the n-octadecyl triethoxysilane clusion that the smallest breaks in the final fragment (NOTS) or glycidyloxypropyl trimethoxysilane length distribution are formed early in the test when (GOPS), with the GOPS surface treatment performed the critical transfer length is shortest. This assump- by the procedure given in Ref. 17. The AS-4 carbon tion anchors the filtered distribution concept that fibers were obtained from the Hexcel Corporation. was advanced by Curtin to develop his theory and The mold preparation procedure and curing proce- found to be plausible by Hui et al. in the develop- dure for the E-Glass SFCs made using the diglycidyl ment of their theory. The experimental data ether of bisphenol-A(DGEBA)resin(Epon 828, Shell) published by Kim et al. on E-glass SFCs showed the cured with meta-Phenylenediamine (n-PDA, Fluka, opposite effect, so that the filtered distribution con- or Sigma-Aldrich) have been published previously by cept utilized by the two theories cannot be applied Holmes et al. 5, 16, 18, 9 McDonough et al. have to the kim et al. data described the procedure for preparing the poly In addition to casting doubt upon the universality isocyanurate SFCs, and Kim et al. have described of the theories, the Kim et al. data suggest that the the procedure for preparing the combinatorial micro- physics of the sequential fragmentation process may composite SFC specimens used in this report. The composite failure behavior since the key input Rich et al. using the procedure described in Re 1 not be well-enough understood to reliably predict AS-4 carbon fiber SFC specimens were prepared arameters are obtained from the efft methodolo- The testing protocols for the E-glass SFC speci- gies. As an example, these composite failure models mens are most completely described in Ref. 18 and indicate that the density of fiber breaks increases as the test protocol associated with the AS-4 SFC speci the interfiber distance between fibers decreases. mens is described in Ref. 15. Finally, the automated Results by Li et al. on micromechanics specimens testing procedure used for the combinatorial micro- composed of 2D Nicalon multifiber arrays, and later composites has been previously described by Kim confirmed by Kim and Holmes on 2D E-glass et al. 121 The standard uncertainty in determining multifiber arrays, indicate that the break density the break locations was determined to be 1.1 um along the length of a fiber decreases as the interfiber whereas the standard uncertainty in the reported distance decreases. This result contradicts the predic- fragment lengths is 1.6 um. tion arrived at from shear lag models derived by ano Therefore, the Kim et al. and Li et al. experi RESULTS AND DISCUSSIONS results indicate that additional investigations are The effects of matrix behavior, adhesion strength, required of the EFFT methodologies to determine and testing rate on uniform break formation in the efficacy of these approaches in assessing interfa- E-glass SFCs cial phenomena in composite materials, in providing The locations of the fiber breaks along the length of useful input parameters for composite failure an E-glass fiber embedded in a SFC composed of models, and in assessing critical flaw nucleation in DGEBA/m-PDA epoxy resin conform to a uniform composite materials. In this article, the fragmenta- distribution, where the probability plot correlation tion of embedded E-glass fibers is further investi- coefficients of the break locations for the uniform gated by assessing the impact of the matrix type, distribution from multiple samples were consistently IFSS, and fiber-fiber interactions on the evolution of the sequential fiber fragmentation process. greater than or equal to 0.999(Fig. 1). From the For completeness, the relative break locations that occurred in SFCs tested by the 2nd VAMAS(the Certain commercial materials and equipment are identified Versailles Project on Advanced Materials and in this paper to specify adequately the experimental proce- Standards) Round Robin testing protoco/'5 are fitted dure. In no case does such identification imply recommenda- tion or endorsement by the National Institute of Standards to the uniform distribution function to illuminate and Technology, nor does it imply necessarily that the prod- differences that may arise between the sequential uct is the best available for the purpose Journal of applied Polymer Science DOI 101002/app

physics of the sequential fragmentation process that occurs in the SFFT. It is important to note that the theories and the supporting Monte Carlo simulations assume that the matrix is elastic-perfectly plastic (EPP). Although this assumption has been repeatedly shown to be incorrect8,9 for most polymer matrices, the EPP assumption is generally considered to be a reasona￾ble approximation for capturing the key features of the sequential fragmentation process in the SFFT methodology. The EPP assumption leads to the con￾clusion that the smallest breaks in the final fragment length distribution are formed early in the test when the critical transfer length is shortest. This assump￾tion anchors the filtered distribution concept that was advanced by Curtin6 to develop his theory and found to be plausible by Hui et al.7 in the develop￾ment of their theory. The experimental data published by Kim et al. on E-glass SFCs showed the opposite effect, so that the filtered distribution con￾cept utilized by the two theories cannot be applied to the Kim et al. data. In addition to casting doubt upon the universality of the theories, the Kim et al. data suggest that the physics of the sequential fragmentation process may not be well-enough understood to reliably predict composite failure behavior since the key input parameters are obtained from the EFFT methodolo￾gies. As an example, these composite failure models indicate that the density of fiber breaks increases as the interfiber distance between fibers decreases. Results by Li et al.10 on micromechanics specimens composed of 2D Nicalon multifiber arrays, and later confirmed by Kim and Holmes11 on 2D E-glass multifiber arrays, indicate that the break density along the length of a fiber decreases as the interfiber distance decreases. This result contradicts the predic￾tion arrived at from shear lag models derived by Cox12 and others.13,14 Therefore, the Kim et al. and Li et al. experimental results indicate that additional investigations are required of the EFFT methodologies to determine the efficacy of these approaches in assessing interfa￾cial phenomena in composite materials, in providing useful input parameters for composite failure models, and in assessing critical flaw nucleation in composite materials. In this article, the fragmenta￾tion of embedded E-glass fibers is further investi￾gated by assessing the impact of the matrix type, IFSS, and fiber–fiber interactions on the evolution of the sequential fiber fragmentation process. For completeness, the relative break locations that occurred in SFCs tested by the 2nd VAMAS (the Versailles Project on Advanced Materials and Standards) Round Robin testing protocol15 are fitted to the uniform distribution function to illuminate differences that may arise between the sequential fragmentation of E-glass/DGEBA/m-PDA specimens and carbon fiber/DGEBA/m-PDA specimens, where DGEBA denotes the diglycidyl ether of bisphenol-A and m-PDA denotes meta-phenylenediamine. EXPERIMENTAL Unsized E-glass fibers,  15 lm in diameter, were obtained from Owens Corning.* The fibers were either used as received (bare E-glass fibers) or treated with the n-octadecyl triethoxysilane (NOTS)16 or glycidyloxypropyl trimethoxysilane (GOPS), with the GOPS surface treatment performed by the procedure given in Ref. 17. The AS-4 carbon fibers were obtained from the Hexcel Corporation.15 The mold preparation procedure and curing proce￾dure for the E-Glass SFCs made using the diglycidyl ether of bisphenol-A (DGEBA) resin (Epon 828, Shell) cured with meta-phenylenediamine (m-PDA, Fluka, or Sigma-Aldrich) have been published previously by Holmes et al.9,15,16,18,19 McDonough et al.20 have described the procedure for preparing the poly￾isocyanurate SFCs, and Kim et al.21 have described the procedure for preparing the combinatorial micro￾composite SFC specimens used in this report. The AS-4 carbon fiber SFC specimens were prepared by Rich et al. using the procedure described in Ref. 15. The testing protocols for the E-glass SFC speci￾mens are most completely described in Ref. 18 and the test protocol associated with the AS-4 SFC speci￾mens is described in Ref. 15. Finally, the automated testing procedure used for the combinatorial micro￾composites has been previously described by Kim et al.11,21 The standard uncertainty in determining the break locations was determined to be 1.1 lm, whereas the standard uncertainty in the reported fragment lengths is 1.6 lm.16 RESULTS AND DISCUSSIONS The effects of matrix behavior, adhesion strength, and testing rate on uniform break formation in E-glass SFCs The locations of the fiber breaks along the length of an E-glass fiber embedded in a SFC composed of DGEBA/m-PDA epoxy resin conform to a uniform distribution, where the probability plot correlation coefficients of the break locations for the uniform distribution from multiple samples were consistently greater than or equal to 0.999 (Fig. 1). From the * Certain commercial materials and equipment are identified in this paper to specify adequately the experimental proce￾dure. In no case does such identification imply recommenda￾tion or endorsement by the National Institute of Standards and Technology, nor does it imply necessarily that the prod￾uct is the best available for the purpose. 510 HOLMES ET AL. Journal of Applied Polymer Science DOI 10.1002/app

COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 511 nitial strain tion fit(ppcc >0.99)was initially achieved at less than 12 breaks/cm Analysis of the GOPS SFC specimens showed that uniform distributions of fiber break locations were also achieved at saturation, correlations from(0.9990 Nth strain Tensile stress profile to 0.9997), with break densities at the end of the test varying from(19.4 to 26.3)breaks /cm(Fig. 2). The onset of uniformity in these specimens occurred between 16 and 26 breaks/cm. Therefore, 94.7% of the 19 E-glass samples analyzed yield break loca- tions that are strongly modeled by a uniform distri- bution, with the set with one outlier coming from the NOTS treated E-glass SFC samples Figure 1 Schematic representation of fiber fragments correlation coefficient for the fiber breaks fitted to a occurring in the single fiber fragmentation test uniform distribution of o 9972. These results indicate that the expected outcome from the sequential statistical theory of spacings, this result leads to the fragmentation of E-glass fiber SFCs at saturation are conclusion that the ordered distribution of the break locations that correspond to a uniform distri. spacings (i.e., fragment lengths) produced by a SFFT bution, with standard statistics spacing theories indi- conforms to(1) cating that the ordered spacings (i.e,fragment To assess the generality of the Kim et al. results, lengths) at saturation conform to the distribution single bare (i. e, unsized) E-glass fibers were embed- function given in (1) ded in a polyisocyanurate matrix and tested using These results appear to contradict the experimen- fast, intermediate, and slow test protocols. In con- tal data of Gulino and Phoenix who studied the se- trast to the bare E-glass DGEBA/m-PDA SFC speci- quential fragmentation of carbon fiber hybrid micr mens, the fiber break densities of these specimens composites. However, it is worthwhile noting that were unaffected by the testing rate. However, all of the E-glass SFCs tested by Holmes et al. exhib analyses of the break locations from these specimens ited debonded regions whose total length comprised indicate that they conform, like the DGEBA/m-PDA less than 5% of the total sample length. As an exam- SFC specimens, to a uniform distribution as satura- ple, the NOTS SFC specimens yielded the largest tion is approached with probability plot correlation average debond regions around each fiber break coefficients greater than 0.999. Consistent with the ( 26 um), with the range of the values for the four behavior observed in the bare E-glass DGEBA/m- specimens being between(11 and 37) um. Therefore, PDA SFC specimens, the break locations evolve to a the debond regions occurring in the fracture of uniform distribution at fiber break densities of 21 breaks/cm and remain uniform for the remainder of he test, with break densities on the order of 099 2 30 breaks/ cm being observed To span the range of interfacial shear strengths, ea 1 tests were also performed on E-glass fibers treated 3 a97 with n-octadecyl triethoxysilane(NOTs) and glyci- a9 dyloxypropyl triethoxysilane (GoPS) that were also 2 awns Polyiso cant ae. All Protocols embedded in the DGEBA/m-PDa matrix. As expected the NOTS SFC specimens exhibited a marked reduction in the fiber break density at satu- ration since the n-octadecyl group does not cova- Minimum value for 0.999 correlation coemcient lently bond to the DEGBA/m-PDA matrix. For the four specimens tested, the saturation break densities ged from(13 to 17)breaks/cm, significantly lower than those observed for the bare E-glass fibers Figure 2 Correlation coefficients for probability plot fit of matrices. Despite these low-break densities, the fiber tion (a) Solid symbols: E-Glass fiber SFCs with various sur break locations at saturation conformed in each face treatments(bare, NOTS, and GOPS), test protocols(fast, specimen to a uniform distribution with probability intermediate, and slow), and matrices(DGEBA/m-PDA ep- plot correlation coefficients ranging from 0.9972 to fiber SFCs in DGEBA/m-PDA tested by fast(or VAMAS) 0.9994 for the four specimens tested(Fig. 2). For the testing protocol. 15 [Color figure can be viewed in the online NotsdatadepictedinFigure2,auniformdistribuissuewhichisavailableatwww.interscience.wiley.com.j Journal of applied Polymer Science DOI 10.1002/ app

statistical theory of spacings, this result leads to the conclusion that the ordered distribution of the spacings (i.e., fragment lengths) produced by a SFFT conforms to (1). To assess the generality of the Kim et al. results, single bare (i.e., unsized) E-glass fibers were embed￾ded in a polyisocyanurate matrix and tested using fast, intermediate, and slow test protocols.18 In con￾trast to the bare E-glass DGEBA/m-PDA SFC speci￾mens, the fiber break densities of these specimens were unaffected by the testing rate. However, analyses of the break locations from these specimens indicate that they conform, like the DGEBA/m-PDA SFC specimens, to a uniform distribution as satura￾tion is approached with probability plot correlation coefficients greater than 0.999. Consistent with the behavior observed in the bare E-glass DGEBA/m￾PDA SFC specimens, the break locations evolve to a uniform distribution at fiber break densities of  21 breaks/cm and remain uniform for the remainder of the test, with break densities on the order of 30 breaks/cm being observed (Fig. 2). To span the range of interfacial shear strengths, tests were also performed on E-glass fibers treated with n-octadecyl triethoxysilane (NOTS) and glyci￾dyloxypropyl triethoxysilane (GOPS) that were also embedded in the DGEBA/m-PDA matrix. As expected the NOTS SFC specimens exhibited a marked reduction in the fiber break density at satu￾ration since the n-octadecyl group does not cova￾lently bond to the DEGBA/m-PDA matrix. For the four specimens tested, the saturation break densities ranged from (13 to 17) breaks/cm, significantly lower than those observed for the bare E-glass fibers tested in the DGEBA/m-PDA and polyisocyanurate matrices. Despite these low-break densities, the fiber break locations at saturation conformed in each specimen to a uniform distribution with probability plot correlation coefficients ranging from 0.9972 to 0.9994 for the four specimens tested (Fig. 2). For the NOTS data depicted in Figure 2, a uniform distribu￾tion fit (ppcc > 0.99) was initially achieved at less than 12 breaks/cm. Analysis of the GOPS SFC specimens showed that uniform distributions of fiber break locations were also achieved at saturation, correlations from (0.9990 to 0.9997), with break densities at the end of the test varying from (19.4 to 26.3) breaks/cm (Fig. 2). The onset of uniformity in these specimens occurred between 16 and 26 breaks/cm. Therefore, 94.7% of the 19 E-glass samples analyzed yield break loca￾tions that are strongly modeled by a uniform distri￾bution, with the set with one outlier coming from the NOTS treated E-glass SFC samples, yielding a correlation coefficient for the fiber breaks fitted to a uniform distribution of 0.9972. These results indicate that the expected outcome from the sequential fragmentation of E-glass fiber SFCs at saturation are break locations that correspond to a uniform distri￾bution, with standard statistics spacing theories indi￾cating that the ordered spacings (i.e., fragment lengths) at saturation conform to the distribution function given in (1). These results appear to contradict the experimen￾tal data of Gulino and Phoenix who studied the se￾quential fragmentation of carbon fiber hybrid micro￾composites. However, it is worthwhile noting that all of the E-glass SFCs tested by Holmes et al. exhib￾ited debonded regions whose total length comprised less than 5% of the total sample length. As an exam￾ple, the NOTS SFC specimens yielded the largest average debond regions around each fiber break ( 26 lm), with the range of the values for the four specimens being between (11 and 37) lm. Therefore, the debond regions occurring in the fracture of Figure 1 Schematic representation of fiber fragments occurring in the single fiber fragmentation test. Figure 2 Correlation coefficients for probability plot fit of fiber break locations at saturation to the uniform distribu￾tion. (a) Solid symbols: E-Glass fiber SFCs with various sur￾face treatments (bare, NOTS, and GOPS), test protocols (fast, intermediate, and slow), and matrices (DGEBA/m-PDA ep￾oxy and polyisocyanurate). (b) Open symbols: AS-4 carbon fiber SFCs in DGEBA/m-PDA tested by fast (or VAMAS) testing protocol.15 [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 511 Journal of Applied Polymer Science DOI 10.1002/app

HOLMES ET AI typical E-glass SFC specimens are significantly less pendent and well-defined Ho) afforded by the EPP than the(85 to 130)um debond regions observed by assumption. However, they took issue with the Gulino and Phoenix, even though these researchers value of the maximum achievable packing density achieved break densities at saturation of 25 and along the broken fiber, stating that the value should 15 breaks/ be one rather than the value of 0.7476 used b From a brief review of the literature on the silane Curtin the NOTS SFC composites may be due to a predomi s n a previous publication by Kim et al, it is treatment of glass fibers, the minimal debonding in In nown for a bare E-glass fiber embedded in a nant mechanical interlocking stress transfer mecha- DGEBA/m-PDA matrix that the smallest fragments nism at the fiber matrix interface caused by the in the final fragment length distribution were not interpenetration of the epoxy matrix into the porous formed at the beginning of the test, as theorized by silane surface treatment. Furthermore, the porous Curtin and Hui et al., but rather at the end of the surface treatment that occurs when the glass fiber is test. Consistent with these results, the size of the treated with GOPS is also accompanied by the estab- fragments surviving to saturation were found to lishment of covalent bonds between the matrix and decrease in size as the test progressed, in apparent silane coupling agent, thereby providing a more effi- contradiction to a lo) based on the EPP assumption cient stress transfer mechanism than observed with where the theories indicate that the size of the frag the NOTS surface treatment. In Ref. 19, Holmes et ments surviving to saturation should increase as the al. showed that removal of the mechanical interlock- strain in the sFc is increased. Furthermore, the ing mechanism by treatment of the smooth glass Kelly-Tyson approximation of the critical transfer surfaces with self-assembled monolayers of n-octa- length(c =482 um for the Bare2_9 SFC specimen), decyl trichlorosilane resulted in extensive debonding also suggested that saturation was not achieved, like that observed in carbon fiber composites, where since there were five fragments whose length ranged he mechanical interlocking mechanism is based on from 488 to 527 um. This observation was somewha the surface roughness of the carbon fiber. These surprising since fragments of length 462 and 482 um results and observations suggest that in the absence fractured near the end of the test, with the shorter of extensive debonding, the physics of the sequential fragment being less than the Kelly-Tyson estimate of fiber fragmentation process in E-glass fibers leads to Ic. It should also be noted that the largest surviving a uniform distribution at saturation over a wide fragments were formed early enough to undergo at range of adhesion strengths, with the phenomenon least six additional increases in strain. appearing to be independent of matrix type To better understand the fragmentation observed in the SFC by Kim et al. the SFFT fragment evolu tion data from the SFCs composed of a bare E-glas Nonideal fragmentation behavior in E-glass fiber SFCs fiber embedded in a polyisocyanurate matrix Table )were analyzed. For the representative data shown Although it is known that most matrices used in the in Table I, the Kelly-Tyson estimate of lc is 420 um SFFt do not conform to the EPP assumption, it is which is w 13% smaller than what was observed in generally believed that the actual fragmentation pro- the bare E-Glass/DGEBA/m-PDA SFC analyzed by cess is consistent with this approximation. On the Kim et al. Even with this reduction in the critical basis of assumed behavior, Curtin formulated his transfer length, four fragments were also found to theory of the fiber fragmentation process by viewing exceed lc, with the range being 445 to 508 um. Note fiber fragmentation as occurring in two parts: (i) that the range of unbroken fragments that exceed lc those fragments formed by breaks separated by is comparable to the range of the five fragment more than lo, the critical transfer length at the cur- lengths(488 to 527 um) that exceeded Ic in the bare rent stress level o and (ii)those fragments smaller E-glass DGEBA/m-PDA SFC specimens even though than lo) formed at an earlier stress level o'< o the average fragment length at saturation in the bare where a shorter 1o< lol prevailed. Consequently, E-glass polyisocyanurate SFC specimen is 13% the filtered length distribution of fragment lengths shorter than observed in the bare E-glass DGEBA/ in part (i)that contain all fragments larger than lo) m-PDA SFC specimen. Interestingly, saturation was are viewed as being the same as that for a fiber with indicated for the bare E-glass polyisocyanurate SFC a unique strength, o, whose effective fiber length is specimen by the absence of fracture in three strain Lr LR, where Lr denotes the total length of the increments at the end of the test(4. 41 to 4.80%) fiber and LR represents the combined lengths of all Consistent with the results obtained for the E-glas fragments smaller than l[o]. Hui et al. in the devel- DGEBA/m-PDA SFC specimen, the size of the frag- opment of their theory viewed the filtered length ments surviving until saturation for the polyisocya- distribution approach used by Curtin as plausible, urate SFC specimen decreased wi Increasins since their formulation also relies on the stress de- strain Journal of applied Polymer Science DOI 101002/app

typical E-glass SFC specimens are significantly less than the (85 to 130) lm debond regions observed by Gulino and Phoenix, even though these researchers achieved break densities at saturation of 25 and 15 breaks/cm. From a brief review of the literature on the silane treatment of glass fibers,19 the minimal debonding in the NOTS SFC composites may be due to a predomi￾nant mechanical interlocking stress transfer mecha￾nism at the fiber matrix interface caused by the interpenetration of the epoxy matrix into the porous silane surface treatment. Furthermore, the porous surface treatment that occurs when the glass fiber is treated with GOPS is also accompanied by the estab￾lishment of covalent bonds between the matrix and silane coupling agent, thereby providing a more effi￾cient stress transfer mechanism than observed with the NOTS surface treatment. In Ref. 19, Holmes et al. showed that removal of the mechanical interlock￾ing mechanism by treatment of the smooth glass surfaces with self-assembled monolayers of n-octa￾decyl trichlorosilane resulted in extensive debonding like that observed in carbon fiber composites, where the mechanical interlocking mechanism is based on the surface roughness of the carbon fiber. These results and observations suggest that in the absence of extensive debonding, the physics of the sequential fiber fragmentation process in E-glass fibers leads to a uniform distribution at saturation over a wide range of adhesion strengths, with the phenomenon appearing to be independent of matrix type. Nonideal fragmentation behavior in E-glass fiber SFCs Although it is known that most matrices used in the SFFT do not conform to the EPP assumption,8,9 it is generally believed that the actual fragmentation pro￾cess is consistent with this approximation. On the basis of assumed behavior, Curtin6 formulated his theory of the fiber fragmentation process by viewing fiber fragmentation as occurring in two parts: (i) those fragments formed by breaks separated by more than l{r}, the critical transfer length at the cur￾rent stress level r and (ii) those fragments smaller than l{r} formed at an earlier stress level r0 < r where a shorter l{r0 } < l{r} prevailed. Consequently, the filtered length distribution of fragment lengths in part (i) that contain all fragments larger than l{r} are viewed as being the same as that for a fiber with a unique strength, r, whose effective fiber length is LT
LR, where LT denotes the total length of the fiber and LR represents the combined lengths of all fragments smaller than l{r}. Hui et al.7 in the devel￾opment of their theory viewed the filtered length distribution approach used by Curtin as plausible, since their formulation also relies on the stress de￾pendent and well-defined l{r} afforded by the EPP assumption. However, they took issue with the value of the maximum achievable packing density along the broken fiber, stating that the value should be one rather than the value of 0.7476 used by Curtin. In a previous publication by Kim et al.,2 it is shown for a bare E-glass fiber embedded in a DGEBA/m-PDA matrix that the smallest fragments in the final fragment length distribution were not formed at the beginning of the test, as theorized by Curtin and Hui et al., but rather at the end of the test. Consistent with these results, the size of the fragments surviving to saturation were found to decrease in size as the test progressed, in apparent contradiction to a l{r} based on the EPP assumption where the theories indicate that the size of the frag￾ments surviving to saturation should increase as the strain in the SFC is increased. Furthermore, the Kelly-Tyson approximation of the critical transfer length (lc ¼ 482 lm for the Bare2_9 SFC specimen), also suggested that saturation was not achieved, since there were five fragments whose length ranged from 488 to 527 lm. This observation was somewhat surprising since fragments of length 462 and 482 lm fractured near the end of the test, with the shorter fragment being less than the Kelly-Tyson estimate of lc. It should also be noted that the largest surviving fragments were formed early enough to undergo at least six additional increases in strain. To better understand the fragmentation observed in the SFC by Kim et al., the SFFT fragment evolu￾tion data from the SFCs composed of a bare E-glass fiber embedded in a polyisocyanurate matrix (Table I) were analyzed. For the representative data shown in Table I, the Kelly-Tyson estimate of lc is 420 lm which is  13% smaller than what was observed in the bare E-Glass/DGEBA/m-PDA SFC analyzed by Kim et al. Even with this reduction in the critical transfer length, four fragments were also found to exceed lc, with the range being 445 to 508 lm. Note that the range of unbroken fragments that exceed lc is comparable to the range of the five fragment lengths (488 to 527 lm) that exceeded lc in the bare E-glass DGEBA/m-PDA SFC specimens even though the average fragment length at saturation in the bare E-glass polyisocyanurate SFC specimen is 13% shorter than observed in the bare E-glass DGEBA/ m-PDA SFC specimen. Interestingly, saturation was indicated for the bare E-glass polyisocyanurate SFC specimen by the absence of fracture in three strain increments at the end of the test (4.41 to 4.80%). Consistent with the results obtained for the E-glass DGEBA/m-PDA SFC specimen, the size of the frag￾ments surviving until saturation for the polyisocya￾nurate SFC specimen decreased with increasing strain. 512 HOLMES ET AL. Journal of Applied Polymer Science DOI 10.1002/app

COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 513 TABLE I Fragment Evolution Pattern from PU04E03 Test Specimen 192.5378.712515.019921223.123.724.325.526.828.730.531.2 (breaks/cm) 20789813729.441249.064768.674.576578482486.392.2980100.0 Number of %o Strain 1421.491621.731.791952052262.372552993.163.283.623.784.174.41 Fragment no. Fragment lengths given in um 1016048606443971631163116311632793793446446446446 47347347347 508508508508508508508 508508 330330330330330330330330330330 2766956372 372372372372372372 584584584584584584584245245245 105710571057507507507507507507214214 550550550550550329329329329329 37237237237 2345678 16671667767767 767314314314314314314 314314 453453453453453453453237237 90054854854854854854854854854827127127 277277277 159715 56565333 23356789 232232232232 188118811881 731283283283283283283283283283283 448448448448448448448448448448 65066506230914061406603603603603603603299 491491491491491491 491 31231231231231231231231231231 903903 299299299299299299299299299299 604604604311311311311311311311 293293293 293293 4197 3331 28128 399399399 399399399399399399 538538 307307307 123555789 313www222i 406406406406 406406406 53653 22 852852852 76276276276276 577577577277277 300300 Plots of the average size of fragments surviving to specimens are shown in Figure 3. For the Bare2_9 saturation for the bare E-glass/DGEBA/m-PDa and NOTS_DI plots, one standard deviation error specimen(Bare 2_9), the bare E-Glass/polyisocyanu- bars are shown at the strain increments where multi- rate specimen(PU04E03), and the noTs_ DI Sfc ple fragments survived to saturation. For visual Journal of applied Polymer Science DOI 10.1002/ app

Plots of the average size of fragments surviving to saturation for the bare E-glass/DGEBA/m-PDA specimen (Bare2_9), the bare E-Glass/polyisocyanu￾rate specimen (PU04E03), and the NOTS_D1 SFC specimens are shown in Figure 3. For the Bare2_9 and NOTS_D1 plots, one standard deviation error bars are shown at the strain increments where multi￾ple fragments survived to saturation. For visual TABLE I Fragment Evolution Pattern from PU04E03 Test Specimen Break density (breaks/cm) 0 1.9 2.5 3.7 8.7 12.5 15.0 19.9 21.2 23.1 23.7 24.3 25.5 26.8 28.7 30.5 31.2 % dc 2.0 7.8 9.8 13.7 29.4 41.2 49.0 64.7 68.6 74.5 76.5 78.4 82.4 86.3 92.2 98.0 100.0 Number of fragments 1 4 5 7 15 21 25 33 35 38 39 40 42 44 47 50 51 % Strain 1.42 1.49 1.62 1.73 1.79 1.95 2.05 2.26 2.37 2.55 2.99 3.16 3.28 3.62 3.78 4.17 4.41 Fragment no. Fragment lengths given in lm 10 16048 6064 4397 1631 1631 1631 1632 793 793 446 446 446 446 446 446 446 446 11 347 347 347 347 347 347 347 347 12 508 508 508 508 508 508 508 508 508 508 13 330 330 330 330 330 330 330 330 330 330 14 2766 956 372 372 372 372 372 372 372 372 372 372 372 372 15 584 584 584 584 584 584 584 584 584 245 245 245 16 339 339 339 17 1057 1057 1057 507 507 507 507 507 507 214 214 214 214 18 293 293 293 293 19 550 550 550 550 550 329 329 329 329 329 20 221 221 221 221 221 21 753 753 753 381 381 381 381 381 381 381 381 381 381 22 372 372 372 372 372 372 372 372 372 372 23 1667 1667 767 767 767 767 314 314 314 314 314 314 314 314 314 24 453 453 453 453 453 453 453 237 237 25 216 216 26 900 548 548 548 548 548 548 548 548 548 271 271 271 27 277 277 277 28 352 352 352 352 352 352 352 352 352 352 352 352 29 1597 1597 1597 351 351 351 351 351 351 351 351 351 351 351 351 351 30 1246 681 681 681 329 329 329 329 329 329 329 329 329 31 351 351 351 351 351 351 351 351 351 32 565 565 565 565 565 565 333 333 333 333 333 333 33 232 232 232 232 232 232 34 1881 1881 1881 458 458 458 458 458 458 458 458 458 458 458 458 458 35 692 692 692 692 692 313 313 313 313 313 313 313 313 36 379 379 379 379 379 379 379 379 37 731 283 283 283 283 283 283 283 283 283 283 283 283 38 448 448 448 448 448 448 448 448 448 448 209 209 39 239 239 40 6506 6506 2309 1406 1406 603 603 603 603 603 603 299 299 299 299 299 41 304 304 304 304 304 42 491 491 491 491 491 491 491 491 491 491 491 43 312 312 312 312 312 312 312 312 312 312 312 44 903 903 903 299 299 299 299 299 299 299 299 299 299 45 604 604 604 311 311 311 311 311 311 311 46 293 293 293 293 293 293 293 47 4197 680 281 281 281 281 281 281 281 281 281 281 281 281 48 399 399 399 399 399 399 399 399 399 399 399 399 49 3517 1185 647 341 341 341 341 341 341 341 341 341 341 50 307 307 307 307 307 307 307 307 307 307 51 538 538 538 538 538 538 538 538 538 538 275 52 263 53 2331 1479 536 536 536 536 536 536 536 305 305 305 54 231 231 231 55 406 406 406 406 406 406 406 406 406 406 56 536 536 536 536 536 536 240 240 240 240 57 296 296 296 296 58 852 852 852 276 276 276 276 276 276 276 276 59 577 577 577 577 577 577 277 277 60 300 300 COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 513 Journal of Applied Polymer Science DOI 10.1002/app

514 HOLMES ET AI 三的 fragments surviving to saturation as the strain is increased, both sets of data suggest a rather sharp transition as indicated by the anova analyses where the sizes of the fragments become decidedly 700 smaller as the strain in the SFC specimen is increased. Interestingly, ANOVA analyses of the data from the NOTS- DI SFC specimen shown in Figure 3 indicate that from the three groupings the average size of the fragments lengths surviving to saturation are indistinguishable at the 95% confi- dence level with a p value of 0.48. The sizes of the fragments surviving to saturation appear to remain 0.0150. 0.0250, ami. 350 40.45 constant as saturation is approached. These results suggest that adhesion strength and stress build-up Figure 3 Plots of average sizes of fragments surviving to in the fiber break regions may have a significant saturation from SFC specimens composed of E-glass fibers treated with n-octadecyl triethoxysilane (NOTS_ D1)and Pact on the formation of small fragment lengths bare E-glass fibers embedded in DGEBA/m-PDA (Bare2_9) as saturation is approached ate(PU04E03)matrices. The error bars represent one standard deviation for the multiple fragments formed at a give strain increment. To maintain clarity in the graph, the error bars for the PU04E03 specimen are not The uniform distribution and fiber break locations shown, but are comparable to those for the Bare2_9 speci- from carbon fiber SFC specimens nating between open and solid symbols. For a given data In 2000, the 2nd round robin assessment of the SFFT set,the point where a group becomes distinguishable from was conducted under the auspices of VAMAS the previous group is represented by a change of symbol Approximately 100 AS-4 carbon fiber/DGEBA/ I Color figure can be viewed in the online issue, which is PDA SFC specimens were prepared in five batches availableatwww.interscience.wileycomJ by the michigan State University Composites Labo- ratory. 5 The National Institute of Standards and Technology (NIST)randomized the samples from larity, error bars were omitted for the PU04E03 these batches and distributed them to seven labora specimen, but are comparable to those given for the tories for testing. In Figure 2, the correlation coeffi- Bare2_9 SFC specimen. It should be noted that the cients of the break locations from 12 specimen dispersion data for the pol pecimen is tested in the nist extractable from the fragment evolution data pro- diamond symbols) along with the E-glass data dis- vided in table i cussed above. When the break locations were fitted Analysis of variance(ANOVA)analyses on each to a uniform distribution, 58% of the 12 specimens specimen depicted in Figure 3 were performed by exhibited correlation coefficients at saturation of dividing the fragments into three to five groups. The 0.999 or greater. However, all of the probability plot groupings are indicated in Figure 3 for each speci- correlation coefficients of the carbon fiber SFC speci men by the alternation between open and solid sym- mens were greater than 0.993. To verify the consis- bols as the strain is increased. For the bare 9 tency of these results, the fragment length data at men, the first three groups (upto N 3.5% turation from three additional laboratories were were indistinguishable at the 95% confidence transcribed to yield relative break locations. The with a P value of 0.15. The fourth grouping for the transcribed data are shown in Figure 4 along with Bare2-9 data set, delineated by open squares, was the correlation coefficients obtained from the NIST distinguishable from the third grouping, delineated data. Analyses of these data indicate that 43% of the by solid diamonds, at the same confidence level 42 specimens tested yield correlations greater than with a P value of 0.008. In a similar manner the first 0.999 with all data exhibiting correlations greater three groupings (< 0.2.9% strain) of the polyisocya- than 0.991. It appears that the extensive debonding able with a P val n(PU04E03)were indistinguish- observed in the carbon fiber SFC specimens causes alue of 0.72. However the fourth the fit of the fiber break locations to the uniform grouping was distinguishable from the third at the distribution to be very slightly reduced in these 95% confidence level with a P value of 0.04, whereas specimens. However, greater than 0.99x goodnes the fourth and fifth groupings were indistinguish- of-fit of these data to a uniform distribution able with a p value of 0.10 suggests that the expected distribution of the Although the data from these two specimens indi- ordered fragment lengths at saturation should cate a general downward trend in the size of the conform to(1) Journal of applied Polymer Science DOI 101002/app

clarity, error bars were omitted for the PU04E03 specimen, but are comparable to those given for the Bare2_9 SFC specimen. It should be noted that the dispersion data for the polyisocyanurate specimen is extractable from the fragment evolution data pro￾vided in Table I. Analysis of variance (ANOVA) analyses on each specimen depicted in Figure 3 were performed by dividing the fragments into three to five groups. The groupings are indicated in Figure 3 for each speci￾men by the alternation between open and solid sym￾bols as the strain is increased. For the Bare2_9 speci￾men, the first three groups (upto  3.5% strain) were indistinguishable at the 95% confidence level with a P value of 0.15. The fourth grouping for the Bare2_9 data set, delineated by open squares, was distinguishable from the third grouping, delineated by solid diamonds, at the same confidence level with a P value of 0.008. In a similar manner, the first three groupings (< 0.2.9% strain) of the polyisocya￾nurate SFC specimen (PU04E03) were indistinguish￾able with a P value of 0.72. However, the fourth grouping was distinguishable from the third at the 95% confidence level with a P value of 0.04, whereas the fourth and fifth groupings were indistinguish￾able with a P value of 0.10. Although the data from these two specimens indi￾cate a general downward trend in the size of the fragments surviving to saturation as the strain is increased, both sets of data suggest a rather sharp transition as indicated by the ANOVA analyses where the sizes of the fragments become decidedly smaller as the strain in the SFC specimen is increased. Interestingly, ANOVA analyses of the data from the NOTS_D1 SFC specimen shown in Figure 3 indicate that from the three groupings the average size of the fragments lengths surviving to saturation are indistinguishable at the 95% confi￾dence level with a P value of 0.48. The sizes of the fragments surviving to saturation appear to remain constant as saturation is approached. These results suggest that adhesion strength and stress build-up in the fiber break regions may have a significant impact on the formation of small fragment lengths as saturation is approached. The uniform distribution and fiber break locations from carbon fiber SFC specimens In 2000, the 2nd round robin assessment of the SFFT was conducted under the auspices of VAMAS. Approximately 100 AS-4 carbon fiber/DGEBA/m￾PDA SFC specimens were prepared in five batches by the Michigan State University Composites Labo￾ratory.15 The National Institute of Standards and Technology (NIST) randomized the samples from these batches and distributed them to seven labora￾tories for testing. In Figure 2, the correlation coeffi￾cients of the break locations from 12 specimens tested in the NIST laboratory are plotted (open diamond symbols) along with the E-glass data dis￾cussed above. When the break locations were fitted to a uniform distribution, 58% of the 12 specimens exhibited correlation coefficients at saturation of 0.999 or greater. However, all of the probability plot correlation coefficients of the carbon fiber SFC speci￾mens were greater than 0.993. To verify the consis￾tency of these results, the fragment length data at saturation from three additional laboratories were transcribed to yield relative break locations. The transcribed data are shown in Figure 4 along with the correlation coefficients obtained from the NIST data. Analyses of these data indicate that 43% of the 42 specimens tested yield correlations greater than 0.999, with all data exhibiting correlations greater than 0.991. It appears that the extensive debonding observed in the carbon fiber SFC specimens causes the fit of the fiber break locations to the uniform distribution to be very slightly reduced in these specimens. However, greater than 0.99x goodness￾of-fit of these data to a uniform distribution suggests that the expected distribution of the ordered fragment lengths at saturation should conform to (1). Figure 3 Plots of average sizes of fragments surviving to saturation from SFC specimens composed of E-glass fibers treated with n-octadecyl triethoxysilane (NOTS_D1) and bare E-glass fibers embedded in DGEBA/m-PDA (Bare2_9) and polyisocyanurate (PU04E03) matrices. The error bars represent one standard deviation for the multiple fragments formed at a give strain increment. To maintain clarity in the graph, the error bars for the PU04E03 specimen are not shown, but are comparable to those for the Bare2_9 speci￾men. Groupings for ANOVA analysis are formed by alter￾nating between open and solid symbols. For a given data set, the point where a group becomes distinguishable from the previous group is represented by a change of symbol (e.g., circles change to triangles for the PU04E03 specimen). [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] 514 HOLMES ET AL. Journal of Applied Polymer Science DOI 10.1002/app

COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 515 according to statistical spacing theory should con- °° form to(1). Furthermore, initial analyses of the break evolution process suggest that the onset of uniform break locations occurs at a lower break density than during the sequential fragmentation of single fibers a detailed study of the fiber break evolution process 0.995 Minimum Value for 0.999 correlation coefficient in these multifiber arrays will be discussed in a future report to clarify the potential impact of this DATA S DATA The distribution of fragment lengths and SFC Weibull parame In the theories of Curtin and Hui et al., 7 Weibull Figure 4 Correlation coefficients for fit of fiber break loca- parameters were extracted from the SFFt data tions at saturation to the uniform distribution. All samples Researchers" have suggested that the validity of are AS-4 carbon fiber SFCs in DGEBA/m-PDA tested by these parameters can be checked by fitting the cu- VAMAS testing protocol for 2nd Round Robin Testing for e SFFT. 15 ( Color figure can be viewed in the online issue, mulative distribution of the ordered fragment whichisavailableatwww.interscience.wileycom] lengths at saturation, however, in practice, only fair agreement between predicted and measured results have been obtained. all the results of this article The effect of fiber-fiber interactions on indicate from spacing theory that the cumulative fiber break spacings distribution of the ordered fragment lengths should conform to the distribution function given in(1) To test the effect of fiber-fiber interactions on the This suggests that the goodness-of-fit of the Weibull sequential fragmentation process, combinatorial and lognormal distribution functions that have been microcomposites were analyzed. These specimens used to fit the ordered fragment lengths at satura- are composed of a 2D 6-fiber array sandwiched tion are the result solely of the flexibility of the mod between two single fibers, with all the fibers being els used with no fundamental basis in physics or E-glass sized with aminopropyl trimethoxysilane statistics. As a preliminary check of this conclusion, (APTMS). Consistent with published data, the sep- pooled data from the four E-glass DGEBA/m-PDA aration distance between the array and the single specimens tested by the slow test protocol were fit fibers is greater than 300 um, to minimize interaction using the continuous beta, lognormal, and three-pa- between the 2D array and the single fibers, whose rameter Weibull distribution functions(Fig. 6). The nominal diameter is 15 um, whereas the interfiber results indicate that the fits are comparable, with the spacing in the array is 21 um. Unlike the unsized three-parameter Weibull and lognormal functions fibers, the APTMS sized fibers formed matrix cracks giving slightly higher correlations than the continu- in the DGEBA/m-PDa matrix that led to premature ous beta function. The two-parameter Weibull func failure of the combinatorial specimen through the tion, which is used to extract the Weibull parame- interacting fiber breaks formed in the array. To ters, was not flexible enough to provide a good fit of eliminate matrix crack formation in the combinato- the data rial specimens, 20% of the molar amount of dgEBA was replaced with the same molar amount of digly- cidyl ether of butanediol (DGEBD) Consistent with the SfC test results, the distribu tion of fiber break locations in the multifiber array 丰 were also found to be uniformly distributed with goodness-of-fit correlation coefficients greater than R=09995 0.999(Fig. 5). The fiber breaks in the two embedded single fibers, which were also uniformly distributed with correlation coefficients greater than 0.999 R=09995 fibers. These results are consistent with Li et al.10 8a 4000 8000 12000 16000 2000 240uu exhibited a higher number of breaks than the cluster Measured break position, Fiber 1(um) esults from 2D Nicalon multifiber arrays and indi cate that the expected outcome from the sequential Figure 5 Uniform probability plots of cluster fiber break The solid line represents the empirical fits IColor figure fragmentation of interacting fibers uniform can be viewed in the online issue, which is available at break locations, whose ordered fra Journal of applied Polymer Science DOI 10.1002/ app

The effect of fiber–fiber interactions on fiber break spacings To test the effect of fiber–fiber interactions on the sequential fragmentation process, combinatorial microcomposites were analyzed.21 These specimens are composed of a 2D 6-fiber array sandwiched between two single fibers, with all the fibers being E-glass sized with aminopropyl trimethoxysilane (APTMS). Consistent with published data,10 the sep￾aration distance between the array and the single fibers is greater than 300 lm, to minimize interaction between the 2D array and the single fibers, whose nominal diameter is 15 lm, whereas the interfiber spacing in the array is 21 lm. Unlike the unsized fibers, the APTMS sized fibers formed matrix cracks in the DGEBA/m-PDA matrix that led to premature failure of the combinatorial specimen through the interacting fiber breaks formed in the array.22 To eliminate matrix crack formation in the combinato￾rial specimens, 20% of the molar amount of DGEBA was replaced with the same molar amount of digly￾cidyl ether of butanediol (DGEBD). Consistent with the SFC test results, the distribu￾tion of fiber break locations in the multifiber array were also found to be uniformly distributed with goodness-of-fit correlation coefficients greater than 0.999 (Fig. 5). The fiber breaks in the two embedded single fibers, which were also uniformly distributed with correlation coefficients greater than 0.999, exhibited a higher number of breaks than the cluster fibers. These results are consistent with Li et al.10 results from 2D Nicalon multifiber arrays and indi￾cate that the expected outcome from the sequential fragmentation of interacting fibers is also uniform break locations, whose ordered fragment lengths according to statistical spacing theory should con￾form to (1). Furthermore, initial analyses of the break evolution process suggest that the onset of uniform break locations occurs at a lower break density than during the sequential fragmentation of single fibers. A detailed study of the fiber break evolution process in these multifiber arrays will be discussed in a future report to clarify the potential impact of this last observation. The distribution of fragment lengths and SFC Weibull parameters In the theories of Curtin6 and Hui et al.,7 Weibull parameters were extracted from the SFFT data. Researchers23 have suggested that the validity of these parameters can be checked by fitting the cu￾mulative distribution of the ordered fragment lengths at saturation, however, in practice, only fair agreement between predicted and measured results have been obtained. All the results of this article indicate from spacing theory3,4 that the cumulative distribution of the ordered fragment lengths should conform to the distribution function given in (1). This suggests that the goodness-of-fit of the Weibull and lognormal distribution functions that have been used to fit the ordered fragment lengths at satura￾tion are the result solely of the flexibility of the mod￾els used with no fundamental basis in physics or statistics. As a preliminary check of this conclusion, pooled data from the four E-glass DGEBA/m-PDA specimens tested by the slow test protocol18 were fit using the continuous beta, lognormal, and three-pa￾rameter Weibull distribution functions (Fig. 6). The results indicate that the fits are comparable, with the three-parameter Weibull and lognormal functions giving slightly higher correlations than the continu￾ous beta function. The two-parameter Weibull func￾tion, which is used to extract the Weibull parame￾ters, was not flexible enough to provide a good fit of the data. Figure 5 Uniform probability plots of cluster fiber breaks. The solid line represents the empirical fits. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] Figure 4 Correlation coefficients for fit of fiber break loca￾tions at saturation to the uniform distribution. All samples are AS-4 carbon fiber SFCs in DGEBA/m-PDA tested by VAMAS testing protocol for 2nd Round Robin Testing for the SFFT.15 [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] COMPARISON OF E-GLASS DATA WITH STATISTICAL THEORIES 515 Journal of Applied Polymer Science DOI 10.1002/app

HOLMES ET AI by investigating the effect of nonlinear viscoelastic Lognormal Function (R =o 99 behavior of the polymer matrix using models such as those developed by Thuruthimattam et al. 24 The authors like to thank professor Andrew Rukhin, University of Maryland Baltimore County/ National Institute of Standards and Technology 3-Parameter Weibull (UMBC/NIST) for his many helpful Function (R=0. 998) ng the preparation of this manuscript Beta Function (R 0.995) References 1. Wagner, H. D. Steenbakkers, L. w. J Mater Sci 1989, 24, 3956 2. Kim, J. H. Leigh, S. D. Holmes, G. Compos Sci Technol submitted Fit of normalized ordered framer at 3. Pyke, R J R Stat Soc B 1965, 27,395 n from specimens tested by the slow 4. Read, C. B. In Encyclopedia of Statistical Sciences; Kotz, s Beta, three-Parameter Weibull, and Johnson, N. L, Eds. Wiley: New York, 1988; Vol8, P 566. s I Color figure can be viewed in the 5. Gulino, R; Phoenix, S L J Mater Sci 1991, 26, 3107. whichisavailableatwww.interscience.wileycom.j 6. Curtin, W.A.J Mater Sci 1991, 26, 5239 7. Hui, C. Y i Phoenix, S. L Ibnabdeljalil, M; Smith, R. L. J Mech Phys Solids 1995, 43, 1551. CONCLUSIONS 8. Bascom, w. D. Jensen, R. M.J Adhes 1986, 19, 219. 9. Holmes, G. A Peterson, R C. Hunston, D. L McDonough, w. The experimental results discussed in this report G. Schutte, C. L. In Time Dependent and Nonlinear Effects ongly indicate that the expected outcome from the Polymers and Composites; Schapery, R A, Ed. American Society EFFT methodologies are fiber breaks whose locations for Testing and Materials: West Conshohocken, Pennsylvania conform to a uniform distribution This outcome is 000pp98-117 10. Li, Z. F, Grubb, D. T, Phoenix, S. L. Compos Sci Technol found to be independent of adhesion strength, ma- 1995,54,251 trix type, fiber type, and fiber-fiber interactions. Uni- 11. Kim, J-H; Holmes, G. A. In Proceedings 27th Annual Meeting form break locations from the sequential fracture of of the Adhesion Society, Inc; Chaudhury, M. K, Anderson, G L, Eds. The Adhesion Society: Blackburg, Virginia, 2004; P525 correlations >0.999, whereas those from carbon 12. Cox, H.L. Br J Appl Phys 1952,3.72 13. Rosen, W. B. In Fiber Composite Materials; American Societ fiber SFCs display correlations >0.99. According to of Metals, Ed. American Society of Metals: Metals Park, OH, the theory of uniform spacings, the cumulative dis- 1965: Chapter 3, P 37. tribution of the ordered spacings (i.e, fragment 14. Rosen, W.B. Dow, N F; Hashin, Z Mechanical Properties of lengths) from the uniform break locations conforms Fibrous Composites, NASA CR-31; General Electric Company: to a restricted discrete beta-like function whose exact Philadelphia, PA, 1964. form was derived by Whitworth. 4 Fits of this type 15. Rich, M. J; Drzal, L. T; Hunston, D. L; Holmes, G; McDo- nough, W. G. In Proceedings of the American Society for of data by Weibull or lognormal distributions do not Composites; Sun, C. T, Kim, H, Eds; CRC Press LLC: Boca validate those models but rather reflect the flexibili Raton, FL, 2002; P 158. of those functional forms. As a preliminary check of 16. Holmes, G. A, Peterson, R. C, Hunston, D. L, McDonough, the Whitworth derivation, pooled data from four 17. Holmes, G. A; Feresenbet, E; Raghavan, D. In Proceedings of the samples tested by the same protocol were fit by a 24th Annual Meeting of the Adhesion Society: Emerson, J. A-, E continuous beta, three-parameter Weibull and log The Adhesion Society: Blacksburg VA 24061-0201, 2001; p 62 normal distribution functions. Although all functions 18. Holmes, G. A Peterson, R C, Hunston, D L; McDonough, provided comparable fits of the data, the two-pa- W.G. Polym Compos 2000, 21, 450 rameter Weibull function, which is used to extract 19. Holmes, G. A Feresenbet, E; Raghavan, D. Compos Interfac Weibull parameters from the SFFT methodology, did 20. McDonough, w. G; Holmes, G. A; Peterson, R C In Proceed- not yield an acceptable fit ings of the 13th Technical Conference on Composite Materials Furthermore, the Kim et al. results showing that American Society for Composites: Baltimore, 1998; P 1688 the"exact"theories put forth by Curtin and Hui21.Kim,J-H;Hettenhouser,JW;Moon,CK;Holmes,GA et al. do not accurately predict the fiber break den- Compos Sci Technol, submitted sity upto saturation are verified in this report. It 22. Holmes, G. A; McDonough, W. G. In Proceedings of the 47th International SAMPE Symposium and Exhibition; Rasmussen, shown experimentally that the fragment lengths sur- B. M, Pilato, L. A, Kliger, H. S, Eds Society for the viving to saturation decrease in size as saturation is Advancement of Material and Process Engineers(SAMPE) approached rather than increase in size as predicted Covina, CA, 2002; P 1690 by the theories. Both theories assume that the matrix 23. Zhao, F. M. Takeda, N. Compos Part A Appl Sci Manuf 2000, 31,1215 is elastic perfectly plastic (EPP assumption). Future 24. Thuruthimattam, B. I; Waas, A M; Wineman, A S. IntJNon work will focus on understanding this discrepancy Lin mech 2001. 36, 69 Journal of applied Polymer Science DOI 101002/app

CONCLUSIONS The experimental results discussed in this report strongly indicate that the expected outcome from the EFFT methodologies are fiber breaks whose locations conform to a uniform distribution. This outcome is found to be independent of adhesion strength, ma￾trix type, fiber type, and fiber–fiber interactions. Uni￾form break locations from the sequential fracture of E-glass SFCs were found to exhibit goodness-of-fit correlations > 0.999, whereas those from carbon fiber SFCs display correlations >0.99. According to the theory of uniform spacings, the cumulative dis￾tribution of the ordered spacings (i.e., fragment lengths) from the uniform break locations conforms to a restricted discrete beta-like function whose exact form was derived by Whitworth.3,4 Fits of this type of data by Weibull or lognormal distributions do not validate those models but rather reflect the flexibility of those functional forms. As a preliminary check of the Whitworth derivation, pooled data from four samples tested by the same protocol were fit by a continuous beta, three-parameter Weibull and log￾normal distribution functions. Although all functions provided comparable fits of the data, the two-pa￾rameter Weibull function, which is used to extract Weibull parameters from the SFFT methodology, did not yield an acceptable fit. Furthermore, the Kim et al.2 results showing that the ‘‘exact’’ theories put forth by Curtin6 and Hui et al.7 do not accurately predict the fiber break den￾sity upto saturation are verified in this report. It is shown experimentally that the fragment lengths sur￾viving to saturation decrease in size as saturation is approached rather than increase in size as predicted by the theories. Both theories assume that the matrix is elastic perfectly plastic (EPP assumption). Future work will focus on understanding this discrepancy by investigating the effect of nonlinear viscoelastic behavior of the polymer matrix using models such as those developed by Thuruthimattam et al.24 The authors like to thank Professor Andrew Rukhin, University of Maryland Baltimore County/ National Institute of Standards and Technology (UMBC/NIST) for his many helpful comments dur￾ing the preparation of this manuscript. References 1. Wagner, H. D.; Steenbakkers, L. W. J Mater Sci 1989, 24, 3956. 2. Kim, J. H.; Leigh, S. D.; Holmes, G. Compos Sci Technol, submitted. 3. Pyke, R. J R Stat Soc B 1965, 27, 395. 4. Read, C. B. In Encyclopedia of Statistical Sciences; Kotz, S., Johnson, N. L., Eds.; Wiley: New York, 1988; Vol.8, p 566. 5. Gulino, R.; Phoenix, S. L. J Mater Sci 1991, 26, 3107. 6. Curtin, W. A. J Mater Sci 1991, 26, 5239. 7. Hui, C. Y.; Phoenix, S. L.; Ibnabdeljalil, M.; Smith, R. L. J Mech Phys Solids 1995, 43, 1551. 8. Bascom, W. D.; Jensen, R. M. J Adhes 1986, 19, 219. 9. Holmes, G. A.; Peterson, R. C.; Hunston, D. L.; McDonough, W. G.; Schutte, C. L. In Time Dependent and Nonlinear Effects in Polymers and Composites; Schapery, R. A., Ed.; American Society for Testing and Materials: West Conshohocken, Pennsylvania, 2000; pp 98–117. 10. Li, Z. F.; Grubb, D. T.; Phoenix, S. L. Compos Sci Technol 1995, 54, 251. 11. Kim, J.-H.; Holmes, G. A. In Proceedings 27th Annual Meeting of the Adhesion Society, Inc.; Chaudhury, M. K., Anderson, G. L., Eds.; The Adhesion Society: Blackburg, Virginia, 2004; p 525. 12. Cox, H. L. Br J Appl Phys 1952, 3, 72. 13. Rosen, W. B. In Fiber Composite Materials; American Society of Metals, Ed.; American Society of Metals: Metals Park, OH, 1965; Chapter 3, p 37. 14. Rosen, W. B.; Dow, N. F.; Hashin, Z. Mechanical Properties of Fibrous Composites, NASA CR-31; General Electric Company: Philadelphia, PA, 1964. 15. Rich, M. J.; Drzal, L. T.; Hunston, D. L.; Holmes, G.; McDo￾nough, W. G. In Proceedings of the American Society for Composites; Sun, C. T., Kim, H., Eds.; CRC Press LLC: Boca Raton, FL, 2002; p 158. 16. Holmes, G. A.; Peterson, R. C.; Hunston, D. L.; McDonough, W. G. Polym Compos 2007, 28, 561. 17. Holmes, G. A.; Feresenbet, E.; Raghavan, D. In Proceedings of the 24th Annual Meeting of the Adhesion Society; Emerson, J. A., Ed.; The Adhesion Society: Blacksburg, VA 24061-0201, 2001; p 62. 18. Holmes, G. A.; Peterson, R. C.; Hunston, D. L.; McDonough, W. G. Polym Compos 2000, 21, 450. 19. Holmes, G. A.; Feresenbet, E.; Raghavan, D. Compos Interfac 2003, 10, 515. 20. McDonough, W. G.; Holmes, G. A.; Peterson, R. C. In Proceed￾ings of the 13th Technical Conference on Composite Materials; American Society for Composites: Baltimore, 1998; p 1688. 21. Kim, J.-H.; Hettenhouser, J. W.; Moon, C. K.; Holmes, G. A. Compos Sci Technol, submitted. 22. Holmes, G. A.; McDonough, W. G. In Proceedings of the 47th International SAMPE Symposium and Exhibition; Rasmussen, B. M., Pilato, L. A., Kliger, H. S., Eds.; Society for the Advancement of Material and Process Engineers (SAMPE): Covina, CA, 2002; p 1690. 23. Zhao, F. M.; Takeda, N. Compos Part A Appl Sci Manuf 2000, 31, 1215. 24. Thuruthimattam, B. J.; Waas, A. M.; Wineman, A. S. Int J Non Lin Mech 2001, 36, 69. Figure 6 Fit of normalized ordered fragment lengths at saturation from specimens tested by the slow test protocol using the Beta, three-Parameter Weibull, and Lognormal functions. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.] 516 HOLMES ET AL. Journal of Applied Polymer Science DOI 10.1002/app