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/appCONCLUSIONS 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