Fig 8. The FCG rates in Rice Husk composite. a) in terms of amplitude, √ΔG. b) the √ΔG versus R plots for selected crack growth rates. c) √ΔG vs √Gmax plots at selected crack growth rates showing the L-shaped curve with limiting √ΔG and √Gmax values. d) shows the cack growth trajectory deviating from the pure fatigue line.
the two mechanisms involved. It also compares the Epoxy trajectory data of Osorio discussed earlier.
Fig. 8 shows the FCG behavior of a polymer composite with rice husk fibers as a strengthening phase [24]. The crack growth rates were determined at three R ratios, 0.1, 0.3, and 0.5. Interestingly the authors have used ΔG instead of ΔK, where G corresponds to the crack tip driving force. From our analysis point, it would not make any difference in terms of the two-parametric requirement. In this case, the corresponding two parameters will be √ΔG and √Gmax as the stress intensity factor K is proportional to √G.
Fig. 8a shows the crack growth rates in terms of √ΔG. The √ΔG – R plots for crack growth rates near the threshold are shown in 8b. The linear portion is extended up to R = 0.7, and a horizontal line is drawn with a possible constant √ΔG value at high R-ratios. The corresponding √ΔG - √Gmax plot, Fig. 8c resembles the typical L-shaped plot of ΔK-Kmax defining the limiting values of the parameters for each crack growth rate. The crack growth trajectory of this rice-husk composite is shown in Fig. 8d. The trajectory as expected deviates from the pure fatigue line indicating the √Gmax-controlled mechanism governing the crack growth. The behavior of this composite follows that of other polymer materials. This data further proves that the two parametric nature involving amplitude and peak stress is fundamental to FCG.
4. Summary and Conclusions:
It was well established that fatigue requires two load parameters for proper analysis. For fatigue crack growth these parameters correspond to the amplitude ΔK and the peak stress intensity factor, Kmax. Crack closure is not required to account for the load ratio effects. This two-parametric approach is shown to be applicable to account for the crack growth behavior in several polymeric materials. In contrast to metals and alloys, polymers deform either by crazing, viscoelastic deformation, localized brittle fracture, or their combination. The analysis establishes that the two-parametric nature of fatigue crack growth remains the same and is relevant for all materials.
5. Acknowledgements: The authors acknowledge the funding support from TDA’s ONR contract  # N68335-16-C-0135. Special acknowledgments are due to Mr. Bill Nickerson, Program Officer of ONR, who encouraged this research and provided valuable insights and guidance.
References
1. Gerber, W. Z. Bestimmung der zulässigen Spannungen in Eisen-Constructionen (calculation of the allowable stresses in iron structures). Z. Bayer. Arch. Ing. Ver., 1874, 6(6), 101–110.
2. Goodman J, Mechanics Applied to Engineering, 1stedition, 1989, Longmans Green, London.
3. ASM HandBook, Fatigue and Fracture, Vol. 19, 1996. https://doi.org/10.31399/asm.hb.v19.9781627081931
4. Elber W, 1970, Fatigue crack closure under cyclic tension, Engineering Fracture Mechanics, 1970; 2: 37–45.
5. Suresh, S. Fatigue of Materials, 2nd Edition, October 1998, Cambridge University Press, NY.
6. Budiansky B, Hitchinson JW, Analysis of Crack Closure in Fatigue Crack Growth, J. Appl. Mech., 1978; 45:267-276.
7. Newman, JC Jr, Finite-Element Analysis of Fatigue Crack Propagation-Including the Effects of Crack closure. Ph.D. Thesis, Virginia Polytechnic Institute, and State University, Blacksburg, VA, May 1974.
8. Newman JC Jr, Elver W, Mechanics of Fatigue Crack Closure, ASTM STP, 198: 892
9. Louat N, Sadananda K, Duesbery M, Vasudevan AK. A theoretical evaluation of crack closure. Metall Trans A 1993;24A:2225-32.
10. Riemelmoser FO, Pippan R., Discussion of error in the analysis of the wake dislocation problem, Metall. Materials Trans., A, 1998; 29:1357-1358
11. Sadananda K, Vasudevan AK, Discussion of error in the analysis of the wake dislocation problem, Metall. Materials Trans., A, 1998; 29: 1359-1360
12. Riemelmoser, F.O., Pippan, R., Discussion of ‘reconsideration of error in the analysis of the wake dislocation problem’- Reply, Metall. Mater. Trans.A., 1999, 30: 1452-1457.
13. Sadananda, K., Vasudevan, A.K., Authors Response, Metall. Mater. Trans.A, 1999, 30: 1457-1459.
14. Sadananda K, Sarkar S, Kujawski D, Vasudevan AK, A two-parameter analysis of S–N fatigue life using Δσ and σmax, International Journal of Fatigue, 2009; 31:1648–1659.
16. Sadananda K, Nani Babu M, Vasudevan AK, The Unified Approach to subcritical crack growth and fracture, Eng. Frac. Mech., 2019; 212:238-257.
17. Strombro J, Micro-mechanical mechanisms for deformation in polymer-material structures, Ph.D. Thesis, 2008, KTH School of Engineering Sciences Department of Solid Mechanics Royal Institute of Technology SE-100 44 Stockholm, Sweden.
18. Sternstein S.S. in Properties of Solid Polymeric Materials, (ed. J.M. Schultz), Academic Press, New York, 1977; 541-598.
19. Hertzberg, RW, Hahn MT, Rimnac CM, A laboratory analysis of a lavatory failure. International Journal of Fracture. 1993, 23: R57–R60 https://doi.org/10.1007/BF00042819
20. Hertzberg RW, Manson JA, Fatigue of Engineering Plastics, Academic Press, New York, 1980.
21. Osorio, AMBA, Stress ratio effects on fatigue crack growth in polymers, Ph.D. Thesis, 1981, Department of Mechanical Engineering, Imperial College of Science & Technology, London SW7 2BX.
22. Clark TR, Hertzberg RW, Manson JA, “Influence of Test Methodology on Fatigue Crack Propagation in Engineering Plastics,” Journal of Testing and Evaluation,  1990; 18: 319-327. https://doi.org/10.1520/JTE12493J
23. Hamda, MA, Mai YW, Wu SX, Cotterell, B., Analysis of fatigue crack growth in a rubber-toughened epoxy resin: effect of temperature and stress ratio, J. Polymer, 1993; 34: 4221-4229.
24. Mohamed SAN, Zainudin ES, Sapuan SM, Azaman MD, Arifin AMT, Effects of Different Stress Ratios on Fatigue Crack Growth of Rice Husk Fibre-reinforced Composite. BioResources, 2020;15: 6192-6205
————-