Abstract
Monte-Carlo simulations and theoretical modeling are used to study the statistical failure modes and associated lifetime distribution of unidirectional 2D and 3D fiber-matrix composites under constant load. Within the composite the fibers weaken over time and break randomly, and the matrix undergoes linear viscoelastic creep in shear. The statistics of fiber failure are governed by the breakdown model of Coleman (1958a), which embodies a Weibull hazard functional of fiber load history imparting power-law sensitivity to fiber load with exponent ρ, and Weibull lifetime characteristics with shape parameter β. The matrix has a power-law creep compliance in shear with exponent α. Fiber load redistribution at breaks is calculated using a shear-lag mechanics model, which is much more realistic than idealized rules based on equal, global or local load-sharing. The present study is concerned only with the `avalanche' failure regime discussed by Curtin and Scher (1997) which occurs for sufficiently large ρ, and whereby the composite lifetime distribution follows weakest-link scaling. The present Monte-Carlo failure simulations reveal two distinct failure modes within the avalanche regime: For larger ρ, where fiber failure is very sensitive to load level, the weakest link volume fails in a `brittle' manner by the gradual growth of a cluster of mostly contiguous fiber breaks, which then abruptly transitions into a catastrophic crack. For smaller ρ, where this load sensitivity is much less, the weakest link volume shows `tough' behavior, i.e., distributed damage in terms of random fiber failures until the failure of a critical volume and its catastrophic extension. The transition from brittle to tough failure mode for each ρ within the avalanche regime is gradual and depends on the matrix creep exponent α and Weibull exponent β. Also, as α increases above zero the sensitivity of median composite lifetime to load level increasingly deviates from power-law scaling known to occur in the elastic matrix case, α=0. By probabilistic modeling of the dominant failure modes in each regime we obtain distribution forms and various scalings for damage growth, and for carefully chosen sets of parameter values we analytically extend simulation results on small composites (limited by current computer power) to more realistic sizes. Our analytical strength distributions are applicable for ρ>2 in 2D, and ρ≳4 in 3D. The 2D bound coincides with the avalanche-percolation threshold derived by Curtin and Scher (1997) using entirely different arguments.
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References
Beyerlein, I.J. and Phoenix, S.L. (1997a). Statistics of fracture for an elastic notched composite lamina containing Weibull fibers-I. Engineering Fracture Mechanics pp. 241–265.
Beyerlein, I.J. and Phoenix, S.L. (1997b). Statistics of fracture for an elastic notched composite lamina containing Weibull fibers-II. Engineering Fracture Mechanics 57, 267–299.
Beyerlein, I.J., Phoenix, S.L. and Raj, R. (1998). Time evolution of stress redistribution around multiple fiber breaks in a composite with viscous and viscoelastic matrices. International Journal of Solids and Structures 35, 3177–3211.
Coleman, B.D. (1956). Time dependence of mechanical breakdown phenomena. Journal of Applied Physics 27, 862–866.
Coleman, B.D. (1957a). A stochastic process model for material breakdown. Transactions of the Society of Rheology 1, 153–168.
Coleman, B.D. (1957b). Time dependence of mechanical breakdown of fibers I. Constant total load. Journal of Applied Physics 28, 1058–1064.
Coleman, B.D. (1958a). On the strength of classical fibers and fiber bundles. Journal of the Mechanics and Physics of Solids 7, 60–70.
Coleman, B.D. (1958b). Statistical and time-dependent mechanical breakdown of fibers. Journal of Applied Physics 7, 60–70.
Copson, E.T. (1965). Asymptotic Expansions Cambridge: University Press. Curtin, W.A. (1998). Size scaling of strength in heterogeneous materials. Physical Review Letters 80(7), 1445–1448.
Curtin, W.A., Pamel, M. and Scher, H. (1997). Time-dependent damage evolution and failure in materials. II. Simulations. Physical Review B 55, 12051–12061.
Curtin, W.A. and Scher, H. (1991). Analytic model for scaling of breakdown. Physical Review Letters 67, 2457–2460.
Curtin,W.A. and Scher, H. (1997). Time-dependent damage evolution and failure in materials. I. Theory. Physical Review B 55, 12038–12050.
Goda, K. (2001). Application of Markov process to chain-of-bundles probability model and lifetime distribution analysis for fibrous composites. Materials Science Research International STP-2, 242–249.
Goda, K. (2003). A strength reliability model by Markov process of unidirectional composites with fibers placed in hexagonal arrays. International Journal of Solids and Structures 40, 6813–6833.
Hedgepeth, J.M. (1961). Stress concentrations in filamentary structures. Technical Report TND-882, NASA.
Hedgepeth, J.M. and Dyke, P.V. (1967). Local stress concentrations in imperfect filament composites. Journal of Composite Materials 1, 294–309.
Horn, R.A. and Johnson, C.R. (1985). Matrix Analysis Cambridge University Press, Cambridge.
Ibnabdeljalil, M. and Phoenix, S.L. (1995). Creep rupture of brittle matrix composites reinforced with time dependent fibers: scalings and Monte-Carlo simulations. Journal of the Mechanics and Physics of Solids 43, 897–931.
Lagoudas, D.C., Phoenix, S.L. and Hui, C.-Y. (1989). Time evolution of overstress profiles near broken fibers in a composite with a viscoelastic matrix. International Journal of Solids and Structures 25, 45–66.
Landis, C.M., Beyerlein, I.J. and McMeeking, R.M. (2000). Micromechanical simulation of the failure of fiber reinforced composites. Journal of the Mechanics and Physics of Solids 48, 621–648.
Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. New York: Springer.
Mahesh, S., Beyerlein, I.J. and Phoenix, S.L. (1999). Size and heterogeneity effects on the strength of fibrous composites. Physica D 133, 371–389.
Mahesh, S., Phoenix, S.L. and Beyerlein, I.J. (2002). Strength distributions and size effects for 2D and 3D composites withWeibull fibers in an elastic matrix. International Journal of Fracture 115, 41–85.
Newman,W.I. and Phoenix, S.L. (2001). Time-dependent fiber bundles with local load sharing. Physical Review E 63, 021507.
Phoenix, S.L. (1978). The asymptotic time to failure of a mechanical system of parallel members. SIAM Journal of Applied Mathematics 34, 227–246.
Phoenix, S.L. and Tierney, L. (1983). A statistical model for the time dependent failure of unidirectional composite materials under local elastic load-sharing among fibers. Engineering Fracture Mechanics 18, 193–215.
Smith, R.L. (1980). A probability model for fibrous composites with local load sharing. Proceedings of the Royal Society of London 373, 539–553.
Tierney, L. (1980). Limit theorems for the failure time of bundles of fibers under unequal load sharing. Ph.D. thesis,Cornell University, Ithaca, New York.
Tierney, L. (1982). Asymptotic bounds on the time to fatigue failure of bundles of fibers under local load sharing. Advances in Applied Probability 56, 95–121.
Wu, B.Q. and Leath, P.L. (2000). Similarity of growing cracks in breakdown of heterogeneous planar interfaces. Physical Review B 62, 9338.
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Mahesh, S., Phoenix, S. Lifetime distributions for unidirectional fibrous composites under creep-rupture loading. International Journal of Fracture 127, 303–360 (2004). https://doi.org/10.1023/B:FRAC.0000037675.72446.7c
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DOI: https://doi.org/10.1023/B:FRAC.0000037675.72446.7c