Abstract
This paper presents a simplified probabilistic model for thermally activated nanocrack propagation. In the continuum limit, the probabilistic motion of the nanocrack tip is mathematically described by the Fokker-Planck equation. In the model, the drift velocity is explicitly related to the energy release rate at the crack tip through the transition rate theory. The model is applied to analyze the propagation of an edge crack in a nanoscale element. The element is considered to reach failure when the nanocrack propagates to a critical length. The solution of the Fokker-Planck equation indicates that both the strength and lifetime distributions of the nanoscale element exhibit a power-law tail behavior but with different exponents. Meanwhile, the model also yields a mean stress-life curve of the nanoscale element. When the applied stress is sufficiently large, the mean stress-life curve resembles the nasquin law for fatigue failure. nased on a recently developed finite weakest-link model as well as level excursion analysis of the failure statistics of quasi-brittle structures, it is argued that the simulated power-law tail of strength distribution of the nanoscale element has important implications for the tail behavior of the strength distribution of macroscopic structures. It provides a physical justification for the two-parameter Weibull distribution for strength statistics of large-scale quasi-brittle structures.
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Bažant, Z.P. and Pang, S.D., Activation Energy Based Extreme Value Statistics and Size Effect in Brittle and Quasi-brittle Fracture, J. Mech. Phys. Solids, 2007, vol. 55, pp. 91–134.
Bažant, Z.P., Le, J.-L., and Bažant, M.Z., Scaling of Strength and Lifetime Distributions of Quasi-brittle Structures Based on Atomistic Fracture Mechanics, Proc. Natl Acad. Sci., 2009, vol. 106, pp. 11484–11489.
Le, J.-L., Bažant, Z.P., and Bažant, M.Z., Unified Nano-mechanics Based Probabilistic Theory of Quasi-brittle and Brittle Structures: I. Strength, Crack Growth, Life-time and Scaling, J. Mech. Phys. Solids, 2011, vol. 59, pp. 1291–1321.
Le, J.-L. and Bažant, Z.P., Unified Nano-Mechanics Based Probabilistic Theory of Quasi-brittle and Brittle Structures: II. Fatigue Crack Growth, Lifetime and Scaling, J. Mech. Phys. Solids, 2011, vol. 59, pp. 1322–1337.
Genet, M., Couégnat, G., Tomsia, A.P., and Ritchie, R.O., Scaling Strength Distributions in Quasi-brittle Materials from Micro- to Macroscales: A Computational Approach to Modeling Natureinspired Structural Ceramics, J. Mech. Phys. Solids, 2014, vol. 68, pp. 93–106.
Xu, Z. and Le, J.-L., A First Passage Model for Probabilistic Failure of Polycrystalline Silicon MEMS Structures, J. Mech. Phys. Solids, vol. 99, pp. 225–241.
Bažant, Z.P. and Le, J.-L., Probabilistic Mechanics of Quasi-brittle Structures: Strength, Lifetime, and Size Effect, Cambridge: Cambridge University Press, 2017.
Omeltchenko, A., Yu, J., Kalia, R.K., and Vashishta, P., Crack Front Propagation and Fracture in a Graphite Sheet: A Molecular Dynamics Study on Parallel Computers, Phys. Rev. Lett., 1997, vol. 78, no. 11, pp. 2148–2151.
Marder, M., Effect of Atoms on Brittle Fracture, Int. J. Frac., 2004, vol. 130, pp. 517–555.
Khare, R., Mielke, S.L., Paci, J.T., Zhang, S.L., Ballarini, R., Schatz, G.C., and Belytschko, T., Coupled Quantum Mechanical/Molecular Mechanical Modeling of the Fracture of Defective Carbon Nanotubes and Graphene Sheets, Phys. Rev. B, 2007, vol. 75, no. 7, p. 075412.
Khare, R., Mielke, S.L., Schatz, G.C., and Belytschko, T., Multiscale Coupling Schemes Spanning the Quantum Mechanical, Atomistic Forcefield, and Continuum Regimes, Comp. Meth. Appl. Mech. Engrg., 2008, vol. 197, pp. 3190–3202.
Buehler, M.J., Atomistic Modeling of Materials Failure, New York: Spinger, 2008.
Tadmor, E.B. and Miller, R.E., Modeling Materials: Continuum, Atomistic and Multiscale Techniques, Cambridge: Cambridge University Press, 2011.
Abraham, F.F., Broughton, J.Q., Bernstein, N., and Kaxiras, E., Spanning the Continuum to Quantum Length Scales in a Dynamical Simulation of Brittle Fracture, Europhys. Lett., 1998, vol. 44, no. 6, pp. 783–787.
Broughton, J.Q., Abraham, F.F., Bernstein, N., and Kaxiras, E., Concurrent Coupling of Length Scales: Methodology and Application, Phys. Rev. B, 1999, vol. 60, pp. 2391–2403.
Xu, M., Tabarraei, A., Paci, J.T., Oswald, J., and Belytschko, T., A Coupled Quantum/Continuum Mechanics Study of Graphene Fracture, Int. J. Frac., 2012, vol. 173, no. 2, pp. 163–173.
Krausz, A.S. and Krausz, K., Fracture Kinetics of Crack Growth, Netherlands: Kluwer Academic Publisher, 1988.
Kramers, H.A., Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reaction, Physica, 1940, vol. 7, pp. 284–304.
Risken, H., The Fokker—Planck Equation, Berlin: Springer-Verlag, 1989.
Barenblatt, G.I., The Formation of Equilibrium Cracks during Brittle Fracture, General Ideas and Hypothesis, Axially Symmetric Cracks, Prikl. Mat. Mekh., 1959, vol. 23, no. 3, pp. 434–444.
Barenblatt, G.I., The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Adv. Appl. Mech., 1962, vol. 7, pp. 55–129.
Bažant, Z.P. and Planas, J., Fracture and Size Effect in Concrete and Other Quasi-brittle Materials, Boca Raton: CRC Press, 1998.
Redner, V., A Guide to First-Passage Processes, Cambridge: Cambridge University Press, 2001.
Basquin, O.H., The Exponential Law of Endurance Tests, Proc. Am. Soc. Test. Mater. ASTE, 1910, vol. 10, pp. 625–630.
Kawakubo, T., Fatigue Crack Growth Mechanics in Ceramics, Cyclic Fatigue in Ceramics, Kishimoto, H., Hoshide, T., Okabe, N., Eds., New York: Elsevier, 1995, pp. 123–137.
Le, J.-L., Manning, J., and Labuz, J.F., Scaling of Fatigue Crack Growth in a Rock, Int. J. Rock Mech. Min. Sci., 2014, vol. 72, pp. 71–79.
Salviato, M. and Bažant, Z.P., The Asymptotic Stochastic Strength of Bundles of Elements Exhibiting General Stress—Strain Laws, Prob. Engrg Mech., 2014, vol. 36, pp. 1–7.
dos Santos, C., Strecker, K., Piorino Neto, F., de Macedo Silva, O.M., Baldacum, V.A., and da Silva, C.R.M., Evaluation of the Reliability of Si3N4—Al2O3—CTR2O3 Ceramics Through Weibull Analysis, Mater. Res., 2003, vol. 6, no. 4, pp. 463–467.
Salem, J.A., Nemeth, N.N., Powers, L.P., and Choi, V.R. Reliability Analysis of Uniaxially Ground Brittle Materials, J. Engrg Gas Turbines Power, 1996, vol. 118, pp. 863–871.
Gross, B., Least Squares Best Fit Method for the Three Parameter Weibull Distribution: Analysis of Tensile and Bend Specimens with Volume or Surface Flaw Failure, NASA Technical Report, 1996, TM-4721, pp. 1–21.
Le, J.-L., Cannone Falchetto, A., and Marasteanu, M.O., Determination of Strength Distribution of Quasi-brittle Structures from Mean Size Effect Analysis, Mech. Mater., 2013, vol. 66, pp. 79–87.
Vanmarcke, E., Random Fields Analysis and Synthesis, Singapore: World Vcientific Publishers, 2010.
Xu, Z. and Le, J.-L., On Power-Law Tail Distribution of Strength Statistics of Brittle and Quasi-brittle Structures, Eng. Frac. Mech., 2018, vol. 197, pp. 80–91.
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Le, JL., Xu, Z. A Simplified Probabilistic Model for Nanocrack Propagation and Its Implications for Tail Distribution of Structural Strength. Phys Mesomech 22, 85–95 (2019). https://doi.org/10.1134/S1029959919020012
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DOI: https://doi.org/10.1134/S1029959919020012