A nonlinear differential kinetic model is derived for describing dynamical behaviours of an atom at a fatigue crack tip using the Newton’s second principle. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. Chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the non-linear stochastic differential kinetic system in contrast to the homoclinic divergence of the non-linear deterministic differential kinetic system. By using the randomization of deterministic fatigue damage equation, the stochastic differential equation and the Fokker-Planck equation of fatigue damage affected by random fluctuation are derived. By means of the solution of equation, the probability distributions of fatigue crack formation and propagation with time are obtained.
KeywordsFatigue Crack Hopf Bifurcation Fatigue Crack Growth Fatigue Damage Atomic Motion
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