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Effects of dissipation and driving on chaotic dynamics in an atomistic crack-tip model

An analysis of the dynamic behaviour associated with a simple atomistic model of a crack tip is presented. The model consists of a linear array of four atoms, with nearest neighbours interacting via a Morse-function potential. Energy exchange with the surroundings is simulated by driving the two outer atoms, that is, forcing them to oscillate sinusoidally with time, and by allowing the motion of the two inner atoms to be damped. The damping may be at least qualitatively related to dissipative mechanisms that may occur at the tip of a crack. Emphasis is placed on determining conditions under which the two inner atoms, which represent the crack tip, oscillate in a chaotic or aperiodic manner in response to the driving and damping. The amplitude of tile driving oscillation and the damping constant are treated as variable parameters in this regard. As expected, chaotic behaviour is to be found associated with larger driving amplitudes and smaller damping constants. Extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. The temperature associated with the two inner atoms, calculated in terms of their time-averaged kinetic energy, can undergo a large increase when the transition from periodic to chaotic oscillations takes place. In addition, the non-linear dynamics is, under some conditions, found to be characterized by hyperchaos, that is, by the existence of two positive Lyapunov exponents. Qualitative implications of the results, relative to actual fracture processes, are discussed.