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

Abstract

Abslraci 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 patential. 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 Elated to dissipative mechanisms that may m r 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 the driving oscillalion 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 timeaveraged 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 hypemhs, that is, by the existence of two positive Lyapunov exponents. Qualitltive implications of the mults. relative to achlal fracture processes. are discussed.

Nonlinear motion of crack tip atoms during cleavage processes

“…Mohan et al [ 12] extended the above model to study the effects of dissipation and excitation on the chaotic dynamics in a cleavage process. To model the energy exchange with the surroundings, they oscillate the two outer atoms according to a sinusoidal function and damp the two inner atoms.…”

“…Chaotic phenomena occurring in this time scope cause micron-scale fractal structures of the fracture surface. The studies of [11] and [12], as well as the present work, relate to the atomic chaos, with the Lyapunov exponent to be several bits per pico-second. Atomic chaos relates to the intrinsic brittleness or ductility of the materials.…”

“…The models used in [11] and [12], however, only consist of a one dimensional array of four atoms, connected by nearest-neighbor interatomic bonds. The central bond represents the crack tip and the two out bonds simulate the compliance of the lattice.…”

“…(?t; [k~i)= Ic(ao-l-2u)-F( K,)u + -~ku (12). Figure 2shows the U(u, KI) versus u curves at different KI values, giving a catastrophic picture of a cleavage process.…”