A nonlinear fracture differential kinetic model to depict chaotic atom motions at a fatigue crack tip based on the differentiable manifold methodology

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. 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. As demonstrated in the paper, 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 nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.