Atomistic modelling of fracture
Maciej Buze · Warwick Research Archive Portal (University of Warwick) · 2019
This thesis is devoted to the mathematical analysis of atomistic modelling of fracture in a crystalline solid. In particular, we focus on a single Mode III crack defect in an infinite two-dimensional square lattice under anti-plane displacements and nearest neighbour interactions, show that the associated lattice equilibration problem is well-defined over a suitable function space and discuss different regimes of the key parameter known as the (rescaled) stress intensity factor k _ 0, which in continuum fracture mechanics characterises the strength of the stress singularity at the crack tip and more broadly acts as a loading parameter on the crack. In the first part of the work, we focus on the small-loading regime with k sufficiently small and, under the assumption that interactions across the crack are disregarded, prove existence, local uniqueness and stability of atomistic solutions and further establish their qualitatively sharp far-field decay estimates. The latter result requires establishing existence and decay estimates for the corresponding lattice Green’s function in the anti-plane crack geometry, which constitutes the main technical result of the thesis. In the final part, we go beyond the small-loading regime and focus on capturing crack propagation in a quasi-static analysis aided by bifurcation theory. We provide evidence that k is a natural bifurcation parameter and that the resulting bifurcation diagram is a periodic “snaking curve”. Subsequently we investigate cell size effects in a finite-cell approximation to the infinite problem by proving sharp convergence rates and obtaining a superconvergence result for critical values of k. This enables us to capture the phenomenon of lattice trapping and how it is significantly influenced by the computational domain size.