Analysis of cell size effects in atomistic crack propagation

Abstract

We consider crack propagation in a crystalline material in terms of bifurcation analysis. We provide evidence that the stress intensity factor is a natural bifurcation parameter, and that the resulting bifurcation diagram is a periodic "snaking curve". We then prove qualitative properties of the equilibria and convergence rates of finite-cell approximations to the "exact" bifurcation diagram.

Numerical-continuation-enhanced flexible boundary condition scheme applied to mode-I and mode-III fracture

“…Two things are apparent: firstly, the standard flexible scheme yields a rate of convergence of order O(R −1 ), which improves upon a known rate of convergence O(R −1/2 ) of the static scheme proven in Ref. 18. A mathematically rigorous proof of the improved rate of convergence will be a subject of further study.…”

“…Relatedly, a mathematically rigorous numerical analysis of domain size effects for a static boundary condition scheme coupled with numerical continuation techniques has been conducted in Ref. 18. It provides a basic framework in which proving convergence rates to the infinite limit is possible and is the principal motivation for the current work.…”

“…While it might be tempting to optimise over K by requiring that ∂ E ∂ K = 0, as was the case for α, we stress that the nature of K is fundamentally different to α -the value of K specifies the dominant behaviour at infinity, scaling like ∼ r 1/2 in the displacement and ∼ r −1/2 in the strain, where r is the distance from the crack tip. Due to this dominant behaviour at infinity, it was shown in [18] that optimising over K is not mathematically sound. On the other hand, α specifies the next order behaviour, with its contributions ∼ r −1/2 in the displacement and ∼ r −3/2 in the strain.…”

A stochastic framework for atomistic fracture

“…Due to the inherent nonlinearity of the atomistic model, it is not possible to obtain an analytic characterization of atomistic equilibrium configurations around a crack. Away from the crack tip, however, the CLE model outlined in Section 2.1, which can be obtained via the Cauchy-Born coupling, as discussed in Section 3.3, approximates the atomistic model well [3].…”

“…We refer to [3,4] for a rigorous derivation of the infinite lattice model. As will be noted in Section 5, in the present work, we restrict our attention to the case where (3.13) is satisfied through setting u(m) = 0 for all m ∈ Λ, such that |m| > R 0 , for some suitably chosen R 0 .…”