Mathematical analysis of fracture and related phenomena in atomistic modelling of crystalline materials

Dr Maciej Buze (Heriot-Watt University)

Thursday 19th October 14:00-15:00 Maths 311B/ZOOM (ID: 821 4273 9503)


The atomistic modelling of fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial and exciting challenges, both on a theoretical and a practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms such as (i) crack propagation; (ii) near-crack tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results.

In this talk I will aim to summarise on-going efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will begin by introducing a framework giving rise to well-defined models for which regularity and stability of solutions can be discussed, followed by describing how the theory can be used to set up practical simulations, such as Mode I crack propagation in silicon on the (111) cleavage plane, using state-of-the-art interatomic potentials. Subsequently I will outline how this framework can be used to rigorously derive upscaled models of near-crack-tip plasticity and, if time permits, touch upon recent work on trying to applying similar methods to study other atomistic phenomena, such as dislocation nucleation.

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