Abstract
An atomistic model of near-crack-tip plasticity on a square lattice under anti-plane shear kinematics is formulated and studied. The model is based upon a new geometric and functional framework of a lattice manifold complex, which ensures that the crack surface is fully taken into account, while preserving the crucial notion of duality. As a result, existence of locally stable equilibrium configurations containing both a crack opening and dislocations is established. Notably, with the boundary in the form of a crack surface accounted for, no minimum separation between a dislocation core and the crack surface or the crack tip is required. The work presented here constitutes a foundation for several further studies aiming to put the phenomenon of near-crack-tip plasticity on a rigorous footing.
Highlights
This rearrangement most prominently comes in the form of topological defects known as dislocations, which are carriers of plastic deformation [HL82]
Such Green’s function is proven to exist and its decay properties are characterised. This paves the way for the results concerning existence of dislocations-only equilibrium configurations, which are stated in section 4.2 and centre around an explicit construction relying on duality and the aforementioned Green’s function
The results presented here achieves a similar feat in the context of a lattice manifold complex representing a cracked lattice domain and heavily rely on theorem 4.2
Summary
In a cracked crystalline body, the term near-crack-tip plasticity refers to the phenomenon of atoms rearranging themselves in the vicinity of the crack tip due to stresses accumulated therein. With cracks and dislocations initiating and propagating via primarily atomistic mechanisms [BBC06, BKG15, HL82, SOCK15], the modelling of near-crack-tip plasticity should ideally take the atomistic scale correctly into account This poses a major challenge, as the inherently nonlinear nature of interactions between atoms renders the interplay between defects in a crystal a highly complex phenomenon. While there is a wealth of recent work about atomistic modelling of materials and in particular atomistic approaches to defects in crystals (cf, among many other, [ADLGP14, AO05, BC07, BDMG99, BLO06, EM07, EOS16, HO14, Pon07]), many of the mathematical techniques employed rely at least in part on exploiting crystal symmetries This renders most of them fundamentally inadequate for the study of near-crack-tip plasticity, as the crack breaks the translational symmetry, meaning that the domain under study is discrete and spatially inhomogeneous. A separate clear future direction is to extend the framework to the case of a moving crack tip, allowing a rigorous study of the influence dislocations exert on crack propagation
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