Lattice Green's function method for Atomistic/Continuum coupling

Metal plasticity is an inherently multiscale phenomenon due to the complex long-range field of atomistic dislocations that are the primary mechanism for plastic deformation in metals. Atomistic/Continuum (A/C) coupling methods are computationally efficient ways to model dislocation interactions with other defects. This coupling utilizes robust but computationally expensive atomistic models in the spatial regions where short-range nonlinear atomistic resolution is required and the less expensive linear continuum elasticity mesoscale models (e.g., discrete dislocations) for regions where long-range effects dominate. Traditional A/C coupling methods use finite element methods in the continuum region, which makes them computationally prohibitive for 3D boundary value problems due to volumetric scaling of degrees of freedom. Here, a lattice Green's function (LGF)-based A/C coupling method is developed to efficiently solve 3D boundary value problems in which the number of degrees of freedom scales with the surface area. Such A/C boundary value problems involve an internal boundary between the atomistic and continuum domains, and an outer boundary where surface forces and displacements are applied. Using LGF, the flexible boundary condition method (FBCM) can be used at the internal boundary. The FBCM is analyzed for the infinite domain A/C problems using several 1D and 2D example problems to highlight the influence of the initial solutions and the numerical LGF on the accuracy of results. Next, a discrete counterpart of the continuum boundary element method, the lattice Green's function method (LGFM), is developed for the outer boundary. The LGFM requires an atomistically-resolved boundary which is computationally prohibitive for large realistic 3D problems. A coarse-graining method for the LGFM on the outer boundary is introduced to address this, where slowly varying surface forces/displacements are interpolated using local shape functions. The coarse-grain LGFM reduces the degrees of freedom on the outer boundary, making the simulations computationally tractable. Validation and application of the LGFM for the non-trivial mechanics problem of a dislocation loop in a 3D FCC box with externally applied surface displacements are shown. Finally, the FBCM and LGFM are integrated to solve the full coupled A/C boundary value problems using LGF.