We develop a method which permits the analysis of problems requiring the simultaneous resolution of continuum and atomistic length scales--and associated deformation processes--in a unified manner. A finite element methodology furnishes a continuum statement of the problem of interest and provides the requisite multiple-scale analysis capability by adaptively refining the mesh near lattice defects and other highly energetic regions. The method differs from conventional finite element analyses in that interatomic interactions are incorporated into the model through a crystal calculation based on the local state of deformation. This procedure endows the model with crucial properties, such as slip invariance, which enable the emergence of dislocations and other lattice defects.; We assess the accuracy of the theory in the atomistic limit by way of several examples: surface energies of exposed (111),(110) and (001) planes, a stacking fault on the (111) plane, and edge dislocations residing on (111) and (100) planes of an aluminum single crystal. The method correctly predicts the splitting of the (111) edge dislocation into Shockley partials and predicts no splitting of the Lomer dislocation, in keeping with observation and the results of direct atomistic simulation. In both cases, the core structures are found to be in very good agreement with direct lattice statics calculations, which attests to the accuracy of the method at the atomistic scale.; Our main focus, is in nano-scale phenomena involving the cooperative behavior of multiple defects. A prime example, and one which illustrates the strengths of the method, is nanoindentation, where the penetration of the indenter is accomodated by the nucleation at the surface--and subsequent propagation into the crystal--of a small number of discrete dislocations. The multiple-scale character of this boundary value problem renders it awkward for analysis by either atomistic or continuum methods. By contrast, our combined atomistic/continuum approach permits following the individual dislocations as they are nucleated under the indenter and driven into the crystal, while, simultaneously, yielding the response of the system at the nano-scale, e.g., in the form of the relation between applied force and depth of indentation. In this class of applications lies the primary scope of our method.
展开▼
