| Any theoretical description of the mechanical behaviour of a material, whether it is a simple analytic model or a complicated computer simulation, is ultimately only as good as its constitutive law governing material response. Traditional continuum mechanics approaches to materials modeling use experiments as their guide to choosing the appropriate constitutive behaviour, but recent trends towards modeling microscopic systems mean that many material properties of interest cannot be obtained from direct experimental observation. Atomistic models provide a means to accurately describe material behaviour on the scale of Angstroms, but are severely limited in that they can model only a small number of atoms. This thesis explores ways to combine the strengths of continuum mechanics with those of atomistics by way of three different examples. First, we consider cohesive zone models and their application to fracture and plasticity. We use atomistics to obtain the appropriate constitutive law, and then quantify the limitations of a specific type of cohesive zone model--the Peierls dislocation. Next, non-local continuum mechanics is considered, with a specific example of a non-local dislocation model. Again, it is demonstrated how atomistics can provide accurate constitutive information to the model. Finally, the quasicontinuum method is introduced. This method, which builds on the earlier model of Tadmor, Ortiz and Phillips, is a finite-element based numerical method which uses a fully atomistic constitutive law. The model takes advantage of a reduced number of degrees of freedom where fields are slowly varying, but automatically includes all atomistic degrees of freedom in regions where it is necessary. Crystal symmetry, slip invariance, and the ability to support crystal defects are automatic consequences of the formulation. The method is used to compute grain boundary structures, to simulate cracks in perfect crystals, and to study the interaction between cracks and grain boundaries. |
