On Aspects of Mixed Continuum-Atomistic Material Modelling
- In the present work, various aspects of the mixed continuum-atomistic modelling of materials are studied, most of which are related to the problems arising due to a development of microstructures during the transition from an elastic to plastic description within the framework of continuum-atomistics. By virtue of the so-called Cauchy-Born hypothesis, which is an essential part of the continuum-atomistics, a localization criterion has been derived in terms of the loss of infinitesimal rank-one convexity of the strain energy density. According to this criterion, a numerical yield condition has been computed for two different interatomic energy functions. Therewith, the range of the Cauchy-Born rule validity has been defined, since the strain energy density remains quasiconvex only within the computed yield surface. To provide a possibility to continue the simulation of material response after the loss of quasiconvexity, a relaxation procedure proposed by Tadmor et al. leading necessarily to the development of microstructures has been used. Thereby, various notions of convexity have been overviewed in details. Alternatively to the above mentioned criterion, a stability criterion has been applied to detect the critical deformation. For the study in the postcritical region, the path-change procedure proposed by Wagner and Wriggers has been adapted for the continuum-atomistic and modified. To capture the deformation inhomogeneity arising due to the relaxation, the Cauchy-Born hypothesis has been extended by assumption that it represents only the 1st term in the Taylor's series expansion of the deformation map. The introduction of the 2nd, quadratic term results in the higher-order materials theory. Based on a simple computational example, the relevance of this theory in the postcritical region has been shown. For all simulations including the finite element examples, the development tool MATLAB 6.5 has been used.