Validity of Cauchy–Born hypothesis in multi-scale modeling of plastic deformations

Abstract

The Cauchy–Born (CB) hypothesis has been widely used in multi-scale modeling of crystalline nano-structures. The violation of CB hypothesis in stress space and the transition to plasticity, which is equivalent to the violation of CB hypothesis in strain space, are generally confused and it becomes crucial to differentiate between the two distinct phenomena; the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading. In this paper, a novel technique is developed to investigate the validity of CB hypothesis in stress space, in the presence of plastic deformations. The method is based on the definition of a validity surface in the space of stress and the identification of those violated regions of the domain that do not satisfy the validity criterion at each step of the analysis. The stress of violated regions is corrected so that the stress over the whole domain remains within the validity surface. The procedure is similar to the return mapping algorithm extensively used in the theory of plasticity. In this manner, a quantitative measure is provided that indicates how improperly the CB hypothesis is violated. Numerical simulations for Gold nano-structures illustrate that for certain configurations the violation of CB hypothesis is limited to those parts of the domain with severe surface effects. However, for general configurations, the violation of CB hypothesis depends on the type of loading and the complexity of the domain; in such cases, the implementation of stress correction procedure is important for the accuracy of solution. Finally, the results of numerical simulations are compared with those of Molecular Statics (MS) analyses to illustrate the efficiency of proposed computational model.