Stability and size-dependency of Cauchy–Born hypothesis in three-dimensional applications

The Cauchy–Born hypothesis (CB) provides a hierarchical approach in the molecular theory of crystal elasticity to relate the continuum and atomic deformations. This kinematic theory has been extensively used as the constitutive law of continuum regions in multi-scale models. In these models, the fine scale is proposed to describe the real behavior of crystalline structure wherever the continuum description fails. The main objective of this article is to investigate the stability and size-dependency of CB hypothesis in three-dimensional applications by direct comparison of information between atomistic and continuous description of a medium. The Sutton–Chen many-body potential is used for the gold metal to consider the real metallic behavior in numerical simulations. Two failure criteria are introduced in the strain and stress domains; the validity surfaces are derived for the Cauchy–Born hypothesis; and the size effect of specimens is investigated on the convergency of results. It is shown that the gold crystal deforms homogeneously inside the validity surface, in which the material is elastic and the CB has remained valid. It is observed that although the deformation is inhomogeneous and the CB is invalid outside the validity surface, the crystalline structure may exhibit elastic or plastic behavior in this region. Moreover, it is numerically shown that the size-dependency of validity surface decreases with the increase of the size of specimens. These observations are meticulously investigated by loading and unloading several cubic specimens using molecular dynamics simulation.