Strain localization analysis for planar polycrystals based on bifurcation theory

In the present paper, an efficient numerical tool is developed to investigate the ductility limit of polycrystalline aggregates under in-plane biaxial loading. These aggregates are assumed to be representative of very thin sheet metals (with typically few grains through the thickness). Therefore, the plane-stress assumption is naturally adopted to numerically predict the occurrence of strain localization. Furthermore, the initial crystallographic texture is assumed to be planar. Considering the latter assumptions, a two-dimensional single-crystal model is advantageously chosen to describe the mechanical behavior at the microscopic scale. The mechanical behavior of the planar polycrystalline aggregate is derived from that of single crystals by using the full-constraint Taylor scale-transition scheme. To predict the occurrence of localized necking, the developed multiscale model is coupled with bifurcation theory. As will be demonstrated through various numerical results, in the case of biaxial loading under plane-stress conditions, the planar single-crystal model provides the same predictions as those given by the more commonly used three-dimensional single-crystal model. Moreover, the use of the two-dimensional model instead of the three-dimensional one allows dividing the number of active slip systems by two and, hence, significantly reducing the CPU time required for the integration of the constitutive equations at the single-crystal scale. Furthermore, the planar polycrystal model seems to be more suitable to study the ductility of very thin sheet metals, as its use allows us to rigorously ensure the plane-stress state, which is not always the case when the fully three-dimensional polycrystalline model is employed. Consequently, the adoption of this planar formulation, instead of the three-dimensional one, allows us to simplify the computational aspects and, accordingly, to considerably reduce the CPU time required for the numerical predictions.