An FE2-X1 approach for multiscale localization phenomena

In this paper, a new methodology based on the Hill–Mandel lemma in an FE2 sense is proposed that is able to deal with localized deformations. This is achieved by decomposing the displacement field of the fine scale model into a homogeneous part, fluctuations, and a cracking part based on additional degrees of freedom (X1)—the crack opening in normal and tangential directions. Based on this decomposition, the Hill–Mandel lemma is extended to relate coarse and fine scale energies using the assumption of separation of scales such that the fine scale model is not required to have the same size as the corresponding macroscopic integration point. In addition, a procedure is introduced to mimic periodic boundary conditions in the linear elastic range by adding additional shape functions for the boundary nodes that represent the difference between periodic boundary conditions and pure displacement boundary conditions due to the same macroscopic strain. In order to decrease the computational effort, an adaptive strategy is proposed allowing different macroscopic integration points to be resolved in different levels on the fine scale.

► Extension of the FE2-approach to problems with localization.
► Decomposition of the macroscopic strain into a homogeneous contribution and a localized part.
► Additional DOFs crack opening and homogeneous strain describe the boundary displacements.
► Additional DOFs are implicitly solved for in the solution procedure.
► Approximation of periodic boundary conditions with prescribed displacements by using additional shape functions.