Circular Microstructural Volume Elements with Periodic Boundary Conditions for Strain Localization Problems

Abstract

A common choice for multiscale modeling of the mechanical response of composites is to use periodic boundary conditions (BCs) on square representative volume elements (RVEs). However, these periodic BCs over-constrain the response when strain localization takes place in bands that are not compatible with the imposed periodic constraints. Previously developed improvements are based on aligning the periodic BCs with an evolving localization band. This is either done by applying a rotation to the periodicity frame or by imposing the periodic BCs in a weak sense and then applying a shift to the function that couples points. However, with matrix-inclusion RVEs, this change of the periodicity frame may cause a mis-alignment of inclusions that cross opposing edges, resulting in an artificial reinforcement along the RVE edge, which limits the number of supported localization angles and fails to provide a transversely isotropic response. It is the objective of this thesis to develop a micromodel with transversely isotropic response for strain localization problems. It is shown that circular RVEs with straightforward application of periodic BCs provide a response which is independent of the load orientation but fail to predict full softening behavior. This is due to over-constraining when cracks reach the boundary. Therefore, a modification to the periodic BCs on a circular RVE is proposed, which allows for cracks to cross the edges. This is achieved by adding an unknown jump to the periodic constraint equations, which does not affect the response before localization. Moreover, a parameter that depends on the RVE size is used to make sure that the kinematics are consistent between scales in an average sense. The performance of the formulation is tested with a series of simulations where macroscopic strain rates are imposed under varying angles. Additionally, a circular heterogeneous RVE with periodic material is presented where inclusions are allowed to cross the edge. It is demonstrated that the circular RVE with the modified periodic BCs successfully predicts an isotropic response with full softening. In the simulations that were performed, a priori knowledge of the localization angle was used. However, the framework can be extended such that the BCs adapt to support an orientation of a localization band that is detected during the simulations.