An Adaptive $$\hbox {FE}^2$$ FE 2 Approach for Fiber–matrix Composites

The contribution is concerned with an adaptive scheme for the finite-element square (\(\hbox {FE}^2\)) method. The \(\hbox {FE}^2\) method allows a continuous homogenization considering the current deformation state of the heterogeneous material structure. The micro-structure is represented by a representative volume element, which differs in the fiber distribution. The fiber material behavior is assumed as elasto-plastic. The non-linear response of the fiber necessitates a numerical homogenization for every load step at each integration point, which leads to an increased computational effort. An indicator for a nested \(\hbox {FE}^2\) homogenization is introduced. It takes advantage of the fact that a accompanying homogenization is only necessary in the regions of non-linear material behavior. The present work deals with an adaptive scheme for fiber–matrix composites to reduce the computational cost. Numerical examples show the capability of the proposed scheme.