On the numerical homogenization of real polycrystalline microstructures

Abstract

AbstractThe FE2 method, cf. [1], a direct micro‐macro homogenization approach, has become a standard procedure for scale‐transition applications. Therein, the modeling of a micro‐heterogeneous material described by a representative volume element (RVE) based on realistic microstructures can give rise to a barely unmanageable computational effort. Alternatively, statistically similar RVEs (SSRVEs) can be used, which are constructed based on morphological information of the real microstructure and lead to a reduction of computational cost, see [2]. In their construction, a least‐square functional is used to minimize the deviation of statistical properties, such as volume fraction, spectral density and lineal‐path function, of the SSRVE and the real microstructure. The application of SSRVEs has been shown to lead to an adequate representation of the mechanical behavior of the real microstructure. The first part of the talk will give an overview on the construction of SSRVEs and present examples of multiscale analyses using the FE2 approach with simplified microstructures in different engineering applications of steel material.The paper also focusses on the details of the microstructure and discusses crystal plasticity models, see e.g. [3], in order to account for the material anisotropy induced by the texture of the crystalline structure of steel. It is well known that for rate independent single crystal plasticity, the ambiguity of the choice of active slip systems and linear dependency of slip criteria may cause instabilities in the algorithm. Classical perturbation methods are often used to solve the problem as well as rate dependent algorithms which model the rate independent case as the limit of vanishing viscosity. However, this leads to stiff constitutive equations and thus requires small time increments. In [4], an alternative approach has been proposed recently which is based on handling the constrained optimization problem in the framework of infeasible primal‐dual interior point methods (IPDIPM). We modify the original constrained optimization problem using slack variables in order to stabilize the algorithm and allow for temporary violation of the constraints. Numerical examples are presented for crystalline structures with face centered cubic properties.