Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions

Abstract

For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE 2 -method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a "large" RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples.

Simulation of Two‐Phase Steels based on Statistically Similar Representative Volume Elements

“…Therefore, the moving-frame algorithm proposed in [2] is used. In [7] is it shown that the combination of the volume fraction (V), the spectral density (SD) and the lineal-path function (LP) as statistical measures leads to promising results in the context of two-phase materials, whose macroscopic mechanical response is mainly governed by the microstructural morpohology. Therefore, the three individual least-square functionals…”

From High Accuracy to High Efficiency in Simulations of Processing of Dual-Phase Steels

“…This led to the construction of SSRVE, which are characterized by a lower complexity than the smallest possible substructure. [14] A schematic illustration of the idea of the SSRVE is presented in Figure 8.…”

Nano to Micro – Perspectives for Homogenization in Crystalline Solids

“…Applying them to large random microstructures leads in most cases to too complex structures, which require high amount of memory and high computation time. Therefore, in [6], [8] the construction of statistically similar RVEs (SSRVEs) with a reduced complexity compared to the real microstructure is proposed, which still represent the mechanical response of the material up to a certain accuracy. Consequently, the number of finite elements required for the discretization of the microscopic boundary value problem is reduced and leads to a decreased computational cost.…”

“…[7]. In [8] it turns out that the combination of the volume fraction, the spectral density and the lineal-path function as statistical measures leads to promising results for twodimensional microstructures. For the construction of three-dimensional SSRVEs we consider the description of the inclusion by nonintersecting ellipsoidal inclusions.…”