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Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations

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Abstract

In this paper a method is presented for the construction of two- and three-dimensional statistically similar representative volume elements (SSRVEs) that may be used in computational two-scale calculations. These SSRVEs are obtained by minimizing a least-square functional defined in terms of deviations of statistical measures describing the microstructure morphology and mechanical macroscopic quantities computed for a random target microstructure and for the SSRVE. It is shown that such SSRVEs serve as lower bounds in a statistical sense with respect to the difference of microstructure morphology. Moreover, an upper bound is defined by the maximum of the least-square functional. A staggered optimization procedure is proposed enabling a more efficient construction of SSRVEs. In an inner optimization problem we ensure that the statistical similarity of the microstructure morphology in the SSRVE compared with a target microstructure is as high as possible. Then, in an outer optimization problem we analyze mechanical stress–strain curves. As an example for the proposed method two- and three-dimensional SSRVEs are constructed for real microstructure data of a dual-phase steel. By comparing their mechanical response with the one of the real microstructure the performance of the method is documented. It turns out that the quality of the SSRVEs improves and converges to some limit value as the microstructure complexity of the SSRVE increases. This converging behavior gives reason to expect an optimal SSRVE at the limit for a chosen type of microstructure parameterization and set of statistical measures.

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Notes

  1. Here, re-binning means the standard procedure of changing the resolution, which is well-known from image processing.

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Acknowledgments

The authors greatly appreciate financial funding by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG), as part of the research group “Microplast” (FOR 797) on Analysis and Computation of Microstructures in Finite Plasticity, project SCHR 570/8-2. In addition to that the authors thank Prof. D. Raabe (Max-Planck-Institute for Iron Research, Düsseldorf, Germany) for providing the three-dimensional microstructure of a dual-phase steel based on 3D EBSD/FIB measurements. Furthermore, assistance regarding the application of the mystic optimization framework by M. McKerns is appreciated.

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Balzani, D., Scheunemann, L., Brands, D. et al. Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations. Comput Mech 54, 1269–1284 (2014). https://doi.org/10.1007/s00466-014-1057-6

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  • DOI: https://doi.org/10.1007/s00466-014-1057-6

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