Abstract
For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE2-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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Balzani, D., Brands, D., Schröder, J., Carstensen, C.: Sensitivity analysis of statistical measures for the reconstruction of microstructures based on the minimization of generalized least-square functionals. In: Technische Mechanik (in press, 2010)
Balzani, D., Schröder, J., Brands, D.: FE2-Simulation of microheterogeneous steels based on statistically similar RVE’s. In: Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, September 22–26, 2008, Bochum, Germany, (in press, 2009)
Beran M.: Statistical continuum theories. Wiley, New York (1968)
Brekelmans W.A.M., Smit R.J.M., Meijer H.E.H.: Prediction of the mechanical behaviour of nonlinear heterogeneous systems by multi-level finite element modelling. Comput. Methods Appl. Mech. Eng. 155, S181–S192 (1998)
Bresenham, J.E.: Algorithm for computer control of a digital plotter. In: IBM Systems Journal 4 (1965), Nr. 1, S. 25–30. Reprinted in Interactive Computer Graphics, Herbert Freeman ed., 1980, and Seminal Graphics: Pioneering Efforts That Shaped The Field, Rosalee Wolfe ed., ACM SIGGRAPH, 1998
Brown W.F.: Solid mixture permettivities. J. Chem. Phys. 23, S1514–S1517 (1955)
Geers M.G.D., Kouznetsova V., Brekelmans W.A.M.: Multi-scale first-order and second-order computational homogenization of microstructures towards continua. Int. J. Multiscale Comput. Eng. 1, S371–S386 (2003)
Hill R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, S357–S372 (1963)
Klinkel, S.O.: Theorie und Numerik eines Volumen-Schalen-Elementes bei finiten elastischen und plastischen Verzerrungen, Universität Fridericiana zu Karlsruhe, Dissertation (2000)
Kröner E.: Allgemeine Kontinuumstheorie der Versetzung und Eigenspannung. Arch. Rational Mech. Anal. 4, S273–S334 (1960)
Kröner E.: Statistical continuum mechanics. CISM Courses and Lectures Bd. 92. Springer, Wien (1971)
Lee E.H.: Elasto-plastic deformation at finite strains. J. Appl. Mech. 36, S1–S6 (1969)
Lu B.L., Torquato S.: Lineal-path function for random heterogeneous materials. Phys. Rev. A 45, S922–S929 (1992)
Miehe, C.: Kanonische Modelle multiplikativer Elasto-Plastizität. Thermodynamische Formulierung und Numerische Implementation. Universität Hannover, Institut für Baumechanik und Numerische Mechanik, Bericht-Nr. F93/1, Habilitationsschrift (1993)
Miehe C., Schotte J., Schröder J.: Computational micro-macro-transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci. 16, S372–S382 (1999)
Miehe C., Schröder J., Becker M.: Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput. Methods Appl. Mech. Eng. 191, S4971–S5005 (2002)
Miehe C., Schröder J., Schotte J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput. Methods Appl. Mech. Eng. 171, S387–S418 (1999)
Miehe C., Stein E.: A canonical model of multiplicative elasto-plasticity formulation and aspects of the numerical implementation. Eur. J. Mech. A/Solids 11, S25–S43 (1992)
Ohser J., Mücklich F.: Statistical analysis of microstructures in materials science. Wiley, New York (2000)
Peric D., Owen D.R.J., Honnor M.E.: A model for finite strain elasto-plasticity based on logarithmic strains: computational issues. Comput. Methods Appl. Mech. Eng. 94, S35–S61 (1992)
Povirk G.L.: Incorporation of microstructural information into models of two-phase materials. Acta Metallurgica 43(8), S3199–S3206 (1995)
Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen, Bericht aus der Forschungsreihe des Institut für Mechanik (Bauwesen), Lehrstuhl I, Habilitationsschrift (2000)
Schröder, J., Balzani, D., Richter, H., Schmitz, H.P., Kessler, L.: Simulation of microheterogeneous steels based on a discrete multiscale approach. In: Hora, P. (Hrsg.): Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, pp. 379–383 (2008)
Simo J.C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Comput. Methods Appl. Mech. Eng. 66, S199–S219 (1988)
Simo J.C.: Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput. Methods Appl. Mech. Eng. 99, S61–S112 (1992)
Simo J.C., Miehe C.: Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput. Methods Appl. Mech. Eng. 96, S133–S171 (1992)
Smit R.J.M., Brekelmans W.A.M., Meijer H.E.H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput. Methods Appl. Mech. Eng. 155, S181–S192 (1998)
Temizer I., Wriggers P.: On the computation of the macroscopic tangent for multiscale volumetric homogenization problems. Comput. Methods Appl. Mech. Eng. 198, S495–S510 (2008)
Torquato S.: Random heterogeneous materials. Microstructure and macroscopic properties. Springer, Berlin (2002)
Torquato S., Stell G.: Microstructures of two-phase random media I. The n-point probability functions. J. Chem. Phys. 77, S2071–S2077 (1982)
Voce E.: A practical strain hardening function. Metallurgica 51, S219–S226 (1955)
Weber G., Anand L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoelastic solids. Comput. Methods Appl. Mech. Eng. 79, S173–S202 (1990)
Zeman, J.: Analysis of Composite Materials with Random Microstructure, University of Prague, Dissertation (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schröder, J., Balzani, D. & Brands, D. Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions. Arch Appl Mech 81, 975–997 (2011). https://doi.org/10.1007/s00419-010-0462-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-010-0462-3