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Construction of Statistically Similar RVEs

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Analysis and Computation of Microstructure in Finite Plasticity

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 78))

Abstract

In modern engineering, micro-heterogeneous materials are designed to satisfy the needs and challenges in a wide field of technical applications. The effective mechanical behavior of these materials is influenced by the inherent microstructure and therein the interaction and individual behavior of the underlying phases. Computational homogenization approaches, such as the FE2 method have been found to be a suitable tool for the consideration of the influences of the microstructure. However, when real microstructures are considered, high computational costs arise from the complex morphology of the microstructure. Statistically similar RVEs (SSRVEs) can be used as an alternative, which are constructed to possess similar statistical properties as the realmicrostructure but are defined by a lower level of complexity. These SSRVEs are obtained from a minimization of differences of statistical measures and mechanical behavior compared with a real microstructure in a staggered optimization scheme, where the inner optimization ensures statistical similarity and the outer optimization problem controls themechanical comparativity of the SSRVE and the real microstructure. The performance of SSRVEs may vary with the utilized statistical measures and the parameterization of the microstructure of the SSRVE.With regard to an efficient construction of SSRVEs, it is necessary to consider statistical measures which can be computed in reasonable time and which provide sufficient information of the real microstructure.Minkowski functionals are analyzed as possible basis for statistical descriptors of microstructures and compared with other well-known statistical measures to investigate the performance. In order to emphasize the general importance of considering microstructural features by more sophisticated measures than basic ones, i.e. volume fraction, an analysis of upper bounds on the error of statistical measures and mechanical response is presented.

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References

  1. Ambrozinski, M., Bzowski, K., Rauch, L., Pietrzyk, M.: Application of statistically similar representative volume elements in numerical simulations of crash box stamping. Archives of Civil and Mechanical Engineering 12, 126–132 (2012)

    Article  Google Scholar 

  2. Arns, C., Knackstedt, M., Mecke, K.: 3d structural analysis: sensitivity of Minkowski functionals. Journal of Microscopy 240, 181–196 (2010)

    Article  MathSciNet  Google Scholar 

  3. Balzani, D., Brands, D., Schröder, J.: Construction of statistically similar representative volume elements. In: Schröder, J., Hackl, K. (eds.) Plasticity and Beyond - Microstructures, Crystal-Plasticity and Phase Transitions. CISM Lecture notes, vol. 550, pp. 355–412. Springer (2014)

    Google Scholar 

  4. Brands, D., Balzani, D., Scheunemann, L., Schröder, J., Richter, H., Raabe, D.: Computational modeling of Dual-Phase steels based on representative three-dimensional microstructures obtained from EBSD data. Archive of Applied Mechanics (2014) (submitted)

    Google Scholar 

  5. 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. Technische Mechanik 30, 297–315 (2010)

    Google Scholar 

  6. Brands, D., Balzani, D., Schröder, J., Raabe, D.: Simulation of DP-steels based on statistically similar representative volume elements and 3D EBSD data. In: Computational Plasticity XI - Fundamentals and Applications, Barcelona, Spain, September 7-9, pp. 1552–1563 (2011)

    Google Scholar 

  7. Beisbart, C., Dahlke, R., Mecke, K., Wagner, H.: Vector- and Tensor-valued Descriptors for Spatial Patterns. In: Mecke, K., Stoyan, D. (eds.) Lecture Notes in Physics, vol. 600 (2002)

    Google Scholar 

  8. Beran, M.: Statistical continuum theories. Wiley (1968)

    Google Scholar 

  9. Baniassadi, M., Mortazavi, B., Hamedani, H.A., Garmestani, H., Ahzi, S., Fathi-Torbaghan, M., Ruch, D., Khaleel, M.: Three-dimensional reconstruction and homogenization of heterogeneous materials using statistical correlation functions and {FEM}. Computational Materials Science 51, 372–379 (2012)

    Article  Google Scholar 

  10. Brown, W.: Solid micture permettivities. Journal of Chemical Physics 23, 1514–1517 (1955)

    Article  Google Scholar 

  11. Balzani, D., Schröder, J., Brands, D.: FE2-simulation of microheterogeneous steels based on statistically similar RVEs. In: Proceedings of the IUTAM Symposium on Variational Concepts with Application to Mechanics of Materials, Bochum, Germany, September 22-26 (2009)

    Google Scholar 

  12. Balzani, D., Scheunemann, L., Brands, D., Schröder, J.: Construction of Two- and Three-Dimensional Statistically Similar RVEs for Coupled Micro-Macro Simulations. Computational Mechanics (2014)

    Google Scholar 

  13. Drugan, W., Willis, J.: A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44, 497–524 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eisenlohr, P., Diehl, M., Lebensohn, R., Roters, F.: A spectral method solution to crystal elasto-viscoplasticity at finite strains. International Journal of Plasticity 46, 37–53 (2013)

    Article  Google Scholar 

  15. Exner, H., Hougardy, H.: Einführung in die quantitative Gefügeanalyse. Deutsche Gesellschaft für Metallkunde (1986)

    Google Scholar 

  16. Feyel, F., Chaboche, J.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering 183, 309–330 (2000)

    Article  MATH  Google Scholar 

  17. Feyel, F.: Multiscale FE2 elastoviscoplastic analysis of composite structures. Computational Materials Science 16, 344–354 (1999)

    Article  Google Scholar 

  18. Fish, J., Shek, K.: Finite deformation plasticity for composite structures: computational models and adaptive strategies. Computational Mechanics in Applied Mechanics and Engineering 172, 145–174 (1999)

    Article  MATH  Google Scholar 

  19. Geers, M., Kouznetsova, V., Brekelmans, W.: Multi-scale first order and second order computational homogenization of microstructures towards continua. International Journal of Multiscale Computational Engineering 1, 371–386 (2003)

    Article  Google Scholar 

  20. Golanski, D., Terada, K., Kikuchi, N.: Macro and micro scale modeling of thermal residual stresses in metal matrix composite surface layers by the homogenization method. Computational Mechanics 19, 188–201 (1997)

    Article  Google Scholar 

  21. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963)

    Article  MATH  Google Scholar 

  22. Kouznetsova, V., Brekelmans, W., Baaijens, F.: An approach to micro-macro modeling of heterogeneous materials. Computational Mechanics 27, 37–48 (2001)

    Article  MATH  Google Scholar 

  23. Kumar, H., Briant, C.L., Curtin, W.A.: Using microstructure reconstruction to model mechanical behavior in complex microstructures. Mechanics of Materials 38, 818–832 (2006)

    Article  Google Scholar 

  24. Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. International Journal of Solids and Structures (2003)

    Google Scholar 

  25. Klinkel, S.: Theorie und Numerik eines Volumen-Schalen-Elementes bei finiten elastischen und plastischen Verzerrungen. PhD thesis, Institut für Baustatik, Universität Karlsruhe (2000)

    Google Scholar 

  26. Kapfer, S.C., Mickel, W., Schaller, F.M., Spanner, M., Goll, C., Nogawa, T., Ito, N., Mecke, K., Schröder-Turk, G.E.: Local Anisotropy of Fluids using Minkowski Tensors. Journal of Statistical Mechanics: Theory and Experiments (2010)

    Google Scholar 

  27. Kröner, E.: Allgemeine Kontinuumstheorie der Versetzung und Eigenspannung. Archive for Rational Mechanics and Analysis 4, 273–334 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lee, E.: Elastic-plastic deformation at finite strains. Journal of Applied Mechanics 36, 1–6 (1969)

    Article  MATH  Google Scholar 

  29. Lee, S., Lebensohn, R., Rollett, A.: Modeling the viscoplastic micromechanical response of two-phase materials using fast fourier transforms. International Journal of Plasticity 27, 707–727 (2011)

    Article  MATH  Google Scholar 

  30. Lu, B., Torquato, S.: Lineal-path function for random heterogeneous materials. Physical Reviews A 45, 922–929 (1992)

    Article  Google Scholar 

  31. McKerns, M., Hung, P., Aivazis, M.: mystic: a simple model-independent inversion framework (2009)

    Google Scholar 

  32. Mantz, H., Jacobs, K., Mecke, K.: Utilizing Minkowski functionals for image analysis: a marching square algorithm. Journal of Statistical Mechanics: Theory and Experiment 12(15), 2–28 (2008)

    Google Scholar 

  33. Miehe, C., Stein, E.: A canonical model of multiplicative elasto-plasticity formulation and aspects of the numerical implementation. European Journal of Mechanics 11, 25–43 (1992)

    Google Scholar 

  34. Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computational Mechanics in Applied Mechanics and Engineering 157, 69–94 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mecke, K., Stoyan, D.: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation. Springer Lecture Notes in Physics, vol. 554. Springer, Heidelberg (2000)

    Book  Google Scholar 

  36. Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 171, 387–418 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. McKerns, M.M., Strand, L., Sullivan, T., Fang, A., Aivazis, M.A.G.: Building a framework for predictive science. In: Proceedings of the 10th Python in Science Conference (2011)

    Google Scholar 

  38. Ohser, J., Mücklich, F.: Statistical analysis of microstructures in materials science. J. Wiley & Sons (2000)

    Google Scholar 

  39. Pelissou, C., Baccou, J., Monerie, Y., Perales, F.: Determination of the size of the representative volume element for random quasi-brittle composites. International Journal of Solids and Structures (2009)

    Google Scholar 

  40. Povirk, G.L.: Incorporation of microstructural information into models of two-phase materials. Acta Metall. Mater. 43, 3199–3206 (1995)

    Article  Google Scholar 

  41. Rauch, L., Kuziak, R., Pietrzyk, M.: From High Accuracy to High Efficiency in Simulations of Processing of Dual-Phase Steels. Metallurgical and Materials Transactions B (2014)

    Google Scholar 

  42. Ramazani, A., Mukherjee, K., Prahl, U., Bleck, W.: Modelling the effect of micromechanical banding on the flow curve behaviour of dual-phase (dp) steels. Computational Materials Science (2012)

    Google Scholar 

  43. Rauch, L., Pernach, M., Bzowski, K., Pietrzyk, M.: On application of shape coefficients to creation of the statistically similar representative element of DP steels. Computer Methods in Materials Science 11, 531–541 (2011)

    Google Scholar 

  44. Schröder, J., Balzani, D., Brands, D.: Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions. Archive of Applied Mechanics 81, 975–997 (2011)

    Article  MATH  Google Scholar 

  45. Scheunemann, L., Balzani, D., Brands, D., Schröder, J.: Statistically Similar Rve Construction based on 3D dual-phase steel microstructures. In: Research and Application in Structural Engineering, Mechanics and Computation (2013)

    Google Scholar 

  46. Scheunemann, L., Balzani, D., Brands, D., Schröder, J.: Design of 3d statistically similar representative volume elements based on minkowski functionals. Submitted to Mechanics of Materials (2014)

    Google Scholar 

  47. Smit, R., Brekelsmans, W., Meijer, H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Computer Methods in Applied Mechanics and Engineering 155, 181–192 (1998)

    Article  MATH  Google Scholar 

  48. Schröder, J.: A numerical two-scale homogenization scheme: the FE2 method. In: Schröder, K.H.J. (ed.) Plasticity and Beyond - Microstructures, Crystal-Plasticity and Phase Transitions. CISM Lecture notes, vol. 550. Springer (2014)

    Google Scholar 

  49. Sporer, S., Goll, C., Mecke, K.: Motion by stopping: rectifying Brownian motion of nonspherical particles. Phys. Rev. E 78, 11917 (2008)

    Article  Google Scholar 

  50. Simo, J.: Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping scheme of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  51. Schröder-Turk, G.E., Kapfer, S., Breidenbach, B., Beisbart, C., Mecke, K.: Tensorial minkowski functionals and anisotropy measures for planar patterns. Journal of Microscopy 238, 57–74 (2010)

    Article  MathSciNet  Google Scholar 

  52. Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Klatt, M.A., Schaller, F.M., Hoffmann, M.J.F., Kleppmann, N., Armstrong, P., Inayat, A., Hug, D., Reichelsdorfer, M., Peukert, W., Schwieger, W., Mecke, K.: Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures. Advanced Materials 23, 2535–2553 (2011)

    Article  Google Scholar 

  53. Schröder-Turk, G.E., Mickel, W., Kapfer, S.C., Schaller, F.M., Breidenbach, B., Hug, D., Mecke, K.: Minkowski Tensors of Anisotropic Spatial Structure. New Journal of Physics 15 (2013)

    Google Scholar 

  54. Torquato, S.: Random heterogeneous materials. Microstructure and macroscopic properties. Springer (2002)

    Google Scholar 

  55. Temizer, I., Wriggers, P.: On the computation of the macroscopic tangent for multiscale volumetric homogenization problems. Computational Mechanics in Applied Mechanics and Engineering 198, 495–510 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  56. Voce, E.: A practical strain hardening function. Metallurgica 51, 219–226 (1955)

    Google Scholar 

  57. Weber, G., Anand, L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoelastic solids. Computer Methods in Applied Mechanics and Engineering 79, 173–202 (1990)

    Article  MATH  Google Scholar 

  58. Yeong, C.L.Y., Torquato, S.: Reconstructing random media. Physical Review E 57, 495–506 (1998)

    Article  MathSciNet  Google Scholar 

  59. Zeman, J.: Analysis of composite materials with random microstructure. PhD thesis, University of Prague (2003)

    Google Scholar 

  60. Zaafarani, N., Raabe, D., Singh, R., Roters, F., Zaefferer, S.: Three -dimensional investigation of the texture and microstructure below a nanoindent in a cu single crystal using 3D EBSD and crystal plasticity finite element simulations. Acta Materialia 54, 1863–1876 (2006)

    Article  Google Scholar 

  61. Zodhi, T., Wriggers, P.: Introduction to computational micromechanics. Springer (2005)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mechanik, Abteilung Bauwissenschaften, Univeristät Duisburg-Essen, 45117, Essen, Germany

    Lisa Scheunemann, Dominik Brands & Jörg Schröder

  2. Institut für Mechanik und Flächentragwerke, TU Dresden, 01062, Dresden, Germany

    Daniel Balzani

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  1. Lisa Scheunemann
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  2. Daniel Balzani
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  3. Dominik Brands
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  4. Jörg Schröder
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Correspondence to Lisa Scheunemann .

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Editors and Affiliations

  1. Inst. für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    Sergio Conti

  2. Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany

    Klaus Hackl

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Scheunemann, L., Balzani, D., Brands, D., Schröder, J. (2015). Construction of Statistically Similar RVEs. In: Conti, S., Hackl, K. (eds) Analysis and Computation of Microstructure in Finite Plasticity. Lecture Notes in Applied and Computational Mechanics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-18242-1_9

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  • DOI: https://doi.org/10.1007/978-3-319-18242-1_9

  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-18242-1

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