×

zbMATH — the first resource for mathematics

Reduced basis hybrid computational homogenization based on a mixed incremental formulation. (English) Zbl 1286.74081
Summary: A new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed. Key features of the Nonuniform Transformation Field Analysis (NTFA) are taken as a point of departure. A short-coming of the NTFA method is its limitation to simple constitutive models on the microscale. A second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension. Both issues are related to the simple evolution law for the mode activity coefficients, which are the new macroscopic internal variables of the homogenized material. In the present contribution a generalization of the NTFA is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation. The derivation is based on purely micro-mechanical considerations. The modification allows for arbitrary Generalized Standard Materials on the microscopic scale including, e.g., crystal visco-plasticity. Additionally, the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field. Numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi-scale method.

MSC:
74Q05 Homogenization in equilibrium problems of solid mechanics
74S25 Spectral and related methods applied to problems in solid mechanics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Michel, J.; Suquet, P., Nonuniform transformation field analysis, International Journal of Solids and Structures, 40, 6937-6955, (2003) · Zbl 1057.74031
[2] Fritzen, F.; Böhlke, T., Three-dimensional finite element implementation of the nonuniform transformation field analysis, International Journal for Numerical Methods in Engineering, 84, 7, 803-829, (2010) · Zbl 1202.74170
[3] Halphen, N.; Nguyen, Q., Sur LES matériaux standards generalisés, Journal de Mécanique, 14, 508-520, (1975) · Zbl 0308.73017
[4] S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, 1999. · Zbl 0924.73006
[5] Torquato, S., Random heterogeneous materials, (2006), Springer, (corr. 2. print.)
[6] Voigt, W., Lehrbuch der kristallphysik, (1910), Teubner Berlin · JFM 54.0929.03
[7] Reuss, A., Berechnung der fliegrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle, Zeitschrift für Angewandte Mathematik und Mechanik, 9, 1, 49-58, (1929) · JFM 55.1110.02
[8] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechanics and Physics of Solids, 11, 127-140, (1963) · Zbl 0108.36902
[9] Willis, J., Bounds and self-consistent estimates for the overall properties of anisotropic composites, Journal of the Mechanics and Physics of Solids, 25, 185-202, (1977) · Zbl 0363.73014
[10] Ponte-Castañeda, P., Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I - theory, Journal of the Mechanics and Physics of Solids, 50, 4, 737-757, (2002) · Zbl 1116.74412
[11] Lahellec, N.; Suquet, P., Effective behavior of linear viscoelastic composites: a time-integration approach, International Journal of Solids and Structures, 44, 2, 507-529, (2007) · Zbl 1123.74041
[12] Danas, K.; Aravas, N., Numerical modeling of elasto-plastic porous materials with void shape effects at finite deformations, Composites Part B: Engineering, 43, 6, 2544-2559, (2012)
[13] Feyel, F., Multiscale FE2 elastoviscoplastic analysis of composite structures, Computational Materials Science, 16, 1-4, 344-354, (1999)
[14] Dvorak, G.; Benveniste, Y., On transformation strains and uniform fields in multiphase elastic media, Proceedings of the Royal Society of London A, 437, 437, 291-310, (1992) · Zbl 0748.73003
[15] Kanouté, P.; Boso, D.; Chaboche, J.; Schrefler, B., Multiscale methods for composites: a review, Archives of Computational Methods in Engineering, 16, 31-75, (2009) · Zbl 1170.74304
[16] Michel, J.; Suquet, P., Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis, Computer Methods in Applied Mechanics and Engineering, 193, 5477-5502, (2004) · Zbl 1112.74471
[17] Roussette, S.; Michel, J.; Suquet, P., Nonuniform transformation field analysis of elastic-viscoplastic composites, Composites Science and Technology, 69, 22-27, (2009)
[18] Fritzen, F.; Böhlke, T., Nonuniform transformation field analysis of materials with morphological anisotropy, Composites Science and Technology, 71, 433-442, (2011)
[19] Michel, J. C.; Suquet, P., Nonuniform transformation field analysis: a reduced model for multiscale nonlinear problems in solid mechanics, (Galvanetto, U.; Aliabadi, F., Multiscale Modelling in Solid Mechanics - Computational Approaches, (2009), Imperial College Press London), 159-206
[20] F. Fritzen, Microstructural modeling and computational homogenization of the physically linear and nonlinear constitutive behavior of micro-heterogeneous materials, Ph.D. Thesis, Karlsruhe, 2011.
[21] Sepe, V.; Marfia, S.; Sacco, E., A nonuniform TFA homogenization technique based on piecewise interpolation functions of the inelastic field, International Journal of Solids and Structures, 50, 5, 725-742, (2013)
[22] Fritzen, F.; Böhlke, T., Reduced basis homogenization of viscoelastic composites, Composites Science and Technology, 76, 84-91, (2013)
[23] Fritzen, F.; Forest, S.; Böhlke, T.; Kondo, D.; Kanit, T., Computational homogenization of elasto-plastic porous metals, International Journal of Plasticity, 29, 102-119, (2012)
[24] P. Armstrong, C. Frederick, A mathematical representation of the multiaxial bauschinger effect, Tech. Rep. RD/B/N731, Berkeley Nuclear Laboratories, Berkeley, U.K., 1966.
[25] Ortiz, L.; Stainier, L., The variational formulation of viscoplastic constitutive updates, Computer Methods in Applied Mechanics and Engineering, 171, 419-444, (1999) · Zbl 0938.74016
[26] Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, Journal for Numerical Methods in Engineering, 55, 1285-1322, (2002) · Zbl 1027.74056
[27] Miehe, C., A multi-field incremental variational framework for gradient-extended standard dissipative solids, Journal of the Mechanics and Physics of Solids, 59, 4, 898-923, (2011) · Zbl 1270.74022
[28] Dvorak, G.; Bahei-El-Din, Y.; Wafa, A., Implementation of the transformation field analysis, Computational Mechanics, 14, 14, 201-228, (1994) · Zbl 0835.73038
[29] Dvorak, G.; Bahei-El-Din, Y.; Wafa, A., The modeling of inelastic composite materials with the transformation field analysis, Modelling and Simulation in Material Science and Engineering, 2, 2, 571-586, (1994) · Zbl 0835.73038
[30] Ryckelynck, D.; Benziane, D., Multi-level a priori hyper reduction of mechanical models involving internal variables, Computer Methods in Applied Mechanics and Engineering, 199, 1134-1142, (2010) · Zbl 1227.74093
[31] Ladevèze, P.; Passieux, J. C.; Néron, D., The Latin multiscale computational method and the proper generalized decomposition, Computer Methods in Applied Mechanics and Engineering, 199, 21-22, 1287-1296, (2010), Multiscale Models and Mathematical Aspects in Solid and Fluid Mechanics · Zbl 1227.74111
[32] Suquet, P., Elements of homogenization for inelastic solid mechanics, (Snachez-Palencia, E.; Zaoui, A., Homogenization Techniques for Composite Media, Lecture Notes in Physics, vol. 272, (1985), Springer Verlag)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.