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Multiple scale eigendeformation-based reduced order homogenization. (English) Zbl 1227.74051
Summary: Multiple scale eigendeformation-based reduced order homogenization method, which provides considerable computational cost saving in comparison to the direct homogenization method, has been developed, verified against the direct homogenization method, and validated against experimental data. The salient feature of the method is in the formulation of the unit cell problem in terms of residual-free stresses, and therefore eliminating the need for costly equilibrium calculations required by the direct homogenization method. This is accomplished by introducing discretized eigendeformation fields at different scales, which are collectively referred to as eigenstrains and eigenseparations, and then precomputing a sequence of elasticity solutions of the unit cell problem subjected to the eigendeformation fields prior to nonlinear analysis.

##### MSC:
 74Q05 Homogenization in equilibrium problems of solid mechanics
##### Keywords:
homogenization model-reduction; eigendeformation
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##### References:
 [1] Hill, R., A theory of the yielding and plastic flow of anisotropic metals, Proc. roy. soc. lond., A193, 281-297, (1948) · Zbl 0032.08805 [2] Babuska, I., Homogenization and application. mathematical and computational problems, () · Zbl 0346.65064 [3] Benssousan, A.; Lions, J.L.; Papanicoulau, G., Asymptotic analysis for periodic structures, (1978), North-Holland [4] Sanchez-Palencia, E., Non-homogeneous media and vibration theory, Lecture notes in physics, vol. 127, (1980), Springer-Verlag Berlin [5] Hill, R., Elastic properties of reinforced solids: some theoretical principles, Journal of the mechanics and physics of solids, 11, 357-372, (1963) · Zbl 0114.15804 [6] Terada, K.; Kikuchi, N., Nonlinear homogenization method for practical applications, (), 1-16 [7] Terada, K.; Kikuchi, N., A class of general algorithms for multi-scale analysis of heterogeneous media, Comput. meth. appl. mech. engrg., 190, 5427-5464, (2001) · Zbl 1001.74095 [8] Matsui, K.; Terada, K.; Yuge, K., Two-scale finite element analysis of heterogeneous solids with periodic microstructures, Comput. struct., 82, 7-8, 593-606, (2004) [9] Smit, R.J.M.; Brekelmans, W.A.M.; Meijer, H.E.H., Prediction of the mechanical behavior of nonlinear heterogeneous systems by multilevel finite element modeling, Comput. meth. appl. mech. engrg., 155, 181-192, (1998) · Zbl 0967.74069 [10] Miehe, C.; Koch, A., Computational micro-to-macro transition of discretized microstructures undergoing small strain, Arch. appl. mech., 72, 300-317, (2002) · Zbl 1032.74010 [11] Kouznetsova, V.; Brekelmans, W.-A.; Baaijens, F.P.-T., An approach to micro – macro modeling of heterogeneous materials, Comput. mech., 27, 37-48, (2001) · Zbl 1005.74018 [12] Feyel, F.; Chaboche, J.-L., FE2 multiscale approach for modeling the elastoviscoplastic behavior of long fiber sic/ti composite materials, Comput. meth. appl. mech. engrg., 183, 309-330, (2000) · Zbl 0993.74062 [13] Ghosh, S.; Lee, K.; Moorthy, S., Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method, Int. J. solids struct., 32, 27-62, (1995) · Zbl 0865.73060 [14] Ghosh, S.; Lee, K.; Moorthy, S., Two scale analysis of heterogeneous elasticplastic materials with asymptotic homogenization and Voronoi cell finite element model, Comput. meth. appl. mech. engrg., 132, 63-116, (1996) · Zbl 0892.73061 [15] Michel, J.-C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. meth. appl. mech. engrg., 172, 109-143, (1999) · Zbl 0964.74054 [16] Geers, M.G.D.; Kouznetsova, V.; Brekelmans, W.A.M., Gradient-enhanced computational homogenization for the micro – macro scale transition, J. phys. IV, 11, 145-152, (2001) [17] McVeigh, C.; Vernerey, F.; Liu, W.K.; Brinson, L.C., Multiresolution analysis for material design, Comput. meth. appl. mech. engrg., 195, 5053-5076, (2006) · Zbl 1118.74040 [18] Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, Int. J. numer. meth. engrg., 55, 1285-1322, (2002) · Zbl 1027.74056 [19] Zohdi, T.I.; Oden, J.T.; Rodin, G.J., Hierarchical modeling of heterogeneous bodies, Comput. meth. appl. mech. engrg., 138, 273-298, (1996) · Zbl 0921.73080 [20] Feyel, F.; Chaboche, J.L., Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials, Comput. meth. appl. mech. engrg., 183, 309-330, (2000) · Zbl 0993.74062 [21] Engquist, E.W.B., Heterogeneous multiscale method: a general methodology for multiscale modeling, Phys. rev. B, 67, 9, 1-4, (2003) [22] Hou, T.Y.; Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 1, 169-189, (1997) · Zbl 0880.73065 [23] Loehnert, S.; Belytschko, T., A multiscale projection method for macro/microcrack simulations, Int. J. numer. meth. engrg., 71, 12, 1466-1482, (2007) · Zbl 1194.74436 [24] Belytschko, T.; Loehnert, S.; Song, J., Multiscale aggregating discontinuities: a method for circumventing loss of material stability, Int. J. numer. meth. engrg., 73, 6, 869-894, (2008) · Zbl 1195.74008 [25] Fish, J.; Shek, K.; Pandheeradi, M.; Shephard, M.S., Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. meth. appl. mech. engrg., 148, 53-73, (1997) · Zbl 0924.73145 [26] Fish, J.; Shek, K.L., Finite deformation plasticity of composite structures: computational models and adaptive strategies, Comput. meth. appl. mech. engrg., 172, 145-174, (1999) · Zbl 0956.74009 [27] Fish, J.; Yu, Q., Multiscale damage modeling for composite materials: theory and computational framework, Int. J. numer. meth. engrg., 52, 1-2, 161-192, (2001) [28] Clayton, J.D.; Chung, P.W., An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization, J. mech. phys. solids, 54, 1604-1639, (2006) · Zbl 1120.74389 [29] Fish, J.; Shek, K.L., Finite deformation plasticity of composite structures: computational models and adaptive strategies, Comput. meth. appl. mech. engrg., 172, 145-174, (1999) · Zbl 0956.74009 [30] Yuan, Z.; Fish, J., Towards realization of computational homogenization in practice, Int. J. numer. meth. engrg., 73, 3, 361-380, (2008) · Zbl 1159.74044 [31] Oskay, C.; Fish, J., Eigendeformation-based reduced order homogenization, Comput. meth. appl. mech. engrg., 196, 1216-1243, (2007) · Zbl 1173.74380 [32] Levin, V.M., Determination of composite material elastic and thermoelastic constants, Izv. AN SSSR, mekhanika tverdogo tela, 11, 6, 137-145, (1976) [33] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta metal, 21, 571-574, (1973) [34] Z. Yuan, Multiscale Design System, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY, January 2008. [35] S.J. Beard, Energy Absorption of Braided Composite Tubes, Thesis, Stanford University, 2001.
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