zbMATH — the first resource for mathematics

Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. (English) Zbl 1112.74469
Summary: The detailed implementation and computational aspects of a novel second-order computational homogenization procedure, which is suitable for a multi-scale modelling of macroscopic localization and size effects. The second-order scheme is an extension of the classical (first-order) computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro-micro scale transition. From the microstructural analysis the macroscopic stress and higher-order stress tensors are obtained, thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum. The higher-order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix. For the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed. As an example, the second-order computational homogenization approach is applied for the multi-scale analysis of simple shear of a constrained heterogeneous strip, where a pronounced boundary size effect appears.

74Q15 Effective constitutive equations in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
PDF BibTeX Cite
Full Text: DOI
[1] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[2] Fish, J.; Shek, K.; Pandheeradi, M.; Shephard, M.S., Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. methods appl. mech. engrg., 148, 53-73, (1997) · Zbl 0924.73145
[3] Suquet, P.M., Local and global aspects in the mathematical theory of plasticity, (), 279-310
[4] Guedes, J.M.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. methods appl. mech. engrg., 83, 143-198, (1990) · Zbl 0737.73008
[5] K. Terada, N. Kikuchi, Nonlinear homogenization method for practical applications, in: S. Ghosh, M. Ostoja-Starzewski (Eds.), Computational Methods in Micromechanics, vol. AMD-212/MD-62, ASME, 1995, pp. 1-16
[6] Ghosh, S.; Lee, K.; Moorthy, S., Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenisation and Voronoi cell finite element model, Comput. methods appl. mech. engrg., 132, 63-116, (1996) · Zbl 0892.73061
[7] Smit, R.J.M.; Brekelmans, W.A.M.; Meijer, H.E.H., Prediction of the mechanical behaviour of non-linear heterogeneous systems by multi-level finite element modeling, Comput. methods appl. mech. engrg., 155, 181-192, (1998) · Zbl 0967.74069
[8] Miehe, C.; Schröder, J.; Schotte, J., Computational homogenization analysis in finite plasticity. simulation of texture development in polycrystalline materials, Comput. methods appl. mech. engrg., 171, 387-418, (1999) · Zbl 0982.74068
[9] 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
[10] Michel, J.C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. methods appl. mech. engrg., 172, 109-143, (1999) · Zbl 0964.74054
[11] Feyel, F.; Chaboche, J.-L., FE^{2} multiscale approach for modelling the elastoviscoplastic behaviour of long fiber sic/ti composite materials, Comput. methods appl. mech. engrg., 183, 309-330, (2000) · Zbl 0993.74062
[12] Terada, K.; Kikuchi, N., A class of general algorithms for multi-scale analysis of heterogeneous media, Comput. methods appl. mech. engrg., 190, 5427-5464, (2001) · Zbl 1001.74095
[13] Ghosh, S.; Lee, K.; Raghavan, P., A multi-level computational model for multi-scale damage analysis in composite and porous materials, Int. J. solids struct., 38, 2335-2385, (2001) · Zbl 1015.74058
[14] Kouznetsova, V.; Brekelmans, W.A.M.; Baaijens, F.P.T., An approach to micro-macro modeling of heterogeneous materials, Comput. mech., 27, 37-48, (2001) · Zbl 1005.74018
[15] Fleck, N.A.; Hutchinson, J.W., Strain gradient plasticity, Adv. appl. mech., 33, 295-361, (1997) · Zbl 0894.73031
[16] Aifantis, E.C., On the microstructural origin of certain inelastic models, Trans. ASME J. engrg. mater. tech., 106, 326-330, (1984)
[17] de Borst, R.; Mühlhaus, H.-B., Gradient-dependent plasticity: formulation and algorithmic aspects, Int. J. numer. meth. engrg., 35, 521-539, (1992) · Zbl 0768.73019
[18] Fleck, N.A.; Hutchinson, J.W., A reformulation of strain gradient plasticity, J. mech. phys. solids, 49, 2245-2271, (2001) · Zbl 1033.74006
[19] Pijaudier-Cabot, G.; Bažant, Z.P., Nonlocal damage theory, J. eng. mech., 113, 1512-1533, (1987)
[20] Peerlings, R.H.J.; de Borst, R.; Brekelmans, W.A.M.; de Vree, J.H.P., Gradient-enhanced damage for quasi-brittle materials, Int. J. numer. meth. engrg., 39, 3391-3403, (1996) · Zbl 0882.73057
[21] Engelen, R.A.B.; Geers, M.G.D.; Baaijens, F.P.T., Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour, Int. J. plasticity, 19, 403-433, (2003) · Zbl 1090.74519
[22] Geers, M.G.D.; Ubachs, R.L.J.M.; Engelen, R.A.B., Strongly nonlocal gradient-enhanced finite strain elastoplasticity, Int. J. numer. meth. engrg., 56, 2039-2068, (2003) · Zbl 1038.74527
[23] Smyshlyaev, V.P.; Fleck, N.A., Bounds and estimates for linear composites with strain gradient effects, J. mech. phys. solids, 42, 12, 1851-1882, (1994) · Zbl 0819.73046
[24] Drugan, W.J.; Willis, J.R., A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites, J. mech. phys. solids, 44, 4, 497-524, (1996) · Zbl 1054.74704
[25] Triantafyllidis, N.; Bardenhagen, S., The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models, J. mech. phys. solids, 44, 11, 1891-1928, (1996) · Zbl 1054.74585
[26] Zhu, H.T.; Zbib, H.M.; Aifantis, E.C., Strain gradients and continuum modeling of size effect in metal matrix composites, Acta mech., 121, 165-176, (1997) · Zbl 0885.73044
[27] van der Sluis, O.; Vosbeek, P.H.J.; Schreurs, P.J.G.; Meijer, H.E.H., Homogenization of heterogeneous polymers, Int. J. solids struct., 36, 3193-3214, (1999) · Zbl 0976.74055
[28] Ostoja-Starzewski, M.; Boccara, S.D.; Jasiuk, I., Couple-stress moduli and characteristic length of a two-phase composite, Mech. res. comm., 26, 4, 387-396, (1999) · Zbl 0954.74049
[29] Smyshlyaev, V.P.; Cherednichenko, K.D., On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media, J. mech. phys. solids, 48, 1325-1357, (2000) · Zbl 0984.74065
[30] Peerlings, R.H.J.; Fleck, N.A., Numerical analysis of strain gradient effects in periodic media, J. phys. IV, 11, 153-160, (2001)
[31] Forest, S.; Pradel, F.; Sab, K., Asymptotic analysis of heterogeneous Cosserat media, Int. J. solids struct., 38, 4585-4608, (2001) · Zbl 1033.74038
[32] 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)
[33] Kouznetsova, V.; Geers, M.G.D.; Brekelmans, W.A.M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, Int. J. numer. meth. engrg., 54, 1235-1260, (2002) · Zbl 1058.74070
[34] V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2002
[35] Hill, R., Elastic properties of reinforced solids: some theoretical principles, J. mech. phys. solids, 11, 357-372, (1963) · Zbl 0114.15804
[36] Nemat-Nasser, S., Averaging theorems in finite deformation plasticity, Mech. mater., 31, 493-523, (1999)
[37] Cook, R.D.; Malkus, D.S.; Plesha, M.E., Concepts and applications of finite element analysis, (1989), Wiley Chichester · Zbl 0696.73039
[38] Miehe, C., Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity, Comput. methods appl. mech. engrg., 134, 223-240, (1996) · Zbl 0892.73012
[39] Miehe, C., Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy, Comput. methods appl. mech. engrg., 192, 559-591, (2003) · Zbl 1091.74530
[40] Toupin, R.A., Elastic materials with couple-stress, Arch. ration. mech. anal., 11, 385-414, (1962) · Zbl 0112.16805
[41] Toupin, R.A., Theories of elasticity with couple-stress, Arch. ration. mech. anal., 17, 85-112, (1964) · Zbl 0131.22001
[42] Mindlin, R.D., Micro-structure in linear elasticity, Arch. ration. mech. anal., 16, 51-78, (1964) · Zbl 0119.40302
[43] Mindlin, R.D.; Eshel, N.N., On first strain-gradient theories in linear elasticity, Int. J. solids struct., 4, 109-124, (1968) · Zbl 0166.20601
[44] Gao, H.; Huang, Y.; Nix, W.D.; Hutchinson, J.W., Mechanism-based strain gradient plasticity–I. theory, J. mech. phys. solids, 47, 1239-1263, (1999) · Zbl 0982.74013
[45] Xia, Z.C.; Hutchinson, J.W., Crack tip fields in strain gradient plasticity, J. mech. phys. solids, 44, 10, 1621-1648, (1996)
[46] Begley, M.R.; Hutchinson, J.W., The mechanics of size-dependent indentation, J. mech. phys. solids, 46, 10, 2049-2068, (1998) · Zbl 0967.74043
[47] Zervos, A.; Papanastasiou, P.; Vardoulakis, I., A finite element displacement formulation for gradient elastoplasticity, Int. J. numer. meth. engrg., 50, 1369-1388, (2001) · Zbl 1047.74073
[48] Herrmann, L.R., Mixed finite elements for couple-stress analysis, (), 1-17
[49] Shu, J.Y.; King, W.E.; Fleck, N.A., Finite elements for materials with strain gradient effects, Int. J. numer. meth. engrg., 44, 373-391, (1999) · Zbl 0943.74072
[50] Amanatidou, E.; Aravas, A., Mixed finite element formulations of strain-gradient elasticity problems, Comput. methods appl. mech. engrg., 191, 1723-1751, (2002) · Zbl 1098.74678
[51] Matsushima, T.; Chambon, R.; Caillerie, D., Large strain finite element analysis of a local second gradient model: application to localization, Int. J. numer. meth. engrg., 54, 499-521, (2002) · Zbl 1098.74705
[52] Shu, J.Y.; Barlow, C.Y., Strain gradient effects on microscopic strain field in a metal matrix composite, Int. J. plasticity, 16, 563-591, (2000) · Zbl 1010.74005
[53] Shu, J.Y.; Fleck, N.A.; van der Giessen, E.; Needleman, N., Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity, J. mech. phys. solids, 49, 1361-1395, (2001) · Zbl 1015.74004
[54] Bittencourt, E.; Needleman, A.; Gurtin, M.E.; van der Giessen, E., A comparison of nonlocal continuum and discrete dislocation plasticity predictions, J. mech. phys. solids, 51, 281-310, (2003) · Zbl 1100.74530
[55] M.G.D. Geers, V.G. Kouznetsova, W.A.M. Brekelmans, Multi-scale first-order and second-order computational homogenization of microstructures towards continua, Int. J. Multiscale Comput. Engrg. in press · Zbl 1112.74469
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.