×

zbMATH — the first resource for mathematics

Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. (English) Zbl 1228.74067
Summary: The homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones. This is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown. At the same time, the prevailing numerical methods, such as concurrent methods, require extensive computational efforts. A simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous materials at small strains. The proposed numerical approach comprises three steps. At the first step, a representative volume element (RVE) for a given nonlinear heterogeneous material is defined, and a loading space consisting of all the boundary conditions to be imposed on the RVE is discretized into a sufficiently large number of points called nodes. At the second step, the boundary condition corresponding to each node is prescribed on the surface of the RVE, and the resulting nonlinear boundary value problem, is solved by the finite element method (FEM) so as to determine the effective response of the heterogeneous material to the loading associated to each node of the loading space. At the third step, the nodal effective responses are interpolated via appropriate interpolation functions, so that the effective strain-energy, stress-strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way. This leads to a non-concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials. The first version of the proposed approach uses multidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank-one tensors to interpolate effective nodal responses and avoid high-rank data. These two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power-law model are involved. The numerical results given by our approach are compared with available analytical estimates, exact results and full FEM or concurrent multilevel FEM solutions.

MSC:
74Q05 Homogenization in equilibrium problems of solid mechanics
74Q15 Effective constitutive equations in solid mechanics
74E30 Composite and mixture properties
74B20 Nonlinear elasticity
Software:
TensorToolbox
PDF BibTeX Cite
Full Text: DOI
References:
[1] Hill, R., Continuum micromechanics of elastoplastic polycrystals, J. mech. phys. solids, 13, 89-101, (1965) · Zbl 0127.15302
[2] Willis, J.R., The overall response of composite materials, ASME J. appl. mech., 50, 1202-1209, (1983) · Zbl 0539.73003
[3] Dvorak, G.J., Transformation field analysis of inelastic composite materials, Proc. roy. soc. lond. A, 437, 311-327, (1992) · Zbl 0748.73007
[4] Qiu, Y.P.; Weng, G.J., A theory of plasticity for porous materials and particle-reinforced composites, Int. J. plast., 59, 261-268, (1992) · Zbl 0825.73037
[5] Ponte Castañeda, P., The effective mechanical properties of nonlinear isotropic composites, J. mech. phys. solids, 39, 45-71, (1991) · Zbl 0734.73052
[6] Hu, G.K., A method of plasticity for general aligned spheroidal void or fiber-reinforced composites, Int. J. plast., 12, 439-449, (1996) · Zbl 0884.73035
[7] Milton, G.W.; Serkov, S.K., Bounding the current in nonlinear conducting composites, J. mech. phys. solids, 48, 1295-1324, (2000) · Zbl 0991.78019
[8] Nemat-Nasser, S., Micromechanics: overall properties of heterogeneous solids, (1993), Elsevier Amsterdam · Zbl 0924.73006
[9] Torquato, S., Random heterogeneous materials: microstructure and macroscopic properties, (2001), Springer · Zbl 0988.74001
[10] Milton, G.W., Theory of composites, (2002), Cambridge University Press · Zbl 0631.73011
[11] Chu, T.; Hashin, Z., Plastic behavior of composites and porous media under isotropic stress, Int. J. engrg. sci., 9, 971-994, (1971) · Zbl 0228.73043
[12] He, Q.-C., Uniform strains field and microstructure-independent relations in nonlinear elastic fibrous composites, J. mech. phys. solids, 47, 8, 1781-1793, (1999) · Zbl 0959.74017
[13] He, Q.-C.; Bary, B., Exact relations for the effective properties of nonlinearly elastic inhomogeneous materials, Int. J. multiscale comput. engrg., 2, 69-83, (2004)
[14] He, Q.-C.; Le Quang, H.; Feng, Z.-Q., Exact results for the homogenization of elastic fiber-reinforced solids at finite strains, J. elast., 83, 153-177, (2006) · Zbl 1103.74049
[15] Le Quang, H.; He, Q.-C., Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading, Int. J. plast., 24, 343-370, (2008) · Zbl 1130.74039
[16] Smit, R.; Brekelmans, W.; Meijer, H., Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Comput. meth. appl. mech. engrg., 155, 181-192, (1998) · Zbl 0967.74069
[17] Feyel, F., Multiscale \(\text{FE}^2\) elastoviscoplastic analysis of composite structure, Comput. mater. sci., 16, 1-4, 433-454, (1999)
[18] Feyel, F.; Chaboche, J.-L., \(\text{FE}^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fiber sic/ti composite materials, Comput. meth. appl. mech. engrg., 183, 309-330, (2000) · Zbl 0993.74062
[19] Feyel, F., A multilevel finite element method \((\text{FE}^2)\) to describe the response of highly non-linear structures using generalized continua, Comput. meth. appl. mech. engrg., 192, 3233-3244, (2003) · Zbl 1054.74727
[20] 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
[21] Ghosh, S.; Lee, K.; Raghavan, P., A multilevel computational model for multi-scale damage analysis in composite and porous media, Int. J. solids struct., 38, 2335-2385, (2001) · Zbl 1015.74058
[22] Yvonnet, J.; He, Q.-C., The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, J. comput. phys., 223, 341-368, (2007) · Zbl 1163.74048
[23] Monteiro, E.; Yvonnet, J.; He, Q.-C., Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction, Comput. mater. sci., 42, 704-712, (2008)
[24] Kouznetsova, V.G.; Geers, M.G.D.; Brekelmans, W.A.M., Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy, Comput. meth. appl. mech. engrg., 193, 5525-5550, (2004) · Zbl 1112.74469
[25] McVeigh, C.; Vernerey, F.; Liu, W.K.; Brinson, C., Multiresolution analysis for material design, Comput. meth. appl. mech. engrg., 195, 5525-5550, (2006)
[26] Ponte Castañeda, P.; Willis, J.R., On the overall properties of nonlinearly viscous composites, Proc. roy. soc. lond. A, 416, 217-244, (1988) · Zbl 0635.73006
[27] Hill, R., Elastic properties of reinforced solids: some theoretical principles, J. mech. phys. solids, 11, 357-372, (1963) · Zbl 0114.15804
[28] Habermann, C.; Kindermann, F., Multidimensional spline interpolation: theory and applications, Comput. econ., 30, 153-169, (2007) · Zbl 1126.65010
[29] Beatson, R.K., On the convergence of some cubic spline interpolation schemes, SIAM J. numer. anal., 23, 4, 903-912, (1986) · Zbl 0592.41009
[30] Hitchkock, F.L., The expression of a tensor or a polyadic as a sum of products, J. math. phys., 6, 164-189, (1927) · JFM 53.0095.01
[31] A. Harshman, Foundations of the PARAFAC Procedure: Models and Conditions for an Explanatory Multi-modal Factor Analysis, UCLA Working Papers in Phonetics, 1970, p. 16.
[32] Carol, J.D.; Chang, J.J., Analysis of individual differences in multidimensional scaling via an N-way generalization of ‘eckart-young’ decomposition, Psychometrika, 35, 283-319, (1970) · Zbl 0202.19101
[33] Möcks, J., Topographic components model for event-related potentials and some biophysical considerations, IEEE trans. biomed. engrg., 35, 482-484, (1988)
[34] Kiers, H.A.L., Toward a standardized notation and terminology in multiway analysis, J. chemometr., 14, 105-122, (2000)
[35] De Lathauwer, L.; De Moor, B.; Vandewalle, J., A multilinear singular value decomposition, SIAM J. matrix anal. appl., 21, 1253-1278, (2000) · Zbl 0962.15005
[36] Kapteyn, A.; Neudecker, H.; Wansbeek, T., An approach to n-mode components analysis, Psychometrika, 51, 269-275, (1986) · Zbl 0613.62078
[37] Tucker, L.R., Some mathematical notes on three-mode factor analysis, Psychometrika, 31, 279-311, (1966)
[38] Muti, D.; Bourennane, S., Multidimensional filtering based on a tensor approach, Signal process., 85, 2338-2353, (2005) · Zbl 1160.94349
[39] Beylkin, G.; Mohlenkamp, J., Algorithms for analysis in high dimensions, SIAM J. sci. comput., 26, 2133-2159, (2005) · Zbl 1085.65045
[40] Zhang, T.; Golub, G.H., Rank-one approximation to high order tensor, SIAM J. matrix anal. appl., 23, 2, 534-550, (2001) · Zbl 1001.65036
[41] B.W. Bader, T.G. Kolda, MATLAB Tensor Toolbox Version 2.2, <http://csmr.ca.sandia.gov/tgkolda/TensorToolbox/>, January 2007.
[42] Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructures, Comput. meth. appl. mech. engrg, 157, 69-94, (1998) · Zbl 0954.74079
[43] Moës, N.; Cloirec, M.; Cartraud, P.; Remacle, J.-F., A computational approach to handle complex microstructure geometries, Comput. meth. appl. mech. engrg., 192, 3163-3177, (2003) · Zbl 1054.74056
[44] deBotton, G.; Hariton, I., High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior, J. mech. phys. solids, 50, 2577-2595, (2002) · Zbl 1100.74541
[45] Idiart, M.I.; Ponte Castañeda, P., Fields in nonlinear composites I. theory, Proc. roy. soc. lond. A, 463, 183-202, (2007) · Zbl 1129.74036
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.