A numerical method for computing the overall response of nonlinear composites with complex microstructure. (English) Zbl 0954.74079

Moulinec, H.; Suquet, P.

Comput. Methods Appl. Mech. Eng. 157, No. 1-2, 69-94 (1998).

Summary: The local and overall responses of nonlinear composites are classically investigated by the finite element method. We propose an alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images. It is based on the exact expression of the Green function of a linear elastic and homogeneous comparison material. First, the case of elastic nonhomogeneous constituents is considered and an iterative procedure is proposed to solve the Lippmann-Schwinger equation which naturally arises in the problem. Then, the method is extended to nonlinear constituents by a step-by-step integration in time. The accuracy of the method is assessed by varying the spatial resolution of the microstructures. The flexibility of the method allows to use it for a large variety of microstructures.

MSC:

74S25	Spectral and related methods applied to problems in solid mechanics
74E30	Composite and mixture properties
74M25	Micromechanics of solids

Keywords:

complex microstructure; linear elastic material; nonlinear composites; Fourier series; Green function; iterative procedure; Lippmann-Schwinger equation; step-by-step integration in time

References:

[1]	Moulinec, H.; Suquet, P., A fast numerical method for computing the linear and nonlinear properties of composites, C.R. Acad. Sc. Paris II, 318, 1417-1423 (1994) · Zbl 0799.73077
[2]	Moulinec, H.; Suquet, P., A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructure, (Pyrz, R., Microstructure-Property Interactions in Composite Materials (1995), Kluwer Academic Publ.,: Kluwer Academic Publ., Dordrecht), 235-246
[3]	Adams, D. F.; Doner, D. R., Transverse normal loading of a unidirectional composite, J. Composite Mat., 1, 152-164 (1967)
[4]	Christman, T.; Needleman, A.; Suresh, S., An experimental and numerical study of deformation in metal-ceramic composites, Acta Metall. Mater., 37, 3029-3050 (1989)
[5]	Swan, C. C., Techniques for stress- and strain-controlled homogenization of inelastic periodic composites, Comput. Methods Appl. Mech. Engrg., 117, 249-267 (1994) · Zbl 0846.73039
[6]	Michel, J. C.; Suquet, P., On the strength of composite materials: variational bounds and numerical aspects, (Mota-Soares, C.; Bendsoe, M. P., Topology Design of Structures (1993), Kluwer Academic Publ.,: Kluwer Academic Publ., Dordrecht), 355-374
[7]	Brockenborough, J. R.; Suresh, S.; Wienecke, H. A., Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape, Acta Metall. Mater., 39, 735-752 (1991)
[8]	Böhm, H. J.; Rammerstoffer, F. G.; Weissenbeck, E., Some simple models for micromechanical investigations of fiber arrangements in MMCs, Comput. Mat. Sci., 1, 177-194 (1993)
[9]	Nakamura, T.; Surech, S., Effects of thermal residual stresses and fiber packing on deformation of metal-matrix composites, Acta Metall. Mater., 41, 1665-1681 (1993)
[10]	Dietrich, Ch.; Poech, M. H.; Fischmeister, H. F.; Schmauder, S., Stress and strain partitioning in Ag-Ni fibre composite under transverse loading. Finite element modeling and experimental study, Comput. Mater. Sci., 1, 195-202 (1993)
[11]	Becker, R.; Richmond, O., Incorporation of microstructural geometry in material modelling, Modelling Simul. Mater. Sci. Engrg., 2, 439-454 (1994)
[12]	Bomert, M., Morphologie structurale et comportement mécanique; caractérisations expérimentales, approach par bornes et estimations autocohérentes généralisées, (Ph.D. Thesis (1996), Ecole Nationale des Ponts et Chaussées: Ecole Nationale des Ponts et Chaussées Paris)
[13]	Garboczi, E. J.; Day, A. R., An algorithm for computing the effective linear properties of heteregeneous materials: three-dimensional results for composites with equal phase Poisson ratios, J. Mech. Phys. Solids, 43, 1349-1362 (1995) · Zbl 0881.73094
[14]	Müller, W. H., Mathematical versus Experimental Stress Analysis of Inhomogeneities in Solids, J. Phys. IV, 6, C1-139-C1-148 (1996)
[15]	Willis, J. R., On methods for bounding the overall properties of nonlinear composites, J. Mech. Phys. Solids, 39, 73-86 (1991) · Zbl 0734.73053
[16]	Ponte, P., Castañeda, New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids, 40, 1757-1788 (1992) · Zbl 0764.73103
[17]	Suquet, P., Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, 41, 981-1002 (1993) · Zbl 0773.73063
[18]	Suquet, P., Elements of homogenization for inelastic solid mechanics, (Sanchez-Palencia, E.; Zaoui, A., Homogenization Techniques for Composite Media. Homogenization Techniques for Composite Media, Lecture Notes in Physics, 272 (1987), Springer Verlag: Springer Verlag Berlin), 193-278
[19]	Guedes, J. M.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptative finite element methods, Comput. Methods Appl. Mech. Engrg., 83, 143-198 (1990) · Zbl 0737.73008
[20]	Kröner, E., Statistical Continuum Mechanics (1972), Springer-Verlag: Springer-Verlag Wien
[21]	Bornen, M.; Hervé, E.; Stolz, C.; Zaoui, A., Self-consistent approaches and strain heterogeneities in two-phase elastoplastic materials, Appl. Mech. Rev., 47, S66-S76 (1994)
[22]	Brault, J. W.; White, O. R., The analysis and restoration of astronomical data via the Fast Fourier Transform, Astron. Astrophys., 13, 169-189 (1971)
[23]	Mura, T., Micromechanics of Defects in Solids (1987), Martinus Nijhoff: Martinus Nijhoff Dordrecht