Properties Of Nonlinear Composites Research Articles

Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations

This paper presents a combined numerical-theoretical study of the macroscopic behavior and local field distributions in a special class of two-dimensional periodic composites with viscoplastic phases. The emphasis is on strongly nonlinear materials containing pores or rigid inclusions. Full-field numerical simulations are carried out using a fast Fourier transform algorithm [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417–1423.], while the theoretical results are obtained by means of the ‘second-order’ nonlinear homogenization method [Ponte Castañeda, P., 2002. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. I. Theory. J. Mech. Phys. Solids 50, 737–757]. The effect of nonlinearity and inclusion concentration is investigated in the context of power-law (with strain-rate sensitivity m) behavior for the matrix phase under in-plane shear loadings. Overall, the ‘second-order’ estimates are found to be in good agreement with the numerical simulations, with the best agreement for the rigidly reinforced materials. For the porous systems, as the nonlinearity increases ( m decreases), the strain field is found to localize along shear bands passing through the voids (the strain fluctuations becoming unbounded) and the effective stress exhibits a singular behavior in the dilute limit. More specifically, for small porosities and fixed nonlinearity m > 0 , the effective stress decreases linearly with increasing porosity. However, for ideally plastic behavior ( m = 0 ) , the dependence on porosity becomes non-analytic. On the other hand, for rigidly-reinforced composites, the strain field adopts a tile pattern with bounded strain fluctuations, and no singular behavior is observed (to leading order) in the dilute limit.

Macroscopic behavior and field fluctuations in viscoplastic composites: Second-order estimates versus full-field simulations

This work presents a combined numerical and theoretical study of the effective behavior and statistics of the local fields in random viscoplastic composites. The full-field numerical simulations are based on the fast Fourier transform (FFT) algorithm [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417–1423], while the theoretical estimates follow from the so-called “second-order” procedure [Ponte Castañeda, P., 2002a. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory. J. Mech. Phys. Solids 50, 737–757]. Two-phase fiber composites with power-law phases are considered in detail, for two different heterogeneity contrasts corresponding to fiber-reinforced and fiber-weakened composites. Both the FFT simulations and the corresponding “second-order” estimates show that the strain-rate fluctuations in these systems increase significantly, becoming progressively more anisotropic, with increasing nonlinearity. In fact, the strain-rate fluctuations tend to become unbounded in the limiting case of ideally plastic composites. This phenomenon is shown to correspond to the localization of the strain field into bands running through the composite along certain preferred orientations determined by the loading conditions. The bands tend to avoid the fibers when they are stronger than the matrix, and to pass through the fibers when they are weaker than the matrix. In general, the “second-order” estimates are found to be in good agreement with the FFT simulations, even for high nonlinearities, and they improve, often in qualitative terms, on earlier nonlinear homogenization estimates. Thus, it is demonstrated that the “second-order” method can be used to extract accurate information not only for the macroscopic behavior, but also for the anisotropic distribution of the local fields in nonlinear composites.

About the improvement of variational bounds for nonlinear composite dielectrics

We deal with the problem of estimating the effective properties of nonlinear composite dielectrics, specially fibrous biphasic composites in which fibers are periodically distributed following a square scheme. We present an integrated approach that combines the derivation of closed-form formulae (via the Asymptotic Homogenization Method, AHM) and bounds for the effective properties of linear composites, and a bounding procedure for the effective properties of nonlinear composites. The consistency of the AHM effective expression against the bounds is shown along with some limit cases. The improvement provided by this methodology becomes more significant in cases near percolation. Numerical results are presented.

A modified affine theory for the overall properties of nonlinear composites

The expressions of the intraphase second moments of stress and strain in heterogeneous linear thermoelastic materials are derived; they are used to quantify the intraphase heterogeneity predicted by the ‘classical affine’ formulation for the homogenization of nonlinear composites. By analogy with the ‘modified secant’ approach, a ‘modified affine’ formulation that accounts for intraphase heterogeneity is proposed.

Improved bounds for the effective properties of a nonlinear two-phase elastic composite

In recent work a nonlinear comparison composite has been used to obtain bounds for the overall properties of nonlinear composites. The bounds involve parameters introduced by G.W. Milton, but even when these parameters are known, the bounds can be widely separated. In this paper improved bounds for a particular microstructure are obtained by using the linear bounds of O.P. Bruno and J. Chaubell. This demonstrates that by incorporating more detailed information about the microgeometry it is possible to significantly improve on the nonlinear bounds which are valid for any microstructure.

Bounds for the Overall Properties of Nonlinear Composites

Bounds for the Overall Properties of Nonlinear Composites

On methods for bounding the overall properties of nonlinear composites: Correction and addition

Willis ( J. Mech. Phys. Solids 39, 73, 1991) concluded that a new bounding method for nonlinear composites, presented by Ponte Castañeda ( J. Mech. Phys. Solids 39, 45, 1991) was equivalent to an earlier method which employed a nonlinear generalization of the Hashin-Shtrikman variational principle. This conclusion was reached by first showing that the nonlinear Hashin-Shtrikman bound is at least as good as the new bound and then that the new bound is at least as good as the older one. A fallacy in the latter part of this demonstration is exposed by considering a simple one-dimensional counter-example, corresponding to a nonlinear laminate. The conditions for coincidence identified by Willis (1991) are incomplete through failure to require explicitly that a stationary point defined by them yields a global minimum. Several cases have been studied previously, for which the two methods do yield the same bound; when they do, Ponte Castañeda's procedure has the potential to give an improvement by the use at an intermediate stage of an improved bound for a linear composite. When the methods yield different bounds, however, that produced by the nonlinear Hashin-Shtrikman procedure is the better.

On methods for bounding the overall properties of nonlinear composites

A new method for bounding the overall properties of nonlinear composites, proposed byPonte Castañeda (J. Mech. Phys. Solids39, 45, 1991), is compared with an older prescription based on a generalization to nonlinear behaviour of the Hashin-Shtrikman procedure. It is demonstrated that the two methods generate precisely the same information, and hence that differences noted by Ponte Castañeda arise from comparing optimal bounds obtained from the new procedure with sub-optimal bounds obtained from the original one. The relative advantages of either procedure are discussed.

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Investigation of nonlinear properties of composites consisting of linear-elastic components

Problems of meridionally axisymmetrical geometrically nonlinear elasticity theory [2] are discussed as applied to the investigation of the properties of composites.