Field distributions and effective-medium approximation for weakly nonlinear media

Pellegrini, Yves-Patrick

doi:10.1103/physrevb.61.9365

Cited by 16 publications

(8 citation statements)

References 42 publications

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“…The method proposed by the authors show that the field distributions are well captured when taking into account first-neighbor interactions between bonds, and treating long-range interactions by an effective medium. Pellegrini [37] has modeled the field distributions in random dielectrics using Gaussian distributions to predict the response of "weakly nonlinear" media.…”

Section: Introductionmentioning

confidence: 99%

Elastostatic field distributions in polycrystals and cracked media

Willot

Brenner

Trumel

2019

Philosophical Magazine

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This work addresses the problem of the reconstruction of the local fields distribution occurring in heterogeneous linear elastic solids. The constitutive heterogeneities are crystals and cracks. Through comparisons with FFT computations, it is shown that self-consistent estimates together with an assumption of normal distribution at the phase scale provide an accurate description of the elastostatic field histograms in polycrystals without cracks. In the case of inter and transgranular cracks, full-field FFT simulations indicate that the field histograms present van Hove singularities. Their natures are determined analytically in the low-density regime, in the case of an homogeneous medium containing cracks.

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Section: Introductionmentioning

confidence: 99%

Elastostatic field distributions in polycrystals and cracked media

Willot

Brenner

Trumel

2019

Philosophical Magazine

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“…A combined experimental and theoretical study of field fluctuations in two-phase elastoplastic solids was given by Bornert et al (1994). In the context of conductivity, Pellegrini (2000) proposed that a Gaussian distribution of the local fields in the constituent phases of random composites with linear constitutive behavior could be a good approximation, and used this type of distribution as an ansatz for random composites with nonlinear constitutive behavior, in an attempt to account for field fluctuations in selfconsistent nonlinear homogenization methods (Pellegrini, 2001). Another objective of this work is to investigate this point further.…”

Section: Introductionmentioning

confidence: 99%

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

Idiart

Moulinec

Castañeda

et al. 2006

Journal of the Mechanics and Physics of Solids

103

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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 Castan˜eda, P., 2002a. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I-Theory. J. Mech. Phys. Solids 50, 737-757]. Twophase 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 ARTICLE IN PRESS www.elsevier.com/locate/jmps 0022-5096/$ -see front matter r (P. Ponte Castan˜eda).''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. r

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“…The path integral (11) together with the approximation (12) can be computed and after some calculations, one is led to…”

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Path-integral approach to strongly nonlinear composites

Barthélemy

2000

Phys. Rev. B

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We study strongly nonlinear disordered media using a functional method. We solve exactly the problem of a nonlinear impurity in a linear host and we obtain a Bruggeman-like formula for the effective nonlinear susceptibility. This formula reduces to the usual Bruggeman effective medium approximation in the linear case and has the following features: (i) It reproduces the weak contrast expansion to the second order and (ii) the effective medium exponent near the percolation threshold are s = 1, t = 1 + κ, where κ is the nonlinearity exponent. Finally, we give analytical expressions for previously numerically calculated quantities.

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Field distributions and effective-medium approximation for weakly nonlinear media

Cited by 16 publications

References 42 publications

Elastostatic field distributions in polycrystals and cracked media

Elastostatic field distributions in polycrystals and cracked media

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

Path-integral approach to strongly nonlinear composites

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