Solutions for effective shear properties in three phase sphere and cylinder models

Christensen, Richard M.; Lo, King Him

doi:10.1016/0022-5096(79)90032-2

Cited by 1,897 publications

(896 citation statements)

References 15 publications

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“…Another homogenization is also performed at the microscale, where the filled matrix is reinforced by a certain volume fraction of aligned glass (carbon) fibers. Here the composite cylinders method (CCM) is utilized (Hashin and Rosen, 1964;Christensen and Lo, 1979;Christensen, , 1990. This second step of homogenization yields the effective properties of a glass (carbon) bundle.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…To resolve this issue, Christensen and Lo (1979) have proposed for this property the use of the generalized self-consistent method.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…This permits to identify analytical solutions as a function of the unknown property, which is calculated with the help of energetic criteria. Computational details for the application of the composite cylinders method can be found in various papers (Christensen and Lo, 1979;Christensen, 1990;Seidel and Lagoudas, 2006;Seidel, 2007;Chatzigeorgiou et al, 2012).…”

Section: Accepted Manuscriptmentioning

confidence: 99%

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Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites

Anagnostou

Chatzigeorgiou

Chemisky

et al. 2018

Composites Part B: Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. AbstractThe aim of this paper is to study, through a multiscale analysis, the viscoelastic behavior of glass reinforced sheet molding compound (SMC) composites and SMC-hybrid composites mixing two types of bundle reinforcement: glass and carbon fibers. SMC exhibit more than two distinct characteristic length scales, so that a sequence of scale transitions is required to obtain the overall behavior of the composite. An analytical procedure is used consisting of properly selected well-established micromechanical methods like the Mori-Tanaka (MTM) and the composite cylinders (CCM) accounting for each scale transition. After selecting a representative volume element (RVE) for each scale, the material response of any given length scale is described on the basis of the homogenized behavior of the next finer one. This hierarchical approach is appropriately extended to the viscoelastic domain to account for the time dependent overall response of the SMC composite material. The anisotropic damage has been introduced through a micromechanical model considering matrix penny-shape microcrack density inside bundles. The capabilities of the hierarchical modeling are illustrated with various parametric studies and simulation of experimental data for glass-based SMC composites.

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Section: Accepted Manuscriptmentioning

confidence: 99%

“…To resolve this issue, Christensen and Lo (1979) have proposed for this property the use of the generalized self-consistent method.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Section: Accepted Manuscriptmentioning

confidence: 99%

See 1 more Smart Citation

Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites

Anagnostou

Chatzigeorgiou

Chemisky

et al. 2018

Composites Part B: Engineering

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show abstract

“…However, neglecting the interaction of particles is an unrealistic assumption of Eshelby for materials with randomly dispersed particulate microstructure, even at a few percent volume fraction [89]. Further proposed models such as Mori-Tanaka [90][91][92], the self-consistent scheme [93][94][95][96][97][98], the generalized self-consistent scheme [99][100][101][102][103], and the differential method [104,105] are mainly based on the mean-field approximation [106] and approximate the interaction between the phases. The extension of these models to account for the electroelastic behavior of composite materials was addressed by Dunn and Taya [107].…”

Section: Introductionmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…The pros and cons of the Mori-Tanaka method have been discussed by Christensen (1990) and Christensen et al (1992). The generalized self-consistent method is a more sophisticated micromechanics approach (Christensen and Lo, 1979;Luo and Weng, 1987;Christensen, 1993;Huang and Hu, 1995;Cheung, 1998, 2001;Riccardi and Montheilet, 1999;. Different from the aforementioned micromechanics methods based on the two-phase model, the generalized self-consistent method is based on a threephase model: an inclusion is embedded in a finite matrix, which in turn is embedded in an infinite composite with the as-yet-unknown effective moduli.…”

Section: Introductionmentioning

confidence: 99%

A generalized self-consistent method accounting for fiber section shape

Jiang

Tong

Cheung

2003

International Journal of Solids and Structures

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A three-phase confocal elliptical cylinder model is proposed for fiber-reinforced composites, in terms of which a generalized self-consistent method is developed for fiber-reinforced composites accounting for variations in fiber section shapes and randomness in fiber section orientation. The reasonableness of the fiber distribution function in the present model is shown. The dilute, self-consistent, differential and Mori-Tanaka methods are also extended to consider randomness in fiber section orientation in a statistical sense. A full comparison is made between various micromechanics methods and with the Hashin and ShtrikmanÕs bounds. The present method provides convergent and reasonable results for a full range of variations in fiber section shapes (from circular fibers to ribbons), for a complete spectrum of the fiber volume fraction (from 0 to 1, and the latter limit shows the correct asymptotic behavior in the fully packed case) and for extreme types of the inclusion phases (from voids to rigid inclusions). A very different dependence of the five effective moduli on fiber section shapes is theoretically predicted, and it provides a reasonable explanation on the poor correlation between previous theory and experiment in the case of longitudinal shear modulus.

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Solutions for effective shear properties in three phase sphere and cylinder models

Cited by 1,897 publications

References 15 publications

Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites

Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

A generalized self-consistent method accounting for fiber section shape

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