On Eshelby's inclusion problem in a three-phase spherically concentric solid, and a modification of Mori-Tanaka's method

Luo, Huiwu; Weng, George J.

doi:10.1016/0167-6636(87)90032-9

Cited by 105 publications

(36 citation statements)

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“…First the elastic parameters of the materials are related to their porosities. Luo and Weng [10] proposed a homogenization method relating the effective bulk and shear moduli to the porosity of the solid. Using this approach, the effective Young's modulus decreases monotonically with the porosity.…”

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confidence: 99%

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Stress-Driven Phase Transformation and the Roughening of Solid-Solid Interfaces

et al. 2008

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The application of stress to multiphase solid-liquid systems often results in morphological instabilities. Here we propose a solid-solid phase transformation model for roughening instability in the interface between two porous materials with different porosities under normal compression stresses. This instability is triggered by a finite jump in the free energy density across the interface, and it leads to the formation of finger-like structures aligned with the principal direction of compaction. The model is proposed as an explanation for the roughening of stylolites -irregular interfaces associated with the compaction of sedimentary rocks that fluctuate about a plane perpendicular to the principal direction of compaction.PACS numbers: 68.35. Ct, 68.35.Rh, 91.60.Hg Morphological instabilities in systems out of equilibrium are central to most research on pattern formation. A host of processes give rise to such instabilities, and among the most intensively studied are the surface diffusion mediated Asaro-Tiller-Grinfeld instability [1], [2], [3] in the surfaces of stressed solids in contact with their melts, surface diffusion mediated thermal grooving and solidification controlled by thermal diffusion in the bulk melt [4]. In sedimentary rocks and other porous materials local stress variations typically promote morphological changes via dissolution in regions of high stress, transport through the fluid saturated pore space and precipitation in regions of low stress. This phenomenon is known as pressure solution or chemical compaction. Such processes are often accompanied by the nucleation and growth of thin irregular sheets, interfaces or seams called stylolites [5]. Stylolites form under a wide range of geological conditions as rough interfaces that fluctuate about a plane perpendicular to the axis of compression. They are common in a variety of rock types, including limestones, dolomites, sandstones and marbles, and they appear on scales ranging from the mineral grain scale to meters or greater. A common feature of stylolitic surfaces is small scale roughness combined with large vertical steps in the direction of the compression. Residual unsoluble minerals (i.e. clays, oxides) often accumulate at the interface as stylolites evolve. Despite the considerable attention given to the rich morphology of stylolites there is still no consensus on the mechanism(s) controlling their formation [6], [7], [8], [9]. Here we demonstrate that even if the stylolite is a consequence of pressure solution alone, porosity or other material property gradients may drive the roughening process. In particular, we demonstrate that a compressional load normal to a solid-solid phase boundary gives rise to a morphological instability.Generally, rocks are heterogeneous bodies with spatially variable porosities. The strain energy densities may be larger in regions of high porosity (i.e. low modulus) than in regions of low porosity. Thermodynamically, the total free energy of the system can be reduced by reducing the porosity variations. ...

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confidence: 99%

“…2, upper panel) was then recorded for different external stresses, σ 0 , and elastic constants (E, ν). The elastic constants were computed from homogenization relations between elasticity and porosity [10].…”

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confidence: 99%

Stress-Driven Phase Transformation and the Roughening of Solid-Solid Interfaces

et al. 2008

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“…It is shown that the results of the generalized self-consistent method have excellent agreement with the experimental data (for example, refer to Huang et al, 1994). The three-phase model is also used to improve the accuracy of the Mori-Tanaka method (Luo and Weng, 1987).…”

Section: Introductionmentioning

confidence: 68%

“…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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“…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

185

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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On Eshelby's inclusion problem in a three-phase spherically concentric solid, and a modification of Mori-Tanaka's method

Cited by 105 publications

References 13 publications

Stress-Driven Phase Transformation and the Roughening of Solid-Solid Interfaces

Stress-Driven Phase Transformation and the Roughening of Solid-Solid Interfaces

A generalized self-consistent method accounting for fiber section shape

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

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