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Research Papers

Strength Homogenization of Double-Porosity Cohesive-Frictional Solids

[+] Author and Article Information
J. Alberto Ortega

Development Engineer,
Schlumberger Technology Center,
Sugar Land, TX 77478
e-mail: jortega9@slb.com

Franz-Josef Ulm

George Macomber Professor
Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: ulm@mit.edu

1Corresponding author.

Manuscript received December 20, 2010; final manuscript received July 7, 2012; accepted manuscript posted October 25, 2012; published online February 4, 2013. Assoc. Editor: Younane Abousleiman.

J. Appl. Mech 80(2), 020902 (Feb 04, 2013) (14 pages) Paper No: JAM-10-1460; doi: 10.1115/1.4007905 History: Received December 20, 2010; Revised July 07, 2012; Accepted July 10, 2012

The strength homogenization of cohesive-frictional solids influenced by the presence of two pressurized pore spaces of different characteristic sizes is addressed in this study. A two-scale homogenization model is developed based on limit analysis and the second-order method (SOM) in linear comparison composite theory, which resolves the nonlinear strength behavior through the use of linear comparison composites with optimally chosen properties. For the scale of the classical configuration of a porous solid, the formulation employs a compressible thermoelastic comparison composite to deliver closed-form expressions of strength criteria. Comparisons with numerical results reveal that the proposed homogenization estimates for drained conditions are adequate except for high triaxialities in the mean compressive strength regime. At the macroscopic scale of the double-porosity material, the SOM results are in agreement with strength criteria predicted by alternative micromechanics solutions for materials with purely cohesive solid matrices and drained conditions. The model predictions for the cohesive-frictional case show that drained strength development in granularlike composites is affected by the partitioning of porosity between micro- and macropores. In contrast, the drained strength is virtually equivalent for single- and double-porosity materials with matrix-inclusion morphologies. Finally, the second-order linear comparison composite approach confirms the applicability of an effective stress concept, previously proposed in the literature of homogenization of cohesive-frictional porous solids, for double-porosity materials subjected to similar pressures in the two pore spaces. For dissimilar pore pressures, the model analytically resolves the complex interplays of microstructure, solid properties, and volume fractions of phases, which cannot be recapitulated by the effective stress concept.

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Figures

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Fig. 1

Representative elementary volumes (rev) for the two-scale double-porosity material. Level I corresponds to the microscopic porous solid formed by a pressure-sensitive solid matrix and microporosity. Level II corresponds to the macroscopic material composed of the (homogenized) material at level I and the macroporosity. The two pore spaces are saturated by fluids sustaining different pressures.

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Fig. 2

(a) Illustrations of strength criteria supported by the SOM method for level I (the porous solid) modeled as a granular composite. (b) Predictions of unconfined compressive strength (UCS) for level I as functions of packing density for matrix-inclusion (MT, Mori–Tanaka) and granular (SC, self-consistent) microstructures. The predictions for the range η < ηcrit are associated with elliptical strength criteria, whereas predictions for the range η > ηcrit are associated with hyperbolic strength criteria.

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Fig. 3

Critical packing densities for (a) the porous solid (level I), and (b,c) the double-porosity material (level II) considering different microstructures. The values calculated for level II correspond to microstructural configurations in which the same estimate is used for both material scales (matrix-inclusion, MT-MT; granular, SC-SC).

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Fig. 4

Strength criteria predicted by the effective strain rate (ESR), linear comparison composite (SOM), and numerical limit analysis (static and kinematic LA) methods for a Drucker–-Prager porous solid. The numerical solutions were determined for the hollow sphere model, and the micromechanics solutions were estimated through the Mori–Tanaka scheme for matrix-inclusion morphologies. The LA numerical predictions for cases (a) and (b) were digitized from figures presented in Thoré et al. [44] and Pastor et al. [47], respectively.

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Fig. 5

Comparison between the Vincent et al. and SOM models for a two-scale double-porosity composite with von Mises solid. The data from Ref. [20] correspond to the results presented in Fig. 9 of their publication. The input parameters are the packing densities η = ζ = 0.9, which correspond to a total porosity of ϕ0 = 0.19.

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Fig. 6

The effect of double-porosity on different strength types for a cohesive-frictional material with matrix-inclusion microstructure (Mori–Tanaka scheme)

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Fig. 7

The effect of double-porosity on different strength types for a cohesive-frictional material with granular microstructure (self-consistent scheme)

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Fig. 8

Predicted strength domains for a double-porosity material with von Mises solid phase and subjected to three pore pressure combinations. The microstructure is modeled with matrix-inclusion (MT-MT) morphology.

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Fig. 9

Predicted strength domains for double-porosity materials with Drucker–Prager solid phase and subjected to three pore pressure combinations. Two microstructures are modeled: (a) matrix-inclusion (MT-MT) and (b) granular (SC-SC).

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Fig. 10

Combinations of critical pore pressures P1,crit and P2,crit that result in macroscopic strength domains with zero hydrostatic tension strength capacity. The estimates are developed for (a) matrix-inclusion (MT-MT) and (b) granular (SC-SC) microstructures and different values of the relative volume fraction of macropores fmp.

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