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.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Coulomb, C. A., 1773, “Essai sur une application des règles de Maximis et Minimis à quelques problèmes de statique relatifs à l'architecture,” Mémoire à l'Académie Royale des Sciences, Paris.
Drucker, D. C., and Prager, W., 1952, “Soil Mechanics and Plastic Analysis or Limit Design,” Quarterly Appl. Math., 10, pp. 157–165.
Schofield, A., and Wroth, P., 1968, Critical State Soil Mechanics, McGraw-Hill, New York.
Hoek, E., and Brown, E. T., 1997, “Practical Estimates of Rock Mass Strength,” Int. J. Rock Mech. Min. Sci., 34(8), pp. 1165–1186. [CrossRef]
SuquetP., 1982, Plasticité et Homogénéisation, Thèse de Doctorat d'Etat dissertation, Université Pierre et Marie Curie, Paris, France.
de BuhanP., 1986, Approche Fondamentale du Calcul à la Rupture des Ouvrages en Sols Renforcés, Thèse de Doctorat d'Etat dissertation, Université de Paris VI, Paris, France.
de Buhan, P., and Taliercio, A., 1991, “A Homogenization Approach to the Yield Strength of Composite Materials,” Eur. J. Mech. A/Solids, 10(2), pp. 129–54.
Lee, B. J., and Mear, M. E., 1992, “Effective Properties of Power-Law Solids Containing Elliptical Inhomogeneities: I. Rigid Inclusions II. Voids,” Mech. Mat., 14(4), pp 313–335, 337–356. [CrossRef]
Suquet, P., 1997, “Effective Behavior of Nonlinear Composites,” Continuum Micromechanics, P.Suquet, ed., Springer-Verlag, Berlin, pp. 197–264.
Barthélémy, J.-F., and Dormieux, L., 2003, “Determination of the Macroscopic Strength Criterion of a Porous Medium by Nonlinear Homogenization,” Comptes Rendus Mécanique, 331(4), pp. 271–276. [CrossRef]
Dormieux, L., Kondo, D., and Ulm, F.-J., 2006, Microporomechanics, Wiley, Chichester, UK.
Cariou, S., Ulm, F.-J., and Dormieux, L., 2008, “Hardness-Packing Density Scaling Relations for Cohesive-Frictional Porous Materials,” J. Mech. Phys. Sol., 56(3), pp. 924–952. [CrossRef]
Lemarchand, E., Ulm, F.-J., and Dormieux, L., 2002, “Effect of Inclusions on Friction Coefficient of Highly Filled Composite Materials,” J. Eng. Mech., 128(8), pp. 876–884. [CrossRef]
Barthélémy, J.-F., and Dormieux, L., 2004, “A Micromechanical Approach to the Strength Criterion of Drucker–Prager Materials Reinforced by Rigid Inclusions,” Int. J. Num. Anal. Meth. Geomech., 28(7–8), pp. 565–582. [CrossRef]
Ponte Castañeda, P., 1991, “The Effective Mechanical Properties of Nonlinear Isotropic Composites,” J. Mech. Phys. Sol., 39(1), pp. 45–71. [CrossRef]
Ponte Castañeda, P., 1992, “New Variational Principles in Plasticity and Their Application to Composite Materials,” J. Mech. Phys. Sol., 40(8), pp. 1757–1788. [CrossRef]
Ponte Castañeda, P., 1996, “Exact Second-Order Estimates for the Effective Mechanical Properties of Nonlinear Composite Materials,” J. Mech. Phys. Sol., 44(6), pp. 827–862. [CrossRef]
Ponte Castañeda, P., 2002, “Second-Order Homogenization Estimates for Nonlinear Composites Incorporating Field Fluctuations: I. Theory,” J. Mech. Phys. Sol., 50(4), pp. 737–757. [CrossRef]
Bilger, N., Auslender, F., Bornert, M., and Masson, R., 2002, “New Bounds and Estimates for Porous Media With Rigid Perfectly Plastic Matrix,” Comptes Rendus Mécanique, 330(2), pp. 127–132. [CrossRef]
Vincent, P.-G., Monerie, Y., and Suquet, P., 2009, “Porous Materials With Two Populations of Voids Under Internal Pressure: I. Instantaneous Constitutive Relations,” Int. J. Sol. Struct., 46(3–4), pp. 480–506. [CrossRef]
Danas, K., and Ponte Castañeda, P., 2009, “A Finite-Strain Model for Anisotropic Viscoplastic Porous Media: I. Theory,” Eur. J. Mech. A/Solids, 28(3), pp. 387–401. [CrossRef]
Gathier, B., and Ulm, F. J., 2008, “Multiscale Strength Homogenization—Application to Shale Nanoindentation,” CEE Research Report R08-01, Massachusetts Institute of Technology, Cambridge, MA.
Ortega, J. A., Gathier, B., and Ulm, F.-J., 2010, “Homogenization of Cohesive-Frictional Strength Properties of Porous Composites: Linear Comparison Composite Approach,” J. Nanomech. Micromech., 1(1), pp. 11–23. [CrossRef]
Gurson, A. L., 1977, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: I. Yield Criteria and Flow Rules for Porous Ductile Media,” J. Eng. Mat. Tech., 99, pp. 2–15. [CrossRef]
Leblond, J.-B., 2003, Mécanique de la Rupture Fragile et Ductile. Etudes en Mécanique des Matériaux et des Structures, Hermes Science, Paris.
Gologanu, M., Leblond, J.-B., Perrin, G., and Devaux, J., 1997, “Recent Extensions of Gurson's Model for Porous Ductile Metals,” Continuum Micromechanics, P.Suquet, ed., Springer-Verlag, Berlin, pp. 61–130.
Trillat, M., and Pastor, J., 2005, “Limit Analysis and Gurson's Model,” Eur. J. Mech. A/Solids, 24(5), pp. 800–819. [CrossRef]
Jeong, H. Y., 2002, “A New Yield Function and a Hydrostatic Stress-Controlled Void Nucleation Model for Porous Solids With Pressure-Sensitive Matrices,” Int. J. Sol. Struct., 39(5), pp. 1385–1403. [CrossRef]
Guo, T. F., Faleskog, J., and Shih, C. F., 2008, “Continuum Modeling of a Porous Solid With Pressure-Sensitive Dilatant Matrix,” J. Mech. Phys. Sol., 56(6), pp. 2188–2212. [CrossRef]
Suquet, P., 1995, “Overall Properties of Nonlinear Composites: A Modified Secant Modulus Theory and Its Link With Ponte Castañeda's Nonlinear Variational Procedure,” C. R. Acad. Sci. Paris, 320(Série IIb), pp. 563–571.
Desrues, J., 2002, “Limitations du Choix de l'Angle de Frottement pour le Critère de Plasticité de Drucker–Prager,” Revue Française de Génie Civil, 6, pp. 853–862.
Salençon, J., 1990, “An Introduction to the Yield Design Theory and Its Application to Soil Mechanics,” Eur. J. Mech. A/Solids, 9(5), pp. 477–500.
Ulm, F. J., and Coussy, O., 2003, Mechanics and Durability of Solids, Vol. 1: Solid Mechanics, Prentice Hall, Upper Saddle River, NJ.
Levin, V. M., 1967, “Thermal Expansion Coefficients of Heterogeneous Materials,” Mekhanika Tverdogo Tela, 2, pp. 83–94.
Laws, N., 1973, “On the Thermostatics of Composite Materials,” J. Mech. Phys. Sol., 21(1), pp. 9–17. [CrossRef]
Perrin, G., and Leblond, J. B., 1990, “Analytical Study of a Hollow Sphere Made of Plastic Porous Material and Subjected to Hydrostatic Tension—Application to Some Problems in Ductile Fracture of Metals,” Int. J. Plast., 6(6), pp. 677–699. [CrossRef]
de Buhan, P., and Dormieux, L., 1996, “On the Validity of the Effective Stress Concept for Assessing the Strength of Saturated Porous Materials: A Homogenization Approach,” J. Mech. Phys. Sol., 44(10), pp. 1649–1667. [CrossRef]
Fabrègue, D., and Pardoen, T., 2008, “A Constitutive Model for Elastoplastic Solids Containing Primary and Secondary Voids,” J. Mech. Phys. Sol., 56(3), pp. 719–741. [CrossRef]
Mori, T., and Tanaka, K., 1973, “Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions,” Acta Metallurgica, 21(5), pp. 571–574. [CrossRef]
Hershey, A. V., 1954, “The Elasticity of an Isotropic Aggregate of Anisotropic Cubic Crystals,” J. Appl. Mech., 21, pp. 226–240.
Kröner, E., 1958, “Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls,” Zeitschrift für Physik A Hadrons and Nuclei, 151(4), pp. 504–518. [CrossRef]
Hashin, Z., 1962, “The Elastic Moduli of Heterogeneous Materials,” ASME J. Appl. Mech., 29(1), pp. 143–150. [CrossRef]
Hashin, Z., and Shtrikman, S., 1963, “A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials, J. Mech. Phys. Sol., 11(2), pp. 127–140. [CrossRef]
Thoré, P., Pastor, F., Pastor, J., and Kondo, D., 2009, “Closed-Form Solutions for the Hollow Sphere Model With Coulomb and Drucker–Prager Materials Under Isotropic Loadings,” Comptes Rendus Mécanique, 337(5), pp. 260–267. [CrossRef]
Trillat, M., Pastor, J., and Thoré, P., 2006, “Limit Analysis and Conic Programming: Porous Drucker–Prager Material and Gurson's Model,” Comptes Rendus Mécanique, 334(10), pp. 599–604. [CrossRef]
Pastor, F., Thoré, P., Loute, E., Pastor, J., and Trillat, M., 2008, “Convex Optimization and Limit Analysis: Application to Gurson and Porous Drucker–Prager Materials,” Eng. Fract. Mech., 75(6), pp. 1367–1383. [CrossRef]
Pastor, J., Thoré, P., and Pastor, F., 2010, “Limit Analysis and Numerical Modeling of Spherically Porous Solids With Coulomb and Drucker–Prager Matrices,” J. Comp. Appl. Math., 234(7), pp. 2162–2174. [CrossRef]
Schweiger, H. F., 1994. “On the Use of Drucker–Prager Failure Criteria for Earth Pressure Problems,” Computers Geotech., 16(3), pp. 223–246. [CrossRef]
Danas, K., and Ponte Castañeda, P., 2009, “A Finite-Strain Model for Anisotropic Viscoplastic Porous Media: II. Applications,” Eur. J. Mech. A/Solids, 28(3), pp. 402–416. [CrossRef]


Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In