Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media

Mechanics of Materials

Volumn 38, Issue 8-10, 2006, Pages 732-747

  Berryman, James G ^a  

^a LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

Microstructures; Polycrystalline material; Variational calculus

Indexed keywords

COMPOSITE MATERIALS; CORRELATION THEORY; ELASTIC MODULI; ELECTRIC CONDUCTIVITY; ESTIMATION; INFORMATION RETRIEVAL; METAL INSULATOR BOUNDARIES; POLYCRYSTALLINE MATERIALS; TRANSPORT PROPERTIES; VARIATIONAL TECHNIQUES;

BERAN-MOLYNEUX BOUNDS; GRAIN-TO-GRAIN CORRELATIONS; HETEROGENEOUS MEDIA; TRANSPORT COEFFICIENTS;

MICROSTRUCTURE;

EID: 33646372185     PISSN: 01676636     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mechmat.2005.06.014     Document Type: Article

References (94)

1. Archie G.E. The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. AIME 146 (1942) 54-62
2. Asaad, Y., 1955. A study of the thermal conductivity of fluid-bearing porous rocks. Ph.D. thesis, University of California-Berkeley, Berkeley, CA.
3. Avellaneda M., and Torquato S. Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media. Phys. Fluids A 3 (1991) 2529-2540
4. Avellaneda M., Cherkaev A.V., Lurie K.A., and Milton G.W. On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality. J. Appl. Phys. 63 (1988) 4989-5003
5. Backus G.E. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res. 67 (1962) 4427-4440
6. Baker Jr. G.A. Essentials of Padé Approximants (1975), Academic Press, San Diego, CA
7. Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater. 6 (1987) 147-157
8. Beran M.J. Use of the variational approach to determine bounds for the effective permittivity in random media. Nuovo Cimento 38 (1965) 771-782
9. Beran M.J. Statistical Continuum Theories (1968), Wiley, New York pp. 181-256
10. Beran M.J., and Molyneux J. Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quart. Appl. Math. 24 (1966) 107-118
11. Berge P.A., Berryman J.G., and Bonner B.P. Influence of microstructure on rock elastic properties. Geophys. Res. Lett. 20 (1993) 2619-2622
12. Berge P.A., Bonner B.P., and Berryman J.G. Ultrasonic velocity-porosity relationships for sandstone analogs made from fused glass-beads. Geophysics 60 (1995) 108-119
13. Bergman D.J. Calculation of bounds for some average bulk properties of composite materials. Phys. Rev. B 14 (1976) 4304-4312
14. Bergman D.J. The dielectric constant of a composite material-a problem of classical physics. Phys. Repts. 43 (1978) 378-407
15. Bergman D.J. Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material. Phys. Rev. Lett. 44 (1980) 1285-1287
16. Bergman D.J. Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Phys. 138 (1982) 78-114
17. Berryman J.G. Long-wavelength propagation in composite elastic media I. Spherical inclusions. J. Acoust. Soc. Am. 68 (1980) 1809-1819
18. Berryman J.G. Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions. J. Acoust. Soc. Am. 68 (1980) 1820-1831
19. Berryman J.G. Effective medium theory for elastic composites. In: Varadan V.K., and Varadan V.V. (Eds). Elastic Wave Scattering and Propagation (1982), Ann Arbor Science, Ann Arbor, Michigan 111-129
20. Berryman J.G. Measurement of spatial correlations functions using image processing techniques. J. Appl. Phys. 57 (1985) 2374-2384
21. Berryman J.G. Variational bounds on elastic constants for the penetrable sphere model. J. Phys. D: Appl. Phys. 18 (1985) 585-597
22. Berryman J.G. Effective stress for transport properties of inhomogeneous porous rock. J. Geophys. Res. 97 (1992) 17409-17424
23. Berryman J.G. Mixture theories for rock properties. In: Ahrens T.J. (Ed). Rock Physics and Phase Relations: American Geophysical Union Handbook of Physical Constants (1995), AGU, New York 205-228
24. Berryman J.G. Poroelastic shear modulus dependence on pore-fluid properties arising in a model of thin isotropic layers. Geophys. J. Int. 157 (2004) 415-425
25. Berryman J.G. Bounds on elastic constants of random polycrystals of laminates. J. Appl. Phys. 96 (2004) 4281-4287
26. Berryman J.G. Thermal conductivity of porous media. Appl. Phys. Lett. 86 (2005) 032905 (online: January 11, 2005)
27. Berryman J.G. Bounds and estimates for elastic constants of random polycrystals of laminates. Int. J. Solids Struct. 42 (2005) 3730-3743
28. Berryman, J.G., 2005c. Bounds and estimates for transport coefficients of random and porous media with high contrasts. J. Appl. Phys. 97 (6), art. no. 063504.
29. Berryman J.G., and Berge P.A. Critique of explicit schemes for estimating elastic properties of multiphase composites. Mech. Mater. 22 (1996) 149-164
30. Berryman J.G., and Milton G.W. Microgeometry of random composites and porous media. J. Phys. D: Appl. Phys. 21 (1988) 87-94
31. Bruggeman D.A.G. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen: I. Dielectrizitätskonstanen und Leitfahigkeiten der Mischkörper aus Isotropen Substanzen. Ann. Phys. (Leipzig) 24 (1935) 636-679
32. Budiansky B. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13 (1965) 223-227
33. Cherkaev A.V. Variational Methods for Structural Optimization (2000), Springer-Verlag, New York pp. 171-212
34. Christensen R.M. Mechanics of Composite Materials (1979), Wiley Interscience, New York pp. 31-72
35. Christensen R.M. A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids 38 (1990) 379-404
36. Cleary M.P., Chen I.-W., and Lee S.-M. Self-consistent techniques for heterogeneous media. ASCE J. Eng. Mech. 106 (1980) 861-887
37. Ferrari M., and Filiponni M. Appraisal of current homogenizing techniques for the elastic response of porous and reinforced glass. J. Am. Ceramic Soc. 74 (1991) 229-231
38. Garboczi E.J., and Berryman J.G. New effective medium theory for the diffusivity or conductivity of a multi-scale concrete microstructure model. Concr. Sci. Eng. 2 (2000) 88-96
39. Garboczi E.J., and Berryman J.G. New differential effective medium theory for the linear elastic moduli of a material containing composite inclusions. Mech. Mater. 33 (2001) 455-470
40. Gibiansky L.V., and Milton G.W. On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus. Proc. Roy. Soc. London A 440 (1993) 163-188
41. Gibiansky L.V., and Torquato S. Rigorous link between the conductivity and elastic moduli of fibre-reinforced composite materials. Philos. Trans. R. Soc. A: Phys. Sci. Eng. 353 (1995) 243-278
42. Gibiansky L.V., Milton G.W., and Berryman J.G. On the effective viscoelastic moduli of two-phase media. III. Rigorous bounds on the complex shear modulus in two dimensions. Proc. Roy. Soc. London A 455 (1999) 2117-2149
43. Gubernatis J.E., and Krumhansl J.A. Macroscopic engineering properties of polycrystalline materials: elastic properties. J. Appl. Phys. 46 (1975) 1875-1883
44. Guéguen Y., and Palciaukas V. Introduction to the Physics of Rocks (1994), Princeton University Press, Princeton, NJ pp. 193-199
45. Hardy G.H., Littlewood J.E., and Pólya G. Inequalities (1952), Cambridge University Press, Cambridge, UK p. 78
46. Hashin Z. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29 (1962) 143-150
47. Hashin Z., and Shtrikman S. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962) 3125-3131
48. Hashin Z., and Shtrikman S. A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11 (1963) 127-140
49. Hashin Z., and Shtrikman S. Conductivity of polycrystals. Phys. Rev. 130 (1963) 129-133
50. Helsing J., and Helte A. Effective conductivity of aggregates of anisotropic grains. J. Appl. Phys. 69 (1991) 3583-3588
51. Hill R. Elastic properties of reinforced solids: some theoretical principles. Proc. Phys. Soc. London A 65 (1952) 349-354
52. Hill R. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13 (1965) 213-222
53. Kirkpatrick S. Classical transport in disordered media: scaling and effective-medium theories. Phys. Rev. Lett. 27 (1971) 1722-1725
54. Kirkpatrick S. Percolation and conduction. Rev. Mod. Phys. 45 (1973) 574-588
55. Korringa J., and LaTorraca G.A. Application of the Bergman-Milton theory of bounds to the permittivity of rocks. J. Appl. Phys. 60 (1986) 2966-2976
56. Korringa J., Brown R.J.S., Thompson D.D., and Runge R.J. Self-consistent imbedding and the ellipsoidal model for porous rocks. J. Geophys. Res. 84 (1979) 5591-5598
57. Kuster G., and Toksöz M.N. Velocity and attenuation of seismic waves in two-phase media: Part I. Theoretical formulations. Geophysics 39 (1974) 587-606
58. Landauer R. The electrical resistance of binary metallic mixtures. J. Appl. Phys. 23 (1952) 779-784
59. McCoy J.J. On the displacement field in an elastic medium with random variations of materials properties. In: Eringen A.C. (Ed). Recent Advances in Engineering Science, Proceedings of the Sixth Annual Meeting of the Society of Engineering Science, Princeton University, Princeton, New Jersey, 11-13 November 1968 (1968), Gordon and Breach, New York 235-254
60. Milton G.W. Bounds on the complex dielectric constant of a composite material. Appl. Phys. Lett. 37 (1980) 300-303
61. Milton G.W. Bounds on the complex permittivity of a two-component composite material. J. Appl. Phys. 52 (1981) 5286-5293
62. Milton G.W. Bounds on the transport and optical properties of a two-component composite material. J. Appl. Phys. 52 (1981) 5294-5304
63. Milton G.W. Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett. 46 (1981) 542-545
64. Milton G.W. Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30 (1982) 177-191
65. Milton G.W. Multicomponent composites, electrical networks, and new types of continued fraction. I. Commun. Math. Phys. 111 (1987) 281-327
66. Milton G.W. The Theory of Composites (2002), Cambridge University Press, Cambridge, UK pp. 77-78, 163, 457-498, 580-581
67. Milton G.W., and Berryman J.G. On the effective viscoelastic moduli of two-phase media. II. Rigorous bounds on the complex shear modulus in three dimensions. Proc. Roy. Soc. London A 453 (1997) 1849-1880
68. Milton G.W., and Phan-Thien N. New bounds on effective elastic moduli of two-component materials. Proc. Roy. Soc. London A 380 (1982) 305-331
69. Mura T. Micromechanics of Defects in Solids (1987), Kluwer Academic Publishers, Dordrecht, The Netherlands
70. Nemat-Nasser S., and Hori M. Micromechanics: Overall Properties of Heterogeneous Materials (1993), North-Holland, Amsterdam pp. 325-388
71. Norris A.N. A differential scheme for the effective moduli of composites. Mech. Mater. 4 (1985) 1-16
72. Norris A.N. An examination of the Mori-Tanaka effective medium approximation for multiphase composites. ASME J. Appl. Mech. 56 (1989) 83-88
73. Olson T., and Avellaneda M. Effective dielectric and elastic constants of piezoelectric polycrystals. J. Appl. Phys. 71 (1992) 4455-4464
74. Peselnick L., and Meister R. Variational method of determining effective moduli of polycrystals: (A) Hexagonal symmetry, (B) trigonal symmetry. J. Appl. Phys. 36 (1965) 2879-2884
75. Prager S. Improved variational bounds on some bulk properties of a two-phase random medium. J. Chem. Phys. 50 (1969) 4305-4312
76. Reuss A. Berechung der Fliessgrenze von Mischkristallenx auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9 (1929) 49-58
77. Schulgasser K. Sphere assemblage model for polycrystals and symmetric materials. J. Appl. Phys. 54 (1983) 1380-1382
78. Silnutzer, N.R., 1972. Effective constants of statistically homogeneous materials. Ph.D. thesis, University of Pennsylvania.
79. Simmons G., and Wang H.F. Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook (1971), MIT Press, Cambridge, Massachusetts pp. 1-300
80. Stroud D., Milton G.W., and De B.R. Analytical model for the dielectric response of brine-saturated rocks. Phys. Rev. B 34 (1986) 5145-5153
81. Thomsen L. Elasticity of polycrystals and rocks. J. Geophys. Res. 77 (1972) 315-327
82. Torquato, S., 1980. Microscopic approach to transport in two-phase random media. Ph.D. thesis, State University of New York at Stony Brook.
83. Torquato S. Effective electrical conductivity of two-phase disordered composite media. J. Appl. Phys. 58 (1985) 3790-3797
84. Torquato S. Bulk properties of two-phase disordered media. V. Effective conductivity of a dilute dispersion of penetrable spheres. J. Chem. Phys. 83 (1985) 4776-4785
85. Torquato S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties (2002), Springer, New York pp. 422-423, 598
86. Torquato S., and Stell G. Microstructure of two-phase random media. I. The n-point probability functions. J. Chem. Phys. 77 (1982) 2071-2077
87. Torquato S., Hyun S., and Donev A. Optimal design of manufacturable three-dimensional composites with multifunctional characteristics. J. Appl. Phys. 94 (2003) 5748-5755
88. Voigt W. Lehrbuch der Kristallphysik (1928), Teubner, Leipzig
89. Walpole L.J. On the overall elastic moduli of composite materials. J. Mech. Phys. Solids 17 (1969) 235-251
90. Watt J.P., and Peselnick L. Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Appl. Phys. 51 (1980) 1525-1531
91. Watt J.P., Davies G.F., and O'Connell R.J. The elastic properties of composite materials. Rev. Geophys. Space Phys. 14 (1976) 541-563
92. Wiener O. Die Theorie des Mischkörpers für das Feld des stationären Strömung. Erste Abhandlung die Mittelswertsätze für Kraft, Polarisation und Energie. Abhandlungen der mathematischphysischen Klasse der Königlich Sächisischen Gesellschaft der Wissenschagten 32 (1912) 509-604
93. Wildenschild D., Roberts J.J., and Carlberg E.D. On the relationship between microstructure and electrical and hydraulic properties of sand-clay mixtures. Geophys. Res. Lett. 27 (2000) 3065-3068
94. Willis J.R. Variational and related methods for the overall properties of composites. In: Yih C.-S. (Ed). Advances in Applied Mechanics (1981), Academic Press, New York 1-78