Dielectric and elastic characterization of nonlinear: Heterogeneous materials

Materials

Volumn 2, Issue 4, 2009, Pages 1417-1479

  Giordano, Stefano ^a,b  

^a UNIVERSITY OF CAGLIARI   (Italy)

^b INFM   (Italy)

Author keywords

Composites (nanosystems embedded in a larger structure); Mixture theory and order parameters; Nonlinear constitutive equations

Indexed keywords

ARBITRARY ASPECT RATIO; ELASTIC CHARACTERIZATION; ELLIPSOIDAL INCLUSIONS; HETEROGENEOUS MATERIALS; MIXTURE THEORY AND ORDER PARAMETERS; NONLINEAR CONSTITUTIVE EQUATIONS; ORIENTATIONAL DISTRIBUTIONS; PSEUDO-ORIENTED PARTICLES;

CHARACTERIZATION; DISPERSIONS; INCLUSIONS; MIXTURES;

DIELECTRIC MATERIALS;

EID: 84866365680     PISSN: None     EISSN: 19961944     Source Type: Journal    
DOI: 10.3390/ma2041417     Document Type: Review

References (107)

1. Torquato, S. Exact expression for the effective elastic tensor of disordered composites. Phys. Rev. Lett. 1997, 79, 681-684.
2. Stroud, D.; Hui, P.M. Nonlinear susceptibilities of granular matter. Phys. Rev. B 1988, 37, 8719-8724.
3. Hashin, Z.; Shtrikman, S. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 1962, 33, 3125-3131.
4. Hashin, Z.; Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids. 1963, 11, 127-140.
5. Brown, W.F. Solid mixture permittivities. J. Chem. Phys. 1955, 23, 1514-1517.
6. Torquato, S. Effective stiffness tensor of composite media-I. Exact series expansions. J. Mech. Phys. Solids 1997, 45,1421-1448.
7. Torquato, S. Effective stiffness tensor of composite media-II. Applications to isotropic dispersions. J. Mech. Phys. Solids 1998, 46, 1411-1440.
8. Bianco, B.; Parodi, M. A unifying approach for obtaining closed-form expressions of mixtures permittivities. J. Electrostat. 1984, 15,183-195.
9. Maxwell, J.C. A Treatise on Electricity and Magnetism; Clarendon: Oxford, UK, 1881.
10. Brosseau, C. Modelling and simulation of dielectric heterostructures: A physical survey from an historical perspective. J. Phys. D: Appl. Phys. 2006, 39, 1277-1294.
11. Fricke, H. The Maxwell-Wagner dispersion in a suspension of ellipsoids. J. Phys. Chem. 1953, 57, 934-937.
12. Fricke, H. A mathematical treatment of the electric conductivity and capacity of disperse systems. Phys. Rev. 1924, 24, 575-587.
13. Sihvola, A.; Kong, J.A. Effective permittivity of dielectric mixtures. IEEE Trans. Geosci. Remo. Sen. 1988, 26, 420-429.
14. Sihvola, A. Electromagnetic Mixing Formulas and Applications; The Institution of Electrical Engineers: London, UK, 1999.
15. Van Beek, L.K.H. Dielectric behaviour of heterogeneous systems. In Progress in Dielectric; Heywood: London, UK 1967, 7, 71-114.
16. Bruggeman, D.A.G. Dielektrizitatskonstanten und Leitfahigkeiten der Mishkorper aus isotropen Substanzen. Ann. Phys. (Leipzig) 1935, 24, 636-664.
17. Giordano, S. Effective medium theory for dispersions of dielectric ellipsoids. J. Electrostat. 2003, 58, 59-76.
18. Bianco, B.; Chiabrera, A.; Giordano, S.D.C.-E.L.F. Characterization of random mixtures of piecewise non-linear media. Bioelectromagnetics 2000, 21, 145-149.
19. Bianco, B.; Giordano, S. Electrical characterization of linear and non-linear random networks and mixtures. Int. J. Circuit Theor. Appl. 2003, 31, 199-218.
20. Giordano, S. Disordered lattice networks: General theory and simulations. Int. J. Circuit Theor. Appl. 2005, 33, 519-540.
21. Giordano, S. Two-dimensional disordered lattice networks with substrate. Physica A 2007, 375, 726-740.
22. Giordano, S. Relation between microscopic and macroscopic mechanical properties in random mixtures of elastic media. J. Eng. Mater. Technol. ASME 2007, 129, 453-461.
23. Goncharenko, A.V.; Popelnukh, V.V.; Venger, E.F. Effect of weak nonsphericity on linear and nonlinear optical properties of small particle composites. J. Phys. D: Appl. Phys. 2002, 35, 1833-1838.
24. Lakhtakia, A.; Mackay, T.G. Size-dependent Bruggeman approach for dielectric-magnetic composite materials. AEU Int. J. Electron. Commun. 2005, 59, 348-351.
25. Mackay, T.G. Homogenization giving rise to unusual metamaterials. In Proc. SPIE; Denver, CO, USA, 12 August 2004; Lakhtakia, A., Maksimenko, S.A., Eds.; Vol. 5509, pp. 34-45.
26. Zharov, A.A.; Shadrivov, I.V.; Kivshar, Y.S. Nonlinear properties of left-handed metamaterials. Phys. Rev. Lett. 2003, 91, 37401:1-37401:4.
27. Giordano, S.; Rocchia,W. Shape dependent effects of dielectrically nonlinear inclusions in heterogeneous media. J. Appl. Phys. 2005, 98, 104101:1-104101:10.
28. Walpole, L.J. Elastic behaviour of composite materials: theoretical foundations. Adv. Appl. Mech. 1981, 11, 169-242.
29. Christensen, R.M. Mechanics of Composite Materials; Dover Publication Inc.: New York, NY, USA, 2005.
30. Hashin, Z. Analysis of composite materials-A survey. J. Appl. Mech. 1983, 50, 481-505.
31. Norris, N. A differential scheme for the effective moduli of composites. Mech. Mater. 1985, 4, 1-16.
32. McLaughlin, R. A study of the differential scheme for composite materials. Int. J. Eng. Sci. 1977, 15, 237-244.
33. Giordano, S. Differential schemes for the elastic characterisation of dispersions of randomly oriented ellipsoids. Eur. J. Mech. A-Solid. 2003, 22, 885-902.
34. Kachanov, M.; Sevostianov, I. On quantitative characterization of microstructures and effective properties. Int. J. Solids Struct. 2005, 42, 309-336.
35. Markov, K.Z., Preziozi, L., Eds. Heterogeneous Media: Micromechanics Modeling Methods and Simulations; Birkhauser: Boston, MA, USA, 2000.
36. Kachanov, M. Effective elastic properties of cracked solids: critical review of some basic concepts. Appl. Mech. Rev. 1992, 45, 305-336.
37. Giordano, S.; Colombo, L. Effects of the orientational distribution of cracks in solids. Phys. Rev. Lett. 2007, 98, 055503:1-055503:4.
38. Giordano, S.; Colombo, L. Effects of the orientational distribution of cracks in isotropic solids. Eng. Frac. Mech. 2007, 74, 1983-2003.
39. Zanda, G. Electromagnetic Properties of Linear and Nonlinear Composite Materials. Master Degree Thesis, University of Cagliari, Italy, 2008.
40. Leung, K.M. Optical bistability in the scattering and absorption of light from nonlinear microparticles. Phys. Rev. A 1986, 33, 2461-2464.
41. Haus, J.W.; Inguva, R.; Bowden, C.M. Effective-medium theory for nonlinear ellipsoidal composites. Phys. Rev. A 1989, 40, 5729-5734.
42. Agarwal, G.S.; Gupta, S.D. T-Matrix approach to the nonlinear susceptibilities of heterogeneous media. Phys. Rev. A 1988, 38, 5678-5687.
43. Stratton, J.A. Electromagnetic Theory; Mc Graw Hill: New York, NY, USA, 1941.
44. Giordano, S.; Palla, P.L.; Colombo, L. Effective permittivity of materials containing graded ellipsoidal inclusions. Eur. Phys. J. B 2008, 66, 29-35.
45. Goncharenko, A.V. Optical properties of core-shell particle composites. I. Linear response. Chem. Phys. Lett. 2004, 386, 25-31.
46. Goncharenko, A.V. Optical properties of core-shell particle composites. II. Nonlinear response. Chem. Phys. Lett. 2007, 439, 121-126.
47. Chen, G.Q.; Wu, Y.M. Effective dielectric response of nonlinear composites of coated metal inclusions. Chin. Phys. Lett. 2007, 24, 1724-1728.
48. Pinchuk, A. Optical bistability in nonlinear composites with coated ellipsoidal nanoparticles. J. Phys. D: Appl. Phys. 2003, 36, 460-464.
49. Gu, L.; Gao, L. Optical bistability of a nondilute suspension of nonlinear coated particles. Physica B 2005, 368, 279-286.
50. Park, J.; Lu, W. Orientation of core-shell nanoparticles in an electric field. Appl. Phys. Lett. 2007, 91, 053113:1-053113:3.
51. Yu, K.W.; Hui, P.M.; Stroud, D. Effective dielectric response of nonlinear composites. Phys. Rev. B 1993, 47, 14150-14156.
52. Bergman, D.J.; Levy, O.; Stroud, D. Theory of optical bistability in a weakly nonlinear composite medium. Phys. Rev. B 1994, 49, 129-134.
53. Levy, O.; Bergman, D.J.; Stroud, D. Harmonic generation, induced nonlinearity, and optical bistability in nonlinear composites. Phys. Rev. E 1995, 52, 3184-3194.
54. Hui, P.M.; Cheung, P.; Stroud, D. Theory of third harmonic generation in random composites of nonlinear dielectrics. J. Appl. Phys. 1998, 84, 3451-3458.
55. Hui, P.M.; Xu, C.; Stroud, D. Second-harmonic generation for a dilute suspension of coated particles. Phys. Rev. B 2004, 69, 014203:1-014203:7.
56. Wei, E.B.; Wu, Z.K. The Nonlinear effective dielectric response of graded composites. J. Phys.: Condens. Matter 2004, 16, 5377-5386.
57. De Gennes, P.G. The Physics of Liquid Crystals; Clarendon: Oxford, UK, 1974.
58. Chandrasekhar, S. Liquid Crystals; Cambridge University Press: Cambridge, UK, 1977.
59. Nagatani, T. Effective permittivity in random anisotropic media. J. Appl. Phys. 1980, 51, 4944-4949.
60. Shafiro, B.; Kachanov, M. Anisotropic effective conductivity of materials with nonrandomly oriented inclusions of diverse ellipsoidal shapes. J. Appl. Phys. 2000, 87, 8561-8569.
61. Giordano, S. Order and disorder in heterogeneous material microstructure: electric and elastic characterization of dispersions of pseudo oriented spheroids. Int. J. Eng. Sci. 2005, 43, 1033-1058.
62. Giordano, S. Equivalent permittivity tensor in anisotropic random media. J. Electrostat. 2006, 64, 655-663.
63. Landau, L.D.; Lifshitz, E.M. Electrodynamics of Continuous Media; Pergamon: Oxford, UK, 1960.
64. Castaneda, P.P.; Suquet P. Nonlinear composites. Adv. Appl. Mech. 1998, 34, 171-302.
65. Talbot, D.R.S.; Willis J.R. Variational principles for inhomogeneous nonlinear media. J. Appl. Math. 1985, 35, 39-54.
66. Talbot, D.R.S.; Willis J.R. Bounds and self-consistent estimates for the overall properties of nonlinear composites. J. Appl. Math. 1987, 39, 215-240.
67. Castaneda, P.P. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 1991, 39, 45-71.
68. Castaneda, P.P. Bounds and estimates for the properties of nonlinear heterogeneous systems. Philos. Trans. R. Soc. London A 1992, 340, 531-567.
69. Castaneda, P.P. A new variational principle and its application to nonlinear heterogeneous systems. J. Appl. Math. 1992, 52, 1321-1341.
70. Castaneda, P.P. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 1996, 44, 827-862.
71. Suquet, P. Overall potentials and extremal surfaces of power law or ideally plastic composites, J. Mech. Phys. Solids 1993, 41, 981-1002.
72. Talbot, D.R.S.;Willis J.R. Some explicit bounds for the overall behaviour of nonlinear composites. Int. J. Solids Struct. 1992, 29, 1981-1987.
73. Beran, M.J. Use of the variational approach to determine bounds for the effective permittivity of random media. Il Nuovo Cimento 1965, 38, 771-782.
74. Catheline, S.; Gennisson, J.L.; Fink, M. Measurement of elastic nonlinearity of soft solid with transient elastography. J. Acoust. Soc. Am. 2003, 114, 3087-3091.
75. Catheline, S.; Gennisson, J.L.; Tanter, M.; Fink, M. Observation of shock transverse waves in elastic media. Phys. Rev. Lett. 2003, 91, 164301:1-164301:4.
76. Buehler, M.J.; Abraham, F.F.; Gao, H. Hyperelasticity governs dynamic fracture at a critical length scale. Nature 2003, 426, 141-146.
77. Buehler, M.J.; Gao, H. Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature 2006, 439, 307-310.
78. Hol'y, V.; Springholz, G.; Pinczolits, M.; Bauer, G. Strain induced vertical and lateral correlations in quantum dot superlattices. Phys. Rev. Lett. 1999, 83, 356-359.
79. Schmidbauer, M.; Seydmohamadi, Sh.; Grigoriev, D.; Wang, Zh. M.; Mazur, Yu. I.; Scḧafer, P.; Hanke, M.; Khler, R.; Salamo, G. J. Controlling Planar and Vertical Ordering in Three-Dimensional (In,Ga)As Quantum Dot Lattices by GaAs Surface Orientation. Phys. Rev. Lett. 2006, 96, 066108:1-066108:4.
80. Mattoni, A.; Colombo L.; Cleri, F. Atomistic study of the interaction between a microcrack and a hard inclusion in SiC. Phys. Rev. B 2004, 70, 094108:1-094108:10.
81. Atkin, R.J.; Fox, N. An Introduction to the Theory of Elasticity; Dover Publication Inc.: New York, NY, USA, 1980.
82. Novozhilov, V.V. Foundations of the Nonlinear Theory of Elasticity; Dover Publication Inc.: New York, NY, USA, 1999.
83. Ballabh, T.K.; Paul, M.; Middya, T.R.; Basu, A.N. Theoretical multiple-scattering calculation of nonlinear elastic constants of disordered solids. Phys. Rev. B 1992, 45, 2761-2771.
84. Mura, T. Micromechanics of Defects in Solids; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1987.
85. Love, A.E.H. A Treatise on the Mathematical Theory of Elasticity; Dover Publication Inc.: New York, NY, USA, 2002.
86. Landau L.D.; Lifschitz, E.M. Theory of Elasticity, Course of Theoretical Physics, Vol. 7, 3rd ed.; Butterworth Heinemann: Oxford, UK, 1986.
87. Sekoyan, S.S. Higher-order moduli of elasticity for an isotropic elastic body. Int. Appl. Mech. 1974, 10, 1259-1262.
88. Hiki, Y. Higher order elastic constants of solids. Ann. Rev. Mater. Sci. 1981, 11, 51-73.
89. Hughes, D.S.; Kelly, J.L. Second-order elastic deformation of solids. Phys. Rev. 1953, 92, 1145-1149.
90. Bateman, T.; Mason, W.P.; McSkimin, H.J. Third-order elastic moduli of germanium. J. Appl. Phys. 1961, 32, 928-936.
91. Cain T.; Ray, J.R. Third-order elastic constants from molecular dynamics: Theory and an example calculation. Phys. Rev. B 1988, 38, 7940-7946.
92. Zhao, J.; Winey, J.M.; Gupta, Y.M. First-principles calculations of second- and third-order elastic constants for single crystals of arbitrary symmetry. Phys. Rev. B 2007, 75, 094105:1-094105:7.
93. Eshelby, J.D. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. London Ser. A 1957, A241, 376-396.
94. Eshelby, J.D. The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. London Ser. A 1959, A252, 561-569.
95. Sevostianov, I.; Vakulenko, A. Inclusion with non-linear properties in elastic medium. Int. J. Fracture 2000, 107, 9-14.
96. Tsvelodub, I.Yu. Determination of the strength characteristics of a physically nonlinear inclusion in a linearly elastic medium. J. Appl. Mech. Tech. Phys. 2000, 41, 734-739.
97. Tsvelodub, I.Yu. Physically nonlinear ellipsoidal inclusion in a linearly elastic medium. J. Appl. Mech. Tech. Phys. 2004, 45, 69-75.
98. Giordano, S.; Colombo, L.; Palla, P.L. Nonlinear elastic Landau coefficients in heterogeneous materials. Europhys. Lett. 2008, 83, 66003:1-66003:5.
99. Giordano, S.; Palla, P.L.; Colombo, L. Nonlinear elasticity of composite materials. Eur. Phys. J. B 2009, 68, 89-101.
100. Christensen, R.M. Effective properties of composite materials containing voids. Proc. R. Soc. London, Ser. A 1993, 440, 461-473.
101. Zimmerman, R.W. Elastic moduli of a solid containing spherical inclusions. Mech. Mater. 1991, 12, 17-24.
102. Zimmerman, R.W. Behaviour of the poisson ratio of a two-phase composite materials in the highconcentration limit. Appl. Mech. Rev. 1994, 47, 38-44.
103. Thorpe, M.F.; Sen, P.N. Elastic moduli of two dimensional composite continua with elliptical inclusions. J. Acoust. Soc. Am. 1985, 77, 1674-1680.
104. Hill, R. Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour. J. Mech. Phys. Solids 1964, 12, 199-212.
105. Hill, R. Theory of mechanical properties of fibre-strengthened materials: II. Inelastic behaviour. J. Mech. Phys. Solids 1964, 12, 213-218.
106. Hill, R. Theory of mechanical properties of fibre-strengthened materials: III. Self-consistent model. J. Mech. Phys. Solids 1965, 13, 189-198.
107. Snyder, K.A.; Garboczi E.J. The elastic moduli of simple two-dimensional isotropic composites: Computer simulation and effective medium theory. J. Appl. Phys. 1992, 72, 5948-5955.