On the effect of interactions of inhomogeneities on the overall elastic and conductive properties

Abstract A simple method of estimating the effect of inhomogeneity interactions on the overall properties (elastic and conductive) is developed. It is formulated in terms of property contribution tensors that give the contribution of an inhomogeneity to the overall properties. The method can be viewed as further development of the approach of Rodin and Hwang (1991) and Rodin (1993) that generalized the method of analysis of crack interactions (Kachanov, 1987) to inhomogeneities. We also extend the method to the conductive properties. Considering the effect of interactions on the property contribution tensors on the example of pores we find that this effect is generally moderate, at most (even when pores touch one another) – in contrast with the effect on local fields. On example of two spheres, we compare the interaction effects on the elastic and the conductive properties, and discuss the impact of interactions on the cross-property connections.

[1]  P. Rangaswamy,et al.  On elastic interactions between spherical inclusions by the equivalent inclusion method , 2006 .

[2]  I. Kunin,et al.  An ellipsoidal crack and needle in an anisotropic elastic medium: PMM vol. 37, n≗3, 1973, pp. 524–531 , 1973 .

[3]  R. Asaro,et al.  The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion , 1975 .

[4]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[5]  V. Kushch.  Elastic equilibrium of a medium containing a finite number of aligned spheroidal inclusions , 1996 .

[6]  Erik H. Saenger,et al.  Effective Elastic Properties of Fractured Rocks: Dynamic vs. Static Considerations , 2006 .

[7]  Y. Y. Earmme,et al.  Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 1: Theory , 1980 .

[8]  V. Levin,et al.  Self-Consistent Methods for Composites , 2008 .

[9]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[10]  S. Nemat-Nasser,et al.  Elastic fields of interacting inhomogeneities , 1985 .

[11]  Tai Te Wu,et al.  The effect of inclusion shape on the elastic moduli of a two-phase material* , 1966 .

[12]  J. R. Bristow.  Microcracks, and the static and dynamic elastic constants of annealed and heavily cold-worked metals , 1960 .

[13]  Andreas Acrivos,et al.  The solution of the equations of linear elasticity for an infinite region containing two spherical inclusions , 1978 .

[14]  G. Rodin.  The overall elastic response of materials containing spherical inhomogeneities , 1993 .

[15]  Y. Murakami.  Stress Intensity Factors Handbook , 2006 .

[16]  G. B. Jeffery,et al.  On a Form of the Solution of Laplace's Equation Suitable for Problems Relating to Two Spheres , 1912 .

[17]  Toshio Mura,et al.  Two-Ellipsoidal Inhomogeneities by the Equivalent Inclusion Method , 1975 .

[18]  Mark Kachanov,et al.  Continuum Model of Medium with Cracks , 1980 .

[19]  J. D. Eshelby,et al.  The elastic field outside an ellipsoidal inclusion , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[20]  C. Christov,et al.  Fast Legendre spectral method for computing the perturbation of a gradient temperature field in an unbounded region due to the presence of two spheres , 2010 .

[21]  Mark Kachanov,et al.  Effective elasticity of rocks with closely spaced and intersecting cracks , 2006 .

[22]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[23]  L. Walpole,et al.  Fourth-rank tensors of the thirty-two crystal classes: multiplication tables , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[24]  T. Zohdi,et al.  On computation of the compliance and stiffness contribution tensors of non ellipsoidal inhomogeneities , 2008 .

[25]  J. Happel,et al.  Low Reynolds number hydrodynamics: with special applications to particulate media , 1973 .

[26]  J. Maxwell.  A Treatise on Electricity and Magnetism , 1873, Nature.

[27]  I. Sevostianov,et al.  Effective elastic properties of the particulate composite with transversely isotropic phases , 2004 .

[28]  G. I. Barenblatt.  THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE , 1962 .

[29]  Gregory J. Rodin,et al.  On the problem of linear elasticity for an infinite region containing a finite number of non-intersecting spherical inhomogeneities , 1991 .

[30]  I. Sevostianov,et al.  Elastic and electric properties of closed-cell aluminum foams Cross-property connection , 2006 .

[31]  I. Sevostianov,et al.  Effective properties of heterogeneous materials: Proper application of the non-interaction and the “dilute limit” approximations , 2012 .

[32]  I. Sevostianov,et al.  Cross-property connections for fiber reinforced piezoelectric materials with anisotropic constituents , 2007 .

[33]  D. Gross,et al.  Spannungsintensitätsfaktoren von Rißsystemen , 1982 .

[34]  W. M. Hicks.  V. The motion of two spheres in a fluid , 1879, Proceedings of the Royal Society of London.

[35]  R. Pyrz,et al.  Cubic inclusion arrangement: Effects on stress and effective properties , 2005 .

[36]  Mark Kachanov,et al.  Effective Elastic Properties of Cracked Solids: Critical Review of Some Basic Concepts , 1992 .

[37]  I. Sevostianov,et al.  Non-interaction Approximation in the Problem of Effective Properties , 2013 .

[38]  J. Rice,et al.  Slightly curved or kinked cracks , 1980 .

[39]  A. Acrivos,et al.  The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations , 1978 .

[40]  I. Sevostianov,et al.  Explicit cross-property correlations for anisotropic two-phase composite materials , 2002 .

[41]  I. Sevostianov,et al.  Connections between Elastic and Conductive Properties of Heterogeneous Materials , 2009 .

[42]  Leon M Keer,et al.  Semi‐analytic solution for multiple interacting three‐dimensional inhomogeneous inclusions of arbitrary shape in an infinite space , 2011 .

[43]  Mark Kachanov,et al.  A simple technique of stress analysis in elastic solids with many cracks , 1985, International Journal of Fracture.

[44]  M. Porfiri,et al.  Analysis of particle-to-particle elastic interactions in syntactic foams , 2011 .

[45]  J. Willis,et al.  THE OVERALL ELASTIC MODULI OF A DILUTE SUSPENSION OF SPHERES , 1976 .

[46]  I. Kunin,et al.  Elastic Media with Microstructure II , 1982 .

[47]  Mark Kachanov,et al.  Elastic solids with many cracks: A simple method of analysis , 1987 .

[48]  D. Jeffrey,et al.  Conduction through a random suspension of spheres , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.