On the calculation of bulk properties of heterogeneous materials
Author:
John J. McCoy
Journal:
Quart. Appl. Math. 37 (1979), 137-149
DOI:
https://doi.org/10.1090/qam/99634
MathSciNet review:
QAM99634
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: Models for calculating the effective, or bulk, properties of heterogeneous materials are considered in the light of a Dyson equation formalism. It is seen that convergence difficulties, of the same nature as those which necessitated the renormalization effected by the Dyson equation in the first place, are frequently reintroduced, but now in calculating the parameters to use in the equation. Thus the validity of the models must be suspect. Fortunately, in most instances the validity of the models, if properly interpreted, can be established. The proper interpretation to be given is considered.
Statistical homogeneity is obtained in a limit in which the test specimen is unboundedly large.
In an alternate, but equivalent, definition one equates the energy stored in a fictitious, homogeneous effective medium and the averaged energy in the randomly heterogeneous test specimen.
The appropriateness of such an appellation might be questioned by some. I use it here since its use is not uncommon; it is a convenient shorthand term, and the precise meaning is made clear by the equation that describes the formalism.
- Mark J. Beran and John J. McCoy, Mean field variation in random media, Quart. Appl. Math. 28 (1970), 245–258. MR 266324, DOI https://doi.org/10.1090/S0033-569X-1970-0266324-1
M. J. Beran and J. J. McCoy, Int. J. Solids Struct. 6, 1035 (1970)
R. Zeller and P. Dederichs, Phys. Stat. Sol. (b)55, 831 (1973)
J. E. Gubernatis and J. A. Krumhansl, J. Math. Phys. 46, 1875 (1975)
G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 401 (1972)
J. M. Peterson and M. Fixman, J. Chem. Phys. 39, 2516 (1963)
G. K. Batchelor, J. Fluid Mech. 52, 245 (1972)
D. J. Jeffrey, Proc. Roy. Soc. Lond. A335, 355 (1973)
- D. J. Jeffrey, Group expansions for the bulk properties of a statistically homogeneous, random suspension, Proc. Roy. Soc. London Ser. A 338 (1974), 503–516. MR 351273, DOI https://doi.org/10.1098/rspa.1974.0099
R. O’Brien, Ph. D. Dissertation, Cambridge University (1977)
- U. Frisch, Wave propagation in random media, Probabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, pp. 75–198. MR 0269192
Some authors, including this one, prefer the use of the notation ${\left \langle L \right \rangle ^{ - 1}}$ as more descriptive than ${G_0}$ . In the interest of keeping my notation consistent with Frisch [14], I have used the ${G_0}$ notation in this paper.
The outlined proof is clearly only a formal proof. To make the proof rigorous would require a demonstration that all the series involved converge, a task that would have to be accomplished on a problem-by-problem basis. This author knows of no attempt to prove convergence for the effective property problem.
All that has been demonstrated here, of course, is that a proper cancellation is achieved in the term of fourth order in ${L_1}$ . In the Appendix we consider terms of higher orders in ${L_1}$ .
M. J. Beran, Statistical continuum theories, Interscience Publishers, New York (1968)
M. A. Elsayed, J. Math. Phys. 15, 2001 (1974)
The formalism can be applied to a finite-sized specimen with variations in $\epsilon \left ( x \right )$ that are “homogeneous” throughout. The effective index operator will exhibit a dependence on the specimen boundary conditions for field points that lie within a neighborhood of the specimen boundary that can be argued to have a thickness that is measured on the microscale. Away from this boundary layer the effective index operator is independent of either the geometry of the boundary surface or the conditions to be satisfied thereon. It is usual to assume that the presence of a boundary layer in the field equations will lead to effects that are similarly localized.
P. Saffman, Studies in Applied Math. 52, 115 (1973)
One might argue that the domain of integration is infinite when viewed on the microscale and that the outer boundary shape is most properly determined by the geometry that can be observed on this scale, i.e. by the statistics of the variations of the property measures. Isotropic statistics can only be compatible with a spherical geometry and, hence, in this case the proper choice for taking the limit is that of a sphere, irrespective of any specimen geometry. While this argument has some appeal since it removes the major source of unease, it is clearly lacking in mathematical rigor.
There is no relationship between $G$ here used to denote the temperature gradient and ${G_0}$ used previously to denote a Green’s function operator.
J. J. McCoy and M. J. Beran, Int. J. Engr. Sci. 14, 7 (1976)
E. Kröner, J. Mech. Phys. Solids 15, 319 (1967)
M. Hori and F. Yonezawa, J. Math. Phys. 15, 2177 (1974)
M. A. Elsayed and J. J. McCoy, J. Comp. Mat. 7, 466 (1973)
M. J. Beran, Il Nuovo Cimento 38, 771 (1965)
M. J. Beran and J. Molyneux, Quart. Appl. Math. 24, 107 (1966)
J. J. McCoy, Il Nuovo Cimento 57, 139 (1968)
J. J. McCoy, Recent advances in engineering science, Pergamon Press (1970)
Statistical homogeneity is obtained in a limit in which the test specimen is unboundedly large.
In an alternate, but equivalent, definition one equates the energy stored in a fictitious, homogeneous effective medium and the averaged energy in the randomly heterogeneous test specimen.
The appropriateness of such an appellation might be questioned by some. I use it here since its use is not uncommon; it is a convenient shorthand term, and the precise meaning is made clear by the equation that describes the formalism.
M. J. Beran and J. J. McCoy, Quart. Appl. Math, 28, 245 (1970)
M. J. Beran and J. J. McCoy, Int. J. Solids Struct. 6, 1035 (1970)
R. Zeller and P. Dederichs, Phys. Stat. Sol. (b)55, 831 (1973)
J. E. Gubernatis and J. A. Krumhansl, J. Math. Phys. 46, 1875 (1975)
G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 401 (1972)
J. M. Peterson and M. Fixman, J. Chem. Phys. 39, 2516 (1963)
G. K. Batchelor, J. Fluid Mech. 52, 245 (1972)
D. J. Jeffrey, Proc. Roy. Soc. Lond. A335, 355 (1973)
D. J. Jeffrey, Proc. Roy. Soc. Lond. A338, 503 (1974)
R. O’Brien, Ph. D. Dissertation, Cambridge University (1977)
U. Frisch, Probabilistic methods in applied mathematics, ed. A. T. Bharucha-Reid, Academic Press, New York (1968)
Some authors, including this one, prefer the use of the notation ${\left \langle L \right \rangle ^{ - 1}}$ as more descriptive than ${G_0}$ . In the interest of keeping my notation consistent with Frisch [14], I have used the ${G_0}$ notation in this paper.
The outlined proof is clearly only a formal proof. To make the proof rigorous would require a demonstration that all the series involved converge, a task that would have to be accomplished on a problem-by-problem basis. This author knows of no attempt to prove convergence for the effective property problem.
All that has been demonstrated here, of course, is that a proper cancellation is achieved in the term of fourth order in ${L_1}$ . In the Appendix we consider terms of higher orders in ${L_1}$ .
M. J. Beran, Statistical continuum theories, Interscience Publishers, New York (1968)
M. A. Elsayed, J. Math. Phys. 15, 2001 (1974)
The formalism can be applied to a finite-sized specimen with variations in $\epsilon \left ( x \right )$ that are “homogeneous” throughout. The effective index operator will exhibit a dependence on the specimen boundary conditions for field points that lie within a neighborhood of the specimen boundary that can be argued to have a thickness that is measured on the microscale. Away from this boundary layer the effective index operator is independent of either the geometry of the boundary surface or the conditions to be satisfied thereon. It is usual to assume that the presence of a boundary layer in the field equations will lead to effects that are similarly localized.
P. Saffman, Studies in Applied Math. 52, 115 (1973)
One might argue that the domain of integration is infinite when viewed on the microscale and that the outer boundary shape is most properly determined by the geometry that can be observed on this scale, i.e. by the statistics of the variations of the property measures. Isotropic statistics can only be compatible with a spherical geometry and, hence, in this case the proper choice for taking the limit is that of a sphere, irrespective of any specimen geometry. While this argument has some appeal since it removes the major source of unease, it is clearly lacking in mathematical rigor.
There is no relationship between $G$ here used to denote the temperature gradient and ${G_0}$ used previously to denote a Green’s function operator.
J. J. McCoy and M. J. Beran, Int. J. Engr. Sci. 14, 7 (1976)
E. Kröner, J. Mech. Phys. Solids 15, 319 (1967)
M. Hori and F. Yonezawa, J. Math. Phys. 15, 2177 (1974)
M. A. Elsayed and J. J. McCoy, J. Comp. Mat. 7, 466 (1973)
M. J. Beran, Il Nuovo Cimento 38, 771 (1965)
M. J. Beran and J. Molyneux, Quart. Appl. Math. 24, 107 (1966)
J. J. McCoy, Il Nuovo Cimento 57, 139 (1968)
J. J. McCoy, Recent advances in engineering science, Pergamon Press (1970)
Additional Information
Article copyright:
© Copyright 1979
American Mathematical Society
