Third-order bounds on the effective bulk and shear modulus of a dispersion of fully penetrable spheres

S. Torquato, G. Stell, J. Beasley

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


We evaluate the third-order Beran-Molyneux bounds on the effective bulk modulus Ke and the third-order McCoy bounds on the effective shear modulus μe of a model of a two-phase composite in which one of the phases consists of spherical inclusions (or voids), with bulk and shear moduli, K2 and μ2, respectively, and volume fraction φ2, dispersed randomly throughout a matrix phase, with bulk and shear moduli, K1 and μ1, respectively, and volume fraction φ1. We tabulate the two fundamental microstructural parameters I1 and L1 required to evaluate the bounds, which depend upon the three-point matrix probability function of the model, for the aforementioned fully-penetrablesphere model. We compare the third-order bounds on Ke and μe to the second-order bounds due to Hashin and Shtrikman and to Walpole. We find that the third-order bounds for our model are always more restrictive than the corresponding second-order bounds. When the moduli of the phases differ by an order of magnitude, the third-order bounds are sharp enough to provide quantitatively useful estimates of Ke and μe for all φ2. The third-order bounds are very restrictive at low φ2 values (e.g., φ2 = 0.1) where they remain useful for cases in which the moduli of the phases differ by two orders of magnitude. Experimental values of μe measured by Corson for a tungstenlead composite are found to lie within the McCoy bounds for our model, with the lower bound giving a good estimate of the data.

Original languageEnglish (US)
Pages (from-to)385-392
Number of pages8
JournalInternational Journal of Engineering Science
Issue number3
StatePublished - 1985
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mechanics of Materials
  • General Engineering
  • Mechanical Engineering
  • General Materials Science


Dive into the research topics of 'Third-order bounds on the effective bulk and shear modulus of a dispersion of fully penetrable spheres'. Together they form a unique fingerprint.

Cite this