Spherical inclusions

Porous glass foam: Walsh et al. (1965) made measurements on the compressiblity (1/K) of a porous glass over a wide range of porosities. The samples they used were fabricated from borosilicate powder die-pressed and then sintered in alumina crucibles. Depending on the thermal history of the samples, they obtained porous glass with porosities ranging from 0.70 to near zero. Porosity measurements were stated to be accurate to $\pm 0.01$.

The resulting glass samples were foam, with microgeometry such that large open dry pores were dispersed in a continuous glass background that thinned to a network of needle-like filaments at the highest porosities. Linear compressibility of the samples was measured by Walsh et al. (1965) using electrical resistance strain gages attached to sample surfaces in a high-pressure medium. The two explicit methods under consideration (MT and KT) produce the same estimates for composites with spherical inclusions and Figure 1 shows that these estimates are quite good at lower concentrations of voids, but they serve more as upper bounds than as estimates for the higher concentrations. Ferrari and Filipponi (1991) and Zimmerman (1991) have also demonstrated that the MT method produces good estimates of the compressibilities of these foam samples.

 

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Figure 1 Bulk moduli of porous glass foam computed from compressibilities measured by Walsh et al.(1965). Measured values are compared to theoretical predictions of the Mori-Tanaka theory and the Kuster-Toksöz (1974) theory, which happen to coincide for composites with spherical inclusions. The same curve is also obtained when either theory assumes the voids are spherical while the glass particles take any other ellipsoidal shape.

Sintered glass beads: Synthetic sandstones were made by sintering glass beads made from soda-lime plate glass at peak temperatures of 700-760$^{\circ}$ C to achieve porosities of $\sim$1-40% (Berge et al., 1993). Samples were cored with diameters of 2.54$\pm$0.002 cm and ground to right circular cylinders with lengths of 2.54$\pm$0.002 cm. The sample volumes, masses, and glass density were used to compute porosities with an uncertainty of $\sim$0.6% for samples with porosities below $\sim$30%, and an uncertainty of $\sim$1% for the higher porosity samples. A pulse transmission technique was used with 0.5 MHz P and 5 MHz S transducers to determine the compressional (P) and shear (S) wave velocities of both dry and saturated samples. Velocity measurements had an uncertainty of $\sim$1%. All measurements were made at atmospheric pressure and room temperature. Berge et al. (1993) showed that the SC approximation produced good estimates of the observed velocities of these synthetic sandstones for all porosities because the granular structure of these synthetic sandstones is compatible with this theory. The results of these calculations are displayed here in Figure 2.

 

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Figure 2 Bulk moduli of sintered glass bead samples measured by Berge et al.(1993). Measured values are compared to predictions of the SC theory (Budiansky, 1965; Hill, 1965; Berryman, 1980) for spherical inclusions (solid line). Dot dash line is the result of the SC method assuming spherical glass particles and needle shaped voids, which is a reasonable model of a granular material with connected porosity.

 

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Figure 3 Bulk moduli of porous glass foam computed from compressibilities measured by Walsh et al.(1965). Measured values are compared to theoretical predictions of the Mori-Tanaka (1973) theory (solid line) and the Kuster-Toksöz theory (dashed line) assuming the voids are needle shaped. Dot dash line is the result of the SC method assuming needle shaped glass and spherical voids.