Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media

Non-triaxial testing geometries

Clearly, it would also be natural to introduce other measures of the $\beta_i$ coefficients as well, especially if the measurements are not being constrained to the triaxial testing configuration. So I might imagine that three such coefficients could be measured according to:

$$-p_f = B\left[\sigma_m + \sum_i A_i(\sigma_{ii} - \sigma_m)\right],$$ (54)

where

$$A_i = \frac{\beta_i}{\beta_1+\beta_2+\beta_3},$$ (55)

for $i = 1,2,3$. In general, no more than two of these $A_i$ coefficients can be independent since $\sum_{i=1,2,3} A_i \equiv 1$. But, for general testing configurations, there could be two useful and distinct measurements to be gathered from deviatoric response testing, although only one was available in the triaxial testing configuration.

In order to be able to deduce the values of the $\beta_i$'s from the $A_i$'s, I need to know the value of the sum $\beta_1 + \beta_2 + \beta_3 = \gamma B$. I also needed to know the value of $B$ to determine any of the $A_i$'s, but the value of $\gamma$ is harder to determine independently. The values of $K_R^u$ and the total $K_R^g$ are both usually easier to determine, so it is likely enough information is available to compute the $\beta_i$ sum this way:

$$\beta_1 + \beta_2 + \beta_3 = \frac{\frac{1}{K_R^u} - \frac{1}{K_R^g}}{1-B} = \gamma B.$$ (56)

If the $\beta_i$ sum has been computed using (58), then clearly I also have

$$\beta_i = A_i\left(\frac{\frac{1}{K_R^u} - \frac{1}{K_R^g}}{1-B}\right).$$ (57)

Once I have computed the $\beta_i$'s, I also find (if I want to, although it is not usually critical information) the values of the partial sums of the grain Reuss modulus:

$$\frac{1}{3\overline{K}_i^g} \equiv s_{i1}^u + s_{i2}^u + s_{i3}^u - \beta_i(1-B).$$ (58)

This additional information is therefore available if needed for some other reason such as determining how well-stirred the particles composing a given granular medium might be.

Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media