Examples

To clarify the situation, we show some examples in Figures 1-8. The details of the analysis that produces these figures are summarized in the Appendix. The main point is that, for the compliance version of the analysis, the contours of constant energy are ellipses when the vector f in (18) is interpreted as a stress. Analogously, when the vector is treated as a strain, the contours of constant energy are ellipses for the dual (or stiffness) formulation. If we choose to think of these figures as diagrams in the complex plane, then we note that -- while circles and lines transform to circles and lines when transforming back and forth between these two planes -- the shapes of ellipses are not preserved (except, of course, in the special case - which is precisely that of isotropy - when the ellipses degenerate to circles). Eigenvectors are determined by the directions in which the points of contact of these two curves lie (indicated by red circles).

 

GlassStrain2Box4
Figure 1 For a glassy porous material having bulk modulus K_dr = 18.52 GPa and shear modulus G_dr = 13.89 GPa, the locus of points $z = R e^{i\theta}$-- see equation (36) -- having constant energy U = 900 GPa, when the linear combination of pure compression and pure uniaxial shear is interpreted as strain field applied to the stiffness matrix (solid black line). The plot is in the complex z-plane, with the inverse of the corresponding expression for the compliance energy superposed for comparison (dashed blue line). Red circles at the two points of intersection correspond to the two eigenvectors of the system of equations. The ellipse (solid black line) in this plane corresponds to the more complex curve in Figure 2.

 

GlassStress
Figure 2 Same parameters as Figure 1, but the linear combination of pure compression and pure uniaxial shear is interpreted as a stress field and is applied to the compliance matrix (dashed blue line). The plot is again in the complex z-plane, with the inverse of the corresponding expression for the stiffness energy superposed for comparison (solid black line). Red circles at the two points of intersection correspond to the two eigenvectors of the system of equations. The ellipse (dashed blue line here) corresponds to the more complex curve in Figure 1.

Figures 1 and 2 present an example based on a glassy material. Typical values for the bulk and shear moduli of glass were used: K_m = 46.3 GPa and G_m = 30.5 GPa, respectively. The value of the Biot-Willis coefficient was arbitrarily chosen as $\alpha = 0.6$,so K_dr = 18.52 GPa. Taking Poisson's ratio as $\nu_{dr} = 0.2$, we have G_dr = 13.89 GPa. Skempton's coefficient was chosen for simplicity to be $B \equiv 1$ in this and all the other examples as well. (This choice is extreme because it implies that K_u = K_m. But, since our interest here is in analysis of the undrained shear modulus, the study of this limit is particularly useful to us.) The most anisotropic choices of $\beta_1$ and $\beta_3$ were used that would not produce absurd (negative) values of the diagonal coefficients for either s^*_ij or c^*_ij, and that also would not produce G_u > G_m. [G_u determined by (32) amd (33) is a type of upper bound - actually the Voigt average. Values of this bound that might exceed G_m need not be considered.] For glass, these values were found to be $\beta_1 = 0.15\alpha/K_{dr}$ and $\beta_3 = 0.70\alpha/K_{dr}$.The value of the energy used for normalization was U = 900.0 GPa. Computed values for the effective and undrained shear moduli were G_eff = 25.43 GPa and G_u = 15.28 GPa.

For the remaining three sets of examples, the values used for the moduli of the samples are taken from results contained in Berryman (2004a), wherein it was shown how certain laboratory data could be fit using an elastic differential effective medium scheme. These results are summarized in the TABLE.

 

SW2Strain
Figure 3 Same as Figure 1 for Sierra White Granite using the parameters from the TABLE.

 

SW2Stress
Figure 4 Same as Figure 2 for Sierra White Granite using the parameters from the TABLE.

Figures 3 and 4 present results for Sierra White granite. Laboratory data on this material were presented by Murphy (1982). The values chosen for $\beta_1$ and $\beta_3$ were $\beta_1 = 0.05\alpha/K_{dr}$ and $\beta_3 = 0.90\alpha/K_{dr}$.The value of the energy used for normalization was $U \simeq 900.0$ GPa. Computed values for the effective and undrained shear moduli were G_eff = 39.8 GPa and G_u = 28.3 GPa.

 

SCV2Strain
Figure 5 Same as Figure 1 for Schuler-Cotton Valley Sandstone using the parameters from the TABLE.

 

SCV2Stress
Figure 6 Same as Figure 2 for Schuler-Cotton Valley Sandstone using the parameters from the TABLE.

Figures 5 and 6 present results for Schuler-Cotton Valley sandstone. Laboratory data on this material were also presented by Murphy (1982). The values chosen for $\beta_1$ and $\beta_3$ were $\beta_1 = 0.20\alpha/K_{dr}$ and $\beta_3 = 0.60\alpha/K_{dr}$.The value of the energy used for normalization was $U \simeq 900.0$ GPa. Computed values for the effective and undrained shear moduli were G_eff = 35.8 GPa and G_u = 17.7 GPa.

 

SRSStrain
Figure 7 Same as Figure 1 for Spirit River Sandstone using the parameters from the TABLE.

 

SRSStress2Box2
Figure 8 Same as Figure 2 for Spirit River Sandstone using the parameters from the TABLE.

Figures 7 and 8 present results for Spirit River sandstone. Laboratory data on this material were presented by Knight and Nolen-Hoeksema (1990). The values chosen for $\beta_1$ and $\beta_3$ were $\beta_1 = 0.25\alpha/K_{dr}$ and $\beta_3 = 0.50\alpha/K_{dr}$.The value of the energy used for normalization was $U \simeq 900.0$ GPa. Computed values for the effective and undrained shear moduli were G_eff = 20.11 GPa and G_u = 12.41 GPa.