Rayleigh wave speed

 

Qrtzsemilogvrvs3b_rho Figure 3 Computation of Rayleigh wave speed in quartz with horizontally aligned cracks for three choices of penny-crack aspect ratio, and a range of values of the crack density $\rho$.If the assumption/prediction that aspect ratio is does not really matter, just crack density, then all three curves should overlap, as they do here at the lower crack densities.

Now, to provide one simple illustration of the use of what has been presented so far, consider the well-known formula for the Rayleigh wave speed v = v_R in an isotropic medium [see Ewing et al. (1957), Al-Husseini et al. (1981), Weertman (1996)]:  

$$\beta_S \beta_P = \beta_{2S}^4,$$ (6)

where $\beta_S = \sqrt{1 - v^2/v_s^2}$, $\beta_P = \sqrt{1 - v^2/v_p^2}$,$\beta_{2S} = \sqrt{1 - v^2/2v_s^2}$, with v_s being the (isotropic) shear wave speed and v_p being the (isotropic) compressional wave speed in the host medium. For an anisotropic medium having the same transversely isotropic (VTI) symmetry that I have been considering (for the case of randomly oriented vertical cracks), Musgrave (1959) shows that the equivalent result for the Rayleigh wave speed v = v_R in the plane perpendicular to the VTI axis of symmetry is determined by the cubic equation  

$$\frac{1}{16}q^3 - \frac{1}{2}q^2 + \left(\frac{3}{2}-\frac{c... ...c_{11}}\right)q + \left(\frac{c_{66}}{c_{11}} - 1\right) = 0,$$ (7)

where $q \equiv \rho_0v^2/c_{66}$ and $\rho_0$ is the density of the medium (which is assumed to be the same as that of the pure material without cracks, since the cracks are very flat and are not introducing any significant amount of porosity). [It is not difficult to check that (6) and (7) are equivalent when the elastic medium is isotropic.]

From the definitions of $\gamma$ and $\epsilon$, it is now straightforward to see that

$$c_{66} = c_{44}(1 + 2\gamma)$$ (8)

and

$$c_{11} = c_{33}(1 + 2\epsilon).$$ (9)

Shear modulus $c_{44} = G/(1+\eta_2 G \rho)$, while c₃₃ is found by inverting $S_{ij}+\Delta S_{ij}$ for the 33 component of the stiffness matrix. So we can now easily compute the Rayleigh wave speed by solving the cubic equation (7). Some results of this type are displayed in Figure 3. In particular, we find that the crack density is indeed a good parameter to use, as all these plots for different choices of crack aspect ratio clearly overlap to numerical accuracy in the low crack density range.