3. Zener or standard linear solid model
The figure shows the standard model used in Sørsdal (2008) taken from Horton. In this outline we will see that what we call model 4 in Hortons figure can be simplified to one rel. time τc that is the inverse of the highest frequency in the seismic band.
We will discuss the linear solid viscoelastic model and follow the lines of Wang (2008):
The Zener (1948) or standard linear solid model is the most general linear equation that links the stress and strain as we have seen in Sørsdal (2008). It defines the attenuation coefficient as:
and the phase velocity as
where c0 is the phase velocity c(w) for w -> 0, and τc and Qc are two constant parameters describing the attenuation property in the standard linear solid model: the attenuation at the peak of the attenuation function with respect to the frequency is Q-1c , and the (angular) frequency at this location is τc-1
Mathematical outline is as following:
The Zener (1948) or standard linear solid model is the most general linear equation in stress a, strain e and their first time derivatives a and s . In the Fourier transform domain (equation 3.4), the stress ∑(w) and the strain E(w) are linked by the complex modulus defined as (Ben-Menahem and Singh, 1981):
where is the strain relaxation time, is the stress relaxation time and MR is a deformation modulus with sub-index R denoting the relaxed modulus.
The mechanical 'relaxation' means that the strain produced by the sudden application of a fixed stress to a material increases asymptotically with time. Similarly, the stress produced when the material is suddenly strained relaxes asymptotically (Kolsky, 1953). It is found that stress waves whose periods are close to the 'relaxation times' of such a medium are severely attenuated when passing through it.
Physically, the 'relaxed' elastic modulus MR is the final value of the ratio of stress to strain after relaxation has taken place, whereas a so-called 'unrelaxed' elastic modulus, Mv, is the initial value of the ratio of stress to strain, before relaxation has time to occur. Mathematically, the relaxed modulus MR is obtained from M(w) for w->0 (see equation 3.5).
The unrelaxed modulus can be given by
MU = lim M(w) = MR τ3/ τ4 (3.6)
Thus, MR and MU are also called the low- and the high-frequency moduli, respectively.
A special case of the standard linear solid model is the Kelvin-Voigt viscoelastic model obtained when = 0, so that
M(w) = MR (1 + iw τ3) (3.7)
In the Kelvin-Voigt viscoelastic model, the stress a in the medium for a given strain s and strain rate s is the superposition of linear elastic and linear viscous stresses
The real part of the complex modulus M(w) in equation (3.5) may be written as
and the attenuation Q-1 is given by
which measures the lag of the strain behind the stress. We assume here τ3 > τ4. Substituting them into the equations for attenuation and dispersion given earlier, we may obtain the attenuation
and the phase velocity
From (3.10) and (3.11) we can draw a very important conclusion about this model. Since τ3 > τ4 the paranthesis on right side of expression (3.11) will always be less than one. That means that the wave number k always will be less than one. Because of that we will never get a causal solution of the wave equation using this model.
We proceed with Wang introducing c0 as the phase velocity for w -> 0, and we have
c0= = limw->∞ (3.12)
Attempting to use a single relaxation time, we may define two parameters as follows:
For the standard linear solid model, considering equation (3.6), this can be written as Qc-1 =1/2∆M, where ∆M is the modulus defect
that is, the normalized difference between the unrelaxed and relaxed moduli. For the standard linear solid model, the attenuation at the peak of the attenuation function with respect to frequency is 1/2∆M and the (angular) frequency at this location is l/τc.
The parameters τ3 and τ4 in (3.10) and (3.11) may be expressed as (Casula and Carcione, 1992):
obtained from the Qc and τc definition in (3.13). Therefore, we may rewrite expressions (3.10) and (3.11) as for the attenuation for the phase velocity. With these two expressions, Zener's Q model may be expressed as
To compare this standard linear solid model with the Kolsky model, may first set
since the highest frequency of the seismic band has the strongest attenuation.
We then use the following two approximations:
Right side of equation 3.6 can be written: