I have a material for which I know both the tensile storage modulus $E'$, and loss modulus $E''$ at a frequency of 1 Hz. As I understand it, $E'$ conceptually describes the elastic properties of the material, while $E''$ is related to viscous properties. Of course these can be combined to form the complex modulus $E^* = E' + iE''$. It appears that people will commonly plot $E^*$ as a vector in the complex plane because the phase angle tells you where on the scale of purely elastic ($0^\circ$) to purely viscous ($90^\circ$) the material is.
That is all starting to make a little sense conceptually, but for my material with known $E'$ and $E''$ at 1 Hz, I would like to be able to figure out its dynamic viscosity, $\eta$. To cut a long story short, I have some equations in terms of $\eta$, and I want to know whether or not it is justifiable to assume that the viscosity is negligible when the material is driven at 1 Hz. But the only data I have are $E'$ and $E''$ at 1 Hz.
Is there any mathematical way of connecting these concepts?
Update: I have found one possible clue. In these lecture notes (p14), the author notes in passing that
$$
\eta' = \frac{G''}{\omega}
$$
where $G''$ is the shear loss modulus, and $\eta'$ is the "effective viscosity." I wonder if this is the answer?
I am fairly new to studying viscoelastic materials, so any guidance would be greatly appreciated!
Best Answer
The typical approach is to compare the magnitudes of the storage modulus $G^\prime(1\,\text{Hz})$ and the loss modulus $G^{\prime\prime}(1\,\text{Hz})$. If the latter (former) is relatively small, then the material resembles an ideal spring (damper), and the viscosity is negligible (predominant). Equivalently, you could check whether the so-called loss tangent $\tan \delta=G^{\prime\prime}/G^\prime$ is small at 1 Hz.
The dynamic viscosity is typically defined in terms of an ideal damper (typical constitutive equations: $\sigma=\eta\dot\varepsilon$ or $\tau=\eta\dot\gamma$). For the more general case of a viscoelastic material with complex modulus ${G^\star}(\omega)=G^\prime(\omega)+iG^{\prime\prime}(\omega)$, we have the complex viscosity $\eta^\star={G^\star}(\omega)/\omega=\eta^\prime(\omega)+i\eta^{\prime\prime}(\omega)$. If you wish, you could call $\eta^{\prime\prime}$ the dynamic or effective viscosity, but that might invite confusion, as $\eta^\star$ is also sometimes called the dynamic viscosity.