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In a paper by Kees Wapenaar titled, "Retrieving the Elastodynamic Green's Function of an Arbitrary Inhomogeneous Medium by Cross Correlation" (2004), the following is stated:

"In the space-frequency domain the particle velocity and stress tensor in an inhomogeneous anisotropic lossless medium obey the equation of motion $j\omega\rho \hat{v}_i - \partial_i\hat{\tau}_{ij} = \hat{f}_i$ and the stress-strain relation $-j\omega s_{ijkl}\hat{\tau}_{kl} + (\partial_j\hat{v}_i + \partial_i\hat{v}_j)/2 = \hat{h}_{ij}$, where $\partial_j$ is the partial derivative in the $x_j$ direction, $\bf\rho(x)$ the mass density of the medium, $s_{ijkl}\bf(x)$ its compliance, $\hat{f}_i(\bf x,\omega)$ the external volume force, and $\hat{h}_{ij}(\bf x,\omega)$ the external deformation rate."

I'm well aware of where the equation of motion comes from, however, I've been going nuts trying to figure out where this stress-strain relationship comes from. I know that for a linear elastic material the strain tensor can be expressed in the usual space-time domain as $\epsilon_{ij}=(\partial_ju_i + \partial_iu_j)/2$. Furthermore, I'm also aware that one may invert Hooke's Law for a linearly elastic material, again, in the space-time domain, to obtain $\epsilon_{ij}=s_{ijkl}\tau_{kl}$. If one differentiates both of these expressions with respect to time and then Fourier transforms them temporally, the resulting relations are ${\hat{h}_{ij}} =(\partial_j \hat{v}_i + \partial_i \hat{v}_j)/2 $ and $\hat{h}_{ij}=j\omega s_{ijkl}\hat{\tau}_{kl}$, respectively.

Anyway, all that to ask, where does such a stress-strain relationship come from? I would appreciate any insight. I can't seem to fit the rest of the pieces together.

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It seems you're exactly right, since you correctly have the stress-strain relation. Perhaps an overzealous editor suspected a typo and changed

$j\omega s_{ijkl}\hat{\tau}_{kl} = (\partial_j\hat{v}_i + \partial_i\hat{v}_j)/2 = \hat{h}_{ij}$

to

$-j\omega s_{ijkl}\hat{\tau}_{kl} + (\partial_j\hat{v}_i + \partial_i\hat{v}_j)/2 = \hat{h}_{ij}$

No further results of the paper rely on this equation, since the Greens function solution is simply posited later.

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  • $\begingroup$ I suspect you're right. I managed to do the rest of the derivation in the paper based on the usual stress-strain relation. I guess published papers can contain mistakes after all! $\endgroup$ – Sigma Jun 4 '15 at 14:23

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