Write the generalized Hooke's law as
<br />
\sigma^{rs}=E^{rsij}\epsilon_{ji}, r,s,i,j \in (1,2,3),<br />
for relating the stress tensor \sigma and the infinitesimal strain tensor \epsilon, where E is the elasticity tensor (by postulating the existence of a
strain energy density has 21 independent coefficients).
For homogeneous isotropic material the elasticity tensor, the generalized Hooke's aw can be expressed using the Lame coefficients as
<br />
\sigma_{ij}=\lambda\delta_{ij}\epsilon_{kk}+2\mu\epsilon_{ji}, <br />
where the Lame coefficients are given as (by introducing the Young's
modulus E and Poisson's ratio \nu )
<br />
\lambda= \frac{\nu E}{(1-2\nu)(1+\nu)} <br />
<br />
\mu=E/2(1+\nu) .<br />
The generalized Hooke's law in a general coordinate system can be written as (using the Lame constants again)
<br />
\sigma^{pq}=\lambda I^{\epsilon}_{1} g^{pq}+2 \mu g^{ip} g^{jq} \epsilon_{ij},<br />
where I^{\epsilon}_{1} is the first invariant of trace(\epsilon).