Write the generalized Hooke's law as
[tex]
\sigma^{rs}=E^{rsij}\epsilon_{ji}, r,s,i,j \in (1,2,3),[/tex]
for relating the stress tensor [itex]\sigma[/itex] and the infinitesimal strain tensor [itex]\epsilon[/itex], 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
[tex]
\sigma_{ij}=\lambda\delta_{ij}\epsilon_{kk}+2\mu\epsilon_{ji}, [/tex]
where the Lame coefficients are given as (by introducing the Young's
modulus E and Poisson's ratio [itex]\nu[/itex] )
[tex]
\lambda= \frac{\nu E}{(1-2\nu)(1+\nu)} [/tex]
[tex]
\mu=E/2(1+\nu) .[/tex]
The generalized Hooke's law in a general coordinate system can be written as (using the Lame constants again)
[tex]
\sigma^{pq}=\lambda I^{\epsilon}_{1} g^{pq}+2 \mu g^{ip} g^{jq} \epsilon_{ij},[/tex]
where [itex]I^{\epsilon}_{1}[/itex] is the first invariant of [itex]trace(\epsilon)[/itex].
