I frequently see the equation $$ \sigma_t = E\alpha \Delta T $$
as the equation for thermal stress.
Where $E$ is Young's modulus, $\alpha$ is the CTE, and $\Delta T$ is the change in temperature.
If we were to write out, for an isotropic material, the stiffness tensor we get the expression below
For thermal stress, the shear components of thermal strain would be zero, so we are left with the normal strains $\varepsilon_{ii}$ as the only potentially non-zero strains. These strains are identical for $i=1,2,3$ since the material is isotropic, and each is equal to $\alpha \Delta T$.
This is where I am confused. If we use the above relationship, one component of the normal stress would be $\sigma_{ii} = (2\mu+3\lambda) \alpha \Delta T$, but this is clearly not equal to $E\alpha \Delta T$, since $2\mu + 3\lambda \neq E$.
I must be misunderstanding something fundamental. Could someone point out what this might be?