# Hooke's Law in 3D and in plane stress problems

Using Lame's constants, the Hooke's law of isotropic materials in 3-dimensions can be written as: \begin{aligned} \sigma_{ij}&=\lambda \varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}=c_{ijk\ell}\varepsilon_{k\ell}\\ c_{ijk\ell}&=\lambda\delta_{ij}\delta_{k\ell}+\mu(\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}) \end{aligned}\tag{1}

While in plane stress problems, Hooke's law seems to be different from the 3D case.

So my question is: Does the relation $(1)$ still hold in plane stress problem? Any help will be greatly appriciated!

• How is it different? Can you provide more details. May 12 '16 at 16:09
• It's in the name. Plane stress means all stresses have 0 in the z direction, but strains don't (due to Poisson's ratio). Plane strain is the opposite. They have equivalents in 3D where either parts are very thin or very thick in the z direction. Aug 25 '19 at 1:07