# Hooke's Law

Hooke's Law expresses the linear relation between stress and strain. It may be written either as

stress = stiffness * strain :
${\displaystyle \sigma _{ij}=\sum _{k,l}^{3}C_{ijkl}\varepsilon _{kl}}$

or as

strain = compliance * stress :
${\displaystyle \varepsilon _{ij}=\sum _{k,l}^{3}S_{ijkl}\sigma _{kl}}$

Both the stiffnesses ${\displaystyle C_{ijkl}}$ and the compliances ${\displaystyle S_{ijkl}}$ have 3x3x3x3=81 elements, but because of symmetries, they are not all independent. The number of independent elements depends upon the symmetry class of the medium:
Isotropic:2
Polar anisotropic: 5
Orthorhombic: 9
Monoclinic: 12