Constitutive laws describe material properties (strains and stresses). Since soft tissues are composed of fibres, which are not aligned, the angles at which the fibres are oriented have an impact on their mechanical properties. Here we consider the contributions from i) along the fibre $ \mathbf{f} $; ii) along the sheet $ \mathbf{s}$; and iii) in normal directions $ \mathbf{n} $.
$ \mathbf{L} = \begin{bmatrix}\mathbf{f} & \mathbf{s} & \mathbf{n}\end{bmatrix} $
Guccione et al. (1991) Transversely isotropic law
$ W = \frac{1}{2} C_1 (e^Q – 1) $
where
$ Q = C_2 E_{ff}^2 + C_3 (E_{ss}^2 +E_{nn}^2 +E_{sn}^2) + 2C_4 (E_{fs}E_{sf}+E_{fn}E_{nf}) $
Nash and Hunter (2000) Pole-zero strain energy function
$ W = k_{11} \frac{E^2_{11}}{|a_{11}-E_{11}|^{b_{11}}}+k_{22} \frac{E^2_{22}}{|a_{22}-E_{22}|^{b_{22}}}+k_{33} \frac{E^2_{33}}{|a_{33}-E_{33}|^{b_{33}}}+ … $
$ +k_{12} \frac{E^2_{12}}{|a_{12}-E_{12}|^{b_{12}}}+k_{13} \frac{E^2_{13}}{|a_{13}-E_{13}|^{b_{13}}}+k_{11} \frac{E^2_{23}}{|a_{23}-E_{23}|^{b_{23}}} $
Holzapfel and Ogden (2000/2009) – Neohookean model
$ W = c(I_1-3) $
Mooney-Rivlin model
$ W = c_1(I_1-3)+c_2(I_2-3) $