Editing Soft tissue (section)

=== Fung-elastic material ===
[[Yuan-Cheng Fung|Fung]] developed a [[constitutive equation]] for preconditioned soft tissues which is 
:<math>W = \frac{1}{2}\left[q + c\left( e^Q -1 \right) \right]</math>
with
:<math>q=a_{ijkl}E_{ij}E_{kl} \qquad Q=b_{ijkl}E_{ij}E_{kl}</math>
quadratic forms of [[strain (mechanics)|Green-Lagrange strains]] <math>E_{ij}</math> and <math>a_{ijkl}</math>, <math>b_{ijkl}</math> and <math>c</math> material constants.<ref name="Fung"/> <math>W</math> is the [[Strain energy density function|strain energy function]] per volume unit, which is the mechanical strain energy for a given temperature.

==== Isotropic simplification ====
The Fung-model, simplified with isotropic hypothesis (same mechanical properties in all directions). This written in respect of the principal stretches (<math>\lambda_i</math>):
:<math>W = \frac{1}{2}\left[a(\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3) + b\left( e^{c(\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3)} -1 \right) \right]</math> , 
where a, b and c are constants.

==== Simplification for small and big stretches ====
For small strains, the exponential term is very small, thus negligible.
:<math>W = \frac{1}{2}a_{ijkl}E_{ij}E_{kl}</math>

On the other hand, the linear term is negligible when the analysis rely only on big strains.
:<math>W = \frac{1}{2}c\left( e^{b_{ijkl}E_{ij}E_{kl}} -1 \right)</math>