Soft tissue

Even though soft tissues have viscoelastic properties, i.e. stress as function of strain rate, it can be approximated by a hyperelastic model after precondition to a load pattern. After some cycles of loading and unloading the material, the mechanical response becomes independent of strain rate.

${\displaystyle \mathbf {S} =\mathbf {S} (\mathbf {E} ,{\dot {\mathbf {E} }})\quad \rightarrow \quad \mathbf {S} =\mathbf {S} (\mathbf {E} )}$

Despite the independence of strain rate, preconditioned soft tissues still present hysteresis, so the mechanical response can be modeled as hyperelastic with different material constants at loading and unloading. By this method the elasticity theory is used to model an inelastic material. Fung has called this model as pseudoelastic to point out that the material is not truly elastic.[8]

In physiological state soft tissues usually present residual stress that may be released when the tissue is excised. Physiologists and histologists must be aware of this fact to avoid mistakes when analyzing excised tissues. This retraction usually causes a visual artifact.[8]

Fung developed a constitutive equation for preconditioned soft tissues which is

${\displaystyle W={\frac {1}{2}}\left[q+c\left(e^{Q}-1\right)\right]}$

with

${\displaystyle q=a_{ijkl}E_{ij}E_{kl}\qquad Q=b_{ijkl}E_{ij}E_{kl}}$

quadratic forms of Green-Lagrange strains ${\displaystyle E_{ij}}$ and ${\displaystyle a_{ijkl}}$, ${\displaystyle b_{ijkl}}$ and ${\displaystyle c}$ material constants.[8] ${\displaystyle W}$ is the strain energy function per volume unit, which is the mechanical strain energy for a given temperature.

The Fung-model, simplified with isotropic hypothesis (same mechanical properties in all directions). This written in respect of the principal stretches (${\displaystyle \lambda _{i}}$):

${\displaystyle W={\frac {1}{2}}\left[a(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3)+b\left(e^{c(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3)}-1\right)\right]}$ ,

where a, b and c are constants.

For small strains, the exponential term is very small, thus negligible.

${\displaystyle W={\frac {1}{2}}a_{ijkl}E_{ij}E_{kl}}$

On the other hand, the linear term is negligible when the analysis rely only on big strains.

${\displaystyle W={\frac {1}{2}}c\left(e^{b_{ijkl}E_{ij}E_{kl}}-1\right)}$

${\displaystyle W=-{\frac {\mu J_{m}}{2}}\ln \left(1-\left({\frac {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3}{J_{m}}}\right)\right)}$

where ${\displaystyle \mu >0}$ is the shear modulus for infinitesimal strains and ${\displaystyle J_{m}>0}$ is a stiffening parameter, associated with limiting chain extensibility.[10] This constitutive model cannot be stretched in uni-axial tension beyond a maximal stretch ${\displaystyle J_{m}}$, which is the positive root of

${\displaystyle \lambda _{m}^{2}+2\lambda _{m}-J_{m}-3=0}$