Fung model

The Fung hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus ${\mu }_m$.
• The exponent $c$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Fung model reduces to the Neo-Hookean model when ${\mu }_m\to 0$ or $c\to 0$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{\mu -{\mu }_m}{2}\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]+\frac{{\mu }_m}{2c}\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{1}{J}\left[\mu +{\mu }_m\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)\right]{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{1}{J^{5/3}}\left[\mu +{\mu }_m\left(e^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}-1\right)\right]\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\frac{2c{\mu }_m}{J^{7/3}}{e}^{c[\mathrm{tr}({\mathbf{B}}^{\ast })-3]}{B}_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. Y.C. Fung, Am. J. Physiol. 213, 1532 (1967). ↩