Fung

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 model when $\mu_m\to 0$ or $c\to 0$.

cauchy_stress(model::Fung, F) -> Matrix{Float64}

\[\boldsymbol{\sigma}(\mathbf{F}) = \frac{1}{J}\left[\mu + \mu_m\left(e^{c[\mathrm{tr}(\mathbf{B}^* ) - 3]} - 1\right)\right]{\mathbf{B}^* }' + \frac{\kappa}{2}\left(J - \frac{1}{J}\right)\mathbf{1}\]

cauchy_tangent_stiffness(
    model::Fung,
    F
) -> Array{Float64, 4}

\[\mathcal{T}_{ijkL}(\mathbf{F}) =\, \frac{1}{J^{5/3}}\left[\mu + \mu_m\left(e^{c[\mathrm{tr}(\mathbf{B}^* ) - 3]} - 1\right)\right]\left(\delta_{ik}F_{jL} + \delta_{jk}F_{iL} - \frac{2}{3}\,\delta_{ij}F_{kL} - \frac{5}{3} \, B_{ij}'F_{kL}^{-T} \right)+ \frac{2c\mu_m}{J^{7/3}}\,e^{c[\mathrm{tr}(\mathbf{B}^* ) - 3]}B_{ij}'B_{km}'F_{mL}^{-T} + \frac{\kappa}{2} \left(J + \frac{1}{J}\right)\delta_{ij}F_{kL}^{-T}\]

first_piola_kirchhoff_stress(
    model::Fung,
    F
) -> Matrix{Float64}

first_piola_kirchhoff_tangent_stiffness(
    model::Fung,
    F
) -> Array{Float64, 4}

second_piola_kirchhoff_stress(
    model::Fung,
    F
) -> Matrix{Float64}

second_piola_kirchhoff_tangent_stiffness(
    model::Fung,
    F
) -> Array{Float64, 4}

helmholtz_free_energy_density(model::Fung, F) -> Float64

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