Mooney-Rivlin

The Mooney-Rivlin hyperelastic constitutive model.^[1]^,^[2]

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus $\mu_m$.

External variables

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

Internal variables

• None.

Notes

• The Mooney-Rivlin model reduces to the Neo-Hookean model model when $\mu_m\to 0$.

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

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

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

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

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

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

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

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

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

\[a(\mathbf{F}) = \frac{\mu - \mu_m}{2}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right] + \frac{\mu_m}{2}\left[I_2(\mathbf{B}^*) - 3\right] + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]\]