Neo-Hookean

The Neo-Hookean hyperelastic constitutive model.^[1]

Parameters

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

External variables

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

Internal variables

• None.

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

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

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

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

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

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

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

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

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

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