Gent

The Gent hyperelastic constitutive model.^[1]

Parameters

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

External variables

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

Internal variables

• None.

Notes

• The Gent model reduces to the Neo-Hookean model model when $J_m\to\infty$.

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

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

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

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

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

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

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

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

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

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