Saint Venant-Kirchhoff

The Saint Venant-Kirchhoff hyperelastic constitutive model.

Parameters

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

External variables

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

Internal variables

• None.

Notes

• The Green-Saint Venant strain measure is given by $\mathbf{E}=\tfrac{1}{2}(\mathbf{C}-\mathbf{1})$.

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

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

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

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

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

\[\mathbf{S}(\mathbf{F}) = 2\mu\mathbf{E}' + \kappa\,\mathrm{tr}(\mathbf{E})\mathbf{1}\]

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

\[\mathcal{G}_{IJkL}(\mathbf{F}) = \mu\,\delta_{JL}F_{kI} + \mu\,\delta_{IL}F_{kJ} + \left(\kappa - \frac{2}{3}\,\mu\right)\delta_{IJ}F_{kL}\]

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

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