Almansi-Hamel

The Almansi-Hamel elastic constitutive model.

Parameters

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

External variables

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

Internal variables

• None.

Notes

• The Almansi-Hamel strain measure is given by $\mathbf{e}=\tfrac{1}{2}(\mathbf{1}-\mathbf{B}^{-1})$.

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

\[\boldsymbol{\sigma}(\mathbf{F}) = \frac{2\mu}{J}\,\mathbf{e}' + \frac{\kappa}{J}\,\mathrm{tr}(\mathbf{e})\mathbf{1}\]

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

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

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

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

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

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