Struct Ogden 

pub struct Ogden {
    pub bulk_modulus: Quantity<Stress>,
    pub shear_moduli: Vec<Quantity<Stress>>,
    pub exponents: Vec<Scalar>,
}

The Ogden hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear moduli $\mu_n$ for $n=1\ldots N$.
• The exponents $\alpha_n$ for $n=1\ldots N$.

External variables

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

Internal variables

• None.

Notes

• The shear modulus is given by $\mu=\boldsymbol{\alpha}\cdot\boldsymbol{\mu}/2$.
• The Ogden model reduces to the Neo-Hookean model when $N=1$ and $\alpha_1\to 2$.

---

1. R.W. Ogden, Proc. R. Soc. Lond. A 326, 565 (1972). ↩

Fields

bulk_modulus: Quantity<Stress>

The bulk modulus $\kappa$.

shear_moduli: Vec<Quantity<Stress>>

The shear moduli $\mu_n$.

exponents: Vec<Scalar>

The exponents $\alpha_n$.

Trait Implementations

impl Clone for Ogden

fn clone(&self) -> Ogden

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Ogden

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Elastic for Ogden

fn cauchy_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<CauchyStress, ConstitutiveError>

\boldsymbol{\sigma}(\mathbf{F}) = \sum_{n=1}^N \frac{\mu_n}{J}\,\mathrm{dev}\left({\mathbf{V}^*}^{\alpha_n}\right) + \frac{\kappa}{2}\left(J - \frac{1}{J}\right)\mathbf{1}

fn cauchy_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<CauchyTangentStiffness, ConstitutiveError>

\mathcal{T}_{ijkL}(\mathbf{F}) = \sum_{n=1}^N\frac{\mu_n}{J^{\alpha_n/3+1}}\left[\mathrm{dev}\left(\frac{\partial\mathbf{V}^{\alpha_n}}{\partial F_{kL}}\right)_{ij} - \left(\frac{\alpha_n}{3}+1\right)\mathrm{dev}(\mathbf{V}^{\alpha_n})_{ij}F_{kL}^{-T}\right] + \frac{\kappa}{2}\left(J + \frac{1}{J}\right)\delta_{ij}F_{kL}^{-T}

fn first_piola_kirchhoff_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<FirstPiolaKirchhoffStress, ConstitutiveError>

Calculates and returns the first Piola-Kirchhoff stress.

fn first_piola_kirchhoff_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<FirstPiolaKirchhoffTangentStiffness, ConstitutiveError>

Calculates and returns the tangent stiffness associated with the first Piola-Kirchhoff stress.

fn second_piola_kirchhoff_stress( &self, deformation_gradient: &DeformationGradient, ) -> Result<SecondPiolaKirchhoffStress, ConstitutiveError>

Calculates and returns the second Piola-Kirchhoff stress.

fn second_piola_kirchhoff_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> Result<SecondPiolaKirchhoffTangentStiffness, ConstitutiveError>

Calculates and returns the tangent stiffness associated with the second Piola-Kirchhoff stress.

impl Hyperelastic for Ogden

fn helmholtz_free_energy_density( &self, deformation_gradient: &DeformationGradient, ) -> Result<Quantity<EnergyDensity>, ConstitutiveError>

a(\mathbf{F}) = \sum_{n=1}^N \frac{\mu_n}{\alpha_n}\big[\mathrm{tr}({\mathbf{V}^*}^{\alpha_n}) - 3\big] + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl Solid for Ogden

fn bulk_modulus(&self) -> Quantity<Stress>

Returns the bulk modulus.

fn shear_modulus(&self) -> Quantity<Stress>

Returns the shear modulus.

fn jacobian<I, J>( &self, deformation_gradient: &DeformationGradientGeneral<I, J>, ) -> Result<Scalar, ConstitutiveError>

Calculates and returns the Jacobian.