Struct SaintVenantKirchhoff

pub struct SaintVenantKirchhoff {
    pub bulk_modulus: Scalar,
    pub shear_modulus: Scalar,
    pub bulk_viscosity: Scalar,
    pub shear_viscosity: Scalar,
}

The Saint Venant-Kirchhoff hyperviscoelastic constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The bulk viscosity $\zeta$.
• The shear viscosity $\eta$.

External variables

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

Internal variables

• None.

Notes

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

Fields

bulk_modulus: Scalar

The bulk modulus $\kappa$.

shear_modulus: Scalar

The shear modulus $\mu$.

bulk_viscosity: Scalar

The bulk viscosity $\zeta$.

shear_viscosity: Scalar

The shear viscosity $\eta$.

Trait Implementations

impl Debug for SaintVenantKirchhoff

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

Formats the value using the given formatter.

impl ElasticHyperviscous for SaintVenantKirchhoff

fn viscous_dissipation( &self, deformation_gradient: &DeformationGradient, deformation_gradient_rate: &DeformationGradientRate, ) -> Result<Scalar, ConstitutiveError>

Calculates and returns the viscous dissipation.

\phi(\mathbf{F},\dot{\mathbf{F}}) = \eta\,\mathrm{tr}(\dot{\mathbf{E}}^2) + \frac{1}{2}\left(\zeta - \frac{2}{3}\,\eta\right)\mathrm{tr}(\dot{\mathbf{E}})^2

fn dissipation_potential( &self, deformation_gradient: &DeformationGradient, deformation_gradient_rate: &DeformationGradientRate, ) -> Result<Scalar, ConstitutiveError>

Calculates and returns the dissipation potential.

impl Hyperviscoelastic for SaintVenantKirchhoff

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

Calculates and returns the Helmholtz free energy density.

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

impl Solid for SaintVenantKirchhoff

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.

impl Viscoelastic for SaintVenantKirchhoff

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

Calculates and returns the second Piola-Kirchhoff stress.

\mathbf{S}(\mathbf{F},\dot\mathbf{F}) = 2\mu\mathbf{E}' + \kappa\,\mathrm{tr}(\mathbf{E})\mathbf{1} + 2\eta\dot{\mathbf{E}}' + \zeta\,\mathrm{tr}(\dot{\mathbf{E}})\mathbf{1}

fn second_piola_kirchhoff_rate_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, _: &DeformationGradientRate, ) -> Result<SecondPiolaKirchhoffRateTangentStiffness, ConstitutiveError>

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

\mathcal{W}_{IJkL}(\mathbf{F}) = \eta\,\delta_{JL}F_{kI} + \eta\,\delta_{IL}F_{kJ} + \left(\zeta - \frac{2}{3}\,\eta\right)\delta_{IJ}F_{kL}

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

Calculates and returns the Cauchy stress.

fn cauchy_rate_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, deformation_gradient_rate: &DeformationGradientRate, ) -> Result<CauchyRateTangentStiffness, ConstitutiveError>

Calculates and returns the rate tangent stiffness associated with the Cauchy stress.

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

Calculates and returns the first Piola-Kirchhoff stress.

fn first_piola_kirchhoff_rate_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, deformation_gradient_rate: &DeformationGradientRate, ) -> Result<FirstPiolaKirchhoffRateTangentStiffness, ConstitutiveError>

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

impl Viscous for SaintVenantKirchhoff

fn bulk_viscosity(&self) -> &Scalar

Returns the bulk viscosity.

fn shear_viscosity(&self) -> &Scalar

Returns the shear viscosity.