Struct SaintVenantKirchhoff

pub struct SaintVenantKirchhoff {
    pub bulk_modulus: Scalar,
    pub shear_modulus: Scalar,
    pub coefficient_of_thermal_expansion: Scalar,
    pub reference_temperature: Scalar,
}

The Saint Venant-Kirchhoff thermohyperelastic constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The coefficient of thermal expansion $\alpha$.
• The reference temperature $T_\mathrm{ref}$.

External variables

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

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 $\mu$.

coefficient_of_thermal_expansion: Scalar

The coefficient of thermal expansion $\alpha$.

reference_temperature: Scalar

The reference temperature $T_\mathrm{ref}$.

Trait Implementations

impl Debug for SaintVenantKirchhoff

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

Formats the value using the given formatter.

impl Solid for SaintVenantKirchhoff

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.

impl Thermoelastic for SaintVenantKirchhoff

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

Calculates and returns the second Piola-Kirchhoff stress.

\mathbf{S}(\mathbf{F}, T) = 2\mu\mathbf{E}' + \kappa\,\mathrm{tr}(\mathbf{E})\mathbf{1} - 3\alpha\kappa(T - T_\mathrm{ref})\mathbf{1}

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

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

\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}

fn coefficient_of_thermal_expansion(&self) -> &Scalar

Returns the coefficient of thermal expansion.

fn reference_temperature(&self) -> &Scalar

Returns the reference temperature.

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

Calculates and returns the Cauchy stress.

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

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

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

Calculates and returns the first Piola-Kirchhoff stress.

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

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

impl Thermohyperelastic for SaintVenantKirchhoff

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

Calculates and returns the Helmholtz free energy density.

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