Struct AlmansiHamel

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

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

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 AlmansiHamel

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

Formats the value using the given formatter.

impl Solid for AlmansiHamel

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.

impl Thermoelastic for AlmansiHamel

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

Calculates and returns the Cauchy stress.

\boldsymbol{\sigma}(\mathbf{F}, T) = \frac{2\mu}{J}\,\mathbf{e}' + \frac{\kappa}{J}\,\mathrm{tr}(\mathbf{e})\mathbf{1} - \frac{3\alpha\kappa}{J}(T - T_\mathrm{ref})\mathbf{1}

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

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

\mathcal{T}_{ijkL}(\mathbf{F}, T) = \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} - \Big[\mathrm{tr}(\mathbf{e}) - 3\alpha(T - T_\mathrm{ref})\Big]\delta_{ij}F_{kL}^{-T}\right\}

fn coefficient_of_thermal_expansion(&self) -> &Scalar

Returns the coefficient of thermal expansion.

fn reference_temperature(&self) -> &Scalar

Returns the reference temperature.

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.

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

Calculates and returns the second Piola-Kirchhoff stress.

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

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