Struct AlmansiHamelEulerian 

pub struct AlmansiHamelEulerian {
    pub bulk_modulus: Quantity<Stress>,
    pub shear_modulus: Quantity<Stress>,
    pub yield_stress: Quantity<Stress>,
    pub hardening_slope: Quantity<Stress>,
    pub rate_sensitivity: Scalar,
    pub reference_flow_rate: Quantity<Rate>,
}

The Eulerian Almansi-Hamel elastic-viscoplastic solid constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The initial yield stress $Y_0$.
• The isotropic hardening slope $H$.
• The rate sensitivity parameter $m$.
• The reference flow rate $d_0$.

External variables

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

Internal variables

• The plastic deformation gradient $\mathbf{F}_\mathrm{p}$.

Notes

• The elastic Almansi-Hamel strain measure is given by $\mathbf{e}_\mathrm{e}=\tfrac{1}{2}(\mathbf{1}-\mathbf{B}_\mathrm{e}^{-1})$.
• The elastic response is Cauchy elastic, so no Helmholtz free energy density exists.

Fields

bulk_modulus: Quantity<Stress>

The bulk modulus $\kappa$.

shear_modulus: Quantity<Stress>

The shear modulus $\mu$.

yield_stress: Quantity<Stress>

The initial yield stress $Y_0$.

hardening_slope: Quantity<Stress>

The isotropic hardening slope $H$.

rate_sensitivity: Scalar

The rate sensitivity parameter $m$.

reference_flow_rate: Quantity<Rate>

The reference flow rate $d_0$.

Trait Implementations

impl Clone for AlmansiHamelEulerian

fn clone(&self) -> AlmansiHamelEulerian

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for AlmansiHamelEulerian

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

Formats the value using the given formatter.

impl ElasticPlasticOrViscoplastic for AlmansiHamelEulerian

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

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

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

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

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

Calculates and returns the first Piola-Kirchhoff stress.

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

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

fn mandel_stress( &self, deformation_gradient: &DeformationGradient, deformation_gradient_p: &DeformationGradientPlastic, ) -> Result<MandelStressElastic, ConstitutiveError>

Calculates and returns the Mandel stress.

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

Calculates and returns the second Piola-Kirchhoff stress.

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

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

impl ElasticViscoplastic<Quantity> for AlmansiHamelEulerian

fn internal_dissipation( &self, deformation_gradient: &DeformationGradient, state_variables: &ViscoplasticStateVariables<Y>, ) -> Result<Quantity<Dissipation>, ConstitutiveError>

Calculates and returns the internal dissipation.

fn state_variables_evolution( &self, deformation_gradient: &DeformationGradient, state_variables: &ViscoplasticStateVariables<Y>, ) -> Result<ViscoplasticEvolution<Y>, ConstitutiveError>

Calculates and returns the evolution of the state variables.

impl Plastic for AlmansiHamelEulerian

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

Returns the initial yield stress.

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

Returns the isotropic hardening slope.

fn yield_stress( &self, equivalent_plastic_strain: Quantity, ) -> Result<Quantity<Stress>, ConstitutiveError>

Calculates and returns the yield stress.

impl Solid for AlmansiHamelEulerian

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.

impl Viscoplastic<Quantity> for AlmansiHamelEulerian

fn initial_state(&self) -> ViscoplasticStateVariables<Quantity>

Returns the initial state of the variables.

fn plastic_evolution( &self, mandel_stress: MandelStressElastic, state_variables: &ViscoplasticStateVariables<Quantity>, ) -> Result<ViscoplasticEvolution<Quantity>, ConstitutiveError>

Calculates and returns the plastic evolution.

fn rate_sensitivity(&self) -> Scalar

Returns the rate_sensitivity parameter.

fn reference_flow_rate(&self) -> Quantity<Rate>

Returns the reference flow rate.

fn plastic_stretching_rate( &self, deviatoric_mandel_stress: MandelStressElastic, yield_stress: Quantity<Stress>, ) -> Result<StretchingRatePlastic, ConstitutiveError>

Calculates and returns the rate of plastic stretching.

fn dissipation_potential( &self, plastic_stretching_rate: StretchingRatePlastic, yield_stress: Quantity<Stress>, ) -> Result<Quantity<Dissipation>, ConstitutiveError>

Calculates and returns the dissipation potential.

fn dual_dissipation_potential( &self, deviatoric_mandel_stress: MandelStressElastic, yield_stress: Quantity<Stress>, ) -> Result<Quantity<Dissipation>, ConstitutiveError>

Calculates and returns the dual dissipation potential.