Struct Hencky 

pub struct Hencky {
    pub bulk_modulus: Scalar,
    pub shear_modulus: Scalar,
    pub yield_stress: Scalar,
    pub hardening_slope: Scalar,
    pub rate_sensitivity: Scalar,
    pub reference_flow_rate: Scalar,
}

The Hencky hyperelastic-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 Hencky strain measure is given by $\mathbf{h}=\tfrac{1}{2}\ln(\mathbf{B})$.

Fields

bulk_modulus: Scalar

The bulk modulus $\kappa$.

shear_modulus: Scalar

The shear modulus $\mu$.

yield_stress: Scalar

The initial yield stress $Y_0$.

hardening_slope: Scalar

The isotropic hardening slope $H$.

rate_sensitivity: Scalar

The rate sensitivity parameter $m$.

reference_flow_rate: Scalar

The reference flow rate $d_0$.

Trait Implementations

impl Clone for Hencky

fn clone(&self) -> Hencky

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Hencky

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

Formats the value using the given formatter.

impl ElasticPlasticOrViscoplastic for Hencky

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{h}_\mathrm{e}' + \frac{\kappa}{J}\,\mathrm{tr}(\mathbf{h}_\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[\frac{\partial\ln B_{ij}^\mathrm{e}}{\partial F_{kM}^\mathrm{e}}\,F_{ML}^{\mathrm{p}-T} - \frac{2}{3}\,\delta_{ij}F_{kL}^{-T} - 2{h_{ij}^\mathrm{e}}'F_{kL}^{-T}\right] + \frac{\kappa}{J}\left[1 - \mathrm{tr}(\mathbf{h}_\mathrm{e})\right]\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 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<f64> for Hencky

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

Calculates and returns the evolution of the state variables.

impl HyperelasticViscoplastic<f64> for Hencky

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

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

impl Plastic for Hencky

fn initial_yield_stress(&self) -> Scalar

Returns the initial yield stress.

fn hardening_slope(&self) -> Scalar

Returns the isotropic hardening slope.

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

Calculates and returns the yield stress.

impl Solid for Hencky

fn bulk_modulus(&self) -> Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> Scalar

Returns the shear modulus.

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

Calculates and returns the Jacobian.

impl Viscoplastic<f64> for Hencky

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

Returns the initial state of the variables.

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

Calculates and returns the plastic evolution.

fn rate_sensitivity(&self) -> Scalar

Returns the rate_sensitivity parameter.

fn reference_flow_rate(&self) -> Scalar

Returns the reference flow rate.

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

Calculates and returns the rate of plastic stretching.