Struct EightChain 

pub struct EightChain<T>
where
    T: SingleChainThermodynamics,
{
    pub bulk_modulus: Quantity<Stress>,
    pub shear_modulus: Quantity<Stress>,
    pub single_chain_model: T,
}

The eight-chain hyperelastic solid constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The single-chain model.

External variables

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

Internal variables

• None.

Notes

• The nondimensional end-to-end length per link of a chain is $\gamma=\sqrt{\mathrm{tr}(\mathbf{B}^*)/3N_b}$.
• The nondimensional force is given by the single-chain model as $\eta=\eta(\gamma)$.
• The initial values are given by $\gamma_0=\sqrt{1/3N_b}$ and $\eta_0=\eta(\gamma_0)$.
• The eight-chain model reduces to the Arruda-Boyce model when $\eta=\mathcal{L}^{-1}(\gamma)$.

Fields

bulk_modulus: Quantity<Stress>

The bulk modulus $\kappa$.

shear_modulus: Quantity<Stress>

The shear modulus $\mu$.

single_chain_model: T

The single-chain model.

Implementations

impl<T> EightChain<T>

where T: SingleChainThermodynamics,

pub fn number_of_links(&self) -> Scalar

Returns the number of links in a single chain.

pub fn single_chain_model(&self) -> &T

Returns the single-chain model.

Trait Implementations

impl<T> Clone for EightChain<T>

where T: SingleChainThermodynamics + Clone,

fn clone(&self) -> EightChain<T>

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<T> Debug for EightChain<T>

where T: SingleChainThermodynamics,

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

Formats the value using the given formatter.

impl<T> Elastic for EightChain<T>

where T: SingleChainThermodynamics,

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

\boldsymbol{\sigma}(\mathbf{F}) = \frac{\mu\gamma_0\eta}{J\gamma\eta_0}\,{\mathbf{B}^*}' + \frac{\kappa}{2}\left(J - \frac{1}{J}\right)\mathbf{1}

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

\begin{aligned}
\mathcal{T}_{ijkL}(\mathbf{F}) =
\,& \frac{\mu\gamma_0\eta}{J^{5/3}\gamma\eta_0}\left(\delta_{ik}F_{jL} + \delta_{jk}F_{iL} - \frac{2}{3}\,\delta_{ij}F_{kL}- \frac{5}{3} \, B_{ij}'F_{kL}^{-T} \right)
\\
&+ \frac{\mu\gamma_0\eta}{3J^{7/3}N_b\gamma^2\eta_0}\left(\frac{\eta'}{\eta} - \frac{1}{\gamma}\right)B_{ij}'B_{km}'F_{mL}^{-T} + \frac{\kappa}{2} \left(J + \frac{1}{J}\right)\delta_{ij}F_{kL}^{-T}
\end{aligned}

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

Calculates and returns the first Piola-Kirchhoff stress.

fn first_piola_kirchhoff_tangent_stiffness( &self, deformation_gradient: &DeformationGradient, ) -> 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, ) -> Result<SecondPiolaKirchhoffStress, ConstitutiveError>

Calculates and returns the second Piola-Kirchhoff stress.

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

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

impl<T> Hyperelastic for EightChain<T>

where T: SingleChainThermodynamics,

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

a(\mathbf{F}) = \frac{3\mu N_b\gamma_0}{\eta_0}\left[\gamma\eta - \gamma_0\eta_0 - \ln\left(\frac{\eta_0\sinh\eta}{\eta\sinh\eta_0}\right) \right] + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl<T> Solid for EightChain<T>

where T: SingleChainThermodynamics,

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.