Struct Carroll 

pub struct Carroll {
    pub bulk_modulus: Quantity<Stress>,
    pub linear_modulus: Quantity<Stress>,
    pub quartic_modulus: Quantity<Stress>,
    pub second_invariant_modulus: Quantity<Stress>,
}

The Carroll hyperelastic solid constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The linear modulus $\mu_1$.
• The quartic modulus $\mu_2$.
• The second-invariant modulus $\mu_3$.

External variables

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

Internal variables

• None.

Notes

• The shear modulus is given by $\mu = 2\mu_1 + 216\mu_2 + \mu_3/\sqrt{3}$.

---

1. M.M. Carroll, J. Elast. 103, 173 (2011). ↩

Fields

bulk_modulus: Quantity<Stress>

The bulk modulus $\kappa$.

linear_modulus: Quantity<Stress>

The linear modulus $\mu_1$.

quartic_modulus: Quantity<Stress>

The quartic modulus $\mu_2$.

second_invariant_modulus: Quantity<Stress>

The second-invariant modulus $\mu_3$.

Implementations

impl Carroll

pub fn quartic_modulus(&self) -> Quantity<Stress>

Returns the quartic modulus.

pub fn second_invariant_modulus(&self) -> Quantity<Stress>

Returns the second-invariant modulus.

Trait Implementations

impl Clone for Carroll

fn clone(&self) -> Carroll

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Carroll

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

Formats the value using the given formatter.

impl Elastic for Carroll

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

\boldsymbol{\sigma}(\mathbf{F}) = \frac{1}{J}\left\{2\left[\mu_1 + 4\mu_2\,\mathrm{tr}(\mathbf{B}^*)^3\right]{\mathbf{B}^* }' - \frac{\mu_3}{\sqrt{I_2(\mathbf{B}^*)}}\left(\mathbf{B}^{* -1}\right)' + \frac{\kappa}{2}\left(J^2-1\right)\mathbf{1}\right\}

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

\begin{aligned}
\mathcal{T}_{ijkL}(\mathbf{F}) =
\,& \frac{2\left[\mu_1+4\mu_2\,\mathrm{tr}(\mathbf{B}^*)^3\right]}{J^{5/3}}\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{\kappa}{2} \left(J + \frac{1}{J}\right)\delta_{ij}F_{kL}^{-T}
\\
&- \frac{\mu_3}{J\sqrt{I_2(\mathbf{B}^*)}}\left[ \frac{2}{3}\,B_{ij}^{* -1}F_{kL}^{-T} - B_{ik}^{* -1}F_{jL}^{-T} - B_{ik}^{* -1}F_{iL}^{-T} + \frac{2}{3}\,\delta_{ij}\left(B_{km}^{* -1}\right)'F_{mL}^{-T} - \left(B_{ij}^{* -1}\right)'F_{kL}^{-T} \right]
\\
&+ \frac{1}{J}\,{B_{ij}^*}'\left(\frac{48\mu_2\,\mathrm{tr}(\mathbf{B}^*)^2}{J^{2/3}}F_{kL} - 16\mu_2\,\mathrm{tr}(\mathbf{B}^*)^3F_{kL}^{-T}\right)
\\
&- \frac{1}{J}\left(B_{ij}^{*-1}\right)'\left[-\frac{\mu_3}{I_2(\mathbf{B}^*)^{3/2}J^{2/3}}\left(\mathrm{tr}(\mathbf{B}^*)\,\delta_{km} - B_{km}^*\right)F_{mL} + \frac{2\mu_3}{3\sqrt{I_2(\mathbf{B}^*)}}F_{kL}^{-T}\right]
\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 Hyperelastic for Carroll

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

a(\mathbf{F}) = \mu_1\left\{\mathrm{tr}(\mathbf{B}^* ) - 3\right\} + \mu_2\left\{\left[\mathrm{tr}(\mathbf{B}^* )\right]^4 - 81\right\} + \mu_3\left\{\sqrt{I_2(\mathbf{B}^*)} - \sqrt{3}\right\} + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl Solid for Carroll

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.