Struct MooneyRivlin

pub struct MooneyRivlin<'a> { /* private fields */ }

The Mooney-Rivlin hyperelastic constitutive model.^1,2

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus $\mu_m$.

External variables

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

Internal variables

• None.

Notes

• The Mooney-Rivlin model reduces to the Neo-Hookean model when $\mu_m\to 0$.

---

1. M. Mooney, J. Appl. Phys. 11, 582 (1940). ↩

2. R.S. Rivlin, Philos. Trans. R. Soc. London, Ser. A 241, 379 (1948). ↩

Trait Implementations

impl<'a> Constitutive<'a> for MooneyRivlin<'a>

fn new(parameters: Parameters<'a>) -> Self

Constructs and returns a new constitutive model.

fn jacobian( &self, deformation_gradient: &DeformationGradient, ) -> Result<Scalar, ConstitutiveError>

Calculates and returns the Jacobian.

impl<'a> Debug for MooneyRivlin<'a>

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

Formats the value using the given formatter.

impl<'a> Elastic<'a> for MooneyRivlin<'a>

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

\boldsymbol{\sigma}(\mathbf{F}) = \frac{\mu - \mu_m}{J}\, {\mathbf{B}^* }' - \frac{\mu_m}{J}\left(\mathbf{B}^{* -1}\right)' + \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-\mu_m}{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_m}{J}\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]
\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.

fn solve( &self, applied_load: AppliedLoad, ) -> Result<(DeformationGradient, CauchyStress), ConstitutiveError>

Solve for the unknown components of the Cauchy stress and deformation gradient under an applied load.

impl<'a> Hyperelastic<'a> for MooneyRivlin<'a>

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

a(\mathbf{F}) = \frac{\mu - \mu_m}{2}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right] + \frac{\mu_m}{2}\left[I_2(\mathbf{B}^*) - 3\right] + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl<'a> Solid<'a> for MooneyRivlin<'a>

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.