Struct Yeoh

pub struct Yeoh<P> { /* private fields */ }

The Yeoh hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra moduli $\mu_n$ for $n=2\ldots N$.

External variables

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

Internal variables

• None.

Notes

• The Yeoh model reduces to the Neo-Hookean model when $\mu_n\to 0$ for $n=2\ldots N$.

---

1. O.H. Yeoh, Rubber Chem. Technol. 66, 754 (1993). ↩

Implementations

impl<P> Yeoh<P>

where P: Parameters,

pub fn moduli(&self) -> &[Scalar]

Returns an array of the moduli.

pub fn extra_moduli(&self) -> &[Scalar]

Returns an array of the extra moduli.

Trait Implementations

impl<P> Constitutive<P> for Yeoh<P>

where P: Parameters,

fn new(parameters: P) -> Self

Constructs and returns a new constitutive model.

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

Calculates and returns the Jacobian.

impl<P: Debug> Debug for Yeoh<P>

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

Formats the value using the given formatter.

impl<P> Elastic for Yeoh<P>

where P: Parameters,

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

\boldsymbol{\sigma}(\mathbf{F}) = \sum_{n=1}^N \frac{n\mu_n}{J}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right]^{n-1}\,{\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}) =
\,& \sum_{n=1}^N \frac{n\mu_n}{J^{5/3}}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right]^{n-1}\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)
\\
&+ \sum_{n=2}^N \frac{2n(n-1)\mu_n}{J^{7/3}}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right]^{n-2}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.

fn root( &self, applied_load: AppliedLoad, ) -> Result<DeformationGradient, OptimizeError>

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

impl<P> Hyperelastic for Yeoh<P>

where P: Parameters,

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

a(\mathbf{F}) = \sum_{n=1}^N \frac{\mu_n}{2}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right]^n + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

fn minimize( &self, applied_load: AppliedLoad, ) -> Result<DeformationGradient, OptimizeError>

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

impl<P> Solid for Yeoh<P>

where P: Parameters,

fn bulk_modulus(&self) -> &Scalar

Returns the bulk modulus.

fn shear_modulus(&self) -> &Scalar

Returns the shear modulus.