Struct Isihara 

pub struct Isihara {
    pub bulk_modulus: Quantity<Stress>,
    pub shear_modulus: Quantity<Stress>,
    pub extra_modulus: Quantity<Stress>,
    pub quadratic_modulus: Quantity<Stress>,
}

The Isihara hyperelastic solid constitutive model.¹

Parameters

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

External variables

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

Internal variables

• None.

Notes

• The Isihara model reduces to the Mooney-Rivlin model when $\mu_q\to 0$.

---

1. A. Isihara, N. Hashitsume, and M. Tatibana, J. Chem. Phys. 19, 1508 (1951). ↩

Fields

bulk_modulus: Quantity<Stress>

The bulk modulus $\kappa$.

shear_modulus: Quantity<Stress>

The shear modulus $\mu$.

extra_modulus: Quantity<Stress>

The extra modulus $\mu_m$.

quadratic_modulus: Quantity<Stress>

The quadratic modulus $\mu_q$.

Implementations

impl Isihara

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

Returns the extra modulus.

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

Returns the quadratic modulus.

Trait Implementations

impl Clone for Isihara

fn clone(&self) -> Isihara

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Isihara

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

Formats the value using the given formatter.

impl Elastic for Isihara

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

\boldsymbol{\sigma}(\mathbf{F}) = \frac{1}{J}\left\{2\left[\frac{\mu-\mu_m}{2} + \mu_q\left(\mathrm{tr}(\mathbf{B}^*) - 3\right)\right]{\mathbf{B}^* }' - \mu_m\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}{J^{5/3}}\left[\frac{\mu-\mu_m}{2} + \mu_q\left(\mathrm{tr}(\mathbf{B}^*) - 3\right)\right]\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]
\\
&+ \frac{1}{J}\,{B_{ij}^*}'\left(\frac{4\mu_q}{J^{2/3}}F_{kL} - \frac{4}{3}\mu_q\,\mathrm{tr}(\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 Isihara

fn helmholtz_free_energy_density( &self, deformation_gradient: &DeformationGradient, ) -> Result<Quantity<EnergyDensity>, 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{\mu_q}{2}\left[\mathrm{tr}(\mathbf{B}^* ) - 3\right]^2 + \frac{\kappa}{2}\left[\frac{1}{2}\left(J^2 - 1\right) - \ln J\right]

impl Solid for Isihara

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.