flavio 0.5.0

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//! Viscoelastic constitutive models.
//!
//! ---
//!
//! Viscoelastic constitutive models cannot be defined by a Helmholtz free energy density function and viscous dissipation function.
//! These constitutive models are therefore defined by a relation for the stress as a function of the deformation gradient and rate.
//! Consequently, the rate tangent stiffness associated with the first Piola-Kirchoff stress is not symmetric for viscoelastic models.
//!
//! ```math
//! \mathcal{U}_{iJkL} \neq \mathcal{U}_{kLiJ}
//! ```

#[cfg(test)]
pub mod test;

use super::{super::fluid::viscous::Viscous, *};

/// Required methods for viscoelastic constitutive models.
pub trait Viscoelastic<'a>
where
    Self: Solid<'a> + Viscous<'a>,
{
    /// Calculates and returns the Cauchy stress.
    ///
    /// ```math
    /// \boldsymbol{\sigma} = J^{-1}\mathbf{P}\cdot\mathbf{F}^T
    /// ```
    fn calculate_cauchy_stress(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<CauchyStress, ConstitutiveError> {
        Ok(deformation_gradient
            * self.calculate_second_piola_kirchoff_stress(
                deformation_gradient,
                deformation_gradient_rate,
            )?
            * deformation_gradient.transpose()
            / deformation_gradient.determinant())
    }
    /// Calculates and returns the rate tangent stiffness associated with the Cauchy stress.
    ///
    /// ```math
    /// \mathcal{V}_{ijkL} = \frac{\partial\sigma_{ij}}{\partial\dot{F}_{kL}} = J^{-1} \mathcal{W}_{MNkL} F_{iM} F_{jN}
    /// ```
    fn calculate_cauchy_rate_tangent_stiffness(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<CauchyRateTangentStiffness, ConstitutiveError> {
        Ok(self
            .calculate_second_piola_kirchoff_rate_tangent_stiffness(
                deformation_gradient,
                deformation_gradient_rate,
            )?
            .contract_first_second_indices_with_second_indices_of(
                deformation_gradient,
                deformation_gradient,
            )
            / deformation_gradient.determinant())
    }
    /// Calculates and returns the first Piola-Kirchoff stress.
    ///
    /// ```math
    /// \mathbf{P} = J\boldsymbol{\sigma}\cdot\mathbf{F}^{-T}
    /// ```
    fn calculate_first_piola_kirchoff_stress(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<FirstPiolaKirchoffStress, ConstitutiveError> {
        Ok(
            self.calculate_cauchy_stress(deformation_gradient, deformation_gradient_rate)?
                * deformation_gradient.inverse_transpose()
                * deformation_gradient.determinant(),
        )
    }
    /// Calculates and returns the rate tangent stiffness associated with the first Piola-Kirchoff stress.
    ///
    /// ```math
    /// \mathcal{U}_{iJkL} = \frac{\partial P_{iJ}}{\partial\dot{F}_{kL}} = J \mathcal{V}_{iskL} F_{sJ}^{-T}
    /// ```
    fn calculate_first_piola_kirchoff_rate_tangent_stiffness(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<FirstPiolaKirchoffRateTangentStiffness, ConstitutiveError> {
        Ok(self
            .calculate_cauchy_rate_tangent_stiffness(
                deformation_gradient,
                deformation_gradient_rate,
            )?
            .contract_second_index_with_first_index_of(&deformation_gradient.inverse_transpose())
            * deformation_gradient.determinant())
    }
    /// Calculates and returns the second Piola-Kirchoff stress.
    ///
    /// ```math
    /// \mathbf{S} = \mathbf{F}^{-1}\cdot\mathbf{P}
    /// ```
    fn calculate_second_piola_kirchoff_stress(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<SecondPiolaKirchoffStress, ConstitutiveError> {
        Ok(deformation_gradient.inverse()
            * self.calculate_cauchy_stress(deformation_gradient, deformation_gradient_rate)?
            * deformation_gradient.inverse_transpose()
            * deformation_gradient.determinant())
    }
    /// Calculates and returns the rate tangent stiffness associated with the second Piola-Kirchoff stress.
    ///
    /// ```math
    /// \mathcal{W}_{IJkL} = \frac{\partial S_{IJ}}{\partial\dot{F}_{kL}} = \mathcal{U}_{mJkL}F_{mI}^{-T} = J \mathcal{V}_{mnkL} F_{mI}^{-T} F_{nJ}^{-T}
    /// ```
    fn calculate_second_piola_kirchoff_rate_tangent_stiffness(
        &self,
        deformation_gradient: &DeformationGradient,
        deformation_gradient_rate: &DeformationGradientRate,
    ) -> Result<SecondPiolaKirchoffRateTangentStiffness, ConstitutiveError> {
        let deformation_gradient_inverse = deformation_gradient.inverse();
        Ok(self
            .calculate_cauchy_rate_tangent_stiffness(
                deformation_gradient,
                deformation_gradient_rate,
            )?
            .contract_first_second_indices_with_second_indices_of(
                &deformation_gradient_inverse,
                &deformation_gradient_inverse,
            )
            * deformation_gradient.determinant())
    }
}