Represents the first Piola Kirchhoff stress as known in nonlinear elasticity. The Piola-Kirchhoff stress is \( T = F\sigma = (I+u_x) \sigma(u) \). More...

#include <elastoVariationalFunctionals.hh>

Detailed Description

template<class HyperelasticEnergy, class StrainTensor>
class Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >

Represents the first Piola Kirchhoff stress as known in nonlinear elasticity. The Piola-Kirchhoff stress is \( T = F\sigma = (I+u_x) \sigma(u) \).

This is essentially a view on a given elastic energy function in current state.

Definition at line 542 of file elastoVariationalFunctionals.hh.

Public Types
using	EnergyFunctional = HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >

using	Scalar = typename HyperelasticEnergy::Scalar

using	Tensor = typename EnergyFunctional::Tensor

Public Member Functions
	FirstPiolaKirchhoffStress (EnergyFunctional const &ef)
	FirstPiolaKirchhoffStress constructs first Piola Kirchhoff stress object from current state of elastic energy. More...

Tensor	deformationGradient () const
	DEPRECATED, use deformationDerivative instead. More...

Tensor	deformationDerivative () const
	deformationGradient provides access to the current deformation derivative \( F = I + u_x \). More...

Tensor	d0 () const
	d0 provides access to the current first Piola Kirchhoff stress. More...

Tensor	d1 (VariationalArg< Scalar, dim, dim > const &arg) const
	d1 provides access to the first directional derivative (in direction of some displacement derivative) of the Piola Kirchhoff stress. More...

Tensor	d2 (VariationalArg< Scalar, dim, dim > const &argv, VariationalArg< Scalar, dim, dim > const &argw) const
	d2 provides access to the second directional derivative (in directions of some displacement derivatives) of the Piola Kirchhoff stress. Warning: The current implementation is only correct for quadratic stored energy functions, i.e. linear materials. Todo: Fix this. More...

Public Attributes
EnergyFunctional const &	energy

Static Public Attributes
static constexpr int	dim = HyperelasticEnergy::dim

Protected Member Functions
Tensor	spkStressD1 (VariationalArg< Scalar, dim, dim > const &arg) const

Tensor	energyD2De (Tensor const &e) const

Member Typedef Documentation

◆ EnergyFunctional

template<class HyperelasticEnergy , class StrainTensor >

using Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::EnergyFunctional = HyperelasticVariationalFunctional<HyperelasticEnergy, StrainTensor>

Definition at line 545 of file elastoVariationalFunctionals.hh.

◆ Scalar

template<class HyperelasticEnergy , class StrainTensor >

using Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::Scalar = typename HyperelasticEnergy::Scalar

Definition at line 546 of file elastoVariationalFunctionals.hh.

◆ Tensor

template<class HyperelasticEnergy , class StrainTensor >

using Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::Tensor = typename EnergyFunctional::Tensor

Definition at line 547 of file elastoVariationalFunctionals.hh.

Constructor & Destructor Documentation

◆ FirstPiolaKirchhoffStress()

template<class HyperelasticEnergy , class StrainTensor >

Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::FirstPiolaKirchhoffStress ( EnergyFunctional const & ef )

inline

FirstPiolaKirchhoffStress constructs first Piola Kirchhoff stress object from current state of elastic energy.

Parameters

ef	is the elastic energy in current state (and at fixed displacement gradient)

Definition at line 554 of file elastoVariationalFunctionals.hh.

Member Function Documentation

◆ d0()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d0 ( ) const

inline

d0 provides access to the current first Piola Kirchhoff stress.

Definition at line 578 of file elastoVariationalFunctionals.hh.

◆ d1()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d1 ( VariationalArg< Scalar, dim, dim > const & arg ) const

inline

d1 provides access to the first directional derivative (in direction of some displacement derivative) of the Piola Kirchhoff stress.

Parameters

arg	is direction of differentiation.

Definition at line 588 of file elastoVariationalFunctionals.hh.

◆ d2()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d2	(	VariationalArg< Scalar, dim, dim > const &	argv,
		VariationalArg< Scalar, dim, dim > const &	argw
	)		const

inline

d2 provides access to the second directional derivative (in directions of some displacement derivatives) of the Piola Kirchhoff stress. Warning: The current implementation is only correct for quadratic stored energy functions, i.e. linear materials. Todo: Fix this.

Parameters

argv	is the first direction of differentiation.
argv	is the second direction of differentiation.

Definition at line 602 of file elastoVariationalFunctionals.hh.

◆ deformationDerivative()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::deformationDerivative ( ) const

inline

deformationGradient provides access to the current deformation derivative \( F = I + u_x \).

Definition at line 567 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d0(), and Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::deformationGradient().

◆ deformationGradient()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::deformationGradient ( ) const

inline

DEPRECATED, use deformationDerivative instead.

Definition at line 559 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d1(), and Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d2().

◆ energyD2De()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::energyD2De ( Tensor const & e ) const

inlineprotected

Definition at line 621 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d2(), and Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::spkStressD1().

◆ spkStressD1()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::spkStressD1 ( VariationalArg< Scalar, dim, dim > const & arg ) const