General base class for variational functionals defined in terms of hyperelastic stored energies. More...

#include <elastoVariationalFunctionals.hh>

Detailed Description

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

General base class for variational functionals defined in terms of hyperelastic stored energies.

This defines the interface for stored energies working on displacement derivatives \( W = W(u_x) \). The interface is intended to be used for defining the domain cache of variational functionals.

The class has as a member an object of the hyperelastic energy defining the stored energy \( W(E) \) depending on the Green-Lagrange strain tensor.

Template Parameters

HyperelasticEnergy	the hyperelastic energy class
StrainTensor	the strain tensor class, usually one of GreenLagrangeTensor or LinearizedGreenLagrangeTensor

Example for linear elastomechanics of steel:

HyperelasticVariationalFunctional<MaterialLaws::StVenantKirchhoff<dim>,LinearizedGreenLagrangeTensor<double,dim>> energy(ElasticModulus::material("steel"));

energy.setLinearizationPoint(du);

Kaskade::Elastomechanics::ElasticModulus::material

static ElasticModulus const & material(std::string const &name)

Returns the material parameters of the given materials.

Kaskade::Elastomechanics::HyperelasticVariationalFunctional

General base class for variational functionals defined in terms of hyperelastic stored energies.

Definition: elastoVariationalFunctionals.hh:44

Kaskade::Elastomechanics::HyperelasticVariationalFunctional::energy

HyperelasticEnergy energy

Definition: elastoVariationalFunctionals.hh:218

Kaskade::Elastomechanics::HyperelasticVariationalFunctional::du

Tensor du

Definition: elastoVariationalFunctionals.hh:217

Kaskade::Elastomechanics::LinearizedGreenLagrangeTensor

Linearized (right) Green-Lagrange strain tensor, the workhorse of linear elastomechanics.

Definition: elasto.hh:280

Definition at line 43 of file elastoVariationalFunctionals.hh.

Public Types
using	Scalar = typename HyperelasticEnergy::Scalar

using	Tensor = Dune::FieldMatrix< Scalar, dim, dim >

Public Member Functions
template<typename... Args>
	HyperelasticVariationalFunctional (Args &&... args)
	Constructor. More...

	HyperelasticVariationalFunctional (HyperelasticEnergy const &energy_, StrainTensor const &strain_)
	Constructor. More...

void	setLinearizationPoint (Tensor const &du0)
	Defines the displacement derivative around which to linearize. More...

Scalar	d0 () const
	Computes the hyperelastic stored energy density. More...

Dune::FieldVector< Scalar, dim >	d1 (VariationalArg< Scalar, dim > const &arg) const
	Computes the directional derivative of the hyperelastic stored energy density. More...

Tensor	d2 (VariationalArg< Scalar, dim > const &arg1, VariationalArg< Scalar, dim > const &arg2) const
	Computes the second directional derivative of the hyperelastic stored energy density. More...

HyperelasticEnergy &	getHyperelasticEnergy ()
	Provides access to the stored energy density. More...

HyperelasticEnergy const &	getHyperelasticEnergy () const
	getHyperelasticEnergy provides read access to the stored energy density. More...

Tensor const &	getDu () const
	getDu provides read access to the current displacement derivative. More...

StrainTensor const &	getStrain () const
	getStrain provides read access to the current strain tensor. More...

Tensor	stress () const
	Returns the 2nd Piola-Kirchhoff stress tensor \( S = \frac{\partial W}{\partial E} \) corresponding to the current displacement derivative. More...

Tensor	cauchyStress () const
	Returns the Cauchy stress tensor \( \sigma = \frac{1}{J} F S F^T \) corresponding to the current displacement derivative. More...

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

Protected Attributes
Tensor	du

HyperelasticEnergy	energy

StrainTensor	strain

Member Typedef Documentation

◆ Scalar

template<class HyperelasticEnergy , class StrainTensor >

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

Definition at line 46 of file elastoVariationalFunctionals.hh.

◆ Tensor

template<class HyperelasticEnergy , class StrainTensor >

using Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::Tensor = Dune::FieldMatrix<Scalar,dim,dim>

Definition at line 48 of file elastoVariationalFunctionals.hh.

Constructor & Destructor Documentation

◆ HyperelasticVariationalFunctional() [1/2]

template<class HyperelasticEnergy , class StrainTensor >

template<typename... Args>

Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::HyperelasticVariationalFunctional ( Args &&... args )

inline

Constructor.

The constructor arguments are forwarded to the hyperelastic stored energy constructor. The strain tensor is default initialized.

Definition at line 57 of file elastoVariationalFunctionals.hh.

◆ HyperelasticVariationalFunctional() [2/2]

template<class HyperelasticEnergy , class StrainTensor >

Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::HyperelasticVariationalFunctional	(	HyperelasticEnergy const &	energy_,
		StrainTensor const &	strain_
	)

inline

Constructor.

Parameters

energy	a hyperelastic energy functional, will be copied to an internal energy member variable
strain	a strain tensor functional, will be copied to an internal strain variable

Definition at line 67 of file elastoVariationalFunctionals.hh.

Member Function Documentation

◆ cauchyStress()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::cauchyStress ( ) const

inline

Returns the Cauchy stress tensor \( \sigma = \frac{1}{J} F S F^T \) corresponding to the current displacement derivative.

Here, \( S \) is the second Piola-Kirchhoff stress tensor and \( F = I + u_x \) the deformation derivative.

The Cauchy stress \( \sigma \), also known as "true" stress, describes the tractions in the deformed coordinates, and therefore the "true" force density on a (cutting) surface.

Definition at line 207 of file elastoVariationalFunctionals.hh.

◆ d0()

template<class HyperelasticEnergy , class StrainTensor >

Scalar Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::d0 ( ) const

inline

Computes the hyperelastic stored energy density.

Definition at line 91 of file elastoVariationalFunctionals.hh.

◆ d1()

template<class HyperelasticEnergy , class StrainTensor >

Dune::FieldVector< Scalar, dim > Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::d1 ( VariationalArg< Scalar, dim > const & arg ) const

inline

Computes the directional derivative of the hyperelastic stored energy density.

Definition at line 99 of file elastoVariationalFunctionals.hh.

◆ d2()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::d2	(	VariationalArg< Scalar, dim > const &	arg1,
		VariationalArg< Scalar, dim > const &	arg2
	)		const

inline

Computes the second directional derivative of the hyperelastic stored energy density.

For scalar functions \( \phi, \psi \), the values and derivatives at a certain point are provided in arg1 and arg 2, this computes the \( d\times d \) matrix \( A \) with \( A_{ij} = \epsilon((u_i)_x)^T \mathcal{C} \epsilon((v_j)_x) \), where \( u_i = \phi e_i, v_j = \psi e_j \) and \( e_k \) is the \( k \)-th unit vector.

This is exactly what is to be returned from the d2 method of a a variational functional's domain cache implementing the Lame-Navier equations.

Definition at line 127 of file elastoVariationalFunctionals.hh.

◆ getDu()

template<class HyperelasticEnergy , class StrainTensor >

Tensor const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::getDu ( ) const

inline

getDu provides read access to the current displacement derivative.

Definition at line 177 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::deformationDerivative().

◆ getHyperelasticEnergy() [1/2]

template<class HyperelasticEnergy , class StrainTensor >

HyperelasticEnergy & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::getHyperelasticEnergy ( )

inline

Provides access to the stored energy density.

The stored energy function has the linearization (and evaluation) point given by the latest call to the setLinearizationPoint() method.

Definition at line 161 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::energyD2De().

◆ getHyperelasticEnergy() [2/2]

template<class HyperelasticEnergy , class StrainTensor >

HyperelasticEnergy const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::getHyperelasticEnergy ( ) const

inline

getHyperelasticEnergy provides read access to the stored energy density.

Definition at line 169 of file elastoVariationalFunctionals.hh.

◆ getStrain()

template<class HyperelasticEnergy , class StrainTensor >

StrainTensor const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::getStrain ( ) const

inline

getStrain provides read access to the current strain tensor.

Definition at line 185 of file elastoVariationalFunctionals.hh.

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

◆ setLinearizationPoint()

template<class HyperelasticEnergy , class StrainTensor >

void Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::setLinearizationPoint ( Tensor const & du0 )

inline

Defines the displacement derivative around which to linearize.

This method shall be called when the evaluation of a stored energy and its derivatives is intended for a different displacement derivative, in particular in the "evaluateAt" method of a domain cache.

Parameters

du0	the current displacement derivative \( u_x \)

Definition at line 81 of file elastoVariationalFunctionals.hh.

◆ stress()

template<class HyperelasticEnergy , class StrainTensor >

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::stress ( ) const

inline

Returns the 2nd Piola-Kirchhoff stress tensor \( S = \frac{\partial W}{\partial E} \) corresponding to the current displacement derivative.

This is the stress in the reference (material) coordinates, and does not provide the actual force density in the deformed configuration. See cauchyStress.

Definition at line 197 of file elastoVariationalFunctionals.hh.

Referenced by Kaskade::Elastomechanics::HyperelasticVariationalFunctional< HyperelasticEnergy, StrainTensor >::cauchyStress(), Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d0(), and Kaskade::Elastomechanics::FirstPiolaKirchhoffStress< HyperelasticEnergy, StrainTensor >::d1().