Convenience class for handling linear elastomechanics. More...

#include <elastoVariationalFunctionals.hh>

Detailed Description

template<int dim_, class Scalar_ = double>
class Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >

Convenience class for handling linear elastomechanics.

This class defines the variational formulation of the Lame-Navier operator \( u \mapsto -2\mu\Delta u - \lambda \nabla \mbox{div} u\) for vector-valued displacements \( u \), i.e. the variational functional \( u \mapsto \frac{\lambda}{2} \mbox{tr}(E)^2 + \mu(E:E) \) with \( E = \frac{1}{2}(u_x + u_x^T) \).

Definition at line 234 of file elastoVariationalFunctionals.hh.

Inheritance diagram for Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >:

Public Types
using	Scalar = Scalar_

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

Public Member Functions
	LameNavier ()
	Constructor. Both Lame constants are initialized to 1, the linearization point to \( (u_0)_x = 0 \). More...

	LameNavier (ElasticModulus const &moduli)

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

Base::Tensor	d2 (VariationalArg< Scalar, dim > const &arg1, VariationalArg< Scalar, dim > const &arg2) const
	d2 As the base method, computes the second directional derivative of the hyperelastic stored energy density. But this transformed version (only valid in the completely linear case) is more efficient. More...

void	setElasticModuli (ElasticModulus const &modulus)
	Define material properties. More...

ElasticModulus	getElasticModuli ()
	Retrieve current material properties. More...

Dune::FieldMatrix< Scalar, dim (dim+1)/2, dim (dim+1)/2 >	strainToStressMatrix () const
	Returns the matrix \( C \) mapping the strain tensor to the stress tensor: \( \sigma = C \epsilon \). More...

Dune::FieldMatrix< Scalar, dim (dim+1)/2, dim dim >	displacementDerivativeToStrainMatrix () const
	returns the matrix \( E \) mapping the displacement derivative to the strain tensor: \( \epsilon = E u_x \) 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...

MaterialLaws::StVenantKirchhoff< dim_, double > &	getHyperelasticEnergy ()
	Provides access to the stored energy density. More...

MaterialLaws::StVenantKirchhoff< dim_, double > 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...

LinearizedGreenLagrangeTensor< double, dim_ > 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 Member Functions
static Dune::FieldVector< Scalar, dim >	mv (Dune::FieldVector< Scalar, dim *(dim+1)/2 > const &A, Dune::FieldVector< Scalar, dim > const &x)
	Computes the symmetric matrix times vector product, when the matrix is compactly stored in a dim*(dim+1)/2-vector. More...

Static Public Attributes
static int const	dim = dim_

Protected Attributes
Tensor	du

MaterialLaws::StVenantKirchhoff< dim_, double >	energy

LinearizedGreenLagrangeTensor< double, dim_ >	strain

Member Typedef Documentation

◆ Scalar

template<int dim_, class Scalar_ = double>

using Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::Scalar = Scalar_

Definition at line 242 of file elastoVariationalFunctionals.hh.

◆ Tensor

using Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::Tensor = Dune::FieldMatrix<Scalar,dim,dim>

inherited

Definition at line 48 of file elastoVariationalFunctionals.hh.

Constructor & Destructor Documentation

◆ LameNavier() [1/2]

template<int dim_, class Scalar_ = double>

Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::LameNavier ( )

inline

Constructor. Both Lame constants are initialized to 1, the linearization point to \( (u_0)_x = 0 \).

Definition at line 254 of file elastoVariationalFunctionals.hh.

◆ LameNavier() [2/2]

template<int dim_, class Scalar_ = double>

Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::LameNavier ( ElasticModulus const & moduli )

inline

Definition at line 257 of file elastoVariationalFunctionals.hh.

Member Function Documentation

◆ cauchyStress()

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::cauchyStress ( ) const

inlineinherited

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()

Scalar Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::d0 ( ) const

inlineinherited

Computes the hyperelastic stored energy density.

Definition at line 91 of file elastoVariationalFunctionals.hh.

◆ d1() [1/2]

Dune::FieldVector< Scalar, dim > Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::d1 ( VariationalArg< Scalar, dim > const & arg ) const

inlineinherited

Computes the directional derivative of the hyperelastic stored energy density.

Definition at line 99 of file elastoVariationalFunctionals.hh.

◆ d1() [2/2]

template<int dim_, class Scalar_ = double>

Dune::FieldVector< Scalar, dim > Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::d1 ( VariationalArg< Scalar, dim > const & arg ) const

inline

d1 As the base method, computes the directional derivative of the hyperelastic stored energy density.

But this transformed version (only valid in the completely linear case) is probably more efficient.

Definition at line 267 of file elastoVariationalFunctionals.hh.

◆ d2() [1/2]

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::d2	(	VariationalArg< Scalar, dim > const &	arg1,
		VariationalArg< Scalar, dim > const &	arg2
	)		const

inlineinherited

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.

◆ d2() [2/2]

template<int dim_, class Scalar_ = double>

Base::Tensor Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::d2	(	VariationalArg< Scalar, dim > const &	arg1,
		VariationalArg< Scalar, dim > const &	arg2
	)		const

inline

d2 As the base method, computes the second directional derivative of the hyperelastic stored energy density. But this transformed version (only valid in the completely linear case) is more efficient.

Definition at line 280 of file elastoVariationalFunctionals.hh.

◆ displacementDerivativeToStrainMatrix()

template<int dim_, class Scalar_ = double>

Dune::FieldMatrix< Scalar, dim *(dim+1)/2, dim *dim > Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::displacementDerivativeToStrainMatrix ( ) const

inline

returns the matrix \( E \) mapping the displacement derivative to the strain tensor: \( \epsilon = E u_x \)

In this context, the symmetric matrix \( \epsilon \) is interpreted as vector in \( \mathbb{R}^{d(d+1)/2} \):

\[ \epsilon = \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\ \epsilon_{12} \\ \epsilon_{13} \\ \epsilon_{23} \end{bmatrix} \quad\mbox{or}\quad \epsilon = \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{12} \end{bmatrix} \]

The displacement derivative \( u_x \) is interpreted as vector in \( \mathbb{R}^{d^2} \) (column-major format):

\[ \frac{\partial u_i}{\partial x_j} = (u_x)_{i+dj} \]

Definition at line 387 of file elastoVariationalFunctionals.hh.

◆ getDu()

Tensor const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::getDu ( ) const

inlineinherited

getDu provides read access to the current displacement derivative.

Definition at line 177 of file elastoVariationalFunctionals.hh.

◆ getElasticModuli()

template<int dim_, class Scalar_ = double>

ElasticModulus Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::getElasticModuli ( )

inline

Retrieve current material properties.

Definition at line 316 of file elastoVariationalFunctionals.hh.

◆ getHyperelasticEnergy() [1/2]

MaterialLaws::StVenantKirchhoff< dim_, double > & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::getHyperelasticEnergy ( )

inlineinherited

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.

◆ getHyperelasticEnergy() [2/2]

MaterialLaws::StVenantKirchhoff< dim_, double > const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::getHyperelasticEnergy ( ) const

inlineinherited

getHyperelasticEnergy provides read access to the stored energy density.

Definition at line 169 of file elastoVariationalFunctionals.hh.

◆ getStrain()

LinearizedGreenLagrangeTensor< double, dim_ > const & Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::getStrain ( ) const

inlineinherited

getStrain provides read access to the current strain tensor.

Definition at line 185 of file elastoVariationalFunctionals.hh.

◆ mv()

template<int dim_, class Scalar_ = double>

static Dune::FieldVector< Scalar, dim > Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::mv	(	Dune::FieldVector< Scalar, dim *(dim+1)/2 > const &	A,
		Dune::FieldVector< Scalar, dim > const &	x
	)

inlinestatic

Computes the symmetric matrix times vector product, when the matrix is compactly stored in a dim*(dim+1)/2-vector.

This is useful, e.g., for computing normal stresses \( \sigma n \).

Definition at line 355 of file elastoVariationalFunctionals.hh.

◆ setElasticModuli()

template<int dim_, class Scalar_ = double>

void Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::setElasticModuli ( ElasticModulus const & modulus )

inline

Define material properties.

This invalidates previously set linearization points.

Definition at line 306 of file elastoVariationalFunctionals.hh.

◆ setLinearizationPoint()

void Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::setLinearizationPoint ( Tensor const & du0 )

inlineinherited

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.

◆ strainToStressMatrix()

template<int dim_, class Scalar_ = double>

Dune::FieldMatrix< Scalar, dim *(dim+1)/2, dim *(dim+1)/2 > Kaskade::Elastomechanics::LameNavier< dim_, Scalar_ >::strainToStressMatrix ( ) const

inline

Returns the matrix \( C \) mapping the strain tensor to the stress tensor: \( \sigma = C \epsilon \).

In this contex, the symmetric matrices \( \sigma \) and \( \epsilon \) are interpreted as vectors in \( \mathbb{R}^{d(d+1)/2} \):

\[ \sigma = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{13} \\ \sigma_{23} \end{bmatrix} \quad\mbox{or}\quad \sigma = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} \]

Definition at line 328 of file elastoVariationalFunctionals.hh.

◆ stress()

Tensor Kaskade::Elastomechanics::HyperelasticVariationalFunctional< MaterialLaws::StVenantKirchhoff< dim_, double > , LinearizedGreenLagrangeTensor< double, dim_ > >::stress ( ) const

inlineinherited

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.