Implementation of the Neo-Hookean material model.The energy is computed as. More...

#include <ikarus/finiteelements/mechanics/materials/hyperelastic/neohooke.hh>

Inheritance diagram for Ikarus::Materials::NeoHookeT< ST >:

Public Types
using	ScalarType = ST

using	StrainMatrix = Eigen::Matrix< ScalarType, dim, dim >

using	StressMatrix = StrainMatrix

using	MaterialTensor = Eigen::TensorFixedSize< ScalarType, Eigen::Sizes< dim, dim, dim, dim > >

using	MaterialParameters = LamesFirstParameterAndShearModulus

using	MaterialImpl = NeoHookeT< ST >
	Type of material implementation. More...

Public Member Functions
	NeoHookeT (const MaterialParameters &mpt)
	Constructor for NeoHookeT. More...

MaterialParameters	materialParametersImpl () const
	Returns the material parameters stored in the material. More...

template<typename Derived >
ScalarType	storedEnergyImpl (const Eigen::MatrixBase< Derived > &C) const
	Computes the stored energy in the Neo-Hookean material model. More...

template<bool voigt, typename Derived >
StressMatrix	stressesImpl (const Eigen::MatrixBase< Derived > &C) const
	Computes the stresses in the Neo-Hookean material model. More...

template<bool voigt, typename Derived >
MaterialTensor	tangentModuliImpl (const Eigen::MatrixBase< Derived > &C) const
	Computes the tangent moduli in the Neo-Hookean material model. More...

template<typename STO >
auto	rebind () const
	Rebinds the material to a different scalar type. More...

template<typename Derived >
auto	materialInversionImpl (const Eigen::MatrixBase< Derived > &Sraw) const
	Computes the strain measure and inverse material tangent for a given stress state. More...

constexpr const MaterialImpl &	impl () const
	Const accessor to the underlying material (CRTP). More...

constexpr MaterialImpl &	impl ()
	Accessor to the underlying material (CRTP). More...

auto	materialParameters () const
	Returns the material parameters stored in the implemented material. More...

auto	storedEnergy (const Eigen::MatrixBase< Derived > &Eraw) const
	Return the stored potential energy of the material. More...

auto	stresses (const Eigen::MatrixBase< Derived > &Eraw) const
	Get the stresses of the material. More...

auto	tangentModuli (const Eigen::MatrixBase< Derived > &Eraw) const
	Get the tangentModuli of the material. More...

auto	materialInversion (const Eigen::MatrixBase< Derived > &Sraw, const Eigen::MatrixBase< Derived > &EstartRaw=Derived::Zero().eval(), const double tol=1e-12, const int maxIter=20) const
	Computes the corresponding strain measure and inverse material tangent for a given stress state. More...

Static Public Member Functions
static constexpr std::string	nameImpl () noexcept

static constexpr std::string	name ()
	Get the name of the implemented material. More...

Static Public Attributes
static constexpr int	dim = 3

static constexpr auto	strainTag = StrainTags::rightCauchyGreenTensor

static constexpr auto	stressTag = StressTags::PK2

static constexpr auto	tangentModuliTag = TangentModuliTags::Material

static constexpr bool	energyAcceptsVoigt = false

static constexpr bool	stressToVoigt = false

static constexpr bool	stressAcceptsVoigt = false

static constexpr bool	moduliToVoigt = false

static constexpr bool	moduliAcceptsVoigt = false

static constexpr double	derivativeFactorImpl = 2

static constexpr bool	isReduced
	Static constant for determining if the material has vanishing stress or strain components (is reduced). More...

static constexpr bool	isLinear

static constexpr double	derivativeFactor
	This factor denotes the differences between the returned stresses and moduli and the passed strain. More...

Detailed Description

template<typename ST>
struct Ikarus::Materials::NeoHookeT< ST >

\[ \psi(\BC) = \frac{\mu}{2} (\tr \BC-3- 2 \log \sqrt{\det \BC}) + \frac{\la}{2} (\log \sqrt{\det \BC})^2 ,\]

where $ \BC $ denotes the right Cauchy-Green strain tensor.

The second Piola-Kirchhoff stresses are computed as

\[ \BS(\BC) =\fracpt{\psi(\BC)}{\BC} = \mu (\BI-\BC^{-1}) + \la \log \sqrt{\det \BC} \BC^{-1},\]

and the material tangent moduli are computed as

\[ \BBC(\BC) =\fracpt{^2\psi(\BC)}{\BC^2} = \la \BC^{-1} \otimes \BC^{-1} + 2 (\mu- \la \log \sqrt{\det \BC} ) \CI,\]

where $ \CI_{IJKL} = \frac{1}{2}({(\BC^{-1})}^{IK}{(\BC^{-1})}^{JL}+{(\BC^{-1})}^{IL} {(\BC^{-1})}^{JK}).$

Remarks: See [4], Section 6.4.3 for a discussion of this material

Template Parameters

ST	The underlying scalar type.

Member Typedef Documentation

◆ MaterialImpl

using Ikarus::Materials::Material< NeoHookeT< ST > >::MaterialImpl = NeoHookeT< ST >

inherited

◆ MaterialParameters

template<typename ST >

using Ikarus::Materials::NeoHookeT< ST >::MaterialParameters = LamesFirstParameterAndShearModulus

◆ MaterialTensor

template<typename ST >

using Ikarus::Materials::NeoHookeT< ST >::MaterialTensor = Eigen::TensorFixedSize<ScalarType, Eigen::Sizes<dim, dim, dim, dim> >

◆ ScalarType

template<typename ST >

using Ikarus::Materials::NeoHookeT< ST >::ScalarType = ST

◆ StrainMatrix

template<typename ST >

using Ikarus::Materials::NeoHookeT< ST >::StrainMatrix = Eigen::Matrix<ScalarType, dim, dim>

◆ StressMatrix

template<typename ST >

using Ikarus::Materials::NeoHookeT< ST >::StressMatrix = StrainMatrix

Constructor & Destructor Documentation

◆ NeoHookeT()

template<typename ST >

Ikarus::Materials::NeoHookeT< ST >::NeoHookeT ( const MaterialParameters & mpt )

inlineexplicit

Parameters

mpt	The Lame's parameters (first parameter and shear modulus).

Member Function Documentation

◆ impl() [1/2]

constexpr MaterialImpl & Ikarus::Materials::Material< NeoHookeT< ST > >::impl ( )

inlineconstexprinherited

Returns: Reference to the underlying material.

◆ impl() [2/2]

constexpr const MaterialImpl & Ikarus::Materials::Material< NeoHookeT< ST > >::impl ( ) const

inlineconstexprinherited

Returns: Const reference to the underlying material.

◆ materialInversion()

auto Ikarus::Materials::Material< NeoHookeT< ST > >::materialInversion	(	const Eigen::MatrixBase< Derived > &	Sraw,
		const Eigen::MatrixBase< Derived > &	EstartRaw = `Derived::Zero().eval()`,
		const double	tol = `1e-12`,
		const int	maxIter = `20`
	)		const

inlineinherited

This assumes the existence of a complementary stored energy function $\chi(\BS)$, such that $$ \partial_{\BS} \chi(\BS) := \BE$$. Except for linear materials, this is not just the inverse of the material tangent, but needs the inversion of the materials stored energy function. For SVK and Linear Elasticity, the inverse of $\BC$ is taken. For NeoHooke an analytical solution exists, and for the general hyperelastic framework (and for all materials that don't implement the material inversion, for that a strain energy function exists) a numerical approach is used.

Template Parameters

tag	Strain tag indicating which strain tensor components are expected as result.
voigt	Boolean indicating whether to return Voigt-shaped result.
useNumeric	forces the function to use the generic numerical approach
Derived	the type of the stress matrix

Parameters

Sraw	input stress matrix
EstartRaw	optionally define a starting value for the numerical algorithm (applies only to numerical inversion)
tol	tolerance for the Newton-Raphson solver (applies only to numerical inversion).
maxIter	maximum number of iterations for the Newton-Raphson solver (applies only to numerical inversion).

Returns: pair of inverse material tangent and strain tensor

◆ materialInversionImpl()

template<typename ST >

template<typename Derived >

auto Ikarus::Materials::NeoHookeT< ST >::materialInversionImpl ( const Eigen::MatrixBase< Derived > & Sraw ) const

inline

Analytical solution is taken from R. Pfefferkorn, S. Bieber, B. Oesterle, M. Bischoff, and P. Betsch, “Improving efficiency and robustness of enhanced assumed strain elements for nonlinear problems,” Numerical Meth Engineering, vol. 122, no. 8, pp. 1911–1939, Apr. 2021, doi: 10.1002/nme.6605.

Template Parameters

Derived The derived type of the input matrix.

Parameters

Sraw	the input stress matrix.

Returns: pair of inverse material tangent and right Cauchy-Green strain tensor in voigt notation.

◆ materialParameters()

auto Ikarus::Materials::Material< NeoHookeT< ST > >::materialParameters ( ) const

inlineinherited

Returns: Material parameter.

◆ materialParametersImpl()

template<typename ST >

MaterialParameters Ikarus::Materials::NeoHookeT< ST >::materialParametersImpl ( ) const

inline

◆ name()

static constexpr std::string Ikarus::Materials::Material< NeoHookeT< ST > >::name ( )

inlinestaticconstexprinherited

Returns: Name of the material.

◆ nameImpl()

template<typename ST >

static constexpr std::string Ikarus::Materials::NeoHookeT< ST >::nameImpl ( )

inlinestaticconstexprnoexcept

◆ rebind()

template<typename ST >

template<typename STO >

auto Ikarus::Materials::NeoHookeT< ST >::rebind ( ) const

inline

Template Parameters

STO	The target scalar type.

Returns: NeoHookeT<ScalarTypeOther> The rebound NeoHooke material.

◆ storedEnergy()

auto Ikarus::Materials::Material< NeoHookeT< ST > >::storedEnergy ( const Eigen::MatrixBase< Derived > & Eraw ) const

inlineinherited

This function return the free Helmholtz energy of the material

Template Parameters

tag	Strain tag indicating which strain tensor components are passed.
Derived	The underlying Eigen type.

Parameters

Eraw	The strain tensor components passed in Voigt notation or matrix notation.

Returns: Scalar return of stored energy.

◆ storedEnergyImpl()

template<typename ST >

template<typename Derived >

ScalarType Ikarus::Materials::NeoHookeT< ST >::storedEnergyImpl ( const Eigen::MatrixBase< Derived > & C ) const

inline

Template Parameters

Derived The derived type of the input matrix.

Parameters

C	The right Cauchy-Green tensor.

Returns: ScalarType The stored energy.

◆ stresses()

auto Ikarus::Materials::Material< NeoHookeT< ST > >::stresses ( const Eigen::MatrixBase< Derived > & Eraw ) const

inlineinherited

Template Parameters

tag	Strain tag indicating which strain tensor components are passed.
voigt	Boolean indicating whether to return Voigt-shaped result.
Derived	The underlying Eigen type.

Parameters

Eraw	The strain tensor components passed in Voigt notation or matrix notation.

Returns: Vectorial or Matrix return of stresses.

◆ stressesImpl()

template<typename ST >

template<bool voigt, typename Derived >

StressMatrix Ikarus::Materials::NeoHookeT< ST >::stressesImpl ( const Eigen::MatrixBase< Derived > & C ) const

inline

Template Parameters

voigt	A boolean indicating whether to return stresses in Voigt notation.
Derived	The derived type of the input matrix.

Parameters

C	The right Cauchy-Green tensor.

Returns: StressMatrix The stresses.

◆ tangentModuli()

auto Ikarus::Materials::Material< NeoHookeT< ST > >::tangentModuli ( const Eigen::MatrixBase< Derived > & Eraw ) const

inlineinherited

Template Parameters

tag	Strain tag indicating which strain tensor components are passed.
voigt	Boolean indicating whether to return Voigt-shaped result.
Derived	The underlying Eigen type.

Parameters

Eraw	The strain tensor components passed in Voigt notation or matrix notation.

Returns: Tangent moduli in Voigt notation or as fourth-order tensor.

◆ tangentModuliImpl()

template<typename ST >

template<bool voigt, typename Derived >

MaterialTensor Ikarus::Materials::NeoHookeT< ST >::tangentModuliImpl ( const Eigen::MatrixBase< Derived > & C ) const

inline

Template Parameters

voigt	A boolean indicating whether to return tangent moduli in Voigt notation.
Derived	The derived type of the input matrix.

Parameters

C	The right Cauchy-Green tensor.

Returns: MaterialTensor The tangent moduli.

Member Data Documentation

◆ derivativeFactor

constexpr double Ikarus::Materials::Material< NeoHookeT< ST > >::derivativeFactor

staticconstexprinherited

For neoHooke the inserted quantity is $C$ the Green-Lagrangian strain tensor, the function relation between the energy and the stresses is $S = 1 \dfrac{\partial\psi(E)}{\partial E}$. This factor is the pre factor, which is the difference to the actual derivative, which is written here

◆ derivativeFactorImpl

template<typename ST >

constexpr double Ikarus::Materials::NeoHookeT< ST >::derivativeFactorImpl = 2

staticconstexpr

◆ dim

template<typename ST >

constexpr int Ikarus::Materials::NeoHookeT< ST >::dim = 3

staticconstexpr

◆ energyAcceptsVoigt

template<typename ST >

constexpr bool Ikarus::Materials::NeoHookeT< ST >::energyAcceptsVoigt = false

staticconstexpr

◆ isLinear

constexpr bool Ikarus::Materials::Material< NeoHookeT< ST > >::isLinear

staticconstexprinherited

◆ isReduced

constexpr bool Ikarus::Materials::Material< NeoHookeT< ST > >::isReduced

staticconstexprinherited

◆ moduliAcceptsVoigt

template<typename ST >

constexpr bool Ikarus::Materials::NeoHookeT< ST >::moduliAcceptsVoigt = false

staticconstexpr

◆ moduliToVoigt

template<typename ST >

constexpr bool Ikarus::Materials::NeoHookeT< ST >::moduliToVoigt = false

staticconstexpr

◆ strainTag

template<typename ST >

constexpr auto Ikarus::Materials::NeoHookeT< ST >::strainTag = StrainTags::rightCauchyGreenTensor

staticconstexpr

◆ stressAcceptsVoigt

template<typename ST >

constexpr bool Ikarus::Materials::NeoHookeT< ST >::stressAcceptsVoigt = false

staticconstexpr

◆ stressTag

template<typename ST >

constexpr auto Ikarus::Materials::NeoHookeT< ST >::stressTag = StressTags::PK2

staticconstexpr

◆ stressToVoigt

template<typename ST >

constexpr bool Ikarus::Materials::NeoHookeT< ST >::stressToVoigt = false

staticconstexpr

◆ tangentModuliTag

template<typename ST >

constexpr auto Ikarus::Materials::NeoHookeT< ST >::tangentModuliTag = TangentModuliTags::Material

staticconstexpr

The documentation for this struct was generated from the following file:

neohooke.hh

Public Types

Public Member Functions

Static Public Member Functions

Static Public Attributes

Detailed Description

Member Typedef Documentation

◆ MaterialImpl

◆ MaterialParameters

◆ MaterialTensor

◆ ScalarType

◆ StrainMatrix

◆ StressMatrix

Constructor & Destructor Documentation

◆ NeoHookeT()

Member Function Documentation

◆ impl() [1/2]

◆ impl() [2/2]

◆ materialInversion()

◆ materialInversionImpl()

◆ materialParameters()

◆ materialParametersImpl()

◆ name()

◆ nameImpl()

◆ rebind()

◆ storedEnergy()

◆ storedEnergyImpl()

◆ stresses()

◆ stressesImpl()

◆ tangentModuli()

◆ tangentModuliImpl()

Member Data Documentation

◆ derivativeFactor

◆ derivativeFactorImpl

◆ dim

◆ energyAcceptsVoigt

◆ isLinear

◆ isReduced

◆ moduliAcceptsVoigt

◆ moduliToVoigt

◆ strainTag

◆ stressAcceptsVoigt

◆ stressTag

◆ stressToVoigt

◆ tangentModuliTag