Ikarus::Materials::ArrudaBoyceT< ST_ > Struct Template Reference

Implementation of the ArrudaBoyce material model (also referred as Eight-Chain model). More...

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

Public Types
using	ScalarType = ST_
template<typename ST = ScalarType>
using	PrincipalStretches = Eigen::Vector< ST, 3 >
template<typename ST = ScalarType>
using	Invariants = PrincipalStretches< ST >
template<typename ST = ScalarType>
using	FirstDerivative = Eigen::Vector< ST, dim >
template<typename ST = ScalarType>
using	SecondDerivative = Eigen::Matrix< ST, dim, dim >
using	MaterialParameters = ArrudaBoyceMatParameters

Public Member Functions
	ArrudaBoyceT (const MaterialParameters &matPar)
	Constructor for ArrudaBoyceT. More...
MaterialParameters	materialParametersImpl () const
	Returns the material parameters stored in the material. More...
template<typename ST >
ST	storedEnergyImpl (const PrincipalStretches< ST > &lambda) const
	Computes the stored energy in the ArrudaBoyce material model. More...
template<typename ST >
FirstDerivative< ST >	firstDerivativeImpl (const PrincipalStretches< ST > &lambda) const
	Computes the first derivative of the stored energy function w.r.t. the total principal stretches. More...
template<typename ST >
SecondDerivative< ST >	secondDerivativeImpl (const PrincipalStretches< ST > &lambda) const
	Computes the second derivatives of the stored energy function w.r.t. the total principal stretches. More...
template<typename STO >
auto	rebind () const
	Rebinds the material to a different scalar type. More...

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

Static Public Attributes
static constexpr int	dim = 3
static constexpr int	numTerms = 5
static constexpr auto	stretchTag = PrincipalStretchTags::deviatoric

Detailed Description

template<typename ST_>
struct Ikarus::Materials::ArrudaBoyceT< ST_ >

The energy is computed as

\[ \hat{\Psi}(\la_1, \la_2, \la_3) = \mu \sum_{p=0}^4{\al_p \beta^p} (W_1^{p+1} - 3^{p+1}), \]

with \( \beta = \frac{1}{\la_m^2} , \al_0 = \frac{1}{2}, \al_1 = \frac{1}{20}, \al_2 = \frac{11}{1050}, \al_3 = \frac{19}{7000}, \al_4 = \frac{519}{673750} \).

The first derivatives w.r.t the total principal stretches are

\[ \fracpt{\Psi}{\la_i} = \mu \sum_{p=0}^4{\al_p \beta^p W_1^p (p+1) \fracpt{W_1}{\la_i}}. \]

The second derivatives w.r.t the total principal stretches are

\[ \fracpt{^2 \Psi}{\la_i\partial\la_j} = \mu \sum_{p=0}^4{ \al_p \beta^p (W_1^p (p+1) \fracpt{^2 W_1}{\la_i\partial\la_j} + W_1^{p-1} p (p+1) \fracpt{W_1}{\la_i} \fracpt{W_1}{\la_j} - \delta_{ij} \frac{1}{\la_i} W_1^p (p+1) \fracpt{W_1}{\la_i}) } \]

Remarks

See [6] and [1] for details on this material. For information on the deviatoric invariant \( W_1 \), see DeviatoricInvariants.

Template Parameters

ST_	The underlying scalar type.

Member Typedef Documentation

FirstDerivative

template<typename ST_ >

template<typename ST = ScalarType>

using Ikarus::Materials::ArrudaBoyceT< ST_ >::FirstDerivative = Eigen::Vector<ST, dim>

Invariants

template<typename ST_ >

template<typename ST = ScalarType>

using Ikarus::Materials::ArrudaBoyceT< ST_ >::Invariants = PrincipalStretches<ST>

MaterialParameters

template<typename ST_ >

using Ikarus::Materials::ArrudaBoyceT< ST_ >::MaterialParameters = ArrudaBoyceMatParameters

PrincipalStretches

template<typename ST_ >

template<typename ST = ScalarType>

using Ikarus::Materials::ArrudaBoyceT< ST_ >::PrincipalStretches = Eigen::Vector<ST, 3>

ScalarType

template<typename ST_ >

using Ikarus::Materials::ArrudaBoyceT< ST_ >::ScalarType = ST_

SecondDerivative

template<typename ST_ >

template<typename ST = ScalarType>

using Ikarus::Materials::ArrudaBoyceT< ST_ >::SecondDerivative = Eigen::Matrix<ST, dim, dim>

Constructor & Destructor Documentation

ArrudaBoyceT()

template<typename ST_ >

Ikarus::Materials::ArrudaBoyceT< ST_ >::ArrudaBoyceT ( const MaterialParameters &  matPar )

inlineexplicit

Parameters

matPar

The material parameters for the Arruda-Boyce material model.

Member Function Documentation

firstDerivativeImpl()

template<typename ST_ >

template<typename ST >

FirstDerivative< ST > Ikarus::Materials::ArrudaBoyceT< ST_ >::firstDerivativeImpl ( const PrincipalStretches< ST > &  lambda ) const

inline

Parameters

lambda

principal stretches

Template Parameters

ST	the scalartype of the principal stretches

Returns

FirstDerivative The first derivatives of the stored energy function w.r.t. the total principal stretches.

materialParametersImpl()

template<typename ST_ >

MaterialParameters Ikarus::Materials::ArrudaBoyceT< ST_ >::materialParametersImpl ( ) const

inline

name()

template<typename ST_ >

static constexpr std::string Ikarus::Materials::ArrudaBoyceT< ST_ >::name ( )

inlinestaticconstexprnoexcept

rebind()

template<typename ST_ >

template<typename STO >

auto Ikarus::Materials::ArrudaBoyceT< ST_ >::rebind ( ) const

inline

Template Parameters

STO	The target scalar type.

Returns

ArrudaBoyceT<ScalarTypeOther> The rebound ArrudaBoyce material.

secondDerivativeImpl()

template<typename ST_ >

template<typename ST >

SecondDerivative< ST > Ikarus::Materials::ArrudaBoyceT< ST_ >::secondDerivativeImpl ( const PrincipalStretches< ST > &  lambda ) const

inline

Parameters

lambda

principal stretches

Template Parameters

ST	the scalartype of the principal stretches

Returns

SecondDerivative The second derivatives of the stored energy function w.r.t. the total principal stretches.

storedEnergyImpl()

template<typename ST_ >

template<typename ST >

ST Ikarus::Materials::ArrudaBoyceT< ST_ >::storedEnergyImpl ( const PrincipalStretches< ST > &  lambda ) const

inline

Using only the first five terms of the inverse Langevin function.

Parameters

lambda

principal stretches

Template Parameters

ST	the scalartype of the principal stretches

Returns

ScalarType

Member Data Documentation

dim

template<typename ST_ >

constexpr int Ikarus::Materials::ArrudaBoyceT< ST_ >::dim = 3

staticconstexpr

numTerms

template<typename ST_ >

constexpr int Ikarus::Materials::ArrudaBoyceT< ST_ >::numTerms = 5

staticconstexpr

stretchTag

template<typename ST_ >

constexpr auto Ikarus::Materials::ArrudaBoyceT< ST_ >::stretchTag = PrincipalStretchTags::deviatoric

staticconstexpr

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

• arrudaboyce.hh