Matrix Invariants (principal and mixed, modified (isochoric) invariants and deviatoric invariants). More...

Classes
struct	RFFGen::LinearAlgebra::InvariantTraits< Invariant::PRINCIPAL >
	Traits class for access of (shifted) principal invariants. More...

struct	RFFGen::LinearAlgebra::InvariantTraits< Invariant::MODIFIED >
	Traits class for access of (shifted) modified principal invariants. More...

struct	RFFGen::LinearAlgebra::InvariantTraits< Invariant::MIXED >
	Traits class for access of (shifted) mixed invariants. More...

struct	RFFGen::LinearAlgebra::InvariantTraits< Invariant::MODIFIED_MIXED >
	Traits class for access of (shifted) modified mixed invariants. More...

struct	RFFGen::LinearAlgebra::FirstModifiedMixedInvariant< Matrix, class >
	First modified mixed invariant $\bar\iota_4=\iota_4\iota_3^{-1/3}$ . More...

struct	RFFGen::LinearAlgebra::SecondModifiedMixedInvariant< Matrix, class >
	Second modified mixed invariant $\bar\iota_5=\iota_5\iota_3^{-2/3}$ . More...

struct	RFFGen::LinearAlgebra::ThirdModifiedMixedInvariant< Matrix, class >
	Third modified mixed invariant $\bar\iota_6=\iota_6\iota_3^{-1/3}$ . More...

struct	RFFGen::LinearAlgebra::FirstModifiedPrincipalInvariant< Matrix, class >
	Isochoric (volume-preserving), first modified principal invariant $\bar\iota_1(A)=\iota_1\iota_3^{-1/3}$ , where $\iota_1$ is the first and $\iota_3$ is the third principal invariant. More...

struct	RFFGen::LinearAlgebra::SecondModifiedPrincipalInvariant< Matrix, class >
	Isochoric (volume-preserving), second modified principal invariant $\bar\iota_2(A)=\iota_2\iota_3^{-1/3}$ , where $\iota_2$ is the second and $\iota_3$ is the third principal invariant. More...

class	RFFGen::LinearAlgebra::SecondPrincipalInvariant< Matrix, class >
	Second principal invariant $\iota_2(A)=\mathrm{tr}(\mathrm{cof}(A))$ for $A\in\mathbb{R}^{n,n}$ . More...

Typedefs
template<class Matrix >
using	RFFGen::LinearAlgebra::SecondDeviatoricInvariant = MathematicalOperations::Chain< MatrixNorm< Matrix >, Deviator< Matrix > >
	Second deviator invariant $J_2(\sigma)=\sqrt{\bar\sigma\negthinspace:\negthinspace\bar\sigma}$ with $\bar\sigma = \sigma - \frac{\mathrm{tr}(\sigma)}{n}I$ and $\sigma\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::FirstMixedInvariant = MathematicalOperations::Chain< FirstPrincipalInvariant< Matrix >, MathematicalOperations::Product< Identity< Matrix >, Constant< Matrix > > >
	First mixed invariant $\iota_4=\iota_1(AM)$ of a matrix $A\in\mathbb{R}^{n,n}$ with respect to the structural tensor $M\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::SecondMixedInvariant = MathematicalOperations::Chain< FirstPrincipalInvariant< Matrix >, MathematicalOperations::Product< MathematicalOperations::Squared< Identity< Matrix > >, Constant< Matrix > > >
	Second mixed invariant $\iota_5=\iota_1(A^2M)$ of a matrix $A\in\mathbb{R}^{n,n}$ with respect to the structural tensor $M\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ThirdMixedInvariant = MathematicalOperations::Chain< FirstPrincipalInvariant< Matrix >, MathematicalOperations::Product< Identity< Matrix >, MathematicalOperations::Squared< Constant< Matrix > > > >
	Third mixed invariant $\iota_6=\iota_1(AM^2)$ of a matrix $A\in\mathbb{R}^{n,n}$ with respect to the structural tensor $M\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedFirstMixedInvariant = ShiftedInvariant< FirstMixedInvariant< Matrix >, 1 >
	Shifted first mixed invariant $\iota_4 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedSecondMixedInvariant = ShiftedInvariant< SecondMixedInvariant< Matrix >, 1 >
	Shifted second mixed invariant $\iota_5 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedThirdMixedInvariant = ShiftedInvariant< ThirdMixedInvariant< Matrix >, 1 >
	Shifted third mixed invariant $\iota_6 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedFirstModifiedMixedInvariant = ShiftedInvariant< FirstModifiedMixedInvariant< Matrix >, 1 >
	Shifted first modified mixed invariant $\bar\iota_4 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedSecondModifiedMixedInvariant = ShiftedInvariant< SecondModifiedMixedInvariant< Matrix >, 1 >
	Shifted second modified mixed invariant $\bar\iota_5 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedThirdModifiedMixedInvariant = ShiftedInvariant< ThirdModifiedMixedInvariant< Matrix >, 1 >
	Shifted third modified mixed invariant $\bar\iota_6 - 1$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ThirdModifiedPrincipalInvariant = ThirdPrincipalInvariant< Matrix >
	Third modified principal invariant is the same as the third principal invariant. This invariant describes volumetric changes.

template<class Matrix , int offset = LinearAlgebra::dimension<Matrix>()>
using	RFFGen::LinearAlgebra::ShiftedFirstModifiedPrincipalInvariant = ShiftedInvariant< FirstModifiedPrincipalInvariant< Matrix >, offset >
	Shifted first modified principal invariant $\bar\iota_1(A) - n$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix , int offset = LinearAlgebra::dimension<Matrix>()>
using	RFFGen::LinearAlgebra::ShiftedSecondModifiedPrincipalInvariant = ShiftedInvariant< SecondModifiedPrincipalInvariant< Matrix >, offset >
	Shifted second modified principal invariant $\bar\iota_2(A) - n$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedThirdModifiedPrincipalInvariant = ShiftedInvariant< ThirdModifiedPrincipalInvariant< Matrix >, 1 >
	Shifted third modified principal invariant $\bar\iota_3(A) - 1$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::FirstPrincipalInvariant = Trace< Matrix >
	First principal invariant $\iota_1(A)=\mathrm{tr}(A)$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ThirdPrincipalInvariant = Determinant< Matrix >
	Third principal invariant $\iota_3(A)=\det(A)$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix , int offset = dimension<Matrix>()>
using	RFFGen::LinearAlgebra::ShiftedFirstPrincipalInvariant = ShiftedInvariant< FirstPrincipalInvariant< Matrix >, offset >
	Shifted first principal invariant $\iota_1(A) - n$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix , int offset = dimension<Matrix>()>
using	RFFGen::LinearAlgebra::ShiftedSecondPrincipalInvariant = ShiftedInvariant< SecondPrincipalInvariant< Matrix >, offset >
	Shifted second principal invariant $\iota_2(A) - n$ for $A\in\mathbb{R}^{n,n}$ .

template<class Matrix >
using	RFFGen::LinearAlgebra::ShiftedThirdPrincipalInvariant = ShiftedInvariant< ThirdPrincipalInvariant< Matrix >, 1 >
	Shifted third principal invariant $\iota_3(A) - 1$ for $A\in\mathbb{R}^{n,n}$ .

Enumerations
enum	RFFGen::LinearAlgebra::Invariant { PRINCIPAL, MIXED, MODIFIED, MODIFIED_MIXED }
	Enum to statically choose invariant traits.

Detailed Description

Matrix Invariants (principal and mixed, modified (isochoric) invariants and deviatoric invariants).

Classes

Typedefs

Enumerations

Detailed Description