MathLib::KelvinVector::Invariants< KelvinVectorSize > Struct Template Reference

Detailed Description

template<int KelvinVectorSize>
struct MathLib::KelvinVector::Invariants< KelvinVectorSize >

Invariants used in mechanics, based on Kelvin representation of the vectors and matrices. The invariants are computed at process creation time.

Definition at line 93 of file KelvinVector.h.

#include <KelvinVector.h>

Public Member Functions
double	determinant (Eigen::Matrix< double, 6, 1 > const &v)
double	determinant (Eigen::Matrix< double, 4, 1 > const &v)

Static Public Member Functions
static double	determinant (Eigen::Matrix< double, KelvinVectorSize, 1 > const &v)
	Determinant of a matrix in Kelvin vector representation.
static double	equivalentStress (Eigen::Matrix< double, KelvinVectorSize, 1 > const &deviatoric_v)
static double	FrobeniusNorm (Eigen::Matrix< double, KelvinVectorSize, 1 > const &deviatoric_v)
	Get the norm of the deviatoric stress.
static double	J2 (Eigen::Matrix< double, KelvinVectorSize, 1 > const &deviatoric_v)
static double	J3 (Eigen::Matrix< double, KelvinVectorSize, 1 > const &deviatoric_v)
static double	trace (Eigen::Matrix< double, KelvinVectorSize, 1 > const &v)
	Trace of the corresponding tensor.
static Eigen::Vector3d	diagonal (Eigen::Matrix< double, KelvinVectorSize, 1 > const &v)

Static Public Attributes
static Eigen::Matrix< double, KelvinVectorSize, KelvinVectorSize > const	deviatoric_projection
static Eigen::Matrix< double, KelvinVectorSize, KelvinVectorSize > const	spherical_projection
static Eigen::Matrix< double, KelvinVectorSize, 1 > const	identity2
	Kelvin mapping of 2nd order identity tensor.

Member Function Documentation

determinant() [1/3]

double MathLib::KelvinVector::Invariants< 4 >::determinant

(

Eigen::Matrix< double, 4, 1 > const &

v

)

Definition at line 28 of file KelvinVector.cpp.

{
    return v(2) * (v(0) * v(1) - v(3) * v(3) / 2.);
}

References MathLib::v.

determinant() [2/3]

double MathLib::KelvinVector::Invariants< 6 >::determinant

(

Eigen::Matrix< double, 6, 1 > const &

v

)

Definition at line 20 of file KelvinVector.cpp.

{
    return v(0) * v(1) * v(2) + v(3) * v(4) * v(5) / std::sqrt(2.) -
           v(3) * v(3) * v(2) / 2. - v(4) * v(4) * v(0) / 2. -
           v(5) * v(5) * v(1) / 2.;
}

References MathLib::v.

determinant() [3/3]

template<int KelvinVectorSize>

static double MathLib::KelvinVector::Invariants< KelvinVectorSize >::determinant ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & v )

static

Determinant of a matrix in Kelvin vector representation.

Referenced by MathLib::KelvinVector::inverse(), and MathLib::KelvinVector::inverse().

diagonal()

template<int KelvinVectorSize>

Eigen::Vector3d MathLib::KelvinVector::Invariants< KelvinVectorSize >::diagonal ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & v )

static

Diagonal of the corresponding tensor which is always of length 3 in 2D and 3D cases.

Definition at line 54 of file KelvinVector-impl.h.

{
    return v.template topLeftCorner<3, 1>();
}

References MathLib::v.

equivalentStress()

template<int KelvinVectorSize>

double MathLib::KelvinVector::Invariants< KelvinVectorSize >::equivalentStress ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & deviatoric_v )

static

The von Mises equivalent stress.

Note

The input vector must have trace equal zero.

Definition at line 18 of file KelvinVector-impl.h.

{
    assert(std::abs(trace(deviatoric_v)) <=
           4e-12 * diagonal(deviatoric_v).norm());
    return std::sqrt(3 * J2(deviatoric_v));
}

FrobeniusNorm()

template<int KelvinVectorSize>

double MathLib::KelvinVector::Invariants< KelvinVectorSize >::FrobeniusNorm ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & deviatoric_v )

static

Get the norm of the deviatoric stress.

Definition at line 27 of file KelvinVector-impl.h.

{
    return std::sqrt(deviatoric_v.transpose() * deviatoric_v);
}

J2()

template<int KelvinVectorSize>

double MathLib::KelvinVector::Invariants< KelvinVectorSize >::J2 ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & deviatoric_v )

static

Second invariant of deviatoric tensor.

Note

The input vector must have trace equal zero.

Definition at line 34 of file KelvinVector-impl.h.

{
    assert(std::abs(trace(deviatoric_v)) <=
           4e-12 * diagonal(deviatoric_v).norm());
    return 0.5 * deviatoric_v.transpose() * deviatoric_v;
}

J3()

template<int KelvinVectorSize>

double MathLib::KelvinVector::Invariants< KelvinVectorSize >::J3 ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & deviatoric_v )

static

Third invariant, equal to determinant of a deviatoric tensor.

Note

The input vector must have trace equal zero.

Definition at line 45 of file KelvinVector-impl.h.

{
    assert(std::abs(trace(deviatoric_v)) <=
           4e-12 * diagonal(deviatoric_v).norm());
    return determinant(deviatoric_v);
}

trace()

template<int KelvinVectorSize>

double MathLib::KelvinVector::Invariants< KelvinVectorSize >::trace ( Eigen::Matrix< double, KelvinVectorSize, 1 > const & v )

static

Trace of the corresponding tensor.

Definition at line 62 of file KelvinVector-impl.h.

{
    return diagonal(v).sum();
}

References MathLib::v.

Member Data Documentation

deviatoric_projection

template<int KelvinVectorSize>

const Eigen::Matrix< double, KelvinVectorSize, KelvinVectorSize > MathLib::KelvinVector::Invariants< KelvinVectorSize >::deviatoric_projection

static

Initial value:

=
        KelvinVector_detail::initDeviatoricProjection<KelvinVectorSize>()

Kelvin mapping of deviatoric projection tensor. \(A_{\rm dev} = P_{\rm dev}:A\) for \(A\) being a second order tensor.

Definition at line 101 of file KelvinVector.h.

identity2

template<int KelvinVectorSize>

const Eigen::Matrix< double, KelvinVectorSize, 1 > MathLib::KelvinVector::Invariants< KelvinVectorSize >::identity2

static

Initial value:

=
        KelvinVector_detail::initIdentity2<KelvinVectorSize>()

Kelvin mapping of 2nd order identity tensor.

Definition at line 107 of file KelvinVector.h.

Referenced by ProcessLib::ThermoMechanicalPhaseField::IntegrationPointData< BMatricesType, ShapeMatrixType, DisplacementDim >::updateConstitutiveRelation().

spherical_projection

template<int KelvinVectorSize>

Eigen::Matrix< double, KelvinVectorSize, KelvinVectorSize > const MathLib::KelvinVector::Invariants< KelvinVectorSize >::spherical_projection

static

Initial value:

=
        KelvinVector_detail::initSphericalProjection<KelvinVectorSize>()

Kelvin mapping of spherical projection tensor. \(A_{\rm sph} = P_{\rm sph}:A\) for \(A\) being a second order tensor.

Definition at line 105 of file KelvinVector.h.

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

• MathLib/KelvinVector.h
• MathLib/KelvinVector-impl.h