MarmotFiniteStrainMechanicsCore

template<int nDim>
struct AlgorithmicModuli

#include <MarmotMaterialFiniteStrain.h>

Algorithmic tangent moduli of a material.

Contains the algorithmic tangent moduli \(\frac{\partial \boldsymbol{\tau}}{\partial \boldsymbol{F}}\) with respect to the deformation gradient \(\boldsymbol{F}\).

Template Parameters:

nDim – Number of spatial dimensions (2 or 3).

Public Members

Fastor::Tensor<double, nDim, nDim, nDim, nDim> dTau_dF

tangent operator w.r.t. deformation gradient

template<int nDim>
struct ConstitutiveResponse

#include <MarmotMaterialFiniteStrain.h>

Constitutive response of a material at given state.

Contains stress, density and elastic energy density.

Template Parameters:

nDim – Number of spatial dimensions (2 or 3).

Public Members

Fastor::Tensor<double, nDim, nDim> tau

Kirchhoff stress.

double rho

mass density

double elasticEnergyDensity

elastic energy per unit volume

template<int nDim>
struct Deformation

#include <MarmotMaterialFiniteStrain.h>

Represents the deformation state of a material.

This struct holds the deformation gradient \(\boldsymbol{F}\) which describes the local deformation of a material.

Public Members

Fastor::Tensor<double, nDim, nDim> F

deformation gradient

class MarmotMaterialFiniteStrain : public MarmotMaterial

Inheritance diagram for MarmotMaterialFiniteStrain:

Collaboration diagram for MarmotMaterialFiniteStrain:

Abstract basic class for mechanical materials in the finite strain regime.

Public Functions

virtual void computeStress(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3>&, const TimeIncrement&) = 0

Updates the material state.

Computes the Kirchhoff \(\boldsymbol{\tau}\) stress from the deformation gradient \(\boldsymbol{F}\) and the time increment \(\Delta t\). It further updates the mass density \(\rho\) and the elastic energy density. Additionally, computes the algorithmic tangent moduli \(\frac{\partial \boldsymbol{\tau}}{\partial \boldsymbol{F}}\).

Parameters:

• response – [inout] ConstitutiveResponse instance

• tangents – [out] AlgorithmicModuli instance

• deformation – [in] Deformation instance

• timeIncrement – [in] TimeIncrement instance

virtual void computeStress(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement, const std::tuple<double, double, double> &eigenDeformation)

Computes the Kirchhoff stress given the deformation, time increment, and eigen deformation.

Parameters:

• response – [inout] ConstitutiveResponse instance

• tangents – [out] AlgorithmicModuli instance

• deformation – [in] Deformation instance

• timeIncrement – [in] TimeIncrement instance

• eigenDeformation – [in] Tuple representing eigen deformation in each spatial direction.

virtual void computePlaneStrain(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &algorithmicModuli, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Compute stress under plane strain conditions.

It uses the general 3D computeStress function for a plane strain Deformation. The algorithmic tangent is modified according to plane strain conditions.

Parameters:

• response – [inout] ConstitutiveResponse instance

• algorithmicModuli – [out] AlgorithmicModuli instance

• deformation – [in] Deformation instance

• timeIncrement – [in] TimeIncrement instance

virtual void computePlaneStrain(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &algorithmicModuli, const Deformation<3> &deformation, const TimeIncrement &timeIncrement, const std::tuple<double, double, double> &eigenDeformation)

Compute stress under plane strain conditions with eigen deformation.

It uses the general 3D computeStress function for a plane strain Deformation. The algorithmic tangent is modified according to plane strain conditions.

Parameters:

• response – [inout] ConstitutiveResponse instance

• algorithmicModuli – [out] AlgorithmicModuli instance

• deformation – [in] Deformation instance

• timeIncrement – [in] TimeIncrement instance

• eigenDeformation – [in] Tuple representing eigen deformation in each spatial direction.

virtual void computePlaneStress(ConstitutiveResponse<2> &response, AlgorithmicModuli<2> &algorithmicModuli, const Deformation<2> &deformation, const TimeIncrement &timeIncrement)

Compute stress under plane stress conditions.

It uses the general 3D computeStress function and iteratively finds the out-of-plane deformation. The algorithmic tangent is modified according to plane stress conditions.

Parameters:

• response – [inout] ConstitutiveResponse instance

• algorithmicModuli – [out] AlgorithmicModuli instance

• deformation – [in] Deformation instance

• timeIncrement – [in] TimeIncrement instance

std::tuple<double, double, double> findEigenDeformationForEigenStress(const std::tuple<double, double, double> &initialGuess, const std::tuple<double, double, double> &eigenStress)

Find the eigen deformation that corresponds to a given eigen stress.

This function iteratively finds the eigen deformation that corresponds to a given eigen stress. This is used e.g. for geostatic stress initialization.

Parameters:

• initialGuess – Initial guess for the eigen deformation.

• eigenStress – Target eigen stress.

Returns:

Eigen deformation that corresponds to the given eigen stress.

struct TimeIncrement

#include <MarmotMaterialFiniteStrain.h>

Represents a time increment in a simulation.

This struct holds information about the current time and the time step size (dT).

Public Members

const double time

time at the beginning of the increment

const double dT

size of the time increment

namespace Fastor

namespace FastorIndices

namespace FastorStandardTensors

namespace Marmot

namespace ContinuumMechanics

namespace DeformationMeasures

Functions

template<typename T>
Tensor33t<T> rightCauchyGreen(const Tensor33t<T> &F)

Computes the right Cauchy-Green tensor \(\boldsymbol{C}\).

The right Cauchy-Green deformation tensor is defined as:

\[ \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \]

or in index notation:

\[ C_{IJ} = F_{iI} F_{iJ} \]

Template Parameters:

T – Scalar type (e.g., float, double, autodiff::dual)

Parameters:

F – Deformation gradient tensor

Returns:

The right Cauchy-Green tensor

template<typename T>
Tensor33t<T> leftCauchyGreen(const Tensor33t<T> &F)

Computes the left Cauchy-Green tensor \(\boldsymbol{b}\).

The left Cauchy-Green deformation tensor is defined as:

\[ \boldsymbol{b} = \boldsymbol{F} \boldsymbol{F}^T \]

or in index notation:

\[ b_{ij} = F_{iI} F_{jI} \]

Template Parameters:

T – Scalar type (e.g., float, double, autodiff::dual)

Parameters:

F – Deformation gradient tensor

Returns:

The left Cauchy-Green tensor b

namespace FirstOrderDerived

Functions

template<typename T>
std::pair<Tensor33t<T>, Tensor3333t<T>> rightCauchyGreen(const Tensor33t<T> &F)

Computes the right Cauchy-Green tensor \(\boldsymbol{C}\) and its derivative with respect to \(\boldsymbol{F}\).

The right Cauchy-Green deformation tensor is defined as:

\[ \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \]

or in index notation:

\[ C_{IJ} = F_{iI} F_{iJ} \]

The derivative of \(\boldsymbol{C}\) with respect to \(\boldsymbol{F}\) is given by:

\[ \frac{\partial C_{IJ}}{\partial F_{kK}} = \delta_{IK} F_{kJ} + F_{kI} \delta_{JK} \]

Template Parameters:

T – Scalar type (e.g., float, double, autodiff::dual)

Parameters:

F – Deformation gradient tensor

Returns:

A pair containing the right Cauchy-Green tensor C and its derivative with respect to F

template<typename T>
std::pair<Tensor33t<T>, Tensor3333t<T>> leftCauchyGreen(const Tensor33t<T> &F)

Computes the left Cauchy-Green tensor \(\boldsymbol{b}\) and its derivative with respect to \(\boldsymbol{F}\).

The left Cauchy-Green deformation tensor is defined as:

\[ \boldsymbol{b} = \boldsymbol{F} \boldsymbol{F}^T \]

or in index notation:

\[ b_{ij} = F_{iI} F_{jI} \]

The derivative of \(\boldsymbol{b}\) with respect to \(\boldsymbol{F}\) is given by:

\[ \frac{\partial b_{ij}}{\partial F_{kK}} = \delta_{ik} F_{jK} + F_{iK} \delta_{jk} \]

Template Parameters:

T – Scalar type (e.g., float, double, autodiff::dual)

Parameters:

F – Deformation gradient tensor

Returns:

A pair containing the left Cauchy-Green tensor b and its derivative with respect to F

namespace EnergyDensityFunctions

Functions

template<typename T>
T PenceGouPotentialA(const Tensor33t<T> &C, const double K, const double G)

Hyperelastic Energy Density Function Wa acc. Pence & Gou (2015), Eq. (2.11)

The energy density function \(W_a\) is given as

\[ W_a = \frac{G}{2} (I_1 - 3) + \left(\frac{K}{2} - \frac{G}{3}\right) (J - 1)^2 - G \ln(J) \]

where \( I_1 = \text{tr}(\boldsymbol{C}) \) is the first invariant of the right Cauchy-Green tensor \( \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \), \( J = \sqrt{\det(\boldsymbol{C})} = \det(\boldsymbol{F}) \) is the determinant of the deformation gradient, and \( K, G \) are the bulk and shear modulus, respectively.

Template Parameters:

T – Scalar type, e.g. double, float, etc.

Parameters:

• C – Right Cauchy-Green tensor

• K – Bulk modulus

• G – Shear modulus

Returns:

Energy density

template<typename T>
T PenceGouPotentialB(const Tensor33t<T> &C, const double K, const double G)

Hyperelastic Energy Density Function Wb acc. Pence & Gou (2015), Eq. (2.12)

The energy density function \(W_b\) is given as

\[ W_b = \frac{K}{8} \left(J - \frac{1}{J}\right)^2 + \frac{G}{2} \left(I_1 J^{-\frac{2}{3}} - 3\right) \]

where \( I_1 = \text{tr}(\boldsymbol{C}) \) is the first invariant of the right Cauchy-Green tensor \( \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \), \( J = \sqrt{\det(\boldsymbol{C})} = \det(\boldsymbol{F}) \) is the determinant of the deformation gradient, and \( K, G \) are the bulk and shear modulus, respectively.

Template Parameters:

T – Scalar type, e.g. double, float, etc.

Parameters:

• C – Right Cauchy-Green tensor

• K – Bulk modulus

• G – Shear modulus

Returns:

Energy density

template<typename T>
T PenceGouPotentialC(const Tensor33t<T> &C, const double K, const double G)

Hyperelastic Energy Density Function Wc acc. Pence & Gou (2015), Eq. (2.13)

The energy density function \(W_c\) is given as

\[ W_c = \frac{G}{2} (I_1 - 3) + \frac{3 G^2}{3 K - 2 G} \left(J^{\frac{2}{3} - \frac{K}{G}} - 1\right) \]

where \( I_1 = \text{tr}(\boldsymbol{C}) \) is the first invariant of the right Cauchy-Green tensor \( \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \), \( J = \sqrt{\det(\boldsymbol{C})} = \det(\boldsymbol{F}) \) is the determinant of the deformation gradient, and \( K, G \) are the bulk and shear modulus, respectively.

Template Parameters:

T – Scalar type, e.g. double, float, etc.

Parameters:

• C – Right Cauchy-Green tensor

• K – Bulk modulus

• G – Shear modulus

Returns:

Energy density

namespace FirstOrderDerived

Functions

template<typename T>
std::tuple<T, Tensor33t<T>> PenceGouPotentialB(const Tensor33t<T> &C, const double K, const double G)

Hyperelastic Energy Density Function Wb acc. Pence & Gou (2015), Eq. (2.12) and its first derivative w.r.t. C.

The energy density function \(W_b\) is given as

\[ W_b = \frac{K}{8} \left(J - \frac{1}{J}\right)^2 + \frac{G}{2} \left(I_1 J^{-\frac{2}{3}} - 3\right) \]

where \( I_1 = \text{tr}(\boldsymbol{C}) \) is the first invariant of the right Cauchy-Green tensor \( \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \), \( J = \sqrt{\det(\boldsymbol{C})} = \det(\boldsymbol{F}) \) is the determinant of the deformation gradient, and \( K, G \) are the bulk and shear modulus, respectively.

Additionally, the first derivative w.r.t. C is computed as

\[ \frac{\partial W_b}{\partial \boldsymbol{C}} = \frac{\partial W_b}{\partial J} \frac{\partial J}{\partial \boldsymbol{C}} + \frac{\partial W_b}{\partial I_1} \frac{\partial I_1}{\partial \boldsymbol{C}} \]

where

\[ \frac{\partial J}{\partial \boldsymbol{C}} = \frac{1}{2} J \boldsymbol{C}^{-1} \]

and

\[ \frac{\partial I_1}{\partial \boldsymbol{C}} = \boldsymbol{I} \]

Template Parameters:

T – Scalar type, e.g. double, float, etc.

Parameters:

• C – Right Cauchy-Green tensor

• K – Bulk modulus

• G – Shear modulus

Returns:

A tuple containing energy density and its first derivative w.r.t. C

namespace SecondOrderDerived

Functions

template<typename T>
std::tuple<T, Tensor33t<T>, Tensor3333t<T>> PenceGouPotentialB(const Tensor33t<T> &C, const double K, const double G)

Hyperelastic Energy Density Function Wb acc. Pence & Gou (2015), Eq. (2.12) and its first and second derivative w.r.t. C.

The energy density function \(W_b\) is given as

\[ W_b = \frac{K}{8} \left(J - \frac{1}{J}\right)^2 + \frac{G}{2} \left(I_1 J^{-\frac{2}{3}} - 3\right) \]

where \( I_1 = \text{tr}(\boldsymbol{C}) \) is the first invariant of the right Cauchy-Green tensor \( \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F} \), \( J = \sqrt{\det(\boldsymbol{C})} = \det(\boldsymbol{F}) \) is the determinant of the deformation gradient, and \( K, G \) are the bulk and shear modulus, respectively.

Additionally, the first and second derivative w.r.t. \(\boldsymbol{C}\), i.e.,

\[\frac{\partial W_b}{\partial \boldsymbol{C}} \quad \text{and} \quad \frac{\partial^2 W_b}{\partial \boldsymbol{C} \partial \boldsymbol{C}} \]

are computed.

Template Parameters:

T – Scalar type, e.g. double, float, etc.

Parameters:

• C – Right Cauchy-Green tensor

• K – Bulk modulus

• G – Shear modulus

Returns:

A tuple containing energy density, its first and second derivative w.r.t. C

namespace FiniteStrain

namespace Plasticity

namespace FlowIntegration

Functions

template<typename T>
Tensor33t<T> exponentialMap(const Tensor33t<T> &dGp)

Computes the incremental plastic deformation gradient from the plastic velocity gradient using the exponential map.

The exponential map is given by

\[ \Delta F_{\bar I J} = \exp\left(\Delta \lambda \frac{\partial g}{\partial M_{\bar K L}}\right)_{J \bar{I}} \]

where \( \Delta \lambda \frac{\partial g}{\partial M_{\bar K L}} \) is the incremental plastic velocity gradient.

Template Parameters:

T – Scalar type.

Parameters:

dGp – Incremental plastic velocity gradient.

Returns:

Incremental plastic deformation gradient.

namespace FirstOrderDerived

Functions

std::pair<Tensor33d, Tensor3333d> explicitIntegration(const Tensor33d &deltaGp)

Computes the incremental plastic deformation gradient from the plastic velocity gradient using a first order approximation. The first order approximation is given by

\[ \Delta F_{\bar I J} = \left( \delta_{\bar K L} + \Delta \lambda \frac{\partial f}{\partial M_{\bar K L}}\right)_{J \bar{I}} \]

where \( \Delta \lambda \frac{\partial f}{\partial M_{\bar K L}} \) is the incremental plastic velocity gradient.

Parameters:

deltaGp – Incremental plastic velocity gradient.

Returns:

A pair of the incremental plastic deformation gradient and its derivative w.r.t. the plastic velocity gradient.

std::pair<Tensor33d, Tensor3333d> exponentialMap(const Tensor33d &deltaGp)

Computes the incremental plastic deformation gradient from the plastic velocity gradient using the exponential map. The exponential map is given by

\[ \Delta F_{\bar I J} = \exp\left(\Delta \lambda \frac{\partial f}{\partial M_{\bar I J}}\right)_{J \bar{I}} \]

where \( \Delta \lambda \frac{\partial f}{\partial M_{\bar I J}} \) is the incremental plastic velocity gradient.

Parameters:

deltaGp – Incremental plastic velocity gradient.

Returns:

A pair of the incremental plastic deformation gradient and its derivative w.r.t. the plastic velocity gradient.

namespace StressMeasures

Functions

template<typename T>
Tensor33t<T> KirchhoffStressFromPK2(const Tensor33t<T> &PK2, const Tensor33t<T> &F)

Computes the Kirchhoff stress from the 2nd Piola-Kirchhoff stress and the deformation gradient.

The Kirchhoff stress is computed as

\[ \boldsymbol{\tau} = \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T \]

or in index notation

\[\tau_{ij} = F_{iI} S_{IJ} F_{jJ}. \]

Template Parameters:

T – Scalar type, e.g. double, float

Parameters:

• PK2 – 2nd Piola-Kirchhoff stress

• F – Deformation gradient

Returns:

Kirchhoff stress

namespace FirstOrderDerived

Functions

template<typename T>
std::tuple<Tensor33t<T>, Tensor3333t<T>, Tensor3333t<T>> KirchhoffStressFromPK2(const Tensor33t<T> &PK2, const Tensor33t<T> &F)

Computes the Kirchhoff stress from the 2nd Piola-Kirchhoff stress and the deformation gradient and the respective partial derivatives.

The Kirchoff stress is computed as

\[ \boldsymbol{\tau} = \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T \]

or in index notation

\[\tau_{ij} = F_{iI} S_{IJ} F_{jJ}. \]

Additionally, the derivatives with respect to \( \boldsymbol{S} \) and \( \boldsymbol{F} \) are computed in index notation as

\[ \frac{\partial \tau_{ij}}{\partial S_{IJ}} = F_{iI} F_{jJ} \]

and

\[ \frac{\partial \tau_{ij}}{\partial F_{kL}} = \delta_{ik} S_{LJ} F_{jJ} + F_{iI} S_{IL} \delta_{jk} \]

Template Parameters:

T – Scalar type, e.g. double, float

Parameters:

• PK2 – 2nd Piola-Kirchhoff stress

• F – Deformation gradient

Returns:

A tuple containing Kirchhoff stress and its derivatives w.r.t \(\boldsymbol{S}\) and \(\boldsymbol{F}\)

file MarmotDeformationMeasures.h

#include “Marmot/MarmotFastorTensorBasics.h”

Include dependency graph for MarmotDeformationMeasures.h:

file MarmotEnergyDensityFunctions.h

#include “Fastor/Fastor.h”

#include “Marmot/MarmotFastorTensorBasics.h”

Include dependency graph for MarmotEnergyDensityFunctions.h:

file MarmotFiniteStrainPlasticity.h

#include “Marmot/MarmotFastorTensorBasics.h”

#include “Marmot/MarmotTensorExponential.h”

Include dependency graph for MarmotFiniteStrainPlasticity.h:

file MarmotMaterialFiniteStrain.h

#include “Fastor/Fastor.h”

#include “Marmot/MarmotMaterial.h”

#include <Fastor/tensor/Tensor.h>

Include dependency graph for MarmotMaterialFiniteStrain.h:

file MarmotStressMeasures.h

#include “Marmot/MarmotFastorTensorBasics.h”

#include <Fastor/tensor/Tensor.h>

Include dependency graph for MarmotStressMeasures.h:

dir /home/runner/work/Marmot/Marmot/modules/core

dir /home/runner/work/Marmot/Marmot/modules/core/MarmotFiniteStrainMechanicsCore/include

dir /home/runner/work/Marmot/Marmot/modules/core/MarmotFiniteStrainMechanicsCore/include/Marmot

dir /home/runner/work/Marmot/Marmot/modules/core/MarmotFiniteStrainMechanicsCore

dir /home/runner/work/Marmot/Marmot/modules