Linear Viscoelastic Orthotropic Power Law model

Theory

This is an orthotropic implementation of the isotropic model described in Linear Viscoelastic Power Law model. By contrast to the isotropic model, an additional scaling factor is used for scaling the stiffness tensor. This is helpful because the experimentally determined Young’s moduli and shear moduli are in general not equal to the initial elastic moduli of the viscoelastic model. By using this scaling factor, the experimentally determined Young’s moduli and shear moduli can be used as input parameters for the viscoelastic model and the initial elastic moduli are automatically adjusted. In the following, the scaling factor is denoted as \(s\) which is usually larger than one.

For the formulation of the viscoelastic model in Linear Viscoelastic Power Law model, the initial elastic stiffness tensor \(\Cel\) is replaced by

\[\Cel_0 = s\, \Cel( E_1, E_2, E_3, G_{12}, G_{23}, G_{31}, \nu_{12}, \nu_{23}, \nu_{31} ),\]

where the Young’s moduli \(E_1\), \(E_2\), and \(E_3\) and the shear moduli \(G_{12}\), \(G_{23}\), and \(G_{31}\) are given in the material directions \(x_1\), \(x_2\), and \(x_3\). Additionally, the Poisson’s ratios \(\nu_{12}\), \(\nu_{23}\), and \(\nu_{31}\) are used.

Note

The scaling factor \(s\) is applied to all initial elastic moduli. This means that the ratios between the Young’s moduli and shear moduli remain unchanged. Only the absolute values of the initial elastic moduli are scaled.

Warning

The definition of Poisson’s ratio in Marmot differs from the standard definition. In Marmot, Poisson’s ratio \(\nu_{ij}\) is defined as the ratio of the lateral strain in direction \(x_j\) to the axial strain in direction \(x_i\) when a uniaxial stress is applied in direction \(x_i\). See namespace Marmot::ContinuumMechanics::Elasticity for more details.

Accordingly, the normalized initial elastic compliance tensor now reads

\[\DelNu = s\, E_1\, \CelInv_0,\]

with the Young’s modulus \(E_1\) as the Young’s modulus in the \(x_1\) material direction. Constant Poisson’s ratios are assumed in all material directions.

Implementation

class LinearViscoelasticOrthotropicPowerLaw : public MarmotMaterialHypoElastic

Inheritance diagram for Marmot::Materials::LinearViscoelasticOrthotropicPowerLaw:

Collaboration diagram for Marmot::Materials::LinearViscoelasticOrthotropicPowerLaw:

Implementation of an orthotropic linear viscoelastic material model following a Power Law compliance function and assuming constant Poisson’s ratios generalized for 3D stress states.

Public Functions

LinearViscoelasticOrthotropicPowerLaw(const double *materialProperties, int nMaterialProperties, int materialLabel)

virtual void computeStress(state3D &state, Marmot::Matrix6d &dStressDDStrain, const Marmot::Vector6d &dStrain, const timeInfo &timeInfo) const override

For a given linearized strain increment \(\Delta\boldsymbol{\varepsilon}\) at the old and the current time, compute the Cauchy stress and the algorithmic tangent \(\frac{\partial\boldsymbol{\sigma}^{(n+1)}}{\partial\boldsymbol{\varepsilon}^{(n+1)}}\).

Parameters:

• state – [inout] A state3D instance carrying stress, strain energy, and state variables

• dStress_dStrain – [inout] Algorithmic tangent representing the derivative of the Cauchy stress tensor with respect to the linearized strain

• dStrain – [in] linearized strain increment

• timeInfo – Structure carrying the current (pseudo-)time and the (pseudo-)time increment

virtual double getDensity(const double *stateVars) const override

Get the mass density of the material.

Parameters:

stateVars – Pointer to the state variable array

Returns:

Mass density of the material

inline MarmotMaterialHypoElastic(const double *matProperties_, int nMaterialProperties_, int materialNumber_)

Constructs the material with a given set of material properties and an identifier.

Parameters:

• matProperties_ – [in] Pointer to the array of material properties.

• nMaterialProperties_ – [in] Number of entries in matProperties_.

• materialNumber_ – [in] Integer identifying this material instance.

Private Members

const double &stiffnessScaleFactor

Factor to scale elastic stiffness.

const double &E1

Young’s modulus in x1 direction.

const double &E2

Young’s modulus in x2 direction.

const double &E3

Young’s modulus in x3 direction.

const double &nu12

Poisson’s ratio.

const double &nu23

Poisson’s ratio.

const double &nu13

Poisson’s ratio.

const double &G12

Shear modulus in x1-x2 plane.

const double &G23

Shear modulus in x2-x3 plane.

const double &G13

Shear modulus in x1-x3 plane.

const double &m

power law compliance parameter

const double &n

power law exponent

const int powerLawApproximationOrder

approximation order for the retardation spectrum (must be 2, 3, 4, or 7)

const size_t nKelvin

number of Kelvin units to approximate the viscoelastic compliance

const double &minTau

minimal retardation time used in the viscoelastic Kelvin chain

const double &spacing

log spacing between the retardation times of the Kelvin Chain

const double &timeToDays

ratio of simulation time to days

const Vector3d direction1

direction x1 w.r.t the global coordinate system

const Vector3d direction2

direction x2 w.r.t the global coordinate system

const std::map<std::string, std::pair<int, int>> stateVarInfo = {{"kelvinStateVars", std::make_pair(0, 6 * nKelvin)},}

KelvinChain::Properties elasticModuli

Elastic moduli of the Kelvin chain units.

KelvinChain::Properties retardationTimes

Retardation times of the Kelvin chain units.

double zerothKelvinChainCompliance

Zeroth Kelvin chain compliance.

Matrix6d CInv

Inverse of the initial elastic stiffness.

Matrix6d Cel

Initial elastic stiffness.

Matrix6d CelUnit

Normalized initial elastic stiffness.

It is normalized by the Young’s modulus in x1 direction E1

Matrix6d CelUnitInv

Inverse of the normalized initial elastic stiffness.

Matrix6d CelUnitGlobal

Initial elastic stiffness in the global coordinate system.

Matrix3d localCoordinateSystem

Local coordinate system of the material.