Von Mises model (Automatic Differentiation)

An automatic differentiation implementation of classical J2 plasticity with isotropic hardening.

Index	Model Parameter	Description
0	\(E\)	Young’s modulus
1	\(\nu\)	Poisson’s ratio
2	\(f_\mathrm{y}^{0}\)	Initial yield stress
3	\(H_\mathrm{iso,lin}\)	First hardening parameter [1]
4	\(\Delta f_\mathrm{y}^{0,\infty}\)	Second hardening parameter [1]
5	\(\delta\)	Third hardening parameter [1]

State Variable	Name	Description
\(\kappa\)	kappa	Hardening variable [1]

[1] (1,2,3,4)

see Hardening Law.

Theory

See Von Mises model.

Implementation

Implementation follows the Von Mises model, however, automatic differentiation is used to compute the consistent algorithmic tangent operator.

class ADVonMises : public MarmotMaterialHypoElasticAD

Inheritance diagram for Marmot::Materials::ADVonMises:

Collaboration diagram for Marmot::Materials::ADVonMises:

Implementation of a isotropic J2-plasticity material for 3D stress states using automatic differentiation.

Public Functions

ADVonMises(const double *materialProperties, int nMaterialProperties, int materialNumber)

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

Public Members

const double &E

Young’s modulus.

const double &nu

Poisson’s ratio.

const double &yieldStress

Initial yield stress.

const double &HLin

First hardening parameter.

const double &deltaYieldStress

Second hardening parameter.

const double &delta

Third hardening parameter.

const double G

Shear modulus.

Protected Functions

virtual void computeStressAD(state3DAD &state, const Marmot::Vector6dual &dStrain, const timeInfo &timeInfo) const override

Compute the Cauchy stress tensor \(\boldsymbol{\sigma}\) given an increment of the linearized strain tensor \(\Delta\boldsymbol{\varepsilon}\).

Dual numbers are used for both the Cauchy stress tensor and the linearized strain increment, such that the algorithmic tangent operator \(\frac{\partial\boldsymbol{\sigma}^{(n+1)}}{\partial\boldsymbol{\varepsilon}^{(n+1)}}\) can be obtained by means of automatic differentiation.

Parameters:

• state – [inout] State carrying the dual Cauchy stress, strain energy, and state variables

• dStrain – [in] linearized strain increment

• timeInfo – Old (pseudo-)time

template<typename T>
inline T fy(T kappa_) const

Hardening function.

in] kappa Hardening variable.

Tparam :

Returns:

Current yield stress.

template<typename T>
inline T f(const T rho_, const double kappa_) const

Yield function.

in] rho Deviatoric radius ( \(\rho = ||s||\)).

Tparam :

Parameters:

kappa – [in] Hardening variable.

Returns:

Value of the yield function.

template<typename T>
inline T g(const T rhoTrial, const double kappa, const T deltaKappa) const

Objective function for stress update algorithm.

in] rhoTrial Deviatoric radius.

in] deltaKappa Increment of the hardening variable.

Tparam :

Parameters:

kappa – [in] Current value of the hardening variable.

Tparam :

Returns:

Value of the yield function.