Finite-strain J2 plasticity

Theory

This model is a finite-strain hyperelastic-plastic J2 plasticity formulation with isotropic hardening. The strain energy potential is based on the compressible Neo-Hookean model (Pence–Gou, variant B).

We assume the multiplicative split of the deformation gradient into elastic and plastic parts,

\[\boldsymbol{F} = \boldsymbol{F}^{\rm e} \boldsymbol{F}^{\rm p},\qquad \boldsymbol{C}^{\rm e} = \big(\boldsymbol{F}^{\rm e})^{\mathsf{T}}\boldsymbol{F}^{\rm e},\qquad J^{\rm e} = \det \boldsymbol{F}^{\rm e},\qquad I_1^{\rm e} = \operatorname{tr}\boldsymbol{C}^{\rm e}.\]

The Pence–Gou potential (variant B) applied to \(\boldsymbol{C}^{\rm e}\) reads:

\[\Psi(\boldsymbol{C}^{\rm e}) = \Psi_{\rm d}(I_1^{\rm e}, J^{\rm e}) + \Psi_{\rm h}(J^{\rm e}) = \frac{G}{2} \left(I_1^{\rm e} (J^{\rm e})^{-2/3} - 3\right) + \frac{K}{8} \left(J^{\rm e} - (J^{\rm e})^{-1}\right)^2.\]

From the potential we obtain the second Piola–Kirchhoff stress (elastic configuration)

\[\boldsymbol{S} = 2 \frac{\partial \Psi}{\partial \boldsymbol{C}^{\rm e}},\]

and the Mandel stress in the intermediate configuration:

\[\boldsymbol{M} = \boldsymbol{C}^{\rm e} \boldsymbol{S}.\]

Plastic admissibility is enforced by a J2 yield function in Mandel stress space with a stress-like hardening variable \(\beta_{\rm p}(\alpha_{\rm p})\):

\[f(\boldsymbol{M}, \beta_{\rm p}) = \frac{1}{f_{\rm y}^0} \left(\sqrt{\boldsymbol{M}^{\rm dev}:\boldsymbol{M}^{\rm dev}} - \sqrt{\tfrac{2}{3}} \beta_{\rm p}\right) \le 0,\]

where \(\boldsymbol{M}^{\rm dev} = \boldsymbol{M} - \frac{1}{3}\operatorname{tr}(\boldsymbol{M})\boldsymbol{I}\) is the deviatoric part of the Mandel stress.

The hardening function is:

\[\beta_{\rm p}(\alpha_{\rm p}) = f_{\rm y}^\infty + (f_{\rm y}^0 - f_{\rm y}^\infty) e^{-\eta \alpha_{\rm p}} + H \alpha_{\rm p},\]

with a strain-like hardening internal variable \(\alpha_{\rm p}\).

An associated flow is used:

\[\boldsymbol{L}^{\rm p} = \dot{\lambda} \frac{\partial f}{\partial \boldsymbol{M}}, \qquad \dot{\alpha}_{\rm p} = -\dot{\lambda} \frac{\partial f}{\partial \beta_{\rm p}}.\]

The Karush-Kuhn-Tucker (KKT) conditions for plastic loading are:

\[\dot{\lambda} \geq 0, \quad f \leq 0, \quad \dot{\lambda} f = 0.\]

The Kirchhoff stress follows by push-forward from \(\boldsymbol{S}\) with \(\boldsymbol{F}^{\rm e}\):

\[\boldsymbol{\tau} = \boldsymbol{F}^{\rm e} \, \boldsymbol{S} \, \big(\boldsymbol{F}^{\rm e})^{\mathsf{T}}.\]

Stress update at a material point proceeds by computing the trial elastic split \(\boldsymbol{F}^{\rm e,tr}=\boldsymbol{F}\,\big(\boldsymbol{F}^{\rm p,old}\big)^{-1}\) and evaluating the yield function \(f\). If plastic yielding occurs (\(f > 0\)) a fully implicit return-mapping is solved for the unknown vector

\[\begin{split}\boldsymbol{X} = \begin{bmatrix} \boldsymbol{F}^{\rm e}_{11} \\ \boldsymbol{F}^{\rm e}_{12} \\ \vdots \\ \boldsymbol{F}^{\rm e}_{33} \\ \alpha_{\rm p} \\ \Delta\lambda \end{bmatrix}.\end{split}\]

A residual system \(\boldsymbol{R}(\boldsymbol{X})=\boldsymbol{0}\) enforces (i) elastic–plastic kinematics, (ii) hardening update, and (iii) consistency \(f=0\). The plastic update of \(\boldsymbol{F}^{\rm p}\) uses the matrix exponential of the flow direction in Mandel-space:

\[\Delta\boldsymbol{F}^{\rm p} = \exp\left(\Delta\lambda\,\frac{\partial f}{\partial \boldsymbol{M}}\right),\]

and the strain-like hardening variable is updated as:

\[\Delta\alpha_{\rm p} = -\Delta\lambda\,\frac{\partial f}{\partial \beta_{\rm p}} = \frac{\Delta\lambda}{f_{\rm y}^0} \sqrt{\frac{2}{3}}.\]

Upon convergence of the return-mapping algorithm, the plastic variables are updated according to:

\[\boldsymbol{F}^{\rm p,new} = \Delta\boldsymbol{F}^{\rm p}\,\boldsymbol{F}^{\rm p,old}, \qquad \alpha_{\rm p}^{\rm new} = \alpha_{\rm p}^{\rm old} - \Delta\lambda\,\frac{\partial f}{\partial \beta_{\rm p}} \;=\; \alpha_{\rm p}^{\rm old} + \frac{\Delta\lambda}{f_{\rm y}^0}\,\sqrt{\tfrac{2}{3}}.\]

The elastic deformation gradient follows from the multiplicative split as \(\boldsymbol{F}^{\rm e,new}=\boldsymbol{F}\,(\boldsymbol{F}^{\rm p,new})^{-1}\), and the Kirchhoff stress and consistent algorithmic tangent are then computed from the hyperelastic constitutive law using \(\boldsymbol{F}^{\rm e,new}\).

The consistent tangent \(\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}}\) is computed using the chain rule as:

\[\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}} = \frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{S}} : 2\,\frac{\partial^{2}\Psi}{\partial\boldsymbol{C}^{\rm e}\,\partial\boldsymbol{C}^{\rm e}} : \frac{\partial\boldsymbol{C}^{\rm e}}{\partial\boldsymbol{F}^{\rm e}} : \frac{\partial\boldsymbol{F}^{\rm e}}{\partial\boldsymbol{F}} + \frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}^{\rm e}} : \frac{\partial\boldsymbol{F}^{\rm e}}{\partial\boldsymbol{F}},\]

where the individual terms are:

\[\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{S}} = \boldsymbol{F}^{\rm e} \otimes \boldsymbol{F}^{\rm e},\]

\[\frac{\partial^{2}\Psi}{\partial\boldsymbol{C}^{\rm e}\,\partial\boldsymbol{C}^{\rm e}} = \frac{\partial^{2}\Psi}{\partial (J^{\rm e})^{2}}\, \frac{\partial J^{\rm e}}{\partial \boldsymbol{C}^{\rm e}}\otimes\frac{\partial J^{\rm e}}{\partial \boldsymbol{C}^{\rm e}} + \frac{\partial \Psi}{\partial J^{\rm e}}\, \frac{\partial^{2} J^{\rm e}}{\partial \boldsymbol{C}^{\rm e}\,\partial \boldsymbol{C}^{\rm e}} + \frac{\partial^{2}\Psi}{\partial J^{\rm e}\,\partial I_1^{\rm e}} \left( \frac{\partial J^{\rm e}}{\partial \boldsymbol{C}^{\rm e}}\otimes\frac{\partial I_1^{\rm e}}{\partial \boldsymbol{C}^{\rm e}} + \frac{\partial I_1^{\rm e}}{\partial \boldsymbol{C}^{\rm e}}\otimes\frac{\partial J^{\rm e}}{\partial \boldsymbol{C}^{\rm e}} \right),\]

\[\frac{\partial\boldsymbol{C}^{\rm e}}{\partial\boldsymbol{F}^{\rm e}} = (\boldsymbol{F}^{\rm e})^{\mathsf{T}}\otimes\boldsymbol{I} + \boldsymbol{I}\otimes(\boldsymbol{F}^{\rm e})^{\mathsf{T}},\]

\[\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}^{\rm e}} = \frac{\partial}{\partial\boldsymbol{F}^{\rm e}}\left(\boldsymbol{F}^{\rm e}\,\boldsymbol{S}\,(\boldsymbol{F}^{\rm e})^{\mathsf{T}}\right) = \boldsymbol{I}\otimes(\boldsymbol{S}\boldsymbol{F}^{\rm e})^{\mathsf{T}} + (\boldsymbol{S}\boldsymbol{F}^{\rm e})\otimes\boldsymbol{I},\]

\[\begin{split}\frac{\partial\boldsymbol{F}^{\rm e}}{\partial\boldsymbol{F}} = \begin{cases} \boldsymbol{I} \otimes (\boldsymbol{F}^{\rm p,old})^{-\mathsf{T}} & \text{ (for elastic step)} \\[0.5em] \text{extracted from } \frac{\partial\boldsymbol{X}}{\partial\boldsymbol{F}} & \text{ (for plastic step)} \end{cases}.\end{split}\]

For plastic step, \(\frac{\partial\boldsymbol{X}}{\partial\boldsymbol{F}}\) is computed by solving the linear system \(\frac{\partial\boldsymbol{R}}{\partial\boldsymbol{X}} \frac{\partial\boldsymbol{X}}{\partial\boldsymbol{F}} = -\frac{\partial\boldsymbol{R}}{\partial\boldsymbol{F}}\) using the converged Jacobian from the return mapping. Then \(\frac{\partial\boldsymbol{F}^{\rm e}}{\partial\boldsymbol{F}}\) is extracted from the first 9 rows of \(\frac{\partial\boldsymbol{X}}{\partial\boldsymbol{F}}\).

Stress update algorithm at a quadrature point for current step \(n+1\)

Notation: \((\cdot)^{\rm old} := (\cdot)_n\), \((\cdot)^{\rm new} := (\cdot)_{n+1}\).

Input (known quantities):

• Current deformation gradient: \(\boldsymbol{F}\)

• Plastic deformation gradient from previous step: \(\boldsymbol{F}^{\rm p,old}\)

• Strain-like hardening variable from previous step: \(\alpha_{\rm p}^{\rm old}\)

Trial elastic state:

• Compute trial elastic deformation gradient: \(\boldsymbol{F}^{\rm e,tr}=\boldsymbol{F}\,\big(\boldsymbol{F}^{\rm p,old}\big)^{-1}\)

• Compute trial Mandel stress: \(\boldsymbol{M}^{\rm tr} = \boldsymbol{C}^{\rm e,tr} \boldsymbol{S}^{\rm tr}\)

• Compute stress-like hardening variable: \(\beta_{\rm p} = f_{\rm y}^\infty + (f_{\rm y}^0 - f_{\rm y}^\infty) e^{-\eta \alpha_{\rm p}^{\rm old}} + H \alpha_{\rm p}^{\rm old}\)

• Evaluate yield function: \(f^{\rm tr} = \frac{1}{f_{\rm y}^0}\left(||\boldsymbol{M}^{\rm tr,dev}|| - \sqrt{\frac{2}{3}} \beta_{\rm p}\right)\)

• If \(f^{\rm tr} \leq 0\) (elastic step):

  • Accept the trial state:

    • \(\boldsymbol{F}^{\rm e,new} = \boldsymbol{F}^{\rm e,tr}\)

    • \(\boldsymbol{F}^{\rm p,new} = \boldsymbol{F}^{\rm p,old}\)

    • \(\alpha_{\rm p}^{\rm new} = \alpha_{\rm p}^{\rm old}\)

  • Compute Kirchhoff stress \(\boldsymbol{\tau}\) and consistent tangent \(\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}}\) using \(\boldsymbol{F}^{\rm e,new}\)

  • Return

Return mapping:

• If \(f^{\rm tr} > 0\) (plastic step):

  • Initialize unknowns: \(\boldsymbol{X} = \{\boldsymbol{F}^{\rm e}_{11}, \boldsymbol{F}^{\rm e}_{12}, \ldots, \boldsymbol{F}^{\rm e}_{33}, \alpha_{\rm p}, \Delta\lambda\}^{\mathsf{T}}\)

  • Solve nonlinear system \(\boldsymbol{R}(\boldsymbol{X}) = \boldsymbol{0}\) by Newton-Raphson iteration:

    • Residual equations:

      • \(\boldsymbol{R}_1\): Elastic-plastic kinematics: \(\boldsymbol{F}^{\rm e}\,\Delta\boldsymbol{F}^{\rm p} - \boldsymbol{F}\,\big(\boldsymbol{F}^{\rm p,old}\big)^{-1} = \boldsymbol{0}\)

      • \(\boldsymbol{R}_2\): Hardening evolution: \(\alpha_{\rm p} - \alpha_{\rm p}^{\rm old} - \frac{\Delta\lambda}{f_{\rm y}^0} \sqrt{\frac{2}{3}} = 0\)

      • \(\boldsymbol{R}_3\): Consistency condition: \(f(\boldsymbol{M}, \beta_{\rm p}) = 0\)

    • Newton-Raphson iteration: \(\boldsymbol{X}^{(k+1)} = \boldsymbol{X}^{(k)} - \left[\frac{\partial \boldsymbol{R}}{\partial \boldsymbol{X}}\right]^{-1} \boldsymbol{R}(\boldsymbol{X}^{(k)})\)

    • Convergence criteria: \(||\boldsymbol{R}|| < \text{TOL}\) and \(||\Delta\boldsymbol{X}|| < \text{TOL}\)

  • Upon convergence, update plastic variables:

    • \(\Delta\boldsymbol{F}^{\rm p} = \exp\left(\Delta\lambda \frac{\partial f}{\partial \boldsymbol{M}}\right)\)

    • \(\boldsymbol{F}^{\rm p,new} = \Delta\boldsymbol{F}^{\rm p} \boldsymbol{F}^{\rm p,old}\)

    • \(\alpha_{\rm p}^{\rm new} = \alpha_{\rm p}^{\rm old} + \frac{\Delta\lambda}{f_{\rm y}^0} \sqrt{\frac{2}{3}}\)

  • Compute updated elastic deformation gradient: \(\boldsymbol{F}^{\rm e,new} = \boldsymbol{F}\,(\boldsymbol{F}^{\rm p,new})^{-1}\)

  • Compute Kirchhoff stress \(\boldsymbol{\tau}\) and consistent tangent \(\frac{\partial\boldsymbol{\tau}}{\partial\boldsymbol{F}}\) using \(\boldsymbol{F}^{\rm e,new}\)

  • Return

Primary reference: A. Dummer, M. Neuner, P. Gamnitzer, G. Hofstetter (2024). Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers. Computer Methods in Applied Mechanics and Engineering 426:116987.

Implementation

class FiniteStrainJ2Plasticity : public MarmotMaterialFiniteStrain

Inheritance diagram for Marmot::Materials::FiniteStrainJ2Plasticity:

Collaboration diagram for Marmot::Materials::FiniteStrainJ2Plasticity:

Finite-strain hyperelastic-plastic J2 model with isotropic hardening.

Elastic law: hyperelastic, compressible Neo-Hookean (Pence–Gou potential, variant B).

Material parameters

• K — bulk modulus

• G — shear modulus

• fy — initial yield stress (reference)

• fyInf — saturated (asymptotic) yield stress

• eta — saturation rate parameter

• H — linear hardening modulus

• implementationType — algorithm selector (see below)

• density (optional) — density

State variables

• Fp — plastic deformation gradient

• alphaP — strain-like hardening variable

Implementation variants (implementationType)

• 0: scalar return mapping (not yet implemented)

• 1: Full return mapping; all derivatives computed analytically

• 2: FDAF — Full return mapping; derivatives approximated via forward finite differences

• 3: FDAC — Full return mapping; derivatives approximated via central finite differences

• 4: CSDA — Full return mapping; derivatives via complex-step differentiation

Public Functions

FiniteStrainJ2Plasticity(const double *materialProperties, int nMaterialProperties, int materialLabel)

Construct the finite-strain J2 plasticity model.

Parameters:

• materialProperties – Array with parameters: K, G, fy, fyInf, eta, H, implementationType, density (optional).

• nMaterialProperties – Length of materialProperties.

• materialLabel – Material label.

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

Compute the Kirchhoff stress and the algorithmic tangent for the current step. Performs an elastic trial; if yielding occurs, return mapping is performed.

Template parameter <3> indicates 3D.

Parameters:

• response – [inout]

  • tau - Kirchhoff stress tensor \(\boldsymbol{\tau}\).

  • elasticEnergyDensity - elastic energy density \(\psi\).

  • rho - density (from material parameters if provided).

• tangents – [inout]

  • dTau_dF - algorithmic tangent \(\partial\boldsymbol{\tau}/\partial\boldsymbol{F}\).

• deformation – [in]

  • F - deformation gradient \(\boldsymbol{F}\).

• timeIncrement – [in]

  • t - old (pseudo-)time.

  • dT- (pseudo-)time increment.

void computeStressWithScalarReturnMapping(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Scalar-return mapping variant; would solve a 1D consistency equation for \(\Delta\lambda\).

Note

Not yet implemented.

void computeStressWithFullReturnMapping(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Full return mapping; all derivatives computed analytically.

void computeStressFDAF(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Full return mapping; derivatives via forward finite differences (FDAF). Approximates \(\partial \boldsymbol{R}/\partial \boldsymbol{X}\) and \(\partial^{2}\Psi/\partial \boldsymbol{C}^{\mathrm e}\,\partial \boldsymbol{C}^{\mathrm e}\).

void computeStressFDAC(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Full return mapping; derivatives via central finite differences (FDAC). Approximates \(\partial \boldsymbol{R}/\partial \boldsymbol{X}\) and \(\partial^{2}\Psi/\partial \boldsymbol{C}^{\mathrm e}\,\partial \boldsymbol{C}^{\mathrm e}\).

void computeStressCSDA(ConstitutiveResponse<3> &response, AlgorithmicModuli<3> &tangents, const Deformation<3> &deformation, const TimeIncrement &timeIncrement)

Full return mapping; derivatives via complex-step differentiation approximation (CSDA). Evaluates \(\partial \boldsymbol{R}/\partial \boldsymbol{X}\) and \(\partial^{2}\Psi/\partial \boldsymbol{C}^{\mathrm e}\,\partial \boldsymbol{C}^{\mathrm e}\) using complex-step.

inline virtual int getNumberOfRequiredStateVars()

Get the number of required state variables.

Returns:

10 (9 for Fp and 1 for alphaP).

inline virtual double getDensity()

Return the material density if provided in material parameters.

virtual void assignStateVars(double *stateVars, int nStateVars)

Bind external storage for internal variables (Fp, alphaP).

Creates/updates the internal state manager that maps (Fp, alphaP) onto this buffer.

Parameters:

• stateVars – Pointer to a contiguous array provided by the caller.

• nStateVars – Number of entries in that array.

Throws:

std::invalid_argument – If nStateVars < getNumberOfRequiredStateVars().

virtual StateView getStateView(const std::string &result)

Expose a named view into the state vector.

Parameters:

result – One of: Fp (length 9), alphaP (length 1).

Returns:

A StateView object that provides access to the requested state variable; returns an empty view if result is unknown.

virtual void initializeYourself()

Initialize state (sets \(\boldsymbol F^{\mathrm p} = \boldsymbol I\))

inline std::tuple<double, Tensor33d, double> yieldFunction(const Tensor33d &Fe, const double betaP)

Yield function \(f(\boldsymbol F^{\mathrm e},\beta_{\mathrm p})\) and first derivatives.

Parameters:

• Fe – Elastic deformation gradient \(\boldsymbol F^{\mathrm e}\).

• betaP – Stress-like hardening variable \(\beta_{\mathrm p}\).

Returns:

\(\{\,f,\;\partial f/\partial \boldsymbol F^{\mathrm e},\;\partial f/\partial \beta_{\mathrm p}\,\}\).

template<typename T>
inline T yieldFunctionFromStress(const Tensor33t<T> &mandelStress, const T betaP)

Yield function \(f(\boldsymbol M,\beta_p)\) w.r.t. Mandel stress (templated).

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

• mandelStress – Mandel stress \(\boldsymbol M\).

• betaP – Stress-like hardening variable \(\beta_{\mathrm p}\).

Returns:

\(f\).

inline std::tuple<double, Tensor33d, Tensor3333d, double> yieldFunctionFromStress(const Tensor33d &mandelStress, const double betaP)

Yield function w.r.t. Mandel stress (double version).

Parameters:

• mandelStress – Mandel stress \(\boldsymbol M\).

• betaP – Stress-like hardening variable \(\beta_{\mathrm p}\).

Returns:

\(\{\,f,\;\partial f/\partial \boldsymbol M,\;\partial^2 f/\partial \boldsymbol M\partial \boldsymbol M,\;\partial f/\partial \beta_{\mathrm p}\,\}\).

template<typename T>
inline std::tuple<T, Tensor33t<T>, T> yieldFunctionFromStressFirstOrderDerived(const Tensor33t<T> &mandelStress, const T betaP)

First-order derivatives of \(f(\boldsymbol M,\beta_{\mathrm p})\) w.r.t. \(\boldsymbol M \).

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

• mandelStress – Mandel stress \(\boldsymbol M\).

• betaP – Stress-like hardening variable \(\beta_{\mathrm p}\).

Returns:

\(\{\,f,\;\partial f/\partial \boldsymbol M,\;\partial f/\partial \beta_{\mathrm p}\,\}\).

inline bool isYielding(const Tensor33d &Fe, const double betaP)

Check yield condition \(f(\boldsymbol F^{\mathrm e},\beta_{\mathrm p})>0\).

Parameters:

• Fe – Elastic deformation gradient \(\boldsymbol F^{\mathrm e}\).

• betaP – Stress-like hardening variable \(\beta_{\mathrm p}\).

Returns:

True if yielding occurs, i.e. \(f>0\).

inline std::tuple<Tensor33d, Tensor3333d> computeMandelStress(const Tensor33d &Fe)

Compute Mandel stress \(\boldsymbol M\) and its derivative w.r.t. \(\boldsymbol F^{\mathrm e}\).

Parameters:

Fe – Elastic deformation gradient \(\boldsymbol F^{\mathrm e}\).

Returns:

\(\{\,\boldsymbol M,\;\partial \boldsymbol M/\partial \boldsymbol F^{\mathrm e}\,\}\).

template<typename T>
inline Tensor33t<T> computeMandelStressOnly(const Tensor33t<T> &Fe)

Compute Mandel stress only (templated scalar type).

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

Fe – Elastic deformation gradient \(\boldsymbol F^{\mathrm e}\).

Returns:

Mandel stress \(\boldsymbol M\).

template<typename T>
inline T computeBetaPOnly(const T alphaP)

Isotropic hardening law.

Parameters:

alphaP – Strain-like hardening variable \(\alpha_{\mathrm p}\).

Returns:

Stress-like hardening variable \(\beta_{\mathrm p}\).

inline std::tuple<double, double> computeBetaP(const double alphaP)

Isotropic hardening law and its derivative.

Parameters:

alphaP – Strain-like hardening variable \(\alpha_{\mathrm p}\).

Returns:

\(\{\,\beta_{\mathrm p},\;\partial\beta_{\mathrm p}/\partial\alpha_{\mathrm p}\,\}\)

inline std::tuple<double, double> g(const Tensor33d &Fe_trial, const Tensor33d &df_dS, double dLambda, double betaP)

Scalar consistency function \(g(\Delta\lambda)\) and its derivative for the scalar-return mapping.

template<typename T>
inline Tensor33t<T> computeReturnMappingDirection(Tensor33t<T> Fe, T betaP)

Flow direction for return mapping, \(\partial f/\partial \boldsymbol S\) (templated).

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

• Fe – Elastic deformation gradient \(\boldsymbol F^{\mathrm e}\).

• betaP – Stress-like hardening \(\beta_{\mathrm p}\).

Returns:

\(\partial f/\partial \boldsymbol S\).

template<typename T>
inline std::tuple<Tensor33d, Tensor33d> computePlasticIncrement(Tensor33t<T> df_dS, T dLambda)

Incremental plastic flow via exponential map.

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

• df_dS – Flow direction in Mandel-stress space \(\partial f/\partial \boldsymbol S\).

• dLambda – Plastic multiplier increment \(\Delta\lambda\).

Returns:

\(\{\,\Delta\boldsymbol F^{\mathrm p},\;\partial\Delta\boldsymbol F^{\mathrm p}/\partial\boldsymbol S:\partial f/\partial\boldsymbol S\,\}\).

template<typename T>
inline VectorXt<T> computeResidualVector(const VectorXt<T> &X, const Tensor33d &FeTrial, const double alphaPTrial)

Residual vector \(\boldsymbol R(\boldsymbol X)\) for the full return mapping (templated).

Vector \(\boldsymbol X\) has 11 unknowns: 9 for \(\boldsymbol F^{\mathrm e}\), 1 for \(\alpha_{\mathrm p}\), 1 for \(\Delta\lambda\). Residual components:

• \(R_{0..8}\): elastic consistency (updated \(\boldsymbol F^{\mathrm e}\) vs trial),

• \(R_{9}\): \(\alpha_{\mathrm p}\) update,

• \(R_{10}\): yield \(f\).

Template Parameters:

T – Scalar type (double, hyper-dual, complex).

Parameters:

• X – Current iterate \([\mathrm{vec}(\boldsymbol F^{\mathrm e}),\;\alpha_{\mathrm p},\;\Delta\lambda]^\mathrm T\).

• FeTrial – Trial elastic deformation gradient \(\boldsymbol F^\text{e,trial}\).

• alphaPTrial – Trial strain-like hardening variable \(\alpha_{\mathrm p}^\text{trial}\).

Returns:

\(\boldsymbol R\).

inline std::tuple<Eigen::VectorXd, Eigen::MatrixXd> computeResidualVectorAndTangent(const Eigen::VectorXd &X, const Tensor33d &FeTrial, const double alphaPTrial)

Residual \(\boldsymbol R \) and Jacobian \(\partial\boldsymbol R/\partial\boldsymbol X\) for the full return mapping (double precision).

Parameters:

• X – Current iterate \([\mathrm{vec}(\boldsymbol F^{\mathrm e}),\;\alpha_{\mathrm p},\;\Delta\lambda]^\mathrm T\).

• FeTrial – Trial elastic gradient \(\boldsymbol F^\text{e,trial}\).

• alphaPTrial – Trial strain-like hardening variable \(\alpha_{\mathrm p}^\text{trial}\).

Returns:

\(\{\,\boldsymbol R,\;\partial\boldsymbol R/\partial\boldsymbol X\,\}\).

Public Members

const double K

Bulk modulus (read from materialProperties[0])

const double G

Shear modulus (read from materialProperties[1])

const double fy

Initial yield stress (read from materialProperties[2])

const double fyInf

Saturated (asymptotic) yield stress (read from materialProperties[3])

const double eta

Saturation rate parameter (read from materialProperties[4])

const double H

Linear hardening modulus (read from materialProperties[5])

const int implementationType

Algorithm variant selector (read from materialProperties[6]).

const double density

Density (read from materialProperties[7]) (if provided).

std::unique_ptr<FiniteStrainJ2PlasticityStateVarManager> stateVars

class FiniteStrainJ2PlasticityStateVarManager : public MarmotStateVarVectorManager

Inheritance diagram for Marmot::Materials::FiniteStrainJ2Plasticity::FiniteStrainJ2PlasticityStateVarManager:

Collaboration diagram for Marmot::Materials::FiniteStrainJ2Plasticity::FiniteStrainJ2PlasticityStateVarManager:

State variable manager for Fp and alphaP; provides named views and layout.

Public Functions

inline FiniteStrainJ2PlasticityStateVarManager(double *theStateVarVector)

Bind the manager to an external contiguous buffer holding (Fp, alphaP).

Public Members

Fastor::TensorMap<double, 3, 3> Fp

Plastic deformation gradient \(\boldsymbol F^{\mathrm p}\).

double &alphaP

Strain-like hardening variable \(\alpha_{\mathrm p}\).

Public Static Attributes

static const auto layout = makeLayout( {{ .name = "Fp", .length = 9 },{ .name = "alphaP", .length = 1 },} )

Memory layout of internal variables: 9 for Fp and 1 for alphaP (total 10).