Rate independent plasticity¶

Overview¶

This class implements rate independent plasticity described by:

The elastic trial state:

$\bm{\varepsilon}^{p}_{tr} = \bm{\varepsilon}^{p}_n \bm{\sigma}_{tr} = \mathbf{\mathfrak{C}}_{n+1} : \left( \bm{\varepsilon}_{n+1} - \bm{\varepsilon}_{tr}^p \right) \bm{\alpha}_{tr} = \bm{\alpha}_{n}$

The plastic correction:

$\bm{\sigma}_{n+1} = \mathbf{\mathfrak{C}}_{n+1} : \left( \bm{\varepsilon}_{n+1} - \bm{\varepsilon}_{n+1}^p \right) \bm{\varepsilon}_{n+1}^p = \begin{cases} \bm{\varepsilon}^{p}_{tr} & f\left(\bm{\sigma}_{tr},\bm{\alpha}_{tr}\right)\le0\\ \bm{\varepsilon}^{p}_{tr}+\mathbf{g}\left( \bm{\sigma}_{n+1}, \bm{\alpha}_{n+1}, T_{n+1} \right)\Delta\gamma_{n+1} & f\left(\bm{\sigma}_{tr},\bm{\alpha}_{tr}\right)>0 \end{cases} \bm{\alpha}_{n+1} = \begin{cases} \bm{\alpha}_{tr} & f\left(\bm{\sigma}_{tr},\bm{\alpha}_{tr}\right)\le0\\ \bm{\alpha}_{tr}+\mathbf{h}\left( \bm{\sigma}_{n+1}, \bm{\alpha}_{n+1}, T_{n+1} \right)\Delta\gamma_{n+1} & f\left(\bm{\sigma}_{tr},\bm{\alpha}_{tr}\right)>0 \end{cases}$

Solving for $\Delta \gamma_{n+1}$ such that

$f\left(\bm{\sigma}_{n+1}, \bm{\alpha}_{n+1} \right) = 0$

In these equations $f$ is a yield function, $\mathbf{g}$ is a flow function, evaluated at the next state, and $\mathbf{h}$ is the rate of evolution for the history variables, evaluated at the next state. NEML integrates all three of these functions into a Rate independent flow rule interface.

If the step is plastic the stress update is solved through fully-implicit backward Euler integration. The algorithmic tangent is then computed using an implicit function scheme. The work and energy are integrated with a trapezoid rule from the final values of stress and plastic strain.

This model maintains a vector of history variables defined by the model’s Rate independent flow rule interface.

At the end of the step the model (optionally) checks to ensure the step met the Kuhn-Tucker conditions

$\Delta \gamma_{n+1} \ge 0 f\left(\bm{\sigma}_{n+1}, \bm{\alpha}_{n+1} \right) \le 0 \Delta \gamma_{n+1} f\left(\bm{\sigma}_{n+1}, \bm{\alpha}_{n+1} \right) = 0.$

Parameters¶

Parameter	Object type	Description	Default
`elastic`	`neml::LinearElasticModel`	Temperature dependent elastic constants	No
`surface`	`neml::RateIndependentFlowRule`	Flow rule interface	No
`alpha`	`neml::Interpolate`	Temperature dependent instantaneous CTE	`0.0`
`tol`	`double`	Integration tolerance	`1.0e-8`
`miter`	`int`	Maximum number of integration iters	`50`
`verbose`	`bool`	Print lots of convergence info	`false`
`kttol`	`double`	Tolerance on the Kuhn-Tucker conditions	`1.0e-2`
`check_kt`	`bool`	Flag to actually check KT	`false`

Class description¶

class SmallStrainRateIndependentPlasticity : public neml::SubstepModel_sd¶

Small strain, rate-independent plasticity.

Public Functions

SmallStrainRateIndependentPlasticity(ParameterSet &params)¶: Parameters: elasticity model, flow rule, CTE, solver tolerance, maximum solver iterations, verbosity flag, tolerance on the Kuhn-Tucker conditions check, and a flag on whether the KT conditions should be evaluated

virtual void populate_state(History &h) const¶: Populate internal variables.

virtual void init_state(History &h) const¶: Initialize history at time zero.

virtual TrialState *setup(const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, const double *const s_n, const double *const h_n)¶: Setup the trial state.

virtual bool elastic_step(const TrialState *ts, const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, const double *const s_n, const double *const h_n)¶: Ignore update and take an elastic step.

virtual void update_internal(const double *const x, const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, double *const s_np1, const double *const s_n, double *const h_np1, const double *const h_n)¶: Interpret the x vector.

virtual void strain_partial(const TrialState *ts, const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, const double *const s_np1, const double *const s_n, const double *const h_np1, const double *const h_n, double *const de)¶: Minus the partial derivative of the residual with respect to the strain.

virtual void work_and_energy(const TrialState *ts, const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, double *const s_np1, const double *const s_n, double *const h_np1, const double *const h_n, double &u_np1, double u_n, double &p_np1, double p_n)¶: Do the work calculation.

virtual size_t nparams() const¶: Number of solver parameters.

virtual void init_x(double *const x, TrialState *ts)¶: Setup an iteration vector in the solver.

virtual void RJ(const double *const x, TrialState *ts, double *const R, double *const J)¶: Solver function returning the residual and jacobian of the nonlinear system of equations integrating the model

const std::shared_ptr<const LinearElasticModel> elastic() const¶: Return the elastic model for subobjects.

void make_trial_state(const double *const e_np1, const double *const e_n, double T_np1, double T_n, double t_np1, double t_n, const double *const s_n, const double *const h_n, SSRIPTrialState &ts)¶: Setup a trial state.

Public Static Functions

static std::string type()¶: Type for the object system.

static ParameterSet parameters()¶: Parameters for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params)¶: Setup from a ParameterSet.