Perfect plasticity¶

Overview¶

This class implements rate independent perfect plasticity described by:

The elastic trial state:

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

The plastic correction:

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

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

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

In these equations $f$ is a yield function, parameterized by the yield stress $\sigma_0$ .

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 does not maintain any history variables.

Parameters¶

Parameter	Object type	Description	Default
`elastic`	`neml::LinearElasticModel`	Temperature dependent elastic constants	No
`surface`	`neml::YieldSurface`	The yield surface	No
`ys`	`neml::Interpolate`	The yield stress as a function of T	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`
`max_divide`	`int`	Maximum number of adaptive subdivisions	`8`

Class description¶

class SmallStrainPerfectPlasticity : public neml::SubstepModel_sd¶

Small strain, associative, perfect plasticity.

Public Functions

SmallStrainPerfectPlasticity(ParameterSet &params)¶: Parameters: elastic model, yield surface, yield stress, CTE, integration tolerance, maximum number of iterations, verbosity flag, and the maximum number of adaptive subdivisions

virtual void populate_state(History &h) const¶: Populate the internal variables (nothing)

virtual void init_state(History &h) const¶: Initialize history (nothing to do)

virtual size_t nparams() const¶: Number of nonlinear equations to solve in the integration.

virtual void init_x(double *const x, TrialState *ts)¶: Setup an initial guess for the nonlinear solution.

virtual void RJ(const double *const x, TrialState *ts, double *const R, double *const J)¶: Integration residual and jacobian equations.

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)¶: 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 *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.

double ys(double T) const¶: Helper to return the yield stress.

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, SSPPTrialState &ts)¶: Setup a trial state for the solver from the input information.

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.