Perzyna viscoplastic flow rule

Overview

The Perzyna model is an associative viscoplastic model [P1966]. It is defined by:

$\dot{\gamma} = g\left(\left\langle f\left(\bm{\sigma}, \mathbf{q}\left(\bm{\alpha}\right), T\right)\right\rangle\right) \mathbf{g}_{\gamma} = \frac{\partial f}{\partial \bm{\sigma}} \left( \bm{\sigma}, \mathbf{q}\left(\bm{\alpha}\right), T \right) \mathbf{h}_{\gamma} = \frac{\partial f}{\partial \mathbf{q}} \left( \bm{\sigma}, \mathbf{q}\left(\bm{\alpha}\right), T \right)$

where $f$ is a flow surface, a hardening interface provides the $\mathbf{q}$ function, and $g$ is a Rate function. The notation $\left\langle \right\rangle$ indicates the Macaulay brackets. The implementation uses the default zero values of the time and temperature rate contributions and so the model evolves only as a function of the inelastic strain rate.

The model maintains the set of history variables defined by the hardening model.

Parameters

Parameter	Object type	Description	Default
`surface`	`neml::YieldSurface`	Flow surface interface	No
`hardening`	`neml::HardeningRule`	Hardening rule interface	No
`g`	`neml::GFlow`	Rate sensitivity function	No

Class description

class PerzynaFlowRule : public neml::ViscoPlasticFlowRule 

Perzyna associative viscoplasticity.

Public Functions

PerzynaFlowRule(ParameterSet &params): Parameters: a flow surface, a hardening rule, and the rate sensitivity function

virtual void populate_hist(History &hist) const: Populate a blank history object.

virtual void init_hist(History &hist) const: Initialize history at time zero.

virtual void y(const double *const s, const double *const alpha, double T, double &yv) const: Scalar strain rate.

virtual void dy_ds(const double *const s, const double *const alpha, double T, double *const dyv) const: Derivative of y wrt stress.

virtual void dy_da(const double *const s, const double *const alpha, double T, double *const dyv) const: Derivative of y wrt history.

virtual void g(const double *const s, const double *const alpha, double T, double *const gv) const: Flow rule proportional to the scalar strain rate.

virtual void dg_ds(const double *const s, const double *const alpha, double T, double *const dgv) const: Derivative of g wrt stress.

virtual void dg_da(const double *const s, const double *const alpha, double T, double *const dgv) const: Derivative of g wrt history.

virtual void h(const double *const s, const double *const alpha, double T, double *const hv) const: Hardening rule proportional to the scalar strain rate.

virtual void dh_ds(const double *const s, const double *const alpha, double T, double *const dhv) const: Derivative of h wrt stress.

virtual void dh_da(const double *const s, const double *const alpha, double T, double *const dhv) const: Derivative of h wrt history.

Public Static Functions

static std::string type(): String type for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Default parameters.

static ParameterSet parameters(): Initialize from parameters.

Rate function

These functions define the rate sensitivity of the Peryna flow model. They have the form $g\left(f, T\right)$ where f is the current value of the flow surface.