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.

class GFlow : public neml::NEMLObject ¶

The “g” function in the Perzyna model — often a power law.

Subclassed by neml::GPowerLaw

Public Functions

GFlow(ParameterSet &params)¶

virtual double g(double f, double T) const = 0¶: The value of g.

virtual double dg(double f, double T) const = 0¶: The derivative of g wrt to the flow surface.

Power law¶

This rate function implements simple power law

$g\left(f, T\right) = \left(\frac{f}{\eta}\right)^n$

for some temperature-dependent rate sensitivity exponent $n$ and fluidity $\eta$ .

Parameters¶

Parameter	Object type	Description	Default
`n`	`neml::Interpolate`	Rate sensitivity exponent	No
`eta`	`neml::Interpolate`	Fluidity	No

Class description¶

class GPowerLaw : public neml::GFlow ¶

g is a power law

Public Functions

GPowerLaw(ParameterSet &params)¶: Parameter: the power law exponent.

virtual double g(double f, double T) const¶: g = (f/eta)^n

virtual double dg(double f, double T) const¶: Derivative of g wrt f.

double n(double T) const¶: Helper, just return the power law exponent.

double eta(double T) const¶: Helper, just returns the fluidity.

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.