Walker-Krempl rate sensitivity modification¶

Overview¶

This class modifies an existing Viscoplastic general flow rule to scale between a rate sensitive response (given by the base model) and a rate insensitive response based on a temperature-dependent parameter $\lambda$ . The approach was developed by Walker and Krempl [WK1978] to modify an underlying viscoplastic model to return a rate insensitive response at lower temperatures.

Given the base model scalar inelastic rate

$\dot{\bm{\varepsilon}}_{vp} = \dot{p} \mathbf{g}$

the model modifies the scalar part of the flow rate to

$\dot{p}_{mod} = \kappa \dot{p}$

keeping the flow direction the same. The scaling function is

$\kappa = 1 - \lambda + \frac{\lambda \sqrt{\frac{2}{3}} \left\Vert \dot{\boldsymbol{e}}\right\Vert }{\dot{\varepsilon}_{ref}}$

where $\dot{\varepsilon}_{ref}$ is a parameter and $\left\Vert \dot{\boldsymbol{e}}\right\Vert$ is the total (not inelastic) deviatoric strain rate.

In the limit $\lambda = 0$ the model returns the rate sensitive, viscoplastic flow response from the underlying model. In the limit $\lambda \rightarrow 1$ the model returns an equivalent rate insensitive flow rate, asymptotically approximating an equivalent rate independent model. An interpolation function can change the value of $\lambda$ as a function of temperature, for example to transition from a rate insensitive response at lower temperatures to a rate sensitive response at higher temperatures.

The implementation applies the same scale factor to all time derivatives in the base model’s history evolution equations, maintaining a consistent modified time integration scheme for all internal variables.

The user must determine an appropriate value of $\lambda$ (which cannot be set exactly to $\lambda = 1$ , as this results in numerical instability) and an appropriate reference rate $\dot{\varepsilon}_{ref}$ . A value of $\lambda = 0.99$ seems to work well and the reference rate should be set to a value much slower than the actual strain rates used by the material model.

Parameters¶

Parameter	Object type	Description	Default
`elastic`	`neml::LinearElasticModel`	Elasticity model	No
`flow`	`neml::ViscoPlasticFlowRule`	Viscoplastic flow rule interface	No
`lambda`	`neml::Interpolate`	Scaling parameter	No
`eps0`	`double`	Reference strain rate	No

Class description¶

class WalkerKremplSwitchRule : public neml::GeneralFlowRule ¶

WalkerKremplSwitchRule.

Public Functions

WalkerKremplSwitchRule(ParameterSet &params)¶: Parameters: elastic model and a viscoplastic flow rule.

virtual void populate_hist(History &hist) const¶: Setup internal state.

virtual void init_hist(History &hist) const¶: Initialize history.

virtual void s(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const sdot)¶: Stress rate.

virtual void ds_ds(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_sdot)¶: Partial of stress rate wrt stress.

virtual void ds_da(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_sdot)¶: Partial of stress rate wrt history.

virtual void ds_de(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_sdot)¶: Partial of stress rate wrt strain rate.

virtual void a(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const adot)¶: History rate.

virtual void da_ds(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_adot)¶: Partial of history rate wrt stress.

virtual void da_da(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_adot)¶: Partial of history rate wrt history.

virtual void da_de(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double *const d_adot)¶: Partial of history rate wrt strain rate.

virtual void work_rate(const double *const s, const double *const alpha, const double *const edot, double T, double Tdot, double &p_rate)¶: The implementation needs to define inelastic dissipation.

virtual void elastic_strains(const double *const s_np1, double T_np1, double *const e_np1) const¶: The implementation needs to define elastic strain.

virtual void set_elastic_model(std::shared_ptr<LinearElasticModel> emodel)¶: Set a new elastic model.

void kappa(const double *const edot, double T, double &kap)¶: The kappa function controlling rate sensitivity (public for testing)

void dkappa(const double *const edot, double T, double *const dkap)¶: Derivative of kappa wrt the strain rate (public for testing)

virtual void override_guess(double *const x)¶: Override initial guess.

Public Static Functions

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

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params)¶: Return default parameters.

static ParameterSet parameters()¶: Initialize from parameter set.