IsoKinJ2I1

Overview

This object implements the yield function

$f\left(\bm{\sigma}, \mathbf{q}, T\right) = J_2\left(\bm{\sigma} + \mathbf{X}\right) + \sqrt{\frac{2}{3}}Q + \operatorname{sign}\left(I_1\left(\bm{\sigma}\right)\right) h \left(I_1\left(\bm{\sigma}\right)\right)^l J_2\left(\mathbf{Y}\right) = \sqrt{\frac{3}{2} \operatorname{dev}\left(\mathbf{Y}\right): \operatorname{dev}\left(\mathbf{Y}\right)} I_1\left(\mathbf{Y}\right) = \operatorname{tr}\left(\mathbf{Y}\right).$

It assumes a “stress-like” history vector of

$\mathbf{q}=\left[\begin{array}{cc}Q & \mathbf{X}\end{array}\right]$

where $Q$ is the isotropic hardening stress and $\mathbf{X}$ is the backstress.

Warning

All of the NEML yield surfaces assume the opposite of the standard sign convention for isotropic and kinematic hardening. The hardening model is expected to return a negative value of the isotropic hardening stress and a negative value of the backstress.

Parameters

Parameter	Object type	Description	Default
`h`	`neml::Interpolate`	Power law prefactor	No
`l`	`neml::Interpolate`	Power law exponent	No

Class description

class IsoKinJ2I1 : public neml::YieldSurface 

Combined isotropic/kinematic hardening with some mean stress contribution.

Public Functions

IsoKinJ2I1(ParameterSet &params): Parameters: h prefactor, l exponent.

virtual size_t nhist() const: Expects 7 history variables [isotropic 6-Mandel-vector-backstress].

virtual void f(const double *const s, const double *const q, double T, double &fv) const: J2(stress + backstress) + isotropic + sign(mean_stress) * h * |mean_stress|^l

virtual void df_ds(const double *const s, const double *const q, double T, double *const df) const: Gradient wrt stress.

virtual void df_dq(const double *const s, const double *const q, double T, double *const df) const: Gradient wrt q.

virtual void df_dsds(const double *const s, const double *const q, double T, double *const ddf) const: Hessian dsds.

virtual void df_dqdq(const double *const s, const double *const q, double T, double *const ddf) const: Hessian dqdq.

virtual void df_dsdq(const double *const s, const double *const q, double T, double *const ddf) const: Hessian dsdq.

virtual void df_dqds(const double *const s, const double *const q, double T, double *const ddf) const: Hessian dqds.

Public Static Functions

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

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

static ParameterSet parameters(): Default parameters.