NonlinearJ2

Nonlinear General J2 Plasticity Model

Summary

This is an abstract material class thus cannot be used directly. This class defines a general plasticity model using J2 yielding criterion with associated flow rule and mixed hardening rule. The isotropic/kinematic hardening response can be customized.

To use this model, a derived class shall be defined first.

class YourJ2 final : public NonlinearJ2 {
// class definition
}

The derived class only needs to implement four pure virtual methods that define the isotropic and kinematic hardening rules.

virtual double compute_k(double) const = 0; // isotropic hardening
virtual double compute_dk(double) const = 0; // derivative isotropic
virtual double compute_h(double) const = 0; // kinematic hardening
virtual double compute_dh(double) const = 0; // derivative kinematic

All four methods take equivalent plastic strain as the input argument, on output, the corresponding quantities shall be provided.

The isotropic hardening function \(K(\bar\varepsilon_p)\) defines the isotropic hardening rule, there are some requirements:

1. \(K(\bar\varepsilon_p)\) should be non-negative,
2. \(K(\bar\varepsilon_p=0)=\sigma_y\) where \(\sigma_y\) is the initial yielding stress.

There is no requirement for the kinematic hardening function \(H(\bar\varepsilon_p)\). Both hardening rules can coexist. However, to successfully solve the trial status, there is an additional constraint that shall be applied on the model:

\[ E+H'(\bar\varepsilon_p)+K'(\bar\varepsilon_p)\geqslant0~\text{for all}~\bar\varepsilon_p \]

Otherwise, the local Newton iteration will fail.

Brief On Theory

The NonlinearJ2 abstract class defines an associative plasticity framework using the von Mises yield criterion, which is defined as follows.

\[ F(\sigma,\bar\varepsilon_p)=\sqrt{\dfrac{3}{2}(s-\beta(\bar\varepsilon_p)):(s-\beta(\bar\varepsilon_p))}-\sigma_y( \bar\varepsilon_p) \]

where \(\beta(\bar\varepsilon_p)\) is the back stress depends on the equivalent plastic strain \(\bar\varepsilon_p\) and \(\sigma_y(\bar\varepsilon_p)\) is the yield stress. Note

\[ \sqrt{3J_2}=\sqrt{\dfrac{3}{2}(s-\beta(\bar\varepsilon_p)):(s-\beta(\bar\varepsilon_p))} \]

It is also called J2 plasticity model. A detailed discussion can be seen elsewhere. \(\beta(\bar\varepsilon_p)=H( \bar\varepsilon_p)\) and \(\sigma_y(\bar\varepsilon_p)=K(\bar\varepsilon_p)\).

History Layout

location	paramater
initial_history(0)	accumulated plastic strain
initial_history(1-6)	back stress

Kinematic Hardening

The back stress \(\beta(\bar{\varepsilon}_p)\) defines a kinematic hardening response. For example a linear kinematic hardening could be defined as:

\[ \beta(\bar\varepsilon_p)=E_K\bar\varepsilon_p \]

and the derivative

\[ \dfrac{\mathrm{d}\beta(\bar\varepsilon_p)}{\mathrm{d}\bar\varepsilon_p}=E_K \]

in which \(E_K\) is the kinematic hardening stiffness.

In this case, user shall override the corresponding two methods with such an implmentation.

double SampleJ2::compute_h(const double p_strain) const { return e_kin * p_strain; }
double SampleJ2::compute_dh(const double) const { return e_kin; }

Of course, a nonlinear relationship could also be defined.

Isotropic Hardening

The isotropic hardening is defined by function \(\sigma_y(\bar\varepsilon_p)\). The value \(\sigma_y(0)\) should be the initial yield stress. Also, for a bilinear isotropic hardening response, user shall override the following two methods.

double SampleJ2::compute_k(const double p_strain) const { return e_iso * p_strain + yield_stress; }
double SampleJ2::compute_dk(const double) const { return e_iso; }

Here another polynomial based isotropic hardening function is shown as an additional example. The function is defined as

\[ \sigma_y=\sigma_0(1+\sum_{i=1}^{n}a_i\bar\varepsilon_p^i) \]

where \(a_i\) are material constants. It is easy to see \(\sigma_y|_{\bar\varepsilon_p=0}=\sigma_0\). The derivative is

\[ \dfrac{\mathrm{d}\sigma_y}{\mathrm{d}\bar\varepsilon_p}=\sigma_0\sum_{i=1}^{n}ia_i\bar\varepsilon_p^{i-1} \]

To define such a hardening behavior, a vector shall be used to store all constants.

// PolyJ2.h
const vec poly_para; // poly_para is a vector stores a_i

The methods could be written as follows.

// PolyJ2.cpp
double PolyJ2::compute_k(const double p_strain) const {
 vec t_vec(poly_para.n_elem);

 t_vec(0) = 1.;
 for(uword I = 1; I < t_vec.n_elem; ++I) t_vec(I) = t_vec(I - 1) * p_strain;

 return yield_stress * dot(poly_para, t_vec);
}

double PolyJ2::compute_dk(const double p_strain) const {
 vec t_vec(poly_para.n_elem);

 t_vec(0) = 0.;
 t_vec(1) = 1.;
 for(uword I = 2; I < t_vec.n_elem; ++I) t_vec(I) = (double(I) + 1.) * pow(p_strain, double(I));

 return yield_stress * dot(poly_para, t_vec);
}

Example

A few different models are shown as examples. User can define arbitrary models.