CustomMises1D

J2 Plasticity Model With Custom Hardening

Theory

This model is an implementation of the Mises1D abstract model.

Syntax

material CustomMises1D (1) (2) (3) (4) [5]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) int, isotropic hardening expression tag
# (4) int, kinematic hardening expression tag
# [5] double, density, default: 0.0

Usage

Both the isotropic and kinematic hardening functions are provided by Expression objects.

Both hardening functions shall be defined in terms of the equivalent plastic strain.

The isotropic hardening function evaluates to the yield stress for trivial equivalent plastic strain.

The expressions shall be able to compute derivatives.

Example

Isotropic Hardening

For example, one can define a purely isotropic hardening model as follows:

expression SimpleScalar 1 x 10+.5x
expression SimpleScalar 2 x 0

material CustomMises1D 1 10 1 2

materialTest1D 1 1E-2 150 150 200 200 250

In the above example, the isotropic hardening function is defined as:

\[ y=10+0.5x, \]

in which \(x\) maps to the equivalent plastic strain and \(y\) maps to the shifted stress.

The kinematic hardening function is defined as:

\[ y=0. \]

For elastic modulus of \(E=10\), the isotropic hardening ratio satisfies:

\[ 0.5=E\dfrac{H}{1-H}, \]

solving which yields \(H=0.04762\).

The last point is \(11.190476190476174\), then

\[ \dfrac{10.714285714285706-10}{2.5-1}=0.04762E. \]

Kinematic Hardening

expression SimpleScalar 1 x 10
expression SimpleScalar 2 x .1x

material CustomMises1D 1 10 1 2
materialTest1D 1 1E-2 150 100 150 100 150

In the above example, purely kinematic hardening is defined.

The hardening ratio is \(0.1/(E+0.1)=0.0099\).

The last point is \(10.074626865671641\), then

\[ \dfrac{10.148514851485146-10}{2.5-1}=0.0099. \]

With custom functions, it is possible to define arbitrary hardening rules.