3.1.7.3. J2 Plasticity Material

This command is used to construct an multi dimensional material object that has a von Mises (J2) yield criterion and isotropic hardening.

nDMaterial J2Plasticity $matTag $K $G $sig0 $sigInf $delta $H

Argument	Type	Description
$matTag	integer	unique tag identifying material
$K	float	bulk modulus
$G	float	shear modulus
$sig0	float	initial yield stress
$sigInf	float	final saturation yield stress
$delta	float	exponential hardening parameter
$H	float	linear hardening parameter

Note

The material formulations for the J2 object are “ThreeDimensional,” “PlaneStrain,” “Plane Stress,” “AxiSymmetric,” and “PlateFiber.”

THEORY:

The theory for the non hardening case can be found [[1]]

J2 isotropic hardening material class

Elastic Model

\[\sigma = K*trace(\epsilon_e) + (2*G)*dev(\epsilon_e)\]

Yield Function

\[\phi(\sigma,q) = || dev(\sigma) || - \sqrt(\tfrac{2}{3}*q(xi)\]

Saturation Isotropic Hardening with linear term

\[q(xi) = \sigma_0 + (\sigma_\inf - \sigma_0)*exp(-delta*\xi) + H*\xi\]

Flow Rules

\[ \begin{align}\begin{aligned}\dot {\epsilon_p} = \gamma * \frac{\partial \phi}{\partial \sigma}\\\dot \xi = -\gamma * \frac{\partial \phi}{\partial q}\end{aligned}\end{align} \]

Linear Viscosity: \(\gamma = \frac{\phi}{\eta}\) ( if \(\phi > 0\) )

Backward Euler Integration Routine Yield condition enforced at time n+1

set \(\eta = 0\) for rate independent case

Code Developed by: Ed Love