# 4.15.1.3. J2PlasticityΒΆ

nDMaterial('J2Plasticity', matTag, K, G, sig0, sigInf, delta, H)

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

 matTag (int) integer 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

The material formulations for the J2Plasticity object are:

• 'ThreeDimensional'

• 'PlaneStrain'

• 'Plane Stress'

• 'AxiSymmetric'

• 'PlateFiber'

J2 isotropic hardening material class

Elastic Model

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

Yield Function

$\phi(\sigma,q) = || dev(\sigma) || - \sqrt(\tfrac{2}{3}*q(x_i))$

Saturation Isotropic Hardening with linear term

$q(x_i) = \sigma_0 + (\sigma_\infty - \sigma_0)*exp(-delta*\xi) + H*\xi$

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