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