J2 Plasticity Material

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

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

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$matTag	integer tag identifying material
$E	elastic Modulus
$G	shear Modulus
$sig0	initial yield stress
$sigInf	final saturation yield stress
$delta	exponential hardening parameter
$H	linear hardening parameter

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

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THEORY:

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

J2 isotropic hardening material class

Elastic Model

<math> \sigma = K*trace(\epsilon_e) + (2*G)*dev(\epsilon_e)</math>

Yield Function

<math> \phi(\sigma,q) = || dev(\sigma) || - \sqrt(\tfrac{2}{3}*q(xi)</math>

Saturation Isotropic Hardening with linear term

<math> q(xi) = \sigma_0 + (\sigma_\inf - \sigma_0)*exp(-delta*\xi) + H*\xi </math>

Flow Rules

<math> \dot {\epsilon_p} = \gamma * \frac{\partial \phi}{\partial \sigma} </math>

<math> \dot \xi = -\gamma * \frac{\partial \phi}{\partial q} </math>

Linear Viscosity

<math>\gamma = \frac{\phi}{\eta} </math> ( if <math> \phi > 0</math> )

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

set <math> \eta = 0 </math> for rate independent case

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Code Developed by: Ed Love