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
matTag

integer tag identifying material

K

bulk 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".

Theory

The theory for the non hardening case can be found http://en.wikipedia.org/wiki/Von_Mises_yield_criterion

\(J_2\) isotropic hardening material

  • Elastic Model \[\sigma = K \operatorname{tr}(\epsilon_e) + 2G \operatorname{dev}(\epsilon_e)\]

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

  • Saturation Isotropic Hardening with linear term

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

  • Flow Rules \[\dot {\epsilon_p} = \gamma \frac{\partial \phi}{\partial \sigma} \]

    \[\dot \xi = -\gamma \frac{\partial \phi}{\partial q}\]

  • Linear Viscosity ( if \(\phi \gt 0\) )

    \[\gamma = \frac{\phi}{\eta}\]

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

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

Code developed by: Ed Love