4.15.1.3. J2PlasticityΒΆ
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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