View source for J2 Plasticity Material

{{CommandManualMenu}}

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

{| 
| style="background:yellow; color:black; width:800px" | '''nDmaterial J2Plasticity $matTag $K $G $sig0 $sigInf $delta $H'''
|}


----

{|
|  style="width:150px" | '''$matTag ''' || integer tag identifying material
|-
|  '''$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]]

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                                                                                                         


----

Code Developed by: <span style="color:blue"> Ed Love </span>