PlasticPlaneStrainJ2

Background

The PlasticPlaneStrainJ2 class creates a plane strain material class with the constitutive equation given as

\[ \sigma = \mathbb{C} \left( \epsilon - \epsilon^p \right) \nonumber \]

where \(\sigma\) is the stress vector, \(\epsilon\) is the strain vector, and \(\mathbb{C}\) is the rank-four consistent tangent stiffness tensor, given as

\[ \mathbb{C} = K \, \textbf{1} \otimes \textbf{1} + 2 \, \mu \left( \mathbb{I} - \frac{1}{3} \textbf{1} \otimes \textbf{1} - \frac{\hat{\textbf{n}} \otimes \hat{\textbf{n}}}{1 + \frac{H}{2 \, \mu}} \right) - 2 \mu \, \gamma \left( \mathbb{I} - \frac{1}{3} \textbf{1} \otimes \textbf{1} - \hat{\textbf{n}} \otimes \hat{\textbf{n}} \right)\nonumber \]

REFERENCE:

• Ronaldo .I. Borja, Plasticity Modeling & Computation, Springer, 2013, pp. 47
• J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer, 1997, pp.124

Pre-Analysis

The python Pre-Analysis in the 01-Pre_Process/Method/Attach.py file provides with an interface to populate the Entities dictionary. This file contains several functions to populate specific fields. For example, to create a PlasticPlaneStrainJ2 material using json format, use:

• addMaterial(tag, name='PlasticPlaneStrainJ2', attributes):

  • tag : The identifier of this material, i.e., tag > -1
  • name : Seismo-VLAB material class name
  • attributes : Specific properties for the created material, for example
    • 'K' : is the bulk’s modulus.
      
    • 'G' : is the elastic shear modulus.
      
    • 'rho' : is the material density.
      
    • 'H' : is the hardening modulus.
      
    • 'Sy' : is the yield stress.
      
    • 'beta' : is the Integration parameter or the Kinematic/Hardening ratio.

  Example

  A PLASTICPLANESTRAINJ2 material can be defined using the python interface as follows:
  SVL.addMaterial(tag=1, name='PlasticPlaneStrainJ2', attributes={'K': 133, 'G': 20.0, 'h': 80, 'beta': 1.0, 'Sy': 40})

  Application
  Please refer to the F03-DY_Lin_2DPointLoad_J2PStrain_Quad4.py file located at 03-Validations/01-Debugging/ to see an example on how to define a PlasticPlaneStrainJ2 material.

On the contrary, the 01-Pre_Process/Method/Remove.py file provides with an interface to depopulate the Entities dictionary. For example, to remove an already define Material, use:

• delMaterial(tag):
  • tag : The identifier of the material to be removed, i.e., tag > -1

Run-Analysis

The C++ Run-Analysis in the 02-Run_Process/02-Materials/02-NonLinear/PlasticPlaneStrainJ2.cpp file provides the class implementation. A PlasticPlaneStrainJ2 material is created using the built-in json parse-structure provided in the Driver.hpp. A PlasticPlaneStrainJ2 is defined inside the "Materials" json field indicating its "Tag" as follows,

• {
      "Materials": {
          "Tag": {
              "name" : "PLASTICPLANESTRAINJ2",
              "attributes": {
                  "K": double,
                  "G": double,
                  "rho": double,
                  "h": double,
                  "Sy": double,
                  "beta": double
              }
          }
      }
  }
  

  Variable	Description
  Tag	Unique material object identifier.
  K	Represents the bulk modulus.
  G	Represents the shear modulus.
  rho	Represents the material density.
  h	Represents the hardening modulus.
  beta	Kinematic/Hardening ratio.
  Sy	Represents the yielding stress.

  
  

  Attention

  This material can only be assigned to two-dimensional elements such as: lin2DQuad4, lin2DQuad8, and kin2DQuad4 elements.

  At each gauss point of the elements in the plane strain problem, the strain tensor is given as:

  \[ \epsilon = \begin{bmatrix} 0 & \epsilon_{11} & \epsilon_{22} & 0 & \epsilon_{12} & 0\end{bmatrix} \nonumber \]

  and only the corresponding components are stored for the stress and tangent stiffness tensors.

  Example

  A PLASTICPLANESTRAINJ2 material with bulk modulus 133, shear modulus 80, no density, hardening modulus 80, hardening ratio 1.0, and yield stress 40 is defined as:
  { "Materials": { "1": { "name" : "PLASTICPLANESTRAINJ2", "attributes": { "K": 133, "G": 80, "rho": 0.0, "h": 80, "Sy": 40.0, "beta": 1.0 } } } }