Elastic3DLinear

Background

The Elastic3DLinear class creates a triaxial material class with the constitutive equation given as

\[ \sigma = \mathbb{C} \, \epsilon \nonumber \]

where \(\sigma\) is the stress vector, \(\epsilon\) is the strain vector, and \(\mathbb{C}\) is the stiffness matrix, given as

\[ \mathbb{C} = \frac{E}{(1 + \nu)(1 - 2\nu)} \begin{bmatrix} 1 - \nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1 - \nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1 - \nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1 - 2 \nu}{2}& 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1 - 2 \nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1 - 2 \nu}{2} \end{bmatrix} \nonumber \]

REFERENCE:

• Bathe K. Jurgen, "Finite Element Procedures", Chapter 4: pages 194, Table 4.3, Prentice-Hall, 1996.

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 Elastic3DLinear material using json format, use:

• addMaterial(tag, name='Elastic3DLinear', 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
    • 'E' : is the elastic modulus of elasticity.
      
    • 'nu' : is the Poisson’s ratio.
      
    • 'rho' : is the material density. Can be omitted for static analysis.

  Example

  A ELASTIC3DLINEAR material can be defined using the python interface as follows:
  SVL.addMaterial(tag=1, name='Elastic3DLinear', attributes={'E': 200.0, 'nu': 0.25, 'rho': 0.0})

  Application
  Please refer to the I01-ST_Lin_3DPointLoad_Elastic_Hexa8.py file located at 03-Validations/01-Debugging/ to see an example on how to define a Elastic3DLinear 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/01-Linear/Elastic3DLinear.cpp file provides the class implementation. A Elastic3DLinear material is created using the built-in json parse-structure provided in the Driver.hpp. A Elastic3DLinear is defined inside the "Materials" json field indicating its "Tag" as follows,

• {
      "Materials": {
          "Tag": {
              "name" : "ELASTIC3DLINEAR",
              "attributes": {
                  "E": double,
                  "nu": double,
                  "rho": double
              }
          }
      }
  }
  

  Variable	Description
  Tag	Unique material object identifier.
  E	Represents the elasticity modulus.
  nu	Represents the Poisson's ratio.
  rho	Represents the material density.

  
  

  Attention

  This material can only be assigned to three-dimensional elements such as: lin3DHexa8, lin3DHexa20, and kin3DHexa8 elements.

  Example

  A triaxial ELASTIC3DLINEAR material with elasticity modulus 200, Poisson's ratio 0.25, no density is defined as:
  { "Materials": { "1": { "name" : "ELASTIC3DLINEAR", "attributes": { "E" : 200.0, "nu" : 0.25, "rho": 0.0 } } } }