r5.04.03 Second gradient modeling

Summary

Here are presented the second gradient models from the work of Chambon et al ([1], [], [2], [3]) and the second dilation gradient models from the work of Fernandes et al ([4], [5]), and Gantier ([14]).

These models are part of microstructured environments and have the numerical objective of giving convergent results in relation to the fineness of spatial discretization in order to avoid obtaining localized solutions. They must be used when the mechanical rheological laws in question present a softening of the behavior, which reflects the damage or degradation of a material before cracking.

Second expansion gradient modeling is, in fact, a simplified approach to the second gradient restricted to materials subject to the expansion phenomenon. The numerical objective is to significantly reduce calculation times. It lends itself particularly well to geomaterials.

Note: Second gradient modeling has been removed from the code. Only dilation gradient modeling is available.

Table of contents

• 1. The second gradient model
  • 1.1. Brief presentation of microstructure media
  • 1.2. The second gradient model
• 2. The second expansion gradient model
  • 2.1. Expanding microstructure media
  • 2.2. The second expansion gradient model
  • 2.3. Spatial discretization by finite elements
• 3. Numerical integration of second gradient models
  • 3.1. The field of generalized deformations
  • 3.2. The field of generalized constraints
  • 3.3. The tangent matrix
• 4. Tips/How to use second gradient models
  • 4.1. The linear elasticity second gradient
  • 4.2. Models available in Code_Aster
  • 4.3. Estimation of the regularization parameter A1 (2nd expansion gradient)
• 5. Verification
• 6. Bibliography