r3.07.05 Solid shell elements in geometric nonlinear

Summary:

In this document, we present the theoretical formulation and the numerical implementation of a finite element of a volume shell for geometric nonlinear analyses. This approach should make it possible to take into account large displacements and large rotations of thin structures, whose characteristic thickness-to-length ratio is less than \(1\mathrm{/}10\). Care should be taken to ensure that these rotations remain less than \(2\pi\).

This formulation is based on a 3D continuous medium approach, degenerated by the introduction of shell kinematics into plane stresses in the weak form of equilibrium. The measure of the deformations that we use is that of Green-Lagrange, energetically combined with Piola-Kirchhoff constraints of the second kind. The equilibrium formulation is therefore total Lagrangian.

The entirely nonlinear geometric problem is examined first. The case of linear buckling is treated as a borderline case of the first approach.

• 1. Introduction
• 2. Formulation
  • 2.1. Geometry of solid shell elements
  • 2.2. Kinematics of solid shells
  • 2.3. Law of behavior
• 3. Principle of virtual work
  • 3.1. Internal virtual work
• 4. Numerical discretization of variational formulation based on the principle of virtual work
  • 4.1. Finite elements
  • 4.2. Discretization of the displacement field
  • 4.3. Discretization of the displacement gradient
  • 4.4. Discretization of the variational formulation based on the principle of virtual work
• 5. Stiffness around the transform of normal
  • 5.1. Singularity of the tangent stiffness matrix
  • 5.2. Principle of virtual work for terms associated with rotation around the normal
  • 5.3. note
  • 5.4. Borderline case of geometrically linear analysis
  • 5.5. Determination of the coefficient k
• 6. Linear buckling
• 7. Implanting shell elements in Code_Aster
  • 7.1. Description
  • 7.2. Use
  • 7.3. Geometric nonlinear « elasticity » calculation
  • 7.4. Implantation
  • 7.5. Linear buckling calculation
• 8. Conclusion
• 9. Bibliography
• 10. Description of document versions