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4.2 Shell Elements
Historically, shells have been formulated using two different approaches . The difference between these approaches lies in the way the rotational degrees of freedom are defined. In the first approach, the rotational degrees of freedom are defined as angles. In addition, the plane stress condition needs to be enforced to take thickness variations into account. This approach is very useful for infinitesimal strains, but becomes very difficult to pursue in finite deformation due to the fact that finite rotations do not commute. Another disadvantage of this approach is that it requires a modification to the material formulation to enforce the plane stress condition. For complex materials this modification is very difficult or even impossible to obtain.
The alternative approach is to use an extensible director to describe the rotational degrees of freedom. With this approach it is not necessary to enforce the plane-stress condition and the full 3D constitutive relations can be employed. This approach is adapted in FEBio as described here.
The shell formulation implemented in FEBio is still a work in progress. The goal is to implement an extensible director formulation with strain enhancements to deal with the well-known locking effect in incompressible and bending problems . With the current state of the implementation, it is advised to use quadratic elements in such problems.
Starting with FEBio 2.6, two shell formulations have become available: The original formulation, where nodes are located at the mid-surface through the thickness of the shell, and a new formulation where nodes are located on the front face of the shell. The original formulation uses nodal displacements and directors as degrees of freedom; the new formulation uses front and back face nodal displacements. The new formulation is designed to properly accommodate shells attached to the surface of a solid element, or shells sandwiched between two solid elements, with minimal alterations to the rest of the code. The original formulation does not strictly enforce continuity of all the relevant degrees of freedom in those situations. However, this original formulation is maintained in the code for backward compatibility.
Most of the shell elements available in FEBio use a compatible strain formulation, where the calculation of strain components is based only on nodal displacements, similar to hexahedral or pentrahedral elements. Users should be aware that this compatible strain formulation is very susceptible to element locking when the shell thickness is much smaller than the shell size (e.g., when the aspect ratio is less than 0.01). Therefore, these shell formulations should be used with caution, keeping in mind this important constraint. Conversely, these shell elements perform very well when they are attached to solid elements (e.g., skin over muscle), or sandwiched between shell elements (e.g., cell membrane separating cytoplasm from extra-cellular matrix).
The element-locking limitation of compatible strain shell formulations has motivated the development of specialized shell formulations that attempt to overcome locking. The FE literature on this subject is rather extensive and we refer the reader to the excellent review chapter by Bischoff et al.  on this topic. Methods for overcoming locking include the assumed natural strain (ANS) formulation for transverse shear strains [70, 19] and transverse normal strains [24, 26]. The ANS formulation may be supplemented with the enhanced assumed strain (EAS) method  and extended to large deformations [61, 108, 95]. FEBio includes the ANS (q4ans) and EAS (q4eas) quadrilateral shell element formulations of Vu-Quoc and Tan , using a seven-parameter EAS interpolation, which is otherwise substantially similar to the five-parameter interpolation presented in an earlier study by Klinkel et al. . These shell elements are not suitable for attachment to a solid element, nor sandwiching between two solid elements. Since they don't experience element locking, they should be loaded more slowly than compatible strain shell elements. The formulations presented below are for the compatible strain shell elements.
Table of contents
- 4.2.1 Shell with mid-surface nodal displacements
- 4.2.2 Shells with front and back face nodal displacements