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4.2.1 Shell with mid-surface nodal displacements
We create a shell formulation by reducing a solid element interpolation which is linear along the parametric coordinate . We start with the general interpolation for a solid element, where and is the number of nodes, and specialize it to the case of a shell as where , is the number of shell element nodes, and are the interpolation functions within the mid-shell surface. The description of the mid-shell surface is thus given by where are the nodal positions for the mid-shell surface.
Example of shell elements with four mid-surface nodal positions and directors ( , reduced from a solid element.
We also define the director across the shell surface as where are the nodal directors. Note that the magnitude of the nodal director represents the shell thickness, and the shell thicknesses at the nodes are . With these definitions we find that the interpolation across the parametric space of the shell element is From this relation we can obtain the covariant basis vectors as from which we may evaluate the contravariant basis vectors using the identity . Then, the gradients of the shape functions are given by It follows from (4.2.1-7) that the virtual displacement is and the incremental displacement is
In FEBio, for historical reasons, the nodal director is currently called rotation. This is a misnomer and users should treat this rotation as the vector whose components have units of length. Thus, fixing or prescribing rotation components in the input file effectively places these constraints on the components of the nodal director; similarly, requesting rotation in the output files will produce the components of the director.
When this type of shell is connected face-to-face with a solid element, the nodes located at automatically share their displacement degrees of freedom with the corresponding nodes from the face of the solid element. However, no continuity is enforced between the directors and the solid element deformation. One consequence of this condition is that a shell sandwiched between two solid elements will not detect out-of-plane shear and normal stresses transmitted by the solid element(s). Another consequence is that bending of the solid element(s) will not produce a bending moment in the shell. Therefore, these shell elements are best used as shell-only structures.