Theory Manual Version 3.4
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Subsubsection 4.2.1.3: Triangular shells Up Section 4.2: Shell Elements Subsubsection 4.2.2.1: Elastic Shell

### 4.2.2 Shells with front and back face nodal displacements

We create a shell formulation by reducing a 3D element interpolation which is linear along . The nodal positions at the back of the shell () are denoted by and those on the front of the shell () are denoted by , thus The vector from to is the director, , From this relation we can get the shell covariant basis vectors, from which we may evaluate the contravariant basis vectors . Let the front-face and back-face displacements be denoted by and , respectively. It follows that and , where represents the shell nodal positions in the reference configuration, provided as nodal coordinates in the input file, and is evaluated from the user-defined referential shell thickness, and the surface surface normals evaluated at each node. If the shell surface is not planar in the reference configuration, users must be careful to select shell thicknesses that don't produce inverted elements (negative Jacobians) as a result of this extrapolation.

Example of shell element with front-face nodal positions and back-face nodal positions (, reduced from a solid element.
It follows that the virtual displacement is and the incremental displacement is so that and Note that so that and To evaluate the deformation gradient in this shell element, we use
For this formulation, when a shell element is connected face-to-face with a solid element, the nodal displacements of the solid element face are set to coincide with the back-face nodal displacements of the shell. When a user prescribes displacement components on that shared face, they apply to the front-face displacements . Similarly, prescribed pressures and contact pressures act on the shell front face.
When a shell element is sandwiched between two solid elements, the nodal displacements of the solid element facing the shell back face are set to coincide with the shell back-face nodal displacements , whereas the nodal displacements of the solid element facing the shell front face are set to coincide with the shell front-face nodal displacements . If the shell thickness exceeds the thickness of the solid element connected to its back face, results become unpredictable.
Subsubsection 4.2.1.3: Triangular shells Up Section 4.2: Shell Elements Subsubsection 4.2.2.1: Elastic Shell