Geometric description
At a given stage in the deformation history of the shell, the position of a material point in the shell is defined by
where the subscript i and other Roman subscripts range from 1 to 3. Subscripts α and other lowercase Greek subscripts which describe the quantities in the reference surface of the shell range from 1 to 2. In the above equation t3 is the normal to the reference surface of the shell. The gradient of the position is
where we have neglected derivatives of with respect to . Note that in the above are local surface coordinates that are assumed to be orthogonal and distance measuring in the reference state. is the coordinate in the thickness direction, distance measuring and orthogonal to in the reference state. The thickness increase factor is assumed to be independent of .
In the deformed state we define local, orthonormal shell directions such that
where is the Kronecker delta and is the identity tensor of rank 2. Summation convention is used for repeated subscripts. The in-plane components of the gradient of the position are obtained as
where we have introduced the reference surface deformation gradient
and the reference surface normal gradient
In the original (reference) configuration we denote the position by ( for the reference surface) and the direction vectors by , which yields
The gradient of the position is
and the in-plane components of the gradient are obtained as
where we have assumed that the in-plane direction vectors follow from the surface coordinates with
and defined the original reference surface normal gradient,
The original reference surface normal gradient is obtained in the finite element formulation from the interpolation of the nodal normals with the shape functions. In the deformed configuration it is not derived from the nodal normals but is updated independently based on the gradient of the incremental rotations.