The thickness of the membrane is d, which can vary over the element. The displacements are interpolated by Lagrange shape functions.

This local membrane surface coordinate system is defined by the boundary coordinate system (t1, t2, n).

The deformation gradient F is in general defined as the gradient of the current coordinates with respect to the original coordinates:

Here is a displacement gradient computed using the tangential derivative operator, and N is the normal vector to the undeformed membrane. FT now contains information about the stretching in the plane of the membrane.

Since the tangential deformation gradient does not contain any information about the transversal stretch λn, it must be augmented by the normal deformation gradient FN to define the full deformation gradient tensor.

n is the normal vector to the deformed membrane. The full deformation gradient tensor F is the sum of tangential and normal deformation gradient tensors.

The Right Cauchy-Green tensor C is generally defined as

.From C, the Green-Lagrange strains are computed using the standard expression

In the membrane, only the C tensor is available, so instead the following expression is used:

where the forces (F) can be distributed over a boundary or an edge or be concentrated in a point. In the special case of a follower load, defined by its pressure p, the force intensity is where n is the normal in the deformed configuration.