Spring Approximation
For the Spring approximation, the membrane deformation is neglected, and the deformation gradient reads
where ⊗ is the outer product of two vectors, (a⊗b)ij = aibj. Here, the thin layer extension ue contains information about the normal as well as the transverse shear deformation, but no membrane deformation is taken into account.
In case of predeformation, the extension of the thin layer reads
where u0 is the predeformation of the thin layer.