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Perturbation technique based on forward difference approximation


 I am trying to implement (in Abaqus) the elasticity tensor using a numerical approach.  I have come across a paper by C.Miehe (Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity, 1996).

There are a few things I don't quite understand.  

Perturbation analysis of a wavy film in a multi-layered structure

A free surface in a multi-layer can experience an undulation due to surface diffusion during fabrication or etching process. In order to analyze the undulation, the elasticity solution for the undulating film is needed. Considering the undulation as a perturbation of a flat surface, a boundary value problem for 2D elasticity is formulated. The solution procedure is straightforward, but very lengthy especially for a multi-layer.

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