A more realistic approach to model the oxide and nitride layer is the
assumption of a visco-elastic compressible fluid. In this case the stress is
calculated from both the strain and the strain rate. The commonly used
*Maxwell body* (Fig. 5.1) for the description of
this phenomenon is analogous to a spring and a dashpot in series. The
shape of the oxide is influenced by the history of the sample. This
model must also be solved self consistently because of the close
coupling.

The model is based on the idea that the dilatational components of the
stress which involves the volumetric expansion, respectively compression,
and the deviatoric part which only accounts for shape modifications,
are decoupled. Thus, Hook's law can be written as

where and

The temporal evolution of the deviatoric components can directly
be written as follows [Sen96]:

with

Indeed, the deviatoric part of Maxwell fluid can be written as

- ^{ . }
+ ^{ . }
= 0 |
(5.17) |

The analytical solution of this equation gives the temporal evolution of the
stress as a function of the strain velocity:

(t) = ^{ . }e^{- } + G^{ . }e^{-
. }
d |
(5.18) |

with as the initial stress at time

1998-12-11