7.1 Oxidation Modeling with Stress

There are two parameters in the oxidation model, which are influenced by stress. The first one is the stress dependent diffusion coefficient [109,110]

Here is the low stress diffusion coefficient (3.9), is the pressure in the respective material, is the activation volume, is the Boltzmann's constant, and is the temperature in Kelvin.

The second parameter is the stress dependent strength of a spatial sink

Both parameters are exponentially reduced with pressure, which is only valid for [111].

With these two stress dependent parameters the three main equations in the oxidation model, which describe the oxidant diffusion (3.2), the -dynamics (3.5), and the volume increase (3.8), become

The stress is generally described with the formula

where stands for the desired volume increase

and represents the actual volume expansion, because are the partial derivatives of the actual displacements (3.21). On a finite element the mechanical problem is loaded by the desired volume increase ( -values) which leads to the internal forces

The actual displacements are obtained after solving the mechanical system (see Fig. 5.5). With these results the actual strains can be calculated

and the stress on an element can be determined with (7.7).

A worth mentioning aspect is the visco-elastic stress computation in the FEDOS simulation procedure. For the actual time step the visco-elastic stress is the sum of a dilatation and a deviatoric part, because as depicted in Section 3.2.5.2. Therefore, also the residual stress for the actual time step consists of a dilatation and a deviatoric part so that

The components of the actual residual stress tensor are build up from the previous time steps according to (7.12) for the dilatation and (7.13) for the deviatoric part [112]

An important characteristic of visco-elastic materials is the stress relaxation of the deviatoric stress components over time with the Maxwellian relaxation time constant , as given in (7.13). The recursive form for residual stress calculation in the right hand side of (7.12) and (7.13) offers the benefit that the residual stress parts and at actual time step can be simply computed by adding the components and from the last step to the already existing residual stress parts and determined at previous step .

In contrast to stress the pressure needed for (7.1) and (7.2) is a scalar. It is positive, if the com-pressive stress components which have a negative sign, are predominant. So pressure always has an opposite sign compared to stress. The pressure is the average of the stress tensors trace

Ch. Hollauer: Modeling of Thermal Oxidation and Stress Effects