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]

$$D(p,T) = D_0(T) \mathrm{exp}\Big(\!-\frac{{p V_D}}{{k_B T}}\Big).$$ (7.1)

Here $D_0(T)$ is the low stress diffusion coefficient (3.9), $p$ is the pressure in the respective material, $V_D$ is the activation volume, $k_B$ is the Boltzmann's constant, and $T$ is the temperature in Kelvin.
The second parameter is the stress dependent strength of a spatial sink

$$k(\eta,p)= \eta(\vec{x},t) k_{max} \mathrm{exp}\Big(\!-\frac{{p V_k}}{{k_B T}}\Big).$$ (7.2)

Both parameters are exponentially reduced with pressure, which is only valid for $p \geq 0$ [111].

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

$$D(p,T) \Delta C(\vec{x},t) = k(\eta,p) C(\vec{x},t),$$ (7.3)

$$\frac{\partial \eta(\vec{x},t)} {\partial t} = - \frac{1}{\lambda} k(\eta,p) C(\vec{x},t)/N_1, \qquad\mathrm{and}$$ (7.4)

$$V^{add}_{rel} = \frac{\lambda - 1} {\lambda}\Delta t k(\eta,p) C(\vec{x},t)/N_1.$$ (7.5)

The stress is generally described with the formula

$$\tilde{\sigma} = \mathbf{D} (\tilde{\varepsilon} - \tilde{\varepsilon_0}) + \tilde{\sigma_0},$$ (7.6)

where $\tilde{\varepsilon_0}$ stands for the desired volume increase

$$\varepsilon_{0,xx}=\varepsilon_{0,yy}=\varepsilon_{0,zz}=\frac{{}_1}{{}^3}V^{add}_{rel},$$ (7.7)

and $\tilde{\varepsilon}$ represents the actual volume expansion, because $\varepsilon_{ij}$ are the partial derivatives of the actual displacements (3.21). On a finite element the mechanical problem $\mathbf{K^e} \vec{d^e}=\vec{f^e}$ is loaded by the desired volume increase ( $\varepsilon_{0,ii}$-values) which leads to the internal forces

$$\vec{f^e_{int}}=\mathbf{B^T}\mathbf{D} \tilde{\varepsilon_0}^e V^e.$$ (7.8)

The actual displacements $\vec{d}^e$ are obtained after solving the mechanical system (see Fig. 5.5). With these results the actual strains can be calculated

$$\tilde{\varepsilon}^e = \mathbf{B} \vec{d}^e,$$ (7.9)

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 $n$ the visco-elastic stress $\tilde{\sigma}^n$ is the sum of a dilatation and a deviatoric part, because $\mathbf{D}=\mathbf{D}_{dil}+\mathbf{D}_{dev}$ as depicted in Section 3.2.5.2. Therefore, also the residual stress $\tilde{\sigma}^n_0$ for the actual time step $n$ consists of a dilatation and a deviatoric part so that

$$\tilde{\sigma}^n_0=\tilde{\sigma}^n_{0,dil}+\tilde{\sigma}^n_{0,dev}.$$ (7.10)

The components of the actual residual stress tensor are build up from the $(n-1)$ previous time steps $\Delta t$ according to (7.12) for the dilatation and (7.13) for the deviatoric part [112]

$$\phantom{xxxxxxxxxxxxxxxxxxxxxx}{\sigma}^n_{0,dil} = \sum_{i=1}^{n-1}{\sigma}^i_{dil} = {\sigma}^{n-1}_{dil}+ {\sigma}^{n-1}_{0,dil},$$ (7.11)

$${\sigma}^n_{0,dev} = \sum_{i=1}^{n-1}{\sigma}^i_{dev} \mathrm{exp}\big(-\frac{{}_{(... ...gma}^{n-1}_{0,dev}\big) \mathrm{exp}\big(-\frac{{}_{\Delta t}}{{}^\tau}\big).$$ (7.12)

An important characteristic of visco-elastic materials is the stress relaxation of the deviatoric stress components over time with the Maxwellian relaxation time constant $\tau$, 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 ${\sigma}^n_{0,dil}$ and ${\sigma}^n_{0,dev}$ at actual time step $n$ can be simply computed by adding the components ${\sigma}^{n-1}_{dil}$ and ${\sigma}^{n-1}_{dev}$ from the last step $(n-1)$ to the already existing residual stress parts ${\sigma}^{n-1}_{0,dil}$ and ${\sigma}^{n-1}_{0,dev}$ determined at previous step $(n-1)$.

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

$$p = - \frac{\mathrm{Trace(}\tilde{\sigma}\mathrm{)}}{3}= - \frac{\sigma_{xx}+\sigma_{yy}+\sigma_{zz}}{3}.$$ (7.13)