Analytical Modeling on Stress Assisted Oxidation and its Effect on Creep Response of Metals

Abstract

The aim of this paper was to investigate the effect of external mechanical loads on the kinetics and process of high temperature oxidation of metals or alloys through a modeling approach. In this model, the complicated interplay among the external stress, growth strain and creep strain, the oxide stress and thickness was identified. The stress accumulation in the oxide and the variation of scale thickness along with time and the effect of oxidation behavior on the creep properties of underlying metal during oxidation process can be expected. Generally, the external tensile stress would promote the growth rate of oxidation layer. Introducing a small external stress may lead to obvious changes of magnitude and state of stress in oxidation layer. At a given external stress, the creep strain rate was not kept constant due to the oxidation layer growth and stress transfer between oxidation layer and underlying metal.

Appendix

If the dislocation in the boundary climbs rate during oxidation was dL/dt, then the growth strain rate can be expressed as [6]

$$ \dot{\varepsilon }_{ox}^{g} = \frac{\theta }{d}\frac{dL}{dt} $$

(30)

where $\theta$ represents the angle tilt boundary and d is the grain size, which are stress independent. Climb by a distance of $\varDelta L$ requires the addition of n molecules of oxide per unit length, 1, of the dislocation line, i.e.,

$$ \left| b \right|\varDelta L.1 = n\varOmega_{ox} $$

(31)

where $\varOmega_{ox}$ is the volume of a molecule of the oxide, and b is the Burgers vector of the edge dislocation in the grain boundary. In such a case, the dislocation climbing rate can be expressed as

$$ \frac{dL}{dt} = \frac{{\varOmega_{ox} }}{b.1}\frac{dn}{dt} $$

(32)

Then, the lateral growth strain rate can be rewritten as

$$ \dot{\varepsilon }_{ox}^{g} = \frac{{\varOmega_{ox} \theta }}{bd.1}\frac{dn}{dt} $$

(33)

The climb of the dislocations is limited by the attachment of diffusing species to the dislocation cores, i.e., $dn/1.dt$. The net rate of attachment to each dislocation core will be equal to the rate of the attachment by trapping minus the rate at which ions detach by the thermal excitation. If the traps are spaced an average distance a along the dislocation line, the net rate of attachment can be expressed as [6]

$$ \frac{dn}{1.dt} = \frac{A}{a}J_{v} \exp \left( {\frac{{ - \sigma_{ox} \varDelta \varOmega }}{kT}} \right) - \frac{B}{a}\exp \left( {\frac{{ - E_{T} + \sigma_{ox} \varDelta \varOmega }}{kT}} \right) $$

(34)

where A represents the cross-section for diffusing ion to attach to the dislocation core per unit length of dislocation, B is an effective diffusion jump frequency and E _T is the trap energy. J _v is the diffusion flux without any stresses in the oxide then the item $J_{v} \cdot \exp \left( { - \sigma_{ox} \varDelta \varOmega /kT} \right)$ could be used to express the stress dependent diffusion flux.

This expression seems to conflict with that of Evans, i.e., $J_{v} \cdot \exp \left( {\sigma_{ox} \varDelta \varOmega /kT} \right)$. However, Clarke [6] also mentioned that large compressive stress would decrease the diffusivity. Hence, $\sigma_{ox}$ represents the stress level in Clarke’s expression while it represents the oxide stress in that of the Evans. Thus, their points coincide with each other. According to Clarke’s suggestion, the attachment rate was much large than the detach rate [6]. Thus, neglecting the second item of Eq. 34 and substituting it into Eq. 33, the lateral growth strain rate is proportional to the diffusion flux

$$ \dot{\varepsilon }_{ox}^{g} = \frac{{\varOmega_{ox} \theta A}}{bda}J_{v} \exp \left( {\frac{{ - \sigma_{ox} \varDelta \varOmega }}{kT}} \right) $$

(35)

It is well known that the scale growth rate was proportional to the diffusion flux, i.e.,

$$ \dot{h}_{ox} \propto J_{v} \exp \left( {\frac{{ - \sigma_{ox} \varDelta \varOmega }}{kT}} \right) $$

(36)

Hence, the lateral growth strain rate is proportional to the scaling rate and this relationship can be expressed as

$$ \dot{\varepsilon }_{g} = D_{ox} \dot{h}_{ox} $$

(37)

where the lateral growth constant, D _ox , is determined by some stress independent parameters, including $\theta$, d, $\varOmega_{ox}$, A, a and b et al.