Void Nucleation and Crack Nucleation

The synergistic effects of all microstructural evolution will lead to the weakening of the material and direct it towards cavitation [73]. Cavities will gradually form micro cracks by interlinkage and thus, the rupture of materials under service conditions is heavily dependent on the formation and growth of cavities. The emergence of micro voids means that the material in service is damaged seriously. As voids grow they elongate perpendicular to the tensile axis and develop into cracks, mainly along the grain boundaries. However in a uniform medium the cavities would extend more rapidly in the tensile axis direction than in the transverse direction [4].

There are two groups of cavities that can be observed in a creep damaged material. The first is large cavities (>1 μm) that were present from the manufacturing process. The second type is small cavities (<0.6 μm) that are produced during creep [73].The cavities present in the steel from the manufacture process grow with creep exposure over time. A recent study [73] has stated that the growth rates of these initial cavities is higher than the formation of new cavities, and as a result they have a much higher volume fraction than creep cavities. This result indicates that final fracture may actually be attributed the cavities produced during the manufacture process [73].

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Figure 21- Creep cavities forming on boundaries perpendicular to the applied stress [47]

It is still not well established by which mechanisms cavities nucleate [47]. It has generally been observed that cavities frequently nucleate on grain boundaries, particularly on those transverse to tensile stress, as illustrated in Figure 21. The nucleation theories for cavity formation fall into several categories that are illustrated in Figure 22. Figure 22 (a) theory is that grain boundary sliding can open up a void at triple points at boundaries [73] [47]. Figure 22 (b) shows a void forming from vacancies condensing in highly stressed regions, usually at the transverse grain boundaries [73] [47]. Figure 22 (c) shows cavity formation at the head of a dislocation pileup [47] and there is the possibility of a combination of all three mechanisms occurring.

In commercial alloys these cavities appear to be associated with second phase particles along the grain boundary as shown in Figure 22 (d) [47]. This could be because the interface between hard particles and the matrix act as void nucleation sites due to the lack of cohesion between the two materials [73] and because second phase particles can result in stress concentrations upon application of a stress. Another nucleation mechanism that may be important, as damage progresses in a material, is the stress concentration that arises around existing cavities. Cavity nucleation by vacancy accumulation appears to require significant stress concentration around the hard particles [47]. The initial stress concentration at the cavity ‘tip’ is a factor of three larger than the applied stress, and even after relaxation by diffusion, the stress may still be elevated [74], which leads to increased nucleation of existing cavities.[47]

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Figure 22 - Mechanisms of cavity nucleation: (a) cavitation from boundary sliding at triple points (b) Cavity nucleation from vacancy condensation (c) cavity formation from Zener-Stroh Mechanism (d) Cavity formation around a hard

particle in conjunction with other mechanisms. [47]

Once these voids have nucleated they grow in many different ways: diffusion, grain boundary sliding, constrained diffusional growth, plasticity and a coupled effect of diffusion and plasticity [73].While adding precipitates to boundaries slows down creep, the growth rate of a cavity is unaffected, this suggests at diffusion rather than plasticity as the main controlling mechanism for cavity growth [47].

Growth of Cavities by Grain Boundary Diffusion

The cavity growth process at grain boundaries, at elevated temperatures, has long been suggested to mainly involve vacancy diffusion. Diffusion to the void occurs by the transport of vacancies along a grain boundary and then by surface migration along the cavity wall [47]. The activation energy was found to be similar to that of grain boundary diffusion, resulting in it being the rate controlling mechanism [4]. Hull and Rimmer were one of the first to propose a mechanism by which diffusion leads to cavity growth of an isolated cavity in a material under an external stress. A stress concentration is established just ahead of the cavity. This leads to an initial negative stress gradient, meaning stress decreases away from cavity; however, a positive stress gradient is suggested to be established due to relaxation by plasticity. The equations derived were:

𝐽𝑔𝑏 = − 𝐷𝑔𝑏

Ω𝑘𝑇∇𝑓 (2.47)

Where J is the flux along the grain boundary, 𝛺 is the atomic volume and 𝛻𝑓 is the gradient of chemical potential. The gradient of chemical potential is then given as:

∇𝑓~Ω 𝜆𝑠

(𝜎 −2𝛾𝑚

𝑎 ) (2.48)

Where a is the cavity radius, σ is the applied stress and 𝜆𝑠 is the cavity separation. 𝛺𝜎 would correspond to the chemical potential at the grain boundary, while Ω2𝛾𝑚

42 potential along the cavity surface. In large voids the chemical potential along the surface can be neglected [4].

Combining equations (2.46) and (2.47) above gives the growth rate of voids as: 𝑑𝑎

𝑑𝑡 ≅

𝐷𝑔𝑏𝛿 (𝜎 −2𝛾_{𝑎 ) Ω}𝑚 2𝑘𝑇𝜆𝑠𝑎

(2.49)

Where 𝛿 is the grain boundary width.

This theory assumes that the surface diffusion of a vacancy would be rapid enough for the void to maintain an almost spherical shape [4]. The most basic concept of applying this equation is illustrated by the Figure 23 below, where the separation distance between voids is now the grain size when there is a single void in a grain.

Figure 23– Cavity growth from diffusion along the grain boundaries [47]

Plasticity

For completion sake, cavities can grow exclusively by superplasticity. Initially it was proposed that the creep controlled cavity growth model was based on the idea that cavity growth during creep should be analogous to the model for a cavity growing in a plastic field. Cavity growth, according to this model, occurs as a result of creep deformation of the material surrounding the grain boundary cavities in the absence of vacancy flux. This mechanism becomes important under high strain rate conditions where significant stress is realized. The cavity growth according to this model is given as

𝑑𝑎

𝑑𝑡 = 𝑎𝜀̇ − 𝛾

2𝐺 (2.50)

It has been suggested on occasion that the observed creep cavity growth rates are consistent with plasticity growth but not always obvious that constrained diffusional cavity growth is not occurring, which is also controlled by the plastic deformation. [47]

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Damage

As cavities appear and grow they cause a loss in the load bearing area. This is referred to as damage and is the most common example of creep damage. The tertiary creep stage is due mostly to this increase in damage. The creep damage theory assumes that the extent of creep damage can be represented by a single parameter, ω. The creep rate is then said to a function of the stress and this damage parameter:

𝜀̇ = 𝑓(𝜎, 𝜔) (2.51)

It is then assumed that the accumulation of damage is represented only by the growth of voids and other changes such as size and growth of precipitates are not considered able to affect the damage parameter. Therefore ω is only a representation of the fraction of area occupied by voids, cavities and cracks along the grain boundaries. This then leads to the concentration of stress in the equation below.

𝜎 = 𝜎0

1 − 𝜔 (2.52)

This equation shows that creep cavitation increases the applied stress when the damage parameter evolves from 0 to 1 as these cavities nucleate and grow [4] [22]. This is however not the only form of stress concentration considered in this study, but also the varying stress fields forming around the void as a stress is applied; this is more related to the shape of the void rather than the loss of load bearing area.