Dislocation pile

A more elegant (and mathematically tractable) description of the problem of dislocation pile-ups is to exploit the representation of a group of dislocations as a continuous distribution. This type of thinking, which we have already seen in the context of the Peierls-Nabarro model (see section 8.6.2), will see action in our consideration of cracks as well. The critical idea is that the discrete set of dislocations is replaced by a dislocation density pb (x) such that... 

Fig. 11.13. Representation of dislocation pile-up as a continuous distribution of dislocations.

Cracks as a Superposition of Dislocations. A scheme that will suit our aim of building a synthetic description of cracks and any allied dislocations is to think of a crack as an array of dislocations. Indeed, the majority of our work has already been done earlier in the context of our consideration of dislocation pile-ups in section 11.4.2. In fact, our present analysis will do little more than demonstrate that the solutions written down there are relevant in the crack context as well. The more fundamental significance of the perspective to be offered here is that we will soon want to build up solutions in which cracks and dislocations are equal partners. 

As an application of the ideas on dislocation pile-ups described in section 11.4.2, consider a pile-up of three dislocations in which the leading dislocation i , fixed at (0,0). The other dislocations have coordinates (xi, 0) and (X2, 0) which are to be determined by applying the equilibrium equations presented as eqn (11.30). Assume that the externally applied stress is constant and is denoted by r. 

The stress-strain behavior of ceramic polycrystals is substantially different from single crystals. The same dislocation processes proceed within the individual grains but these must be constrained by the deformation of the adjacent grains. This constraint increases the difficulty of plastic deformation in polycrystals compared to the respective single crystals. As seen in Chapter 2, a general strain must involve six components, but only five will be independent at constant volume (e,=constant). This implies that a material must have at least five independent slip systems before it can undergo an arbitrary strain. A slip system is independent if the same strain cannot be obtained from a combination of slip on other systems. The lack of a sufficient number of independent slip systems is the reason why ceramics that are ductile when stressed in certain orientations as single crystals are often brittle as polycrystals. This scarcity of slip systems also leads to the formation of stress concentrations and subsequent crack formation. Various mechanisms have been postulated for crack nucleation by the pile-up of dislocations, as shown in Fig. 6.24. In these examples, the dislocation pile-up at a boundary or slip-band intersection leads to a stress concentration that is sufficient to nucleate a crack. 

Figure 6.24 Crack nucle-ation as a result of dislocation pile-up a) Zener model b) Cottrell model.

The first explanation for threshold stress was proposed recently by Morita and Hiraga [17, 63, 64], who claimed that w-values as high as 5 and Q-values of 680 kj mol can be related to the existence of a threshold stress for intragranular dislocation motion, with such motion being the accommodation process for GBS. This hypothesis was based on the observation of dislocation pile-ups, as reported elsewhere [64]. 

A dislocation pile-up is not a stable dislocation microstructure, as a strong dislocation repulsion of dislocating each other must occur however, it will remain stable for as long as an external applied stress exists. Such stress can be calculated using the equation [64] ... 

This reaction is correct, as may be seen by checking the components of the Burgers vectors and it is also energetically favorable. A consequence of the above reaction is the formation of a sessile dislocation, beyond which the trailing dislocations pile up. The Burgers vector of the newly-formed partial dislocation, i.e. 

Fig. 3.62 Dislocation pile-ups (paitials and their faults are shown) behind a Lomer-Cottrell lock, acting as obstacles to their movement...

Fig. 3.82 A schematic view of a dislocation pile up at a grain boundary...

L is the length of the dislocation pile-up and Cq is the equilibrium concentration of the vacancies in a dislocation-free crystal. The vacancy concentration at a distance, r, from each pile-up is assumed to be equal to Cq. The rate of climb, X, is given (Garofalo) as ... 

Although this is a discussion on brittle materials, such as ceramics (glass is a perfect, brittle material), several researchers have developed theories of fracture based on dislocation models. More specifically, the shear stress created by dislocation pile-ups at some obstacle, specifically grain boundaries in polycrystaUine materials, reaches a sufficient value for crack formation. The following illustrates Stroh s [52] basic concept of microcrack formation, ultimately leading to the occurrence of fracture in brittle materials. 

Similarly to Zener s model [9] of microcrack formation at a pile up of edge dislocations, Stroh [52] developed a theory of fracture based on the concept of cracks initiated by the stress concentration of a dislocation pile-up. For brittle materials in which crack growth is not damped-out by plastic flow, Stroh calculated that the conditions for crack initiation may be given by ... 

Stress, Ts, is created by the internal pile-up of n dislocations, teff is an effective stress and Ty is the yield stress, y is the surface energy per unit area of the plane, as indicated earlier, and d/2 is the length of the dislocation pile-up. One illustration of Stroh s [52] concept is shown below ... 

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