Pile-ups of dislocations

The relationship of brittle fracture to plastic deformation has, of course, been elaborated in various ways with the aid of dislocation theory, e.g. nucleation of microcracks has been discussed in terms of piling-up of dislocations [124]. Davies [145] has shown that embrittlement requires the presence of islands of martensite (about 1 pm in size) and has suggested that cracks are initiated in the martensite or at the martensite-ferrite interface. 

Fig. 9.3. Pile up of dislocations against a grain boundary in Ti (courtesy of Ian Robertson).

The advent of transmission electron microscopy opened the doors to the direct observation of processes such as the pile-up of dislocations at a grain boundary and the subsequent commencement of dislocation motion in adjacent grains. The process of slip transmission described above may be seen from the perspective... 

Fig. 11.14. Schematic of the pile-up of dislocations at a grain boundary. Enhancement in the stress field in an adjacent grain as a result of the presence of the pile-up is also indicated schematically.

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. 

AH is the activation energy for self-diffusion, v is a frequency factor and S is an entropy term. Equation (6.56) is obtained under the assumption that vacancies are easily destroyed or created and that an equilibrium concentration exists between pile-ups of dislocations. However, the diffusion of flux vacancies may be different in specific climb processes. 

Fig. 11.5. Pile-up of dislocations at obstacles and vacancy diffusion. Dislocation 1 is a vacancy sink, dislocation 2 a vacancy source...

Normally, deformation resistance from the effect of the pile up of dislocations, strain gradient work hardening, and the artifacts due to indentation tip shapes, strain rates, loading scales and directions, etc., involved in the contact mode of indentation would play a role of significance [39]. The surface smoothness has less influence on the measurement [40]. 

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. 

A pile-up of n dislocations along a distance, L, may be considered to be a giant dislocation with Burgers vector nb. The breakdown of a barrier occurs due to ... 

Fig. 3.85 Slip bands in the intermediate region of the indented volume, a Slip band planarity and evidence of profuse pile-ups. The dislocations exhibiting paired lines in the boxed area are not dipoles but dissociated dislocations since the distance between partials is constant whether the dipole is imaged with the g or -g reflecting plane, b Intersecting slip bands, c The slight misalignment and differences in pile-up projected widths indicate that the slip bands are parallel to at least two crystallographically distinct planes [31]. With kind permission of Elsevier...

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 ... 

Matthews, J. W. and Blakeslee, A. E. (1975), Defects in epitaxial multilayers II. Dislocation pile-ups, threading dislocations, slip lines and cracks. Journal of Crystal Growth 29, 273-280. 

Usually, creep deformation of ice single crystals is associated to a steady-state creep regime, with a stress exponent equal to 2 when basal glide is activated . In the torsion experiments performed, the steady-state creep was not reached, but one would expect it to be achieved for larger strain when the immobilisation of the basal dislocations in the pile-ups is balanced by the dislocation multiplication induced by the double cross-slip mechanism. 

Our fundamental assertion concerning the geometry of pile-ups is that they reflect the equilibrium spacing of the various dislocations which are participants in such a pile-up. From a discrete viewpoint, what one imagines is an equilibrium between whatever applied stress is present and the mutual interactions of the dislocations. In simple terms, using the geometry depicted schematically in fig. 11.12, we argue that each dislocation satisfies an equilibrium equation of the form... 

This equation considers the equilibrium of the Z dislocation. In particular, it is nothing more than a dislocation-by-dislocation statement of Newton s first law of motion, namely, = 0, which says that the total force on the dislocation is zero. The factor A is determined by the elastic moduli and differs depending upon whether we are considering dislocations of edge A = fib/27r l — v)), screw(A = fib/lit) or mixed character. The problem of determining the equilibrium distribution of the dislocations in the pile-up has thus been reduced to one of solving nonlinear equations, with the number of such equations corresponding to the number of free dislocations in the pile-up. For further details see problem 3 at the end of the chapter. 

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