Dislocations and cracks

This equation reflects the addition of the externally applied shear stress, — r, as well as the contribution of the pile-up itself as embodied in the integral. Substitution of the dislocation density determined above yields 

What this result demonstrates is the presence of an enhanced stress in front of the pile up (x 0). We forego a discussion of the implications of such stress enhancements for dislocation nucleation until section 11.5.3. 

In chap. 2, we noted that linear elastic fracture mechanics offers powerful insights into the continuum description of fracture processes. On the other hand, we also noted that the occurrence of fracture ultimately depends upon mechanisms that take place in the vicinity of the crack tip, some of which depend upon atomic-level 

11 Points, Lines and Walls Defect Interactions and Material Response 

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Within the cracked and dislocated Real Nonstiochiometric crystals dominate. 

We note that this is one of many examples within the book wherein a fundamental solution is derived and used as the basis of solution for more complex problems using superposition. This same viewpoint will rear its head again in the context of dislocations when we show that the solution for the single straight dislocation may be repeatedly superposed to construct extremely complex solutions such as those for cracks and dislocation pileups. 

Dislocation Based Fracture Mechanics by Johannes Weertman, World Scientific, Singapore Singapore, 1996. What I find especially engaging about this book is its idiosyncratic treatment of a number of problems. Weertman shows repeatedly how in adopting the notion of dislocation solutions as being fundamental, many interesting concepts may be explained concerning cracks and 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. 

For the case of interest here, namely, the fields that arise for a crack in the presence of a dislocation, the procedure described above is disarmingly simple. Through a suitable choice of transformation, the problem of a crack and dislocation can be mapped onto the problem of a single subsurface dislocation beneath a traction-free surface already solved by the method of images in section 8.5.2. 

The subject of fracture has already arisen in several different contexts throughout the book. In chap. 2 we described the rudiments of the theory of linear elastic fracture mechanics. In addition, in the previous chapter we described the interplay of cracks and dislocations. The current discussion is aimed at elucidating yet another feature of fracture, namely, the fact that the study of fracture serves as a paradigmatic example of some of the ideas on bridging scales introduced earlier in the chapter. 

Removal of solid soils by penetration without liquid crystal formation has been reported for tripalmitin, octadecane, and tristearin [143-145]. In these cases penetration of detergents occurred at crack and dislocation sites of soils. 

The difference in the local tsunami wave field between crack and dislocation models is shown by example in Figs. 10 and 11. In both cases, a 270 km long and 70 km wide, pure thrust rupture is used, with a dip of 20° and average slip of 1 m. For the crack model, the slip distribution is that derived from Freund and Barnett (1976) under smooth closure conditions (<7=0.3). Details of the modeling procedure are given in Geist and Dmowska (1999). Marigrams (time series plot of wave amplitude) are displayed for both the... 

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