Split singularities and dislocation injection in strained silicon

It is a very interesting topic. I investigated cracks in three-dimensional domains and corner singularities in the past. It is an important issue. However, I have few questions:

• What about the mode III stress intensity factor? Is it not considered here since it is decoupled from K1 and K2 and has no 'lambda' singularity? 

• You assumed beta=0 (Dundur's parameter). Is it a reasonable assumption? It seems as if it degenerates the problem. It is interesting to observe how the 'oscillatory' parameter beta affects on the results.  

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Yuval

Dear Yuval,

Thank you very much for your interest and great comment.

1. We are also interested in 3D corner singularities since it is such an important issue. May I ask for a copy of your work and the related references about 3D corner singularities? I also suggest you to post your 3D singularity work on iMechanica. There are so many failure modes driven by corner singularities, and 3D corner with large elastic mismatch and CTE mismatch is so common in microelectronics, so I think this is a very interesting topic, at least for the mechanics people working in semiconductor industry.
2. In order avert the complexity of 3D corner singularities, we use a long stripe instead of a square pad, but the idea and the frame work still apply, as Martijn pointed out in his reply. The mode III is not considered because the stress state is plain strain stress state, there is no out-of-plane shearing.
3. Beta effect was studied a lot for cracking problems, such as those by He and Hutchinson in late 1980s and early 1990s. All those works showed the beta effect was no big deal, because beta effect is basically Poisson’s ratio effect. So for simplicity, we also adopt this assumption. Of course, as you pointed out, this assumption needs to be verified or studied.
4. As shown in Figure 2 in our paper, the upper right triangle region is of complex conjugates. In order to avert this complexity, we adopt beta=0, then there is no worry when we use the linear superposition of stronger and weaker singular stress fields. Since for strained silicon structure, such as the typical SiN/silicon system, alpha and beta are rarely in the complex region, therefore we needn’t worry about it.
5. You raise a very point about the “oscillatory” effect. For a material combination such as silicon/polymer, the singularity exponents are commonly a pair of complex conjugates. In such cases, we have to consider the “oscillatory” effect, just as dealing with the interfacial crack.

i need this papers about the use of  subdivision method to solve integral with weak singularity in bem

 - J O Lachat & J O watson " effective numerical treatment of boundary inegral equation " a new formulation for three dimenssional elasostatics Int .Jr.Numer.methods .Eng.211-228(1958)