ahmed.hussein's blog

In this work, the evolution of the dislocations microstructure in single crystal two-phase superalloy microcrystals under monotonic loading has been studied using the three-dimensional discrete dislocation dynamics (DDD) method. The DDD framework has been extended to properly handle the collective behavior of dislocations and their interactions with large collections of arbitrary shaped precipitates. Few constraints are imposed on the initial distribution of the dislocations or the precipitates, and the extended DDD framework can support experimentally-obtained precipitate geometries.

Discrete dislocation dynamics simulations were performed to investigate the dislocation microstructure evolution and cyclic hardening during the early stages of fatigue loading in nickel single crystals. The effects of the crystal size and initial dislocation densities on both the mechanical response and the evolution of dislocation microstructure were quantified.

Three newly identified cross-slip mechanisms from atomistic simulations of fcc crystals, namely surface, bulk and intersection cross-slip types, were hierarchically informed into discrete dislocation dynamics simulations. The influence of each cross-slip type on the evolution of the dislocation microstructure in face-centered cubic microcrystals having different crystal sizes and initial dislocation densities was investigated.

I am working on some boundary integral equation formulation and I am currently stuck with some mathematics. I wish anyone can help me out with this. 

I have an anisotropic (sometimes called generalized) biharmonic differential operator which  takes the form

L = k11 D1^4 +  k12 D1^2 D2^2 + k22 D2^4 

where D1 = d/dx, D2 = d/dy, my problem is two dimensional.

I need to find a fundamental solution (Green's function) for this operator, that is 

L(u) = -delta

where delta is Dirac delta function.