I propose to investigate the behavior of dislocations as the core radius ∈ tends to zero, both at the point-discrete and at the continuum level.
A first topic of research is to study the Gamma convergence of the suitably rescaled energy (1) as ∈ goes to zero. It can be easily seen that rescaling the functional by multiplying it by |1/ log ∈| one obtains an energy counting the number of dislocations. In some sense, we loose informations on the position of the dislocations. We hence prefer to perform a different rescaling by considering the functionals F^∈ := E^∈ – |log ∈|. This renormalization leads to a different Gamma limit whose form and value strongly depends on the position of x and on the Dirichlet boundary datum of the deformation.
Our main goal is the analysis of such Gamma limit, with the final scope of proving that in the Γ-limit the minimality property of dislocations being inside the crystal is preserved.
I will consider the same Gamma limit for a large number of dislocations, in order to describe also their interaction behavior (in the spirit of some other existence approaches). Note that as the number of dislocations increases with ∈ another rescaling is necessary, and a different Gamma limit is expected. Similar results have been obtained in A. Garroni, G. Leoni, M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations , J. Eur. Math. Soc. 12 , 1231-1266, (2010).
As a second and more delicate goal of the project is to perform the limiting analysis in the setting of continuum dislocations, i.e. in a 3-dimensional setting. I have already considered this setting in a series of papers, corresponding to generalizations of a result of existence of solution to a minimization problem where a fixed single loop was studied. Tools such as currents and the theory of Cartesian maps allow to extend such result to many other energies minimization problems for a framework including dislocations as unknown variables. In particular, the minimizing dislocations might form very complex structures, described in their generality by integral currents with coecients in Z^3. A linear elastic energy with the core radius approach (here the core will be represented by tubular neighborhoods of dislocations lines) can be considered. I aim here to study the problem of its minimization with Dirichlet boundary conditions, as for the 2-dimensional case. Furthermore, the corresponding Gamma limit will be object of research. As for the 2-dimensional case, some results in this direction already exist, but with different rescalings. My strategy will be that of rescaling by subtracting the leading term in the asymptotic of the energy as ∈ goes to zero, which is again of order | log ∈| times the total mass of the dislocation density.