Dynamics of disclination dipoles

In materials science, crystal lattice defects play a crucial role in determining the mechanical behavior of solids. These defects are generally classified into two main categories: dislocations and disclinations. Dislocations are related to translational defects, characterized by a Burgers vector, and are commonly divided into screw and edge types, depending on the orientation of the defect line with respect to the Burgers vector. On the other hand, disclinations are rotational defects associated with an angular mismatch, quantified by the Frank angle. Although dislocations have been extensively studied, in recent years disclinations have gained growing attention for their role in plasticity phenomena at small scales. One key result is that the renormalized energy of a disclination dipole is equivalent to that of an edge dislocation under appropriate rescaling. We investigate the dynamics of a finite number of disclinations within a two-dimensional circular domain, assuming that the defects evolve according to the maximum dissipation criterion, in line with the Peierls-Nabarro approach. Our main focus is the study of a dipole of disclinations with equal Frank angles in a unit disk. Due to symmetry, the problem can be reduced to two degrees of freedom: the position of the center of the dipole and the distance between the two defects. Through a detailed force analysis and a numerical implementation based on the explicit Euler scheme, we identify the stationary regimes and, correspondingly, classify the qualitative behavior of the system into three categories: a converging dipole with its center moving toward the origin; a converging dipole with its center approaching the boundary; and a diverging dipole where the defects move away from each other. In the diverging regime, we compute an asymptotic estimate for the dipole separation, which exhibits a polynomial-in-time growth rate of the distance between the defects. In the converging regimes, on the contrary, we observe an exp(-exp) decay profile in the distance. Regardless of the converging regime, a particularly interesting outcome is that the collision of disclinations occurs only asymptotically in time. This behavior contrasts with that of dislocations, which are known to collide in finite time. Furthermore, in a specific case, we obtain an implicit formulation for the evolution of the dipole center. After a suitable transformation and a time renormalization, this formulation turns out to be mathematically equivalent to the dynamics of a screw dislocation in a circular domain, revealing a deep connection between the two types of defects. In the final part of this thesis, we derive the dynamics of the edge dislocation from Eshelby’s equivalence. In particular, to achieve this result, we extract the non-divergent contribution of the energy. By exploiting the maximum dissipation criterion, we obtain the dynamics of an edge dislocation in a circular domain. The obtained dynamics coincide with those derived from the time rescaling of the disclination dipole dynamics. From the analysis of the problem, it emerges that edge dislocations, when located away from the center, tend to reach the boundary in finite time, while those positioned at the center of the domain remain there due to symmetry considerations.