A dislocation is a one-dimensional singularity in an elastic deformation that encodes a mismatch in the underlying displacement field.

Mathematically, it arises as a defect set where strain incompatibility concentrates, and it represents a fundamental example of line defects in continuum models of solids.

In the applied analysis literature, several approaches have been developed; here we focus on the one introduced by Lauteri and Luckhaus in their study of the interaction between two crystalline lattices rotated by an angle [Lauteri–Luckhaus, preprint (2016)].

In their seminal work, they established the scaling predicted by Read and Shockley in the 1950s [Read–Shockley, Phys. Rev. 1950], without assumptions on the defect distribution and introducing several deep ideas.

We propose a Lauteri–Luckhaus–type energy tailored to semi-coherent interfaces.

In this setting, the analogue of the Read–Shockley law is given by the classical van der Merwe model [van der Merwe, Proc. Phys. Soc. 1963], whose energetic behavior we rigorously confirm.

We also introduce a linearized model that enables us to prove the analogous logarithmic scaling in higher dimensions through a slicing procedure.

Based on joint work with Emanuele Spadaro (Sapienza).