Light elastic structures arise naturally in the optimization of designs that carry prescribed loads with vanishing material volume. In this talk, I discuss a relaxed shape-optimization problem for such structures, formulated in terms of a stress measure and a material distribution subject to an equilibrium constraint. The model is motivated by the theory of Michell trusses and by recent Γ-convergence results for elastic energies in the vanishing-mass regime.

I will present existence of minimizers for the relaxed problem and derive first-order optimality conditions using a Lagrangian formulation. The resulting Karush-Kuhn-Tucker conditions provide a measure-theoretic description of the alignment principle for optimal trusses, as studied by Bouchitté and Buttazzo via the Monge–Kantorovich equation and further developed by Bouchitté, Gangbo, and Seppecher in the context of Michell trusses and lines of principal action. I will then show that these conditions provide geometric information on optimal designs. In two space dimensions, the singular part of the optimal stress is concentrated on a countably one-rectifiable set, while in three dimensions both one- and two-dimensional rectifiable components may occur.

This is joint work with Dimitrios Andreakis and Filip Rindler.