Michell truss type theories as a Γ-limit of optimal design in linear elasticity

We show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of Γ-convergence. We improve our results from [H. Olbermann, Michell trusses in two dimensions as a Γ-limit of optimal design problems in linear elasticity, Calc. Var. Partial Differential Equations 56 2017, 6, Article ID 166] in that our treatment here includes the three-dimensional case and that we allow for more general boundary conditions and applied forces.

MICHELL TRUSSES AND LINES OF PRINCIPAL ACTION

We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutually orthogonal families of curves. The method is illustrated on a specific example where uniqueness can be proved by studying an unusual system of hyperbolic PDEs. The questions we address here are of interest in elasticity theory, optimal designs, as well as in functional analysis.