A note on equilibrated stress fields for no-tension bodies under gravity

We study the equilibrium problem for two-dimensional bodies made of a no-tension material under gravity, subjected to distributed or concentrated loads on their boundary. Admissible and equilibrated stress fields are interpreted as tensor-valued measures with distributional divergence represented by a vector-valued measure, as developed by the authors of the present paper. Such stress fields allow us to consider stress concentrations on surfaces and lines. Working in $\mathbb {R}^n,$ we calculate the weak divergence of a stress field that is asymptotically of the form $|\mathbfit{x}|^{-n+1} \mathbfit{T}_0(\mathbfit{x} /|\mathbfit{x}|)$ for $\mathbfit{x} \to \mathbf {0}$ on a region that is asymptotically a cone with vertex $\mathbf {0}$. Such stress fields arise as parts of our solutions for two-dimensional panels. Proceeding to problems in dimension two, we first determine an admissible equilibrated solution for a half-plane under gravity that underlies two subsequent solutions for rectangular panels. For the latter we give solutions for three types of loads.