On the choice of functions spaces in the limit analysis for masonry bodies

The kinematic and static problems of limit analysis of no-tension bodies are formulated. The kinematic problem involves the infimum of kinematically admissible multipliers, and the static problem the supremum of statically admissible multipliers. The central question of the paper is under which conditions these two numbers coincide. This involves choices of function spaces for the competitor displacements and competitor stresses. A whole ordered scale of these spaces is presented. These problems are formulated as convex variational problems considered by Ekeland and Témam. The static problem is unconditionally shown to be the dual problem (in the sense of the mentioned reference) of the kinematic problem. A necessary and sufficient condition, the normality, guarantees that the kinematic and static problems give the same result. The normality is not always satisfied, as examples show (one of which is presented here). The qualification hypothesis of Ekeland and Témam, sufficient for the equality of the static and kinematic problems, is never satisfied in the spaces of admissible displacements of bounded deformation or of functions integrable together with the gradient in the power p. In the cases of lipschitzian displacements and of smooth displacements, the qualification hypothesis is equivalent to simple conditions that can be satisfied in the case of the pure traction problem. However, it is shown thatthen the space of admissible stresses must be enlarged to contain stress fields represented by finitely or countably additive tensor-valued measures.