A historical perspective of Menabrea's "principle of elasticity"

This paper presents the theorem proposed by Luigi Federico Menabrea to study linear elastic systems. The publications on the subject by Menabrea are examined, as well as the criticism and the corrections brought to his first attempts of proof. We consider Menabrea's work in the frame of the studies of his contemporaries; we try to provide a historical and epistemological background for Menabrea's theorem and for its consequences in modern mechanics. Menabrea's theorem is a version of the theorem of least work, or of minimum complementary energy. A rough statement of the latter, for a ssystem free of dislocations, is as follows: A linear elastic system is subjected to a compatible deformation if and only if the elastic complementary energy is a minimum in the space of balanced actions'. These statements were the basis for the resolutions of redundant frames proposed by Castigliano and Muller-Breslau. The theorem was stated more or less in this form in 1858 by Luigi Federico Menabrea in a short memoir. The declared purpose of this memoir was to provide a general tool of solution for the problems of linear elasticity met in engineering practice, and this tool was to be more efficient than the ad hoc procedures at ease then. Such procedures were based on what is nowadays called method of forces and followed the path tracked by Navier.