A historical perspective on Betti's theorem and its connection with the potential theory

In this paper a central result in the theory of linearized elasticity, namely the reciprocity theorem, is outlined for a continuous solid body, performing the exegesis of the original formulation by Enrico Betti (1823–1892), comparing the proofs he provided with those permitted by modern approaches in continuum mechanics. For linear elastic structures under small displacement, rotations and strains, legitimating the superposition of the structural responses under multiple loading systems, Betti's theorem allows one to deduce relations involving only loading and displacements over the boundary surface, without requiring the knowledge of the body's elastic stiffness nor making reference explicitly to the distribution of strains and stresses (as it occurs for the conventional formulation of the virtual work). We discuss the auxiliary use of this theorem made by Betti in his works on continuum mechanics, inspired by the potential theory, in particular involving Green's identities. The outlined theoretical strategies, resting on the rigorous deduction of analytical expressions, can represent nowadays a reference for the scholars prevalently devoted to repetitive numerical simulations, and an indispensable tool to investigate truly innovative materials models.