Genesis and progress of virtual power principle

The virtual power principle (VPP) of continuum mechanics states a celebrated variational equality between external and internal virtual powers for any virtual velocity field conforming with linear kinematic constraints. The topic is here addressed to investigate how the original ideas born in the early XIX century are modelled by modern formulations based on Functional Analysis and Differential Geometry. These notions are able to provide an effective mathematical context for proving existence of Lagrange multipliers associated with the constraint of rigidity on velocity fields. The VPP stands as privileged tool for giving to stress fields a consistent definition based on duality with conforming virtual stretching fields. By complementarity, the VPP generates a variational condition for integrability of stretching fields, with self-equilibrated stresses as test fields. Progress is got by the formulation of the rate virtual power principle (RVPP) by time derivation of the VPP along the motion, with internal virtual power integrated per unit mass. The basic distinction between spatial and material fields according to the geometric paradigm is prompted to replace the one previously adopted in the literature. The need for a non-redundant implicit formulation of the rigidity constraint is emphasised to contrast degeneracy. This logical demand avoids proliferation of multipliers, in the spirit of Ockham’s Razor, a celebrated philosophical motto with multiform applications. The shining mathematical theory set out by Leonhard Euler, Jean-Baptiste Le Rond d’Alembert, Joseph Louis Lagrange, and Augustin Cauchy is in this respect a point of optimality. A geometric rate theory of elasticity meets the call for no-dissipation in push-closed elastic cycles, with non need of any finite strain elastic energy functional, thus leading to a proper statement of rate equilibrium problems, basilar for computational formulations and for investigations about instability phenomena and post-critical behaviours.