On Hill’s lemma in continuum mechanics

Hill’s first-order lemma in continuum mechanics has been derived in the literature for quasi-static cases and for dynamic cases in the absence of body forces. In this manuscript, a generalized first-order Hill identity is first derived in an Eulerian dynamic description including body forces. Then, a second-order Hill identity using a Lagrangian description is obtained, involving the variation in time of the associated kinetic energy. Simple examples, that allow for analytical analysis, of one-dimensional elastic deformation of bars are considered that involve dynamics and body forces to illustrate the generalization of Hill’s identities proposed here.