On conditions of potentialness for geometrically exact shearable beams

Abstract – In classical mechanics, there are different, yet interrelated, formulations which allow studying mechanical systems from various perspectives and deepening. However, it is desirable that the equations describing the dynamics of a system be the same, whichever path is followed to derive them. In this paper, the focus is on the equations of motion of beams in the geometrically exact framework, for which we also require that they have the additional property of the ‘potentialness’ (or potentiality), that is they can be derived from a properly defined potential. Extending results from the literature that focus on unshearable beams, this contribution is devoted to the shearable case. For a planar beam whose stress-free configuration is curvilinear, four different sets of balance equations are considered, where two different choices of the components of the internal resultant force and two different curvatures as bending strain measures are taken into account. The conditions to be imposed on constitutive assumptions, relating generalized static entities to strains, are assessed through Helmholtz conditions. Some constitutive assumptions for nonlinear shearable beams available in the literature are also discussed in light of theoretical results obtained in this contribution.