An introductory treatise on reduced balance laws of Cosserat beams

This paper serves as an introduction to the variational formulation of Cosserat beams. It provides a detailed derivation and treatment of reduced balance laws of Cosserat beams from the Lagrangian differential equation of motion and Hamilton's principle. Emphasis is given to the details of the derivation, maintaining Bernoulli's assumption of the rigid cross-section. Both the strong form and the weak form of the equilibrium equation for Cosserat beams are derived independently from the infinitesimal stress equilibrium equation. The weak form is then validated by obtaining it from the strong form of the reduced law in a purely mathematical sense. Finally, the strong form is obtained using Hamilton's principle. Once the equations are obtained considering an initially straight reference beam configuration, the balance equation for the beam with initial curved (but unstrained) reference configuration is obtained. The D'Alembert forces are interpreted from the non-inertial director frame of reference and conclusions drawn. The energy conservation law and the conditions associated with it are obtained, establishing the relation between the Lagrangian and Hamiltonian functional for Cosserat beams.