Lagrangian theory of extensible elastica with arbitrary undeformed shape

This work presents a consistent formulation of the Lagrangian function for slender elastic bodies with arbitrary initial geometries, within a dynamic framework and under finite displacements. Building upon and extending previous research, we develop a rigorous expression for the kinetic energy, thereby completing the Lagrangian formulation. Our approach ensures consistency across geometric and dynamic nonlinearities. Furthermore, we derive pattern solutions for representative benchmark problems, illustrating the applicability and versatility of the proposed framework. These results open new avenues for the application of our formulation across various domains in applied science and engineering.