A Family of Higher-Order Theories for Wave Propagation in One-Dimensional Structural Elements

This paper proposes a systematic framework for the development and classification of higher-order theories for modeling wave propagation in one-dimensional structural elements. Building upon and organizing the existing higher-order models, a generalized approach is introduced to construct an entire family of such theories with controllable complexity and accuracy. The methodology enables the inclusion of higher-order terms in a physically meaningful and mathematically consistent manner, going beyond the classical polynomial extension of displacement fields. The proposed theories leverage traction-free boundary conditions, leading to accurate wave representation. Preliminary considerations on their implementation in the finite element method are also discussed, laying the foundation for future work focused on advanced finite element formulations.