Advanced beam modeling under finite strains: theoretical and numerical study with experimental validation

Highly slender beam-like structures constitute fundamental components in numerous engineering applications, ranging from civil and mechanical systemsto biomedical devices. The accurate prediction of their mechanical response requires the incorporation of nonlinear effects capable of describing instability phenomena, post-buckling behavior, and large deformations. This thesis is devoted to the development, extension, and validation of advanced beam formulations for the analysis of structures undergoing finite displacements and strains. The work builds upon an efficient finite-difference framework for geometrically exact beams and extends it to include different shear-deformable beam theories. In particular, the Reissner and Ziegler formulations are incorporated within a unified computational framework, enabling a systematic investigation of the influence of different kinematic assumptions on stiffness, stability, and post-critical response. Within this framework, a theoretical stability study is carried out to assess the role of shear deformation and kinematic modeling assumptions on elastic instability phenomena, critical loads,and post-buckling behavior. The resulting models are employed to analyze geometrically nonlinear structural behavior, with particular attention to snap through phenomena, and large-displacement equilibrium paths. The proposed beam formulation is further generalized to account for constitutive nonlinearities through the introduction of hyperelastic material models. Constitutive laws belonging to the Seth–Hill family are considered, together with the incompressible Neo-Hookean and Gent models, allowing the analysis of highly deformable materials over a wide range of strain levels. The interaction between geometric and material nonlinearities is investigated, highlighting the influence of the adopted strain measure and constitutive description on the predicted structural response. Since the framework of the formulation is 2D, cross-sectional deformability enriches the model towards a 3D framework. By relaxing the classical assumption of rigid cross-sections, the formulation becomes suitable for the analysis of soft materials subjected to large strains, for which transverse deformations and lateral expansion significantly affect the internal force resultants and the overall mechanical response. All developed models are formulated through the integrated form of the equilibrium equations coupled with compatibility and constitutive relations. The resulting governing equations are discretized using an explicit finite-difference scheme and solved through a shooting procedure, leading to a computationally efficient framework that avoids the need for conventional finite-element interpolation functions while maintaining high accuracy in the nonlinear regime. The developed methodology is finally applied to the analysis and design of 3D-printed lattice metamaterials for energy-absorption applications. Numerical simulations and experimental investigations are combined to evaluate the effects of topology, bistability, snap-through mechanisms,and energy-absorption capacity. The proposed framework provides an accurate and computationally efficient tool for the analysis of nonlinear structural problems involving both conventional materials and highly deformable metamaterials, offering a practical alternative to more computationally demanding geometrically exact finite-element formulations.