A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory

Among the most studied models in mathematical physics, Timoshenko beam is outstanding for its importance in technological applications. Therefore it has been extensively studied and many discretizations have been proposed to allow its use in the most disparate contexts. However, it seems to us that available discretization schemes present some drawbacks when considering large deformation regimes. We believe these drawbacks to be mainly related to the fact that they are formulated without keeping in mind the mechanical phenomena for describing which Timoshenko continuum model has been proposed. Therefore, aiming to analyze the deformation of complex plane frames and arches in elastic large displacements and deformation regimes, a novel intrinsically discrete Lagrangian model is here introduced whose phenomenological application range is similar to that for which Timoshenko beam has been conceived. While being largely inspired by the ideas outlined by Hencky in his renowned doctoral dissertation, the presented approach overcomes some specific limitations concerning the stretch and shear deformation effects. The proposed model is applied to get the solutions for some relevant benchmark tests, both in the case of arch and frame structures. It is proved that, also when shear deformation effects are of relevance, the enriched, yet simple, model and numerical computation scheme herein proposed can be profitably used for efficient structural analyses of non-linear mechanical systems in rather nonstandard situations.