Mathematical and physical models of beams with cellular microstructure

In recent years, metamaterials have been widely studied as they combine the practical demands for structural optimization and material innovation with the desire to explore new frontiers in the mechanics of materials and structures. In fact, thanks to a proper design, they can exhibit characteristics that overcome those of conventional materials. For their analysis, homogenization techniques have proven to be particularly effective: despite the transition from a discrete model, with a finite number of degrees of freedom, to a homogenized equivalent continuum, with infinite degrees of freedom, they enable a significant reduction in computational effort thanks to the simplification of the equations that describe their behavior. This thesis investigates the mechanical behavior of beam-like structures through the development, according to the homogenization approach, and the validation, through comparisons with Finite Element (FE) models, of one-dimensional models embedded in a 2D space. Two types of metamaterials are studied, i.e. the grain and the grid beam. For both, an equivalent beam model is formulated within the framework of the direct one-dimensional approach, in which the constitutive law is derived through the so-called mixed homogenization procedure. The latter is based on enforcing energy equivalence between a unit cell (the smallest repeatable unit of the periodic model) and a segment of the solid beam (equivalent model) of the same length. The homogenization procedure can be carried out either analytically or numerically, with the numerical approach relying on a FE analysis of the cell. The grain beam is made of a periodic assembly of grain-pairs interconnected with solid bars. This configuration gives rise to a metamaterial that through a coupled axial-shear mechanical response exhibits chirality. The equivalent model is formulated under the assumption of rigid cross-sections, both in plane and out of their plane, and the identification of the elastic coefficients is performed, through a cell analysis, analytically and numerically. The analytical procedure also allows a qualitative investigation of the mechanical response, showing that the elastic coefficients depend on specific non-dimensional parameters, thus enabling the design of the chirality itself. The inertial properties are analytically identified under the assumption of masses lumped at the joints. The grid beam is a framed microstructure made of two orthogonal families of (micro) beams. Unlike the chiral case, this beam-like structure is identified assuming a rigid in plane and deformable out of plane cross-section. In this framework, the warping phenomenon induced by shear is particularly significant. Nevertheless, it is not adequately described by classical beam theories and, moreover, the models available in the literature for grid beams are able to describe the deformation of the cross-section only on average and for particularly simple case studies. To overcome these limitations, this thesis develops a homogenized beam model that extends classical formulations by introducing additional strain measures capable of capturing both distortional and bi-distortional deformation modes. The elastic coefficients are identified analytically and numerically, while the inertial properties are analytically identified under the assumption of masses lumped at the joints. The mechanical response of both metamaterials is investigated under static, free dynamic, and buckling conditions where, in this latter, the equivalent model accounts for both the (classical) global and the (non-conventional) local geometric effects. Results obtained from the homogenized models are compared with those derived from FE analyses and, in the grain beam’s case, from experimental data.