Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements

Inspired by some composite fiber reinforcements used in aeronautical engineering and by the need of conceiving new metamaterials, some discrete models including (quasi-) inextensible elasticae are considered. A class of continuum models approximately describing the macroscopic mechanical behavior of introduced structures is then heuristically proposed. Some of these continuum models can be regarded as a special kind of second-gradient elastic media, in which the higher-gradient elasticity is conferred by the flexural stiffnesses of elasticae constituting the microscopic lattice. The discrete models are studied by means of suitably tailored numerical codes designed to avoid numerical instabilities and locking and a comparison of discrete versus continuum models is attempted. The obtained results show that the theory of generalized continua may be useful in some engineering applications and it plays a relevant role in the mechanics of woven composites. The introduced discrete and continuum models are used to describe the so-called bias extension test on woven fabrics and it is shown that a good choice to correctly reproduce the targeted phenomenology is to use a second gradient continuum theory. However, as discussed throughout the paper, in the context of rigorous micro-macro identification procedures there still remain many open problems to be solved, especially when dealing with systems subjected to particular constraints, such as inextensibility.