Discrete Homogenization Procedure for Estimating the Mechanical Properties of Nets and Pantographic Structures

The characterization of the mechanical response of complex materials, like tissues, woven composites, fibre networks etc. often requires multilevel analyses, since the micro structure strongly influences the observable behaviour. Existing approaches can be subdivided essentially into equivalent continuum or multiscale models. Often non-local continuum models are needed, see for instance [1]. The choice of the appropriate mechanical model and of its physical parameters must rely on the experimental or numerical consideration of the real micro-structure of the material. Asymptotic homogenization techniques can be effectively used for obtaining simplified continuum models, since they can be expanded up to the required order able to include the desired physical effects. In the case of lattice microstructures (networks, pantographic structures, tissues,...), discrete homogenization appears particularly useful [2, 3]. The purpose of the work is to extend the method to general, unbalanced, lattice systems, periodic or quasi-periodic, discussing the forms obtained for the constitutive relations and for the micro-rotations. The presentation will be restricted to a 2D case. Although not specifically presented, the procedure will be set within a framework ready for analysing the geometrical non linear case.