King post truss as a motif for internal structure of (meta)material with controlled elastic properties

One of the most interesting challenges in the modern theory of materials consists in the determination of those microstructures which produce, at the macro-level, a class of metamaterials whose elastic range is many orders of magnitude wider than the one exhibited by 'standard' materials. In dell'Isola et al. (2015 Zeitschrift für angewandte Mathematik und Physik66, 3473-3498. (doi:10.1007/s00033-015-0556-4)), it was proved that, with a pantographic microstructure constituted by 'long' micro-beams it is possible to obtain metamaterials whose elastic range spans up to an elongation exceeding 30%. In this paper, we demonstrate that the same behaviour can be obtained by means of an internal microstructure based on a king post motif. This solution shows many advantages: it involves only microbeams; all constituting beams are undergoing only extension or compression; all internal constraints are terminal pivots. While the elastic deformation energy can be determined as easily as in the case of long-beam microstructure, the proposed design seems to have obvious remarkable advantages: it seems to be more damage resistant and therefore to be able to have a wider elastic range; it can be realized with the same three-dimensional printing technology; it seems to be less subject to compression buckling. The analysis which we present here includes: (i) the determination of Hencky-type discrete models for king post trusses, (ii) the application of an effective integration scheme to a class of relevant deformation tests for the proposed metamaterial and (iii) the numerical determination of an equivalent second gradient continuum model. The numerical tools which we have developed and which are presented here can be readily used to develop an extensive measurement campaign for the proposed metamaterial.