Abstract
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 Physik 66, 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.
Highlights
Recent advances in computer-aided material and structural fabrication, such as rapid prototyping, threedimensional (3D) printing or additive manufacturing, have created avenues for designing materials with tailored behaviour
We show that by combining the king post truss into pantographic microstructures, it is possible to obtain metamaterials with unique and controlled strain energy responses
We have demonstrated in this paper that the king post truss can be combined into pantographic sheet structures that have unprecedented strain-energy response
Summary
Recent advances in computer-aided material and structural fabrication, such as rapid prototyping, threedimensional (3D) printing or additive manufacturing, have created avenues for designing materials with tailored behaviour These technological developments have opened unexpected fields of applications of ancient mathematical problems and methods, whose applications were originally found in a different context. The revolutionary and somehow underestimated contribution by Maxwell [1] allowed for the exact design, already in 1864, of a wide class of beam lattices; an approach used later (owing to the vulgarization due to Mohr [2]) for building bridges and, in general, many kinds of civil engineering structures This approach due to its mathematical generality, can be applied to design lattice microstructures constituting novel modern metamaterials. From the viewpoint of macro-scale modelling the proposed metamaterial exhibits strong second gradient continuum behaviour, such as the predicted zones of quasi-uniform deformation with transition zones (boundary layers) of finite thickness
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