On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

Abstract

The set GU f of possible effective elastic tensors of composites built from two materials with elasticity tensors C1 > 0 and C2 = 0 comprising the set U = {C1, C2} and mixed in proportions f and 1−f is partly characterized. The material with tensor C2 = 0 corresponds to a material which is void. (For technical reasons C2 is actually taken to be nonzero and we take the limit C2 → 0). Specifically, recalling that GU f is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate p-mode material, that is easily compliant to p ≤ 5 independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.

A parametric class of composites with a large achievable range of effective elastic properties

“…Figure 7(C) indicates that limit laminate structures also remain very close to Hashin-Shtrikman bound for volume fractions smaller than 1 (see also Figure 10). These results are consistent with the ones obtained previously in [26] and recent work [18], establishing the attainability of the right Hashin-Shtrikman bound for volume fractions smaller than 1.…”

Accordion-like metamaterials with tunable ultra-wide low-frequency band gaps

“…We have shown that our metastructures are not subject to this restriction and can generate low-frequency band gaps even if the end diameter of inclined bars is larger than the central one. Further interesting possibilities can be envisioned for future extensions to recently proposed 'rank 2' and 'rank 3' walled structures [40]. Table A1.…”

Pantographic metamaterials: an example of mathematically driven design and of its technological challenges

“…The aim of the investigations was more specifically (1) to design novel and exotic architectured metamaterials based on a mathematical understanding of the related mechanical problems and on suitably designed numerical simulations, (2) to build the designed prototypes by using 3D printing technology, (3) to test with sensitive apparatuses the so-built prototypes, (4) to elaborate the obtained data with modern image correlation techniques, (5) to produce a careful model fitting of the experimental data by means of the systematic use of numerical simulations, and (6) to compare the proposed models with experimental evidence.…”

“…With the latest advancements (e.g., 3D-printing technology and, more generally, of rapid prototyping techniques), the small-scale production of materials with complex geometries has become more affordable than ever [1][2][3][4]. The exploitation of these new technologies has made possible the development in the last few years of materials with very different substructures.…”