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Reentrant tensegrity: An auxetic, three-periodic, chiral tensegrity construction

The Π+ cylinder packing in three totally different geometric incarnations. (Left) The Π+ cylinder packing composed of straight cylinders, with chiral space group symmetry P4132, and three distinct cylinder axes. (Centre) A compacted model of Π+ the place the cylinders turn out to be curvilinear, which now has the chiral space group I4132. (Right) The bmn periodic tensegrity construction, the place the incompressible rods are proven in black and the elastic struts coloured just like the cylinder packing above. The periodic unit cell is printed within the thick black strains. Credit: Science Advances, 10.1126/sciadv.abj6737

In a brand new report now printed in Science Advances, Mathias Oster, and a staff of scientists on the Institute for Mathematics on the Berlin Institute of Technology and the School of Engineering on the University of Edinburg within the U.Ok., introduced a three-periodic, chiral tensegrity construction and demonstrated that it’s auxetic, i.e., such supplies turn out to be thicker perpendicular to the utilized drive when stretched. An auxetic construction has a negative Poisson’s ratio and might kind materials with unexpected behavior. The tensegrity construction is a type of tensile structure held collectively by the steadiness of tensile and compression forces performing on them. The scientists constructed the tensegrity construction utilizing chiral symmetry cylinder packing to rework cylinders to elastic parts and cylinder contacts to incompressible rods. The consequence confirmed native re-entrant geometry at its vertices, which they confirmed utilizing finite element modeling. The structure represented a easy three-dimensional (3D) analog to the two-dimensional (2D) re-entrant honeycomb mannequin to kind an fascinating design goal for multifunctional supplies.

Tensegrity within the lab

In this work, Oster et al. proposed a beforehand unknown 3D auxetic construction with auxetic conduct, as an idealized geometric motif and simulated elastic materials. To start with, the staff targeted on tensegrity—a time period that defines integrity beneath stress. The time period originated from the architectural work of Kenneth Snelson and Buckminster Fuller to make use of tensegrity buildings as a mix of stress and compression forces to offer the phantasm of rods floating in space. Tensegrity can mix two sort of design parts generally known as strut and cable parts beneath stress to stabilize the construction, the place cables hold vertices shut collectively, whereas struts maintain them aside. The purity and ease of tensegrity can result in a really pure mathematical description. Mathematically, tensegrity could be described as a set of vertices that fulfill easy distance constraints. Researchers have made an fascinating parallel to the spatial constraints of tensegrity construction by using sphere packings, to discover their configurations and stability.

Reentrant tensegrity – an auxetic, three-periodic, chiral tensegrity structure.
The geometry of the tensegrity construction beneath deformation. (Left) The beginning configuration of our tensegrity construction, proven as a block of two × 2 × 2 unit cells. The black dice body is proven for visualization functions. The construction is in equilibrium and has the complete I4132 symmetry. (Right) The construction is stretched alongside the x axis, and the ensuing equilibrium construction is proven. The symmetry of the construction has been damaged, for instance, the fourfold screwaxes of the helices disappear. The enlargement within the perpendicular y path in response to the stretch could be seen within the measurement of the deformed construction, and the same magnitude of enlargement can be current within the z path. Credit: Science Advances, 10.1126/sciadv.abj6737

Modeling tensegrity

Much like sphere packing, crystalline supplies could be equally described by way of periodic packing of cylinders in 3D space. In this occasion, the cylinders represented rods of strongly bonded atoms or teams of atoms. For occasion, the 3D construction of the mineral garnet is well-known, however using cylinder packings supplied an easier description to grasp the construction. Inspired by the parallel between tensegrities and sphere packings, Oster et al. constructed a tensegrity construction utilizing helical cylindrical packing by reimagining the construction as a collection of inflexible rods suspended in space by a periodic internet of elastic filaments to kind a periodic tensegrity construction. The reentrant geometry of the vertices recommended auxetic conduct, which Oster et al. studied in depth.

Reentrant tensegrity – an auxetic, three-periodic, chiral tensegrity structure.
Measurement of the mechanics of the tensegrity construction beneath deformation. (Top) The size of periodic lattice translation within the y and z instructions (Ly and Lz) on repeated extension and compression cycles (cyclical Lx). One can see that, after an preliminary phase of instability, the construction reaches a gentle state of enlargement within the y and z instructions throughout enlargement within the x path and, likewise, contraction when x is contracted. (Middle) The instantaneous Poisson’s perform (νxy and νxz) on repeated extension and compression cycles of the construction within the x path. The values attain a comparatively regular state of round −1.1 for νxy and − 0.75 for νxz. (Bottom) The Poisson’s ratio (νxy and νxz) computed utilizing the log remodel true pressure on repeated extension and compression cycles of the construction within the x path. The values reached listed below are similar to these seen within the instantaneous Poisson’s perform above. Credit: Science Advances, 10.1126/sciadv.abj6737

Simulating the tensegrity construction

The staff then noticed the equilibrium configurations and quasi-static deformations of the constructed periodic tensegrity construction. At the start of simulated deformations, they analyzed the configurations equivalent to the densest packing inside a set unit cell. Using Newton’s method, Oster et al. confirmed the construction to be an equilibrium configuration. The scientists used a number of strategies to confirm the outcomes. The phase of deformation instantly after the preliminary lack of symmetry of the construction supplied an fascinating viewpoint from the disciplines of each materials science and arithmetic. Thereafter, the staff targeted on the engineering potential to appreciate these idealized geometric constructions by extending the idea of auxetic periodic tensegrity buildings to finite 3D lattices composed of elastic parts. The driving drive towards auxeticity trusted the interaction between geometry and elasticity. To then develop these ideas with supplies, Oster et al. explored 3D printing of a toy mannequin of the construction, which they achieved by printing the constructs utilizing a rubber-like thermoplastic polyurethane to look at delicate auxetic conduct of the construction.

  • Reentrant tensegrity – an auxetic, three-periodic, chiral tensegrity structure.
    Tension and compression of the tensegrity parts. 8 × 8 × 8 lattice for dc/ds = 0.6. The colour map represents the extent of axial stresses σ11, normalized by the Young’s modulus E, alongside the weather’ arc lengths. The inset exhibits a consultant quantity and additional the dissection of the cable and the strut parts to indicate that they’re subjected to stress and compression, respectively. The deformed configurations are proven at 0.025 pressure. Credit: Science Advances, 10.1126/sciadv.abj6737
  • Reentrant tensegrity – an auxetic, three-periodic, chiral tensegrity structure.
    Deformation of a 3D print of a block of the tensegrity construction. The object is printed utilizing rubber-like thermoplastic polyurethane materials. The full construction is printed in the identical materials, so there isn’t any differentiation made between the incompressible bars and the elastic parts. Despite this extremely simplified design, we nonetheless observe auxetic conduct. Credit: Science Advances, 10.1126/sciadv.abj6737


In this fashion, Mathias Oster and colleagues described a way to assemble a chiral, triply periodic tensegrity construction based mostly on excessive symmetry rod packing—a well-known method in structural chemistry. The work displayed native re-entrant geometry in any respect of its vertices to offer the construction an auxetic conduct. The staff confirmed how the auxetic conduct was additionally relevant to real looking materials simulations. They contrasted the quantitative variations between the computation for the idealized construction and that obtained from the finite component technique (FEM). They then introduced a probably easy three-periodic incarnation of the re-entrant honeycomb motif as an fascinating design goal for framework supplies. Since the construction is chiral, too, it may be a goal for metamaterials, the place the chirality is a precursor to an array of performance in supplies helpful for features with electrical, optical, and magnetic properties. The described method opens a design expertise to develop a wide selection of auxetic supplies. While the construction instigated varied explorations throughout the fields of algebraic geometry and optimization, it was too complicated for many obtainable numerical instruments. However, from a supplies science perspective, the tensegrity structure was comparatively easy. The work has already prompted the event of latest mathematical and symbolic approaches with optimism for the way forward for such research.

Team creates new ultralightweight, crush-resistant tensegrity metamaterials

More info:
Mathias Oster et al, Reentrant tensegrity: A 3-periodic, chiral, tensegrity construction that’s auxetic, Science Advances (2021). DOI: 10.1126/sciadv.abj6737

Sten Andersson et al, Body-centred cubic cylinder packing and the garnet construction, Nature (2005). DOI: 10.1038/267605b0

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Reentrant tensegrity: An auxetic, three-periodic, chiral tensegrity construction (2021, December 20)
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