^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

Edited by: Federico Bosia, Università degli Studi di Torino, Italy

Reviewed by: Massimiliano Zingales, Università degli Studi di Palermo, Italy; Giuseppe Puglisi, Politecnico di Bari, Italy

Specialty section: This article was submitted to Mechanics of Materials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study formulates numerical and analytical approaches to the self-equilibrium problem of novel units of tensegrity metamaterials composed of class

Recent research in the area of mechanical metamaterials has revealed several distinctive features of lattice materials formed by tensegrity units and lumped masses, which originate from the peculiar, nonlinear mechanical response of such units. Ordinary engineering materials typically exhibit either elastic stiffening (e.g., crystalline solids) or elastic softening (e.g., foams). More puzzling is the geometrically nonlinear response of structural lattices based on tensegrity units (e.g., tensegrity prisms), which may gradually change their elastic response from stiffening to softening through the modification of mechanical, geometrical, and prestress variables (tensegrity metamaterials (Skelton and de Oliveira,

The tensegrity metamaterials studied in Fraternali et al. (

A primary goal of this work is the study the existence of self-equilibrium configurations of class

Let us consider the class _{n}_{b}_{s}

Reference configurations of class

With the aim of comparing the results of this study with those available in Bieniek (

Illustration of the aspect angles

Limiting configurations corresponding to

Limiting configurations corresponding to

By introducing a Cartesian reference frame with the origin at the centroid of the prism, and letting the superscript

Following the notation presented in Skelton and de Oliveira (_{i}_{B}_{S}]^{T}

The state of stress acting on the structure under examination can be characterized through the force densities acting in all the members (i.e., the members’ forces divided by their current length). We collect such quantities into the following matrix (Donahue et al., _{i}_{i}_{b}_{s}_{i}_{i}

Our current goal is to formulate a numerical approach to the search for solutions of the equilibrium problem (8) under 0 external forces (i.e., when

Let’s start first by examining System 1 in Figure ^{Ⓡ} (Version 11), over the whole search domain

Plots of

We are interested in the local minimum of ^{T}_{0} = [_{0}, _{0}]^{T}^{T}_{0} with increments equal to 10^{−7} of the amplitude of the search interval. In addition, we iteratively solve equation (_{0} = _{0} + Δ

^{−7}. The use of the above iterative procedure led us to detect a (non-degenerate) prestressable configuration of System 1 (cf. Figure

Freestanding configurations obtained for System 1 and System 2 in Figure

The numerical approach to the self-equilibrium problem of class

We now aim at formulating an alternative analytic approach to the same problem that makes use of the symmetry properties of the system under consideration and examines the equilibrium equations of the two nodes attached to an arbitrary bar. Let us examine, e.g., the bar that connects the nodes 2 and 11 (cf. Figure _{t}_{b}_{2–1}, _{2–3}, _{2–11}, and _{2–3}. One can use such equations to obtain _{2–1}, _{2–3}, and _{2–3} as a function of _{2–11} ≡ _{b}_{2–1} = _{2–3} ≡ _{t}_{b}

For what concerns the self-equilibrium equations of node 11, we observe that such equations involve only the three force densities _{2−11} ≡ _{b}_{7−11} ≡ _{c}_{8−11} ≡ _{c}_{7} − _{11} and _{8} − _{11} must be orthogonal to the bar vector _{2} − _{11}, i.e., it results,

Making use of equations (

By grouping equation (_{b}^{Ⓡ} to get the same result obtained in the previous section, i.e., ^{Ⓡ} in correspondence to _{t}

States of self-stress for System 1 and System 2.

1–2 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

2–3 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

3–1 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

4–5 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

5–6 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

6–4 | 1.000 _{t} |
100.0 _{t} |
1.000 _{t} |
100.0 _{t} |

10–7 | 3.335 _{t} |
166.8 _{t} |
1.767 _{t} |
176.721 _{t} |

7–11 | 4.033 _{t} |
201.6 _{t} |
1.931 _{t} |
193.114 _{t} |

11–8 | 3.335 _{t} |
166.8 _{t} |
1.767 _{t} |
176.721 _{t} |

8–12 | 4.033 _{t} |
201.6 _{t} |
1.931 _{t} |
193.114 _{t} |

12–9 | 3.335 _{t} |
166.8 _{t} |
1.767 _{t} |
176.721 _{t} |

9–10 | 4.033 _{t} |
201.6 _{t} |
1.931 _{t} |
193.114 _{t} |

1–4 | 0.974 _{t} |
97.4 _{t} |
0.698 _{t} |
69.836 _{t} |

2–5 | 0.974 _{t} |
97.4 _{t} |
0.698 _{t} |
69.836 _{t} |

3–6 | 0.974 _{t} |
97.4 _{t} |
0.698 _{t} |
69.836 _{t} |

1–10 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

2–11 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

3–12 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

4–7 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

5–8 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

6–9 | 1.720 _{t} |
−203.527 _{t} |
1.175 _{t} |
−193.846 _{t} |

The kinematic problem conjugate to equation (_{m}_{dof}_{m}

Infinitesimal mechanisms from the freestanding configuration of System 1.

Infinitesimal mechanisms from the freestanding configuration of System 2.

Let’s now assume that all the members of the structure under consideration behave as linear elastic springs governed by the following constitutive equations:
_{i}_{b}_{s}_{dof}_{n}_{i}_{j}

Such a matrix can be decomposed as follows:
_{M}_{G}_{i}_{G}

According to the nomenclature given in Guest (_{G}_{M}_{G}_{t}

We have formulated numerical and analytical approaches to the search for freestanding configurations of class

The results obtained in Sects. 4 and 5 lead us to conclude that the freestanding configurations of class

The presence of a high number of infinitesimal mechanisms from the freestanding configurations suggests that the examined systems can be usefully employed as novel units of mechanical metamaterials, which exhibit geometrically nonlinear response, support solitary wave dynamics, and offer a valid alternative to the standard tensegrity prisms studied in Fraternali et al. (

One-dimensional tensegrity metamaterial alternating lumped masses with regular and expanded class

MM has developed the numerical conditions and has taken care of the writing together with IM. IM has employed the numerical codes developed within the project and has contributed to the writing. FF has supervised the work carried out by all the authors. ZB has provided the topic and the motivation.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at