Tensegrity metamaterials are a type of artificial materials that can exploit the tunable nonlinear mechanical behavior of the constituent tensegrity units. Here, we present reduced-order analytical models describing the prestrain-induced bistable effect of two particular tensegrity units. Closed-form expressions of the critical prestrain at which a unit transitions into a bistable regime are derived. Such expressions depend only on the geometry of the units. The predictions of the reduced-order model are verified by numerical simulations and by the realization of physical models. The present results can be generalized to analogous units with polygonal base, and the proposed tensegrity units can be assembled together to form one-dimensional metamaterials with tailorable nonlinear features such as multistability and solitary wave propagation.

Architected metamaterials have been studied extensively in recent years^{1,2} with particular attention to multistable metamaterials obtained by tessellating units with bistable behavior. For instance, multistable metamaterials were proposed for energy trapping and impact mitigation^{3} and for stable transmission of mechanical signals over arbitrary distances.^{4} The propagation of transition waves in one-dimensional lattices composed of concentrated masses and bistable springs was treated analytically,^{5} and typical multistable metamaterials were optimized^{6} and extended to two and three dimensions.^{7}

In the particular class of tensegrity metamaterials, the repeating unit is a tensegrity structure,^{8–10} that is, a prestressed cable-bar framework. Because many tensegrity structures are deployable and/or possess a highly nonlinear response depending on their geometry and level of prestress,^{11–13} these types of structures became of interest for realizing adaptive and tunable structures.^{14} Researchers have used stimulus-responsive polymers to achieve programmable deployment of tensegrity structures^{15} and studied the bandgap tuning of tensegrity chains numerically^{16} and experimentally.^{17} Tensegrity chains can also support the propagation of solitary waves, which was observed in one-, two-, and three-dimensional tessellations of tensegrity units.^{18–22} The mechanical response of three-dimensional tensegrity metamaterials was studied with continuum models,^{23,24} and different regimes of wave propagations were shown to depend on the selfstress level.^{25} In addition, optimal-density^{26} and energy-dissipation^{27} planar tensegrity metamaterials were proposed, and a systematic approach to obtain tensegrity metamaterials with desired properties was devised.^{28} Although additively manufactured tensegrity-like metamaterials with no prestress have been experimentally studied,^{29–32} the additive manufacturing of prestressed tensegrity-like metamaterials has not been attempted yet. Nevertheless, possible prestressing procedures at the microscale could rely on 4D printing,^{33} for example, by using two-photon laser printing of photo-responsive structures.^{34}

Here, we introduce two tensegrity structures, the “six-node” and the “eight-node” units, which demonstrate a monostable-to-bistable transition triggered by changes in geometry and selfstress level. Previous research has shown that some tensegrity structures can exhibit bistable^{35,36} or multistable^{37} behaviors. Our units are the smallest known spatial tensegrity structures with such features. Each unit shows different aspects of the possible bistable regimes with the six-node unit having no infinitesimal mechanisms and the eight-node having three. We developed analytical models for each structure to calculate the critical prestrain, the amount of prestrain necessary for the units to enter a bistable regime. These models were consistent with the structures' symmetry properties and allowed us to derive closed-form expressions for the critical prestrain. We performed numerical simulations on one-dimensional assemblies of the units, which confirmed the expected prestrain-induced multistable behavior. We verified our analytical and numerical results through physical models with different bistable responses. These findings have implications for the design and benchmarking of mechanical metamaterials with adjustable multistable behavior.

**p**

_{i}for the

*i*th node and $p$ denoting the collection of all nodal position vectors, while the edges connect pairs of nodes and are labeled as “bar” or “cable.” The current length, rest length, and spring constant of the edge

*ij*connecting nodes

*i*and

*j*are $ l i j = | p i \u2212 p j | , \u2009 l \xaf i j > 0$, and

*k*> 0, respectively. We consider an elastic energy as

_{ij}^{38}

**K**

_{T}, equal to the Hessian of the elastic energy, $ \u2202 p 2 \u2009 U$, is

**K**

_{M}, which depends on the spring constants, and the geometric stiffness operator,

**K**

_{G}, which depends on the axial forces (see Refs. 36, 39, and 40 for details). We recall that the internal mechanisms of $T$, if any, are the sets of nodal displacements, which do not cause first-order changes of the edge lengths,

^{41,42}excluding rigid-body motions. Internal mechanisms and rigid-body motions lie in the nullspace of

**K**

_{M}.

**K**

_{T}is a sufficient condition for the stability of an equilibrium configuration; however, there are two more specialized stability conditions for tensegrity structures. The notion of prestress stability

^{43}applies to tensegrity structures with non-null self-stress possessing internal mechanisms. A tensegrity structure is said to be prestress stable if, for every internal mechanism $ v \u2009 \u2208 null ( K M )$,

**K**

_{G}has no negative eigenvalues, then we speak of super stability,

^{44}that is, the structure at $ p ( eq )$ is stable independently of material properties and of the level of selfstress. On the contrary, if

**K**

_{G}has negative eigenvalues, whether there are internal mechanisms or not, then it is possible that a stable tensegrity structure at a certain selfstress level becomes unstable at larger selfstress levels.

^{36,40}In fact, given a selfstressed equilibrium configuration where

**K**

_{T}is positive definite and

**K**

_{G}has a negative eigenvalue, since

**K**

_{G}is linear in the axial forces

*t*, it is possible to scale up

_{ij}**K**

_{G}with the selfstress in the elements by suitable changes of rest lengths until

**K**

_{T}is not positive definite anymore. Similar situations in which the (positive) material stiffness is in competition with negative geometric stiffness occur also in typical continuum mechanics problems, such as the buckling of a beam subjected to axial compression, the buckling of thin-walled columns with residual stresses, or the zero stiffness of prestressed rings obtained by bending a initially straight rod with circular cross section with respect to eversion deformations.

^{45}

The case of a **K**_{G} with some negative eigenvalues applies to the two tensegrity units we propose: both are stable in a certain configuration at low to moderate selfstress levels but become unstable when the selfstress level exceeds a certain critical value. This leads to the emergence of two additional stable configurations, indicating a switch from a single- to double-well energy landscape. The eight-node unit features also another bistable regime when its configuration is prestress unstable [ $ K G \u2009 v \u2009 \xb7 \u2009 v < 0$ for some $ v \u2009 \u2208 null ( K M )$]. In the following, selfstress levels are quantified in a dimensionless way in terms of elements' prestrain, here defined as^{40} $ \epsilon 0 : = ( \lambda 0 \u2212 \lambda \xaf ) / \lambda 0$, with $ \lambda \xaf$ and *λ*_{0}, respectively, the rest length of a characteristic element and its length in a reference equilibrium configuration.

Figures 1(a) and 1(b) depict stable configurations for the six-node (a) and the eight-node unit (b), corresponding to different prestrain values. The six-node unit has no internal mechanisms, while the eight-node unit has three internal mechanisms and is prestress-stable when its geometric parameters range in a certain set. Each unit can exhibit configurations with symmetry point group $ D 2 h$ [left configurations in Figs. 1(a) and 1(b)] and with symmetry point group *D*_{2} [right configurations in Figs. 1(a) and 1(b)]. In the former case, symmetry operations correspond to an inversion center, three mirror planes, and three twofold cyclic-symmetry axes, while in the latter case, they correspond to just three twofold cyclic-symmetry axes.

By performing numerical simulations based on the full-order model described above, we found that the structures depicted on the left in Figs. 1(a) and 1(b) become bistable when the prestrain of elastic cables exceeds a certain critical value. By choosing a $ D 2 h$ symmetric configuration as a reference configuration, the bars were considered rigid, while the cables were modeled as elastic springs with same spring constant *k*, rest length $ \lambda \xaf$, and prestrain *ε*_{0}, except for the two vertical cables in the eight-node unit shown in Fig. 1(b), left, which were modeled as inextensible. To enforce rigidity and inextensibility constraints, the corresponding members were assigned a large spring constant relative to *k*. With these choices, the equilibrium condition (2) is satisfied in the reference configuration. The smallest nonzero eigenvalue *ξ* of **K**_{T} is then computed as a function of prestrain in that configuration. The results are shown in Figs. 1(c) and 1(d) and reveal that the smallest nonzero eigenvalue becomes negative when prestrain values become large enough. We observed that the associated eigenvector corresponds to a twisting deformation mode with *D*_{2} symmetry [Figs. 1(e) and 1(f)]. Afterward, we run a number of simulations in which the $ D 2 h$ reference configurations shown in Figs. 1(a) and 1(b) are perturbed by random nodal displacements of small magnitude with no prescribed symmetry, and the energy (1) is minimized by using a standard numerical procedure. For the same values of prestrain determined by the analysis of **K**_{T} in the reference configuration, when the prestrain *ε*_{0} is small, the structures return to the unperturbed $ D 2 h$ symmetric configuration, while for large prestrains, they find either one of two other stable equilibrium configurations, away from the unperturbed one, both possessing *D*_{2} symmetry and mirror images of each other. The stability of each of the *D*_{2} symmetric equilibrium configuration is verified by the positive definiteness of **K**_{T}. No other equilibrium configurations were found in the vicinity of the reference configuration, thus demonstrating the prestrain-induced monostable to bistable transition of the units. The admissibility of axial forces, i.e., cables being in tension, is checked *a posteriori* in all calculations.

We describe next the two reduced-order models of these units. Consider the six-node unit in the reference configuration defined by the parameters *a*, *b*, and *c* shown in Fig. 2(a), consisting of rigid bars and linear springs (the cables). The springs are assumed to have the same spring constant *k*, and their rest length is $ \lambda \xaf N \u2264 \lambda 0 = a 2 + b 2 + c 2$. The system's *D*_{2} symmetric configurations can be identified by the relative rotation angle $ 2 \theta $ about the vertical axis between the bars *AB* and *CD*. In the projected view on the *x*–*y* plane [Fig. 2(b)], the bar *EF* remains orthogonal to the line bisecting the angle $ 2 \theta $. Springs can be grouped in two categories: those whose length increase with *θ*, and those whose length decrease with *θ*, depicted respectively in orange and green in Fig. 2(a).

*θ*= 0, and the corresponding configuration is stable only when the prestrain $ \epsilon 0$ is less than a critical value $ \epsilon crit$, which is determined solely by the geometry and can be expressed as

*α*and $ \epsilon crit$. Figure 2(d) displays the change in elastic energy from the value

*U*

_{0}in the reference configuration, normalized by $ k \lambda 0 2$, for different prestrains, and highlights the shift from a monostable to bistable regime as prestrain increases, along with the ranges in which axial forces in cables are positive.

We consider now the eight-node unit in the reference configuration defined by the parameters *a*, *b*, *c*, and *d* < *c* shown in Fig. 3(a), obtained from the previous structure by doubling the central bar and adding two vertical cables. We assume that bars are rigid and that cables have same spring constant *k* and rest length $ \lambda \xaf < \lambda 0 = a 2 + b 2 + ( c \u2212 d ) 2$, except for *EG* and *FH*, which are inextensible. We require the structure to retain *D*_{2} symmetry during a motion. Therefore, if *AB* rotates with respect to *CD* by an angle $ 2 \theta 1$ about the *z* axis, and *EF* rotates with respect to *GH* by an angle $ 2 \theta 2$ about the same axis, then in the projected view onto the Cartesian *x*–*y* plane the bisecting lines of these angles remains orthogonal to each other [Fig. 3(b)]. As in the previous model, there are two kind of springs, depicted in Fig. 3(a) in orange, with length $ \lambda 1$, and green, with length $ \lambda 2$. The angles *θ*_{1} and *θ*_{2} are the two Lagrangian parameters for the system.

*D*

_{2}symmetry have the form

*ABEF*with respect to the tetrahedron

*CDGH*about the vertical symmetry axis.

*c*>

*d*, we have

Figures 3(c) and 3(d) show the contour plots of the elastic energy $ U ( \theta 1 , \theta 2 )$ for a prestress stable eight-node unit (A), for prestrain values below (c) and above (d) the critical value. The plots display also the region with positive axial forces in cables, the mechanism and eigenvector directions at $ ( \theta 1 , \theta 2 ) = ( 0 , 0 )$, and the stable configurations obtained numerically using the full-order model. It can be observed that, for the monostable structure, the mechanism direction is close to one of the eigenvector directions. Figure 3(e) displays the contour plot of the energy for a prestress-unstable eight-node unit (B). Stable configurations for this unit are located in the direction of the mechanism. Figure 3(f) shows the contour plot of the critical prestrain (14) for $ c / a = 12 / 8$ with the marked positions defining units eight-node A and eight-node B.

One-dimensional assemblies of each unit were analyzed numerically using the full-order model. Adjacent units in an assembly share a subset of elements, as shown in Figs. 4(a) and 4(b). Bars are rigid and cables linearly elastic, except for the cables parallel to the longitudinal axis, which are inextensible. In order to have equal units in geometry and selfstress, we considered *a* = *b*, and same prestrain $ \epsilon 0$ and spring constant *k* for all elastic cables, except for the four cables at each end of the assembly, which have spring constant $ k / 2$.

Starting from prestressed assembly configurations with $ D 2 h$ symmetric units, simulations are conducted in two steps. First, a twisting load is applied to the assembly, and the equilibrium configuration reached under such load is determined. Second, the twisting load is removed, and the final equilibrium configuration is determined. If $ \epsilon 0 < \epsilon crit$, the final equilibrium configuration coincides with the starting configuration. If $ \epsilon 0 > \epsilon crit$, different twisted equilibrium configurations are obtained depending on the applied and removed twisting load, demonstrating the multistable response of such assemblies. Figures 4(c) and 4(d) show simulation results for particular assemblies of both types of units. All units are twisted in the same way in Figs. 4(c), middle, and 4(d), middle, in a periodic overall deformation, while in Figs. 4(c), bottom, and 4(d), bottom, the assembly is twisted partly in one way and partly in the other way.

Physical models of the two units are presented next. The units were built of wooden bars and additively manufactured nodes and cables obtained by fused deposition modeling. Cables were fabricated using polyurethane, except for the two “inextensible cables” in the eight-node unit, which were realized in polylactic acid. To connect the bars and cables that converge into the nodes of the units, custom designed universal joints realized in polylactic acid were used. Five tensegrity units with different prestrain were tested to confirm the monostable to bistable transition, two specimens of the six-node unit with monostable and bistable configurations, and three specimen of the eight-node unit: case A, prestress-stable, which can be either monostable or bistable, and case B, which is prestress-unstable and bistable. Dimensions and prestrain of the units are listed in Table I. Small random perturbations of the equilibrium configurations were manually applied to verify the expected monostable/bistable behavior. Photos of the units are shown in Figs. 5(a)–5(f). Additionally, Figs. 5(g)–5(j) depict models of three-unit and nine-unit tensegrity chains based on the six-node unit.

. | Six-node unit . | Eight-node case A (prestress-stable) . | Eight-node case B (prestress-unstable) . | ||
---|---|---|---|---|---|

. | Monostable . | Bistable . | Monostable . | Bistable . | Bistable . |

a (mm) | 135 | 135 | 135 | 135 | 135 |

b (mm) | 62 | 62 | 130 | 130 | 80 |

c (mm) | 170 | 170 | 235 | 235 | 235 |

d (mm) | ⋯ | ⋯ | 85 | 85 | 85 |

$ \lambda 0$ (mm) | 226 | 226 | 240 | 240 | 217 |

$ \epsilon crit$ (%) | 7.0 | 7.0 | 21.9 | 21.9 | ⋯ |

$ \lambda \xaf$ (mm) | 212 | 190 | 190 | 160 | 200 |

$ \epsilon 0$ (%) | 6.2 | 15.9 | 20.8 | 33.3 | 7.8 |

. | Six-node unit . | Eight-node case A (prestress-stable) . | Eight-node case B (prestress-unstable) . | ||
---|---|---|---|---|---|

. | Monostable . | Bistable . | Monostable . | Bistable . | Bistable . |

a (mm) | 135 | 135 | 135 | 135 | 135 |

b (mm) | 62 | 62 | 130 | 130 | 80 |

c (mm) | 170 | 170 | 235 | 235 | 235 |

d (mm) | ⋯ | ⋯ | 85 | 85 | 85 |

$ \lambda 0$ (mm) | 226 | 226 | 240 | 240 | 217 |

$ \epsilon crit$ (%) | 7.0 | 7.0 | 21.9 | 21.9 | ⋯ |

$ \lambda \xaf$ (mm) | 212 | 190 | 190 | 160 | 200 |

$ \epsilon 0$ (%) | 6.2 | 15.9 | 20.8 | 33.3 | 7.8 |

This study can be extended to similar units with a polygonal base (see the supplementary material) and has potential applications to the design and benchmarking of multistable metamaterials. Future work can regard the additive manufacturing of tensegrity-like structures with cables replaced by bars, using compliant hinges instead of pin-connections,^{30} and employing responsive materials, such as photo-thermal-responsive liquid-crystal elastomers,^{34,46} that have actuation strains up to about 0.2.^{47} The effect of external loads, nodal constraints, and more elaborate constitutive models could also be explored for better predictions of monostable-bistable switching.

## SUPPLEMENTARY MATERIAL

See the supplementary material for detailed calculations on the six-node and eight-node units.

The work of A.M. was supported by the Italian Minister of University and Research through the project “3D printing,” a bridge to the future (Grant No. 2017L7X3CS_004) within the PRIN 2017 Program and by University of Rome Tor Vergata through the project “OPTYMA” (No. CUP E83C22002290005) within the “Ricerca Scientifica di Ateneo 2021” Program. F.A.S. acknowledges the funding by Fundação para a Ciência e a Tecnologia (FCT) in the framework of Project No. UIDB/04625/2020.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Andrea Micheletti:** Conceptualization (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **Filipe Amarante dos Santos:** Investigation (equal); Methodology (equal); Resources (lead); Validation (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). **Simon Guest:** Conceptualization (equal); Formal analysis (supporting); Methodology (equal); Project administration (equal); Supervision (lead); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Tensegrity: Structural Systems for the Future*

*Tensegrity Systems*

*Frameworks, Tensegrities, and Symmetry*

*Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems*

*Rigidity Theory and Applications*