# Design and Testing of Bistable Lattices with Tensegrity Architecture and Nanoscale Features Fabricated by Multiphoton Lithography

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Obtaining Bistable Frameworks from Monostable Tensegrity Structures

#### 2.2. Double Tensegrity Prism to Design a Bistable Unit and Corresponding Assemblies

_{0}; and different sizes a and b of base triangles. The doubled structure possesses two independent bistable mechanisms, which can be amalgamated together by composition to negate the relative rotation between end bases. When the top base is displaced vertically while keeping its rotation blocked, the bistable mechanisms of the two prisms are activated simultaneously, resulting in the rotation of the middle triangle only, as illustrated in Figure 3a. Figure 3b depicts the system in the corresponding stable equilibrium configurations before and after such process, which are addressed here as primary (in blue) and secondary (in grey). This system is the individual bistable unit cell which serves as a building block of the larger modular assemblies shown in Figure 3e,g. The size of the middle triangle in the bistable unit is selected such that there is enough clearance, circumventing collisions between adjacent units during activation of the bistable mechanism. While the single unit has two independent bistable mechanisms, assembling three unit cells side-by-side shown in Figure 3e leads to a single bistable mechanism. Correspondingly, a multi-layer assembly such as the one in Figure 3g has one bistable mechanism per layer.

_{0}= −7 degrees, corresponding to an initial twist angle φ

_{0}= 23 degrees. The Young’s modulus is taken as equal to 1.2 Gpa (the methodology to obtain it is provided in the next subsection) and the stiffness constant of the angular springs has been assigned to be k

_{s}= 1.24 μN μm.

_{0}. The dimensionless force parameter F * is the resultant compressive force F divided by the axial spring constant of the shortest bar, k

_{a}, and by the bars’ diameter. The dimensionless displacement δ * is the actual vertical displacement divided by the unit’s height. The null value of δ * corresponds to the unstable equilibrium configuration.

_{0}= 0, the response is qualitatively the same as the one of the parent tensegrity system. For negative values of θ

_{0}, the plot embosoms the form of typical bistable systems, with the snapping load increasing with the magnitude of the relative twist. Positive values of θ

_{0}are impractical from an experimental viewpoint and are not considered in this work, as they correspond to an upward bistable mechanism which can be activated from the primary stable configuration by an upward displacement imposed on the top nodes. Figure 3d depicts the force-displacement plot for the single unit cell (

**a**) realized with angular springs and for different values of the angular stiffness constant, keeping θ

_{0}= −7 degrees. As the angular stiffness constant increases, the slope of the curve increases. In addition, if the force is never negative, the bistable behavior diminishes. Figure 3f elucidates the force-displacement relationship for a three-unit assembly (e): loading the primary configuration cause the structure to snap on the other equilibrium path; then the structure reaches the secondary equilibrium configuration upon unloading. Figure 3h presents the force-displacement relationship for a twenty-unit-two-layer assembly (g), where one of the two layers has slightly different spring constants. For this structure, two snapping events commence, one for each layer of the structure, and there can be three stable configurations.

#### 2.3. Fabrication by Multiphoton Lithography and Mechanical Testing

^{2}and thickness 20 μm.

_{0}= −7 degrees. After calibrating the fabricating conditions, the following geometric parameters were decided to be suitable for an efficient fabrications of the unit cell used in the arrays: h = 9 μm, a = 6 μm and b = 4.5 μm, and the initial relative twist θ

_{0}= −9 degrees. All structures were fabricated with constant cross sections, since tapering of the cross section at the nodes resulted in inept photopolymerization of the structure. Figure 5 shows two SEM images of a fabricated three-unit array.

## 3. Results

#### 3.1. Three-Point Bending of Double-Clamped Beams

_{Β}= 131.99 ± 0.17 MPa. From the loading-unloading force-displacement plot in Figure 4e, a slight viscoelastic behavior of the material can be observed, characterized by hysteresis, energy dissipation and loading-rate dependency. In addition, from the plot in Figure 4f, two cracking events can be distinguished, highlighted by the two vertical drops in the plot. These are also confirmed by visual inspection of the images (Figure 4d) and the movie of the testing (see Video S1: Three-point bending). The cracking commences at the end sections of the beam, where the bending moment is maximum. This result is consistent with the assumption of clamped boundary conditions. After cracking, the beam can still sustain some loading, and the corresponding slope in the subsequent branch of the plot decreases to about one-fifth of the initial slope, a value which is consistent with simply supported boundary conditions.

#### 3.2. Individual Unit Compression Testing

#### 3.3. Three-Unit Array Conpression Testing

#### 3.4. Two-Layer Twenty-Unit Array Compression Testing

#### 3.5. Cracking and Fracture during the Testings

## 4. Discussion

_{0}and initial angular speed ω

_{0}of the top base, so as to activate the bistable mechanism of the first unit. The simulations shown in Figure 11 correspond to assuming a/h = 0.5, θ

_{0}= −3 deg, v

_{0}(E/ρ)

^{−0.5}= 0.3609, v

_{0}/ω

_{0}= a

^{2}/h, k

_{s}/(a

^{2}k

_{a}) = 0.0041, with ρ denoting the mass density of the material. One observes the propagation of a compression wave localized on a single prism (enclosed by the red dashed rectangle in Figure 11), with negligible motion of the rest of the column, under the examined loading condition. The reader is referred to the Video S7 of Supplementary Materials for an animation of the motion of the structure illustrated in Figure 11. The response of the benchmark bistable structure under examination highlights that the use of highly nonlinear tensegrity systems with nanoscale features may allow the creation of revolutionary types of acoustic lenses, to be used as a noninvasive scalpel to accurately target defects in engineering and biological materials. Micro- and nano-scale tensegrity lattices with bistable responses (acting as phononic crystals) can indeed be employed to generate compact-support waves within tensegrity acoustic lenses [18,49], which may travel and coalesce at a focal point in an adjacent medium (i.e., a material defect or a tumor mass in a host medium). A comprehensive study on this exciting, novel application of micro- and nano-scale tensegrites with a bistable response is addressed to future work. Furthermore, the structures analyzed in this paper utilize marked tunability (due to geometry and prestress) and scalability (size-independent properties) to go beyond conventional systems. The scalability property derives from the geometric nature of the bistable response, and the material nature of the viscous behaviors observed in the experiments. The mechanical modeling presented in this study can be applied down to the scale at which Van der Waals forces can be neglected, (several angstroms, see, e.g., [62], where tensegrity structures with strut length of 65 nm have been studied) A bistable viscous response can also observed in the macro-scale tensegrity structure shown in Video S8 of Supplementary Materials, which shows 20 cm timber struts connected with flexible polyvinyl chloride (PVC) tubes.

## 5. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the analyzed behaviors and fabricated structures. (

**a**) Selfstress and mechanism in the prestress-stable two-element system (top). Deformed configuration under a vertical load and corresponding response (center and bottom). (

**b**) Corresponding bistable system and bistable snapping response under the same vertical load. (

**c**–

**f**) Examples of prestress-stable systems with one seflstress state and one mechanism: (

**c**) A two-dimensional system displaced along its mechanism; (

**d**) triangular tensegrity prism; (

**e**) expanded octahedron or tensegrity icosahedron; (

**f**) an irregular x-tower. Fabricated geometries of structures: (

**g**) individual unit cells, (

**h**) arrays of three unit cells in one layer, (

**i**) arrays of two layers with ten unit cells at each layer.

**Figure 2.**Design process of a tensegrity structure. (

**a**) A bar framework in the shape of a regular right prism. (

**b**) A corresponding right-handed tensegrity prism with cables (single line) and bars (double line). (

**c**) Prestress-stable equilibrium configuration of the tensegrity prism. (

**d**) When the internal mechanism is activated, the top triangle rotates about and translates along the vertical three-fold symmetry axis.

**Figure 3.**Mechanical modeling of the analyzed structures. (

**a**) Geometry of the individual unit cell based on a double tensegrity prism. (

**b**) Top view and side view of the primary (in blue) and secondary (in grey) stable configurations when the bistable mechanisms of the two prisms are activated simultaneously. (

**c**) Static response of an individual unit cell with no angular springs under a vertical load for different values of the relative twist. (

**d**) Static response of an individual unit cell with angular springs for different values of their spring constant. (

**e**) Three-unit array. (

**f**) Static response of a three-unit array. (

**g**) Twenty-unit two-layer array. (

**h**) Static response of a twenty-unit two-layer array.

**Figure 4.**Experimental setup and testing to obtain the mechanical properties. (

**a**) Schematic of the multiphoton lithography experimental setup [46]. (

**b**) Beam structure employed for three-point bending measurements. (

**c**) Beam structure at the beginning of the testing and dimensions. (

**d**) Beam structure after cracking at the end sections. (

**e**) A characteristic force-displacement curve obtained by three-point bending. (

**f**) The force-displacement curve of a beam reaching the failure strength (cf. Video S1: Three-point bending).

**Figure 6.**Mechanical testing on a single unit cell. (

**a**) The force-displacement plot for an individual unit. (

**b**) Snapshots of the structure at different times during testing (cf. Video S2: Individual unit). The white scale bar for each SEM figure is 5 μm.

**Figure 7.**Mechanical testing on a three-unit array. (

**a**) Imposed displacement vs. time and (

**b**) force vs. displacement plots. (

**c**) Snapshots of the sample during testing (cf. Video S3: Three-unit array). The white scale bar for each SEM figure is 8 μm.

**Figure 8.**Mechanical testing on a twenty-unit two-layer array. (

**a**) Imposed displacement–time and (

**b**) force-displacement plots. (

**c**) Snapshots of the sample during testing (cf. Video S4: Twenty-unit two-layer array). Some fracturing beams are enclosed in the green rectangle The black scale bar for each SEM figure is 10 μm.

**Figure 9.**Mechanical responses revealing microcrack and fracture. (

**a**) Imposed displacement-time and force-displacement plots for a three-unit array (cf. Video S5: Three-unit cracking). (

**b**) Imposed displacement-time and force-displacement plots for a twenty-unit two-layer array (cf. Video S6: Twenty-unit two-layer cracking). Arrows indicate fracture events during the deformation.

**Figure 10.**Helium ion microscopy (HIM) images of a three-unit array after testing. (

**a**) HIM imaging of a structure that fracture did not commence. (

**b**–

**d**) HIM images of a fractured sample.

**Figure 11.**Snapshots extracted from the video of the motion of a column of ten bistable prisms impacted with initial vertical and angular speeds at the top base (see also Video S7 of Supplementary Materials).

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**MDPI and ACS Style**

Vangelatos, Z.; Micheletti, A.; Grigoropoulos, C.P.; Fraternali, F.
Design and Testing of Bistable Lattices with Tensegrity Architecture and Nanoscale Features Fabricated by Multiphoton Lithography. *Nanomaterials* **2020**, *10*, 652.
https://doi.org/10.3390/nano10040652

**AMA Style**

Vangelatos Z, Micheletti A, Grigoropoulos CP, Fraternali F.
Design and Testing of Bistable Lattices with Tensegrity Architecture and Nanoscale Features Fabricated by Multiphoton Lithography. *Nanomaterials*. 2020; 10(4):652.
https://doi.org/10.3390/nano10040652

**Chicago/Turabian Style**

Vangelatos, Zacharias, Andrea Micheletti, Costas P. Grigoropoulos, and Fernando Fraternali.
2020. "Design and Testing of Bistable Lattices with Tensegrity Architecture and Nanoscale Features Fabricated by Multiphoton Lithography" *Nanomaterials* 10, no. 4: 652.
https://doi.org/10.3390/nano10040652