On the dynamics and control of mechanical properties of hierarchical rotating rigid unit auxetics

In this work, we investigate the deformation mechanism of auxetic hierarchical rotating square systems through a dynamics approach. We show how their deformation behaviour, hence their mechanical properties and final configuration for a given applied load, can be manipulated solely by altering the resistance to rotational motion of the hinges within the system. This provides enhanced tunability without necessarily changing the geometry of the system, a phenomenon which is not typically observed in other non-hierarchical unimode auxetic systems. This gives this hierarchical system increased versatility and tunability thus making it more amenable to be employed in practical application which may range from smart filtration to smart dressings.

1 Expressions for L x and L y The geometry of a two-level hierarchical rotating square system may be described by the aperture between the Level 0 squares (2θ 0 ), the aperture between the Level 1 square-like units (2θ 1 ) and the number of Level 0 repeat units (N 0 ). This means that the Level 1 building block is made up of N 0 × N 0 Level 0 repeat units which corresponds to 2N 0 × 2N 0 squares. These parameters may be used to dene the linear dimensions of the Level 1 square-like units, u 1 and v 1 , as follows: u 1 = 2N 0 l cos (θ 0 ) + 2(N 0 − 1)l sin (θ 0 ) and v 1 = 2N 0 l (cos (θ 0 ) + sin (θ 0 )), where l stands for the length of the sides of the Level 0 squares. These dimensions can in turn be used to dene the overall on-axis dimensions of the system: L x = 2 (u 1 cos (θ 1 ) + v 1 sin (θ 1 )) (1) L y = 2 (v 1 cos (θ 1 ) + u 1 sin (θ 1 )) (2) 1 2 Dependence of values of θ 0 and θ 1 on strain ε x FIG. S 1: Behaviour of the system plotted with respect to the strain (ε x ). Panels (a)-(c) correspond to the results where the resistance to the rotational motion of hinges was governed by the harmonic potential while panels (d)-(f) correspond to friction-based hinges. All of these results were generated for the analogical parameters as it was the case in the main text. 3 The eect of the variation of the magnitude of the force F in time on the behaviour of the system FIG. S 2: Deformation of the system (expressed in terms of the change in strain) subjected to the force F having its magnitude gradually changed from 2000N to 500N (which value was considered in the main text). Panels (a)-(d) correspond to the system with hinges having their rotational motion governed by the harmonic potential while panels (e)-(h) correspond to friction-based hinges.
The eect of a non-constant force on the deformation of the hierarchical system under study was also investigated. This was done by changing the magnitude of the force  Fig. S 2). This is due to the fact that in this case the initial force is very high and thus, the ratio of the torque generated by the applied force and the resistance torque is signicantly greater than it was the case for a constant force − → F = 500N . This also indicates that one may apply a force which changes in magnitude with time in order to have an additional way of controlling the behaviour of the system. Moreover, the results were generated for hinges governed by friction.
Based on Fig. S 5(a), one can note that irrespective of the initial values of angles θ 0 and θ 1 , the pathway of θ 1 with respect to t is closely resembling the one presented in the main text. The same applies to Fig. S 5(b) and Fig. S 5(c), where analogically to the results in the main text, which shows that upon increasing the value of f the rate of deformation of the angle θ 0 decreases and that the Poisson's ratio becomes increasingly more negative. All of this leads to the conclusion that the results discussed in the main text are not dependent on the initial value of the θ 0 and θ 1 parameters. 6 Dierent loading directions FIG. S 6: The system considered in this study for loading (a) in the x-direction and (b) in the ydirection.
In the main paper only loading in the x-direction (see Fig. S 6(a)) was discussed. This is because for this type of connectivity, upon subjecting the structure to the uniaxial tensile load, both levels can open simultaneously. On the other hand, for loading in the y-direction, the Level 0 squares tend to close when subjected to the same tensile load, while at the same time, Level 1 building blocks rotate in a way promoting the expansion of the structure in the loading direction. This eect can be explained based on the orientation of vectors − → r i and − → F . Based on Fig. S6(a), one can note that in the case of the vast majority of permissible values of θ 0 and θ 1 , the torque corresponding to Level 1 building block and the Level 0 squares to which the force is applied, has the same orientation. Based on the right hand rule, it is easy to note that in both cases a clockwise rotation will occur. This in turn leads to opening of both levels upon the application of the tensile load. In the case of connectivity II, Fig. S6(b), in the majority of cases, the right hand rule implies that the Level 0 and Level 1 building blocks will close and open respectively. 7 Results concerning loading in the y-direction taking friction-based hinge as an example FIG. S 7: Plots showing a variation in (a) θ 0 , (b) θ 1 and (c) Poisson's ratio ν xy as a function of time in the loading direction for systems corresponding to type II connectivity with f values ranging between 0.5 N·m and 2.5 N·m. Graphs shown here were generated for initial angles 2θ 0 and 2θ 1 set to be equal to 10 • and 20 • respectively.
In the case of loading in the y-direction, whenever angle − → r 0 , − → F assumes positive value, the resultant torque associated with Level 0 is directed in a way which promotes a decrease of the θ 0 angle (see Fig. S 7(a)). This is opposite to what can be observed for loading in the x-direction where for positive values of angles − → r 0 , − → F and − → r 1 , − → F both levels of the system are opening.
8 Animations ANIM1.gif : Animation showing the deformation of the system corresponding to K h = 0.035 N·m·deg −1 , where the hinging process is governed by the harmonic potential.
ANIM2.gif : Animation showing the deformation of the system corresponding to K h = 1.396 N·m·deg −1 , where the hinging process is governed by the harmonic potential.