Page 375 - Control Theory in Biomedical Engineering

Tunable stiffness using negative Poisson's ratio 341

applied on both ends. The rotation of the rigid squares about the connecting
hinges leads to the auxetic behavior. We explored this property by testing
different base materials and folding designs and considered the potential
applications. For this experiment, three different base materials were used,
namely, silicon rubber sheets, high-density foam sheets, and cardboard paper
sheets. The behavior of each of the materials is described in the sections that
follow.

7.1.1 Cardboard paper
We used a cardboard paper as the individual squares, as it possesses greater
structural integrity compared to normal paper, which helps in load-bearing
capacity. A 4 12 rectangular matrix comprised of 15mm squares was cut
out and its behavior is shown below. The Poisson’s ratio is 1 in both direc-
tions of stretching and the young’s modulus for a unit thickness of squares is
given to be:
8
E 1 ¼ E 2 ¼ K h ∗ 1 , (1)
l 2 ð 1 sinθÞ
where K ℎ is the stiffness constant of the hinges, θ is the angle between the
squares, and l is the length of the squares.
Fig. 20 shows the auxetic behavior of such a structure. To apply this
property for variable stiffness devices, we fold the rectangular matrix into
a tube-like structure as shown in Fig. 21. Pulling the tube from both ends
causes the squares to rotate and consequently the structure to expand.
In order to simplify the understanding of such structures, a similar cut-
based on rectangular matrix was folded into a triangular prism-like structure.
We observe that as the squares rotate and open, spaces are created between
each element where bending is possible along the cutting hinges (Fig. 22A).
However, when the shape of the tube is restored, the spaces are closed up

Fig. 21 Rotating squares design folded into tubular structures.