Figure 1.
Illustration of a helix (top), a hemihelix with one perversion marked by an arrow (middle) and a hemihelix with multiple perversions (bottom).
The scale bar is 5 cm, and is the same for each image. These different shapes were all produced in the same way as shown in figure 2 with the same value of pre-strain but with decreasing values of the height-to-width ratio of the bi-strip's cross-section.
,
,
).
Figure 2.
Sequence of operations leading to the spontaneous creation of hemihelices and helices.
Starting with two long elastomer strips of different lengths, the shorter one is stretched to be the same length as the other. While the stretching force, P, is maintained, the two strips are joined side-by-side. Then, as the force is slowly released, the bi-strip twists and bends to create either a helix or a hemihelix.
Figure 3.
The number of perversions observed as a function of both the prestrain and the cross-section aspect ratio, .
The data indicates that there is a transition between the formation of helixes at larger aspect ratios and hemihelices at smaller aspect ratios. The precise phase boundary cannot be determined with any precision experimentally and so is shown shaded but there is evidently only a weak dependence on the value of the pre-strain. In some cases, bistrips made the same way produce either one or the other of the two perversion numbers indicated.
Figure 4.
Snapshots recorded from the finite element simulations, illustrating the formation of (A) a helix, (B) a hemihelix with single perversion and (C) a hemihelix with 12 perversions.
The colors represent the local values of the computed von Mises stress. The prestrain was the same in all three cases.
,
, (A)
, (B)
, (C)
. The images are taken when the end to end distances are
. Gravity was included in the simulations and acts from left to right in these images.
Figure 5.
Coordinate system used in the Kirchhoff analysis together with the dimensions and
of the cross-section.
Figure 6.
The critical loads for different buckling modes.
For a small ratio, the critical end-to-end distances
for different modes are very close to one another and difficult to distinguish. Increasing the aspect ratio by increasing the thickness decreases the critical buckling load as well as separating the individual modes. To illustrate this behavior results for four modes and the helix are shown.
Figure 7.
Growth rate as a function of the mode number
for three different strips characterized by
,
,
and
.
The growth rate is determined when the applied force decreases to .
Figure 8.
Contour plots showing the value of for which the growth rate is maximum as function to
and
.
The growth rates are calculated for . Black dotted lines show the boundaries between modes with different number of perversions
, while the red line corresponds to
and separates hemihelices (on its left) from helices (on its right). For clarity not all the higher modes are shown.