Anomalously Soft Non-Euclidean Springs

In this work we study the mechanical properties of a frustrated elastic ribbon spring—the non-Euclidean minimal spring. This spring belongs to the family of non-Euclidean plates: it has no spontaneous curvature, but its lateral intrinsic geometry is described by a non-Euclidean reference metric. The reference metric of the minimal spring is hyperbolic, and can be embedded as a minimal surface. We argue that the existence of a continuous set of such isometric minimal surfaces with different extensions leads to a complete degeneracy of the bulk elastic energy of the minimal spring under elongation. This degeneracy is removed only by boundary layer effects. As a result, the mechanical properties of the minimal spring are unusual: the spring is ultrasoft with a rigidity that depends on the thickness $t$ as $t^{7/2}$ and does not explicitly depend on the ribbon’s width. Moreover, we show that as the ribbon is widened, the rigidity may even decrease. These predictions are confirmed by a numerical study of a constrained spring. This work is the first to address the unusual mechanical properties of constrained non-Euclidean elastic objects.

Figure 1

A realization of the HCF. (a) An example of a few members of the HCF. These embeddings have pitch values of $p=1,0.9,0.7,0.4,0$ (left to right). The number of turns is conserved under this transformation. The red curve in each surface is the midline, which is a helix with a radius $R=\sqrt{{\overline{R}}^2+1-p^2}$. (b) A detailed example of an embedding with $p=0.4$. The radial and azimuthal coordinates ($v$ and $u$, respectively) are denoted along with the radius $R$ and the pitch $p$.

Figure 2

The stretching (top) and bending (bottom) energy densities as functions of $v$, the radial coordinate, for different values of $p$ ($p=1$, circles; $p=0.7$, triangles; $p=0$, squares—see the configurations in Fig. 1). The energy densities are plotted in dimensionless units, that is, normalized by $Y\overline{p}$. The bending energy density of an isometric embedding is indicated (black line at the bottom panel). Both the inner and outer boundary layers are apparent for the lower pitch values and disappear for $p=1$. As predicted, the ratio between the measured bending energy densities on the edges and those of the isometric embedding is 1 for $p=1$ and approaches $\frac{3}{4}$ for $p\to 0$ (inset of the bottom panel). These results were obtained for $t=0.02$, $W=1$, and $\overline{R}=1$.

Figure 3

The dimensionless energy ($E/Y{\overline{p}}^3$) of a minimal spring as a function of the pitch $p$ (here, $t=0.05$, $W=1$). The data are well fitted by a parabola (red curve) in the entire range of deformation. The helical figures illustrate embeddings of $\overline{\mathbf{a}}$ with the relevant different pitch values.

Figure 4

Main graph: the dimensionless rigidity ($\kappa /Y\overline{p}$) of a minimal spring as a function of thickness for five different widths (in the range $1.4<W<4.6$). The data collapse to a single $t^{7/2}$ power law (gray dashed line). The data sets override each other, which is consistent with the prediction that the rigidity depends very weakly on the width. Inset: the rigidity as a function of $v_{\max }$, the outer radius, for $t=0.1$ and $v_{\min }=0.5$. The data indicate that widening the ribbon outwards leads to a decrease of the rigidity. The variation in the rigidity is well fitted by the predicted relation $\kappa \propto {[1/1+(\overline{R}+v_{\max }{)}^2]}^{3/2}+\text{const}$ (black solid line). For large widths ($v_{\max }\gg \overline{R}$), the rigidity is virtually constant.