A cascade of length scales in elastic rings under confinement.

Elastic objects under confinement are common in mechanics and biology. Examples include mitochondria and chromosomes, for which conformation and function are strongly determined by confining forces. When elastic objects grow in a confined space, minimization of elastic energy creates a complex spatial configuration and force network. To simulate a two-dimensional ring growing within a rigid circular boundary with a fixed radius, we take a long strip of elastic material mylar of fixed length, join the ends to form a closed loop, and then shrink the confining ring boundary Fig. 1 . The loop contacts and overlaps itself in many locations Figs. 1, 2 c , and 3 . The loop trajectory is discretized and its curvature computed. Where local maxima in curvature occur, the thickness is measured by counting the number of overlapping layers. Figure 2 a shows a scatter plot of this data for the shape in Fig. 2 c , and we find that the magnitude of a curvature maximum decreases with the number of layers at that point. To quantify this relationship, we consider a region in which two portions of the elastic loop are in contact Fig. 2 b . Because the contact forces balance and are dominated by the curvature maxima, we find also that the elastic energy stored near each maximum is approximately equal. The elastic energy is x B s 2 /2ds. Here x ,x+L is the portion of the loop where the curvature is large, B is bending rigidity, and is curvature. The curvature near the maximum is of order max,N, while the length of the region of the large curvature is approximately 1 / max,N. The integral thus scales as B max,N 2 /2 1 / max,N =B max,N /2. The elastic energy of N layers is NB max,N /2 and, in equilibrium, this term is approximately constant for regions in contact, which implies max,N 1 /N. FIG. 1. Color