Unfolding the Sulcus

Even as we probe physics on ever-smaller scales, materials that can be held and manipulated with our hands often still resist our understanding. Elastic materials, in particular, still confound because of the nonlinear relationship between strain and the displacement of the material needed to maintain the rotational invariance of the elastic energy. The effects of these nonlinearities are often more pronounced at free surfaces, where strain can be alleviated by a large rotation of the surface. When a slab of an elastic material such as rubber is compressed, it develops a sulcus—a sharp furrow in its surface that plunges into the material. First reported for photographic gelatin films over one hundred years ago, they are not just a laboratory curiosity. Sulci create large strains that can lead to material failure. They are also a common motif in the morphogenesis of many organs, most famously in the characteristic folds on the surface of the human brain or, say, the arm of an infant [see Fig. 1(a) and (b)]. Though a mechanism for the formation of a sulcus was proposed almost fifty years ago [1], a complete understanding has remained elusive [2– 6]. Now, in a paper appearing in Physical Review Letters, Evan Hohlfeld from Harvard University and Lawrence Berkeley National Laboratory and L. Mahadevan from Harvard University have proposed that the formation of a sulcus is controlled by a new type of instability dominated by nonlinearities in the elastic energy [7]. Their case is bolstered both by detailed numerics and by experiments. Moreover, they suggest that similar nonlinear instabilities may be lurking behind the formation of many other singular structures found in materials. In the calculation of Biot, a free surface of a compressed elastic material becomes unstable at a critical strain of 45.6% [1]. Indeed, experiments show that a compressed slab forms sharp furrows above some critical strain. Rather than develop as an instability, however, the sulci in experiments nucleate and grow laterally as fully formed furrows. Moreover, this often occurs at a lower strain of 35% [2–4], noticeably smaller FIG. 1: Localized folds, called sulci, induced on soft materials due to compressive stresses are ubiquitous in nature: (a) the arm of an infant, (b) a primate brain. (c) Schematic illustration of a bifurcation diagram showing the scaled height h of a sulcus plotted against the applied strain. A sulcus nucleates at a critical strain ec due to a spontaneous breaking of scale symmetry. (Credit: (a),(b) E. Hohlfield and L. Mahadevan [7])