Surface Sulci in Squeezed Soft Solids

The squeezing of soft solids, the constrained growth of biological tissues, and the swelling of soft elastic solids such as gels can generate large compressive stresses at their surfaces. This causes the otherwise smooth surface of such a solid to become unstable when its stress exceeds a critical value. Previous analyses of the surface instability have assumed two-dimensional plane-strain conditions, but in experiments isotropic stresses often lead to complex three-dimensional sulcification patterns. Here we show how such diverse morphologies arise by numerically modeling the lateral compression of a rigidly clamped elastic layer. For incompressible solids, close to the instability threshold, sulci appear as $\mathsf{I}$-shaped lines aligned orthogonally with their neighbors; at higher compressions they are $\mathsf{Y}$-shaped and prefer a hexagonal arrangement. In contrast, highly compressible solids when squeezed show only one sulcified phase characterized by a hexagonal sulcus network.

Figure 1

Simulated unfolding of a compressed and sulcified solid layer starts from a state with a transversely isotropic stretch $\lambda =0.54$ (a). The layer is then decompressed quasistatically and intermediate states are shown for $\lambda =0.61$ (b), $\lambda =0.67$ (c) corresponding to the Biot threshold, and $\lambda =0.72$ (d) just before unsulcification. Coloring indicates the largest compressive stress at the surface. Experimental sulcal patterns are shown in a swelling gel in (e) and (f), courtesy of J. Yoon and R. C. Hayward.

Figure 2

A square lattice (a) of $\mathsf{I}$-shaped sulci with alternating orientation minimizes energy near the sulcification threshold, whereas at higher compression hexagonal arrangement (b) of $\mathsf{Y}$-shaped sulci is favorable. Surface profiles along the lines indicated in (a) and (b) are shown below the patterns. In (c) the energies of these patterns are compared to that of a freely relaxing layer of Fig. 1 as a function of stretch $\lambda ={\lambda }_x={\lambda }_y$. All energies are normalized by the energy $U_{\mathrm{ref}}$ of an unsulcified reference state. Numbers below the points indicate the optimal spacing of the square (blue) or hexagonal (red) lattice.

Figure 3

Depth averaged energy distribution for (a) an $\mathsf{I}$ pattern and (b) a $\mathsf{Y}$ pattern. (c) Energy density as a function of distance from the base in material coordinates. Energy densities are normalized by the energy density $U_{\mathrm{ref}}$ of the unsulcified reference state.

Figure 4

Layers with various anisotropic strain ratios ${ϵ}_y/{ϵ}_x$ (rows) are shown at modest compression (${\lambda }_x\sqrt{{\lambda }_y}=0.54$, left column) and at high compression (${\lambda }_x\sqrt{{\lambda }_y}=0.4$, right column).

Figure 5

(a) A connected sulcus network in a simulated highly compressible solid ($K=2\mu$, isotropic $\lambda =0.54$). (b) A schematic diagram summarizing morphologies as a function of bulk modulus and isotropic compression. Sulcification threshold is sketched according to simulated points (crosses).