Mechanics of invagination and folding: Hybridized instabilities when one soft tissue grows on another

We address the folding induced by differential growth in soft layered solids via an elementary model that consists of a soft growing neo-Hookean elastic layer adhered to a deep elastic substrate. As the layer-to-substrate modulus ratio is varied from above unity toward zero, we find a first transition from supercritical smooth folding followed by cusping of the valleys to direct subcritical cusped folding, then another to supercritical cusped folding. Beyond threshold, the high-amplitude fold spacing converges to about four layer thicknesses for many modulus ratios. In three dimensions, the instability gives rise to a wide variety of morphologies, including almost degenerate zigzag and triple-junction patterns that can coexist when the layer and substrate are of comparable softness. Our study unifies these results providing understanding for the complex and diverse fold morphologies found in biology, including the zigzag precursors to intestinal villi, and disordered zigzags and triple junctions in mammalian cortex.

Figure 1

Elastic linear instability of a growing slab on a substrate, showing a transition from finite to zero wavelength instability at modulus ratio ${\eta }_b\approx 0.525$. (a) Growth $g$ required for instability at wavelength $l$ relative to a deformed thickness $a$ for a range of moduli ratios. (b) and (c) Wavelength and threshold growth of the first unstable mode as a function of $\eta$. Dashed lines indicate the asymptotic results for stiff layers.

Figure 2

Folding of a uniaxially growing layer in two dimensions. (a) Morphologies as a function of modulus ratio $\eta$ and growth ratio $g=g_x$. The diagram includes the linear instability curve (blue line, dotted for $\eta <{\eta }_s\approx 1.14$, where the sulcification threshold is met before the linear instability) and numerically determined thresholds for cusp formation (black line). (b) Energy-minimizing wavelengths as a function of $g$ for a range of moduli ratios $\eta$. Hollow and solid circles correspond to smooth and cusped folds, respectively. The blue curve is the wavelength of linear instability (the dotted end is above the sulcification threshold $g_s\approx 1.55$). (c) Simulated minimum energy states near threshold (left column) and far from threshold (middle and right column) for $\eta$ = 0.5, 1, and 4. Color indicates isotropic stress: dark red $\le -5\mu$, dark blue $\ge 5\mu$ ($\mu ={\mu }_1$ in the layer and $\mu ={\mu }_2$ in the substrate). (d) Cortical folds in a ferret brain (braincatalogue.org). (e) Cross sections of previllous folds in intestinal epithelium of a chick embryo on days 13 and 14 [40] (credits: A. E. Shyer).

Figure 3

Hysteresis loops for three different modulus ratios, demonstrating the three different folding regimes. (a) When $\eta =0.5$, sulcification sets in supercritically at $g=g_s\approx 1.55$, producing a nonlinear supercritical instability, reminiscent of pure sulcification. (b) When $\eta =1$, sulcification occurs subcritically at $g=g_s$, immediately producing a deep cusped fold, while in unfolding the fold also disappears subcritically at a lower value of $g$. Folding is nonlinear and subcritical. (c) When $\eta =4$, folding is linear and supercritical, reminiscent of pure wrinkling.

Figure 4

Three-dimensional folding of an equibiaxially growing layer. (a) Energy-minimizing periodic patterns near threshold (left) and far from the threshold (right, zigzag, and triple-junction patterns are shown for each $\eta$). (b) Diagram of energy-minimizing ordered patterns as a function of $\eta$ and $g$. The symbols (Z's for zigzags, Y's for triple junctions, o's for smooth dents, etc.) indicate the minimum energy pattern at each simulated point. The blue curve is the threshold for linear instability or sulcification. The dashed lines are contours for growth strains $\epsilon /{\epsilon }_c$. (c) Deformation energies of zigzags and triple junctions relative to the energies of flat reference states. (d) and (e) Energetically optimal half-wavelengths [$l_x, l_y$, and $l_h$ are indicated in (a)] of zigzags and triple junctions, respectively. Thick black curves indicate the wavelength of linear instability. (f) Zigzags in epithelium of a developing chick gut [3]. (g) A simulated soft layer growing in a large simulation domain ($\eta =1$ and $g=g_x=g_y=2$) folds spontaneously into a disordered mixture of Y- and T-shaped triple-junction folds and S-shaped zigzag folds, reminiscent of the folded cerebral cortex (h) [41].