Wrinkling pattern evolution of cylindrical biological tissues with differential growth

Three-dimensional surface wrinkling of soft cylindrical tissues induced by differential growth is explored. Differential volumetric growth can cause their morphological stability, leading to the formation of hexagonal and labyrinth wrinkles. During postbuckling, multiple bifurcations and morphological transitions may occur as a consequence of continuous growth in the surface layer. The physical mechanisms underpinning the morphological evolution are examined from the viewpoint of energy. Surface curvature is found to play a regulatory role in the pattern evolution. This study may not only help understand the morphogenesis of soft biological tissues, but also inspire novel routes for creating desired surface patterns of soft materials.

Figure 1

(a) Cylindrical core-shell model. The surface pattern is characterized by two mode numbers $m_{\mathrm{crit}}$ and $n_{\mathrm{crit}}$, and the circumferential and axial wavelengths are given by ${\Lambda }_1=2\pi R/m_{\mathrm{crit}}$ and ${\Lambda }_2=2\pi /n_{\mathrm{crit}}$, respectively. (b) Schematic of the volumetric tissue growth configurations.

Figure 2

(a) Determination of the critical growth factor at the occurrence of surface wrinkling, where ${\mu }_{\mathrm{s}}/{\mu }_{\mathrm{c}}=20$ and (b) comparison between the theoretical solutions and FEM results under different modulus ratios. Here, the ratio of shell thickness $H$ to outer radius $R_b$ is $0.05$.

Figure 3

Variation in the critical growth factor ${\overline{g}}_{\mathrm{crit}}$ with the modulus ratio ${\mu }_{\mathrm{s}}/{\mu }_{\mathrm{c}}$ for the axial wrinkling mode $m_{\mathrm{crit}}=0$ (solid line) and the circumferential wrinkling mode $n_{\mathrm{crit}}=0$ (dashed line). At the intersection (dot), the two modes have the same ${\overline{g}}_{\mathrm{crit}}$ value.

Figure 4

FEM simulations of surface pattern evolution of a core-shell cylinder where we take $\mathbf{G}=\left(1+\overline{g}\right)\mathbf{I},B=20,H/B=0.05,{\mu }_{\mathrm{s}}/{\mu }_{\mathrm{c}}=20$, and the aspect ratio of the cylinder $L/B=10$. (a)–(d) give the variations in displacement $U_R$ along the radial direction on the shell surface with increasing growth factor $\overline{g}$, and (e)–(h) show the energy density distributions on the shell surface corresponding to (a)–(d), respectively.

Figure 5

Surface pattern transition from the square mode to the hexagonal mode. (a) The square mode is introduced into the system as a geometric imperfection to initiate postbuckling analysis. (b) and (c) show that the crest along the cylinder axis becomes flattened first and then is split into two separate crests with an increasing growth factor. (d) Occurrence of a hexagonal pattern. (e) and (f) represent the square and hexagonal modes on a cylindrical surface, respectively.

Figure 6

Variation in the normalized total elastic strain energy $\overline{U}$ with respect to the growth factor $\overline{g}$. The dashed line represents the theoretical solution of the total elastic strain energy without buckling. Before buckling (Region I), FEM simulations (solid line) agree well with the theoretical solution. In Region II, the hexagonal wrinkling pattern lowers the values of $\overline{U}$. During the secondary buckling (Region III), the labyrinth pattern further lowers the potential energy.

Figure 7

Surface pattern evolution of polydimethy siloxane with the oxidized surface under ethanol vapor from the hexagonal mode to the segmented labyrinth mode [32].