Figure 1

Difference between frozen topography and frozen defects. (a) Frozen topography: There is a fourfold (green square), a fivefold (red pentagon), and a sevenfold disclination (blue heptagon) that are free to move on a fixed surface composed of sixfold sites (gray hexagons), often at the expense of creating additional defects [12]. (b) Frozen defects: The defects are frozen and the shape of the shell can freely change. (c) Phase diagram of buckled membranes as function of $h/R$ and $R/a$.Reuse & Permissions

Figure 2

(a) Difference between ($P$, 0) and ($P$, $P$) shells illustrating the importance of the orientation of defects relative to the crystallographic axes. (b)–(d) Simulations of crystalline shells with different topological defects. The black dotted line shows the path between two disclinations and the number indicates the coordination number. The top (bottom) panel of each box denotes the initial (final) state. The final state is at volume fraction $\approx 0.8$. (b) Icosahedron shell. (c) Dodecahedron shell. (d) Cube shell. (e) Octahedron shell. (f) Tetrahedron shell.Reuse & Permissions

Figure 3

(a) Mechanical response of a crushed icosahedral shell during the loading or unloading (LU) cycle. The labels (1) to (8) show the shape of the shell at different points during the LU cycle. As the pressure is gradually increased, the shell becomes more undulating as reflected in (2). At the upper critical buckling pressure $p_b^u\approx 210$, the icosahedral-like shell undergoes an abrupt collapse into a cubelike shell in (4). As the pressure is decreased, the shell does not return to the inflated condition by the same path at the pressure-volume diagram, but slowly inflates via a different pathway as reflected by Eqs. (6, 7). The dotted arrows denote the direction of the LU cycle. (b) Plot of $Q_{\ell }$ during the LU cycle. We see significant deviation from icosahedral symmetry and the emergence of octahedral symmetry. The asymmetry of the curves about point 4 reflects the hysteretic behavior of the system. (c) Plot of $|W_{\ell }|$ during the LU cycle.Reuse & Permissions