Indentation of Ellipsoidal and Cylindrical Elastic Shells

Thin shells are found in nature at scales ranging from viruses to hens’ eggs; the stiffness of such shells is essential for their function. We present the results of numerical simulations and theoretical analyses for the indentation of ellipsoidal and cylindrical elastic shells, considering both pressurized and unpressurized shells. We provide a theoretical foundation for the experimental findings of Lazarus et al. [following paper, Phys. Rev. Lett. 109, 144301 (2012)] and for previous work inferring the turgor pressure of bacteria from measurements of their indentation stiffness; we also identify a new regime at large indentation. We show that the indentation stiffness of convex shells is dominated by either the mean or Gaussian curvature of the shell depending on the pressurization and indentation depth. Our results reveal how geometry rules the rigidity of shells.

Figure 1

The indentation of ellipsoidal shells. Main figure: Cut away view of an ellipsoid centered on the origin and subjected to a point force at $(0,0,c)$. Inset: Numerically determined force-displacement curve for a spherical shell with $a=b=c=1\mathrm{m}$, $E=70\mathrm{GPa}$, $\nu =0.3$, $t=2\mathrm{mm}$ subject to an internal pressure $p={10}^5\mathrm{Pa}$. The two linear regimes considered here are shown as dashed lines.

Figure 2

The stiffness of unpressurized ellipsoidal shells $k_1$ compared to the stiffness of a sphere with the same mean curvature ${\kappa }_M,k_1^{(s)}$, plotted as a function of the asphericity parameter $ϵ=(R_y-R_x)/(R_x+R_y)$. The asymptotic expression (13) (dashed curve) agrees well with numerical results for $ϵ\ll 1$. However, the fit (14) (solid curve) gives a good account even for $ϵ=O(1)$. Numerical data were obtained in simulations with $t=5\mathrm{mm}$ and $b=c=1\mathrm{m}$.

Figure 3

The small-deformation stiffness of pressurized ellipsoidal shells $k_1$ as a function of the dimensionless pressure $\tau$, given in (16). Results are shown for ellipsoids with $t=0.005$, $b=c=1$ and $a=0.75$, $ϵ\approx 0.28$ (•); $a=2$, $ϵ\approx -0.6$ (▪); and $a=5$, $ϵ\approx -0.92$ (▾) as well as for a horizontal cylinder with radius 1, $ϵ=-1$ (▴), which corresponds to $a=\infty$. Here the length scale ${\ell }_b$ is determined using the mean curvature ${\kappa }_M$ and $\tau =\tau ({\kappa }_M,{\sigma }_M)$, as in (16). The analytic prediction for the corresponding sphere, (7), is shown by the solid curve. Dashed horizontal lines show the corresponding value of $k_1$ for $p=0$ (see Fig. 2).

Figure 4

Force-displacement curves at large displacements. The results of ABAQUS simulations are shown for shells with $a=5\mathrm{m}$, ${\kappa }_G/{\kappa }_M^2=0.148$ (dashed curves); $a=2.5\mathrm{m}$, ${\kappa }_G/{\kappa }_M^2=0.476$ (dash-dotted curves); and $a=1\mathrm{m}$, ${\kappa }_G/{\kappa }_M^2=1$ (solid curves). The simulated shells have thickness $t=2\mathrm{mm}$, $b=c=1\mathrm{m}$, and a variety of internal pressures: $p={10}^5\mathrm{Pa}$ (red), $p={10}^6\mathrm{Pa}$ (green), and $p={10}^7\mathrm{Pa}$ (blue). Here ${\kappa }_M^{-1}$ is used as the radius of curvature in ${\ell }_p$. Numerical (solid black curve) and asymptotic (dotted line) results for a spherical membrane shell [5] are also shown.