Wrapping an Adhesive Sphere with an Elastic Sheet

We study the adhesion of an elastic sheet on a rigid spherical substrate. Gauss’s Theorema Egregium shows that this operation necessarily generates metric distortions (i.e., stretching) as well as bending. As a result, a large variety of contact patterns ranging from simple disks to complex branched shapes are observed as a function of both geometrical and material properties. We describe these different morphologies as a function of two nondimensional parameters comparing, respectively, bending and stretching energies to adhesion. A complete configuration diagram is finally proposed.

Figure 1

(a) Experimental setup: an elastic plate of typical size $L$, Young’s modulus $E$, Poisson’s ratio $\nu$, and thickness $h$ is placed onto rigid sphere of radius $R$ coated with ethanol (surface tension $\gamma =22.4\mathrm{mN}·{\mathrm{m}}^{-1}$). Ethanol totally wets both the plate and the sphere. (b) Typical experimental observation: ($E=2.6\mathrm{GPa}$, $\nu =0.4$, $h=15\mu \mathrm{m}$, $R=60\mathrm{mm}$). In this example the region in contact with the sphere (contact zone) forms branched wavy patterns, while the unstuck parts of the sheet do not touch the sphere. A fluorescent dye allows us to visualize the liquid meniscus that delimits both regions.

Figure 2

(a) Experimental setup: an elastic annulus of radius $L$ and width $b$ is deposited on a sphere coated with ethanol. (b) Height of the blister for a typical experiment ($R=100\mathrm{mm}$, $L_{\mathrm{ec}}=18\mathrm{mm}$, $b=2\mathrm{mm}$, and $16.5\mathrm{mm}<L<44.5\mathrm{mm}$). (c) Side view of the blisters obtained as elastic annuli of increasing radii are successively deposited on an adhesive sphere.

Figure 3

(a) Size of the contact zone $a$, defined as the radius of the largest disk inscribed in the contact zone (inset), as a function of $R(\gamma /Eh{)}^{1/4}$ for four different thicknesses and radii of a sphere spanning 25 to 500 mm. Solid line: linear fit $a=1.9R(\gamma /Eh{)}^{1/4}$. (b) $R=25\mathrm{mm}$, $E=2.8\mathrm{GPa}$, $h=30\mu \mathrm{m}$. (c) $R=25\mathrm{mm}$, $E=2.6\mathrm{GPa}$, $h=15\mu \mathrm{m}$. (d) $R=197\mathrm{mm}$, $E=2.6\mathrm{GPa}$, $h=15\mu \mathrm{m}$. (e) $R=50\mathrm{mm}$, $E=2.6\mathrm{GPa}$, $h=15\mu \mathrm{m}$.

Figure 4

Configuration diagram of the observed patterns of contact as a function of $R/L_{\mathrm{ec}}$ and $L/\xi$, with $\xi =R(\gamma /Eh{)}^{1/4}$. Diamonds correspond to complete contact (case 2), stars to local disk shaped contact (case 3), squares to straight strips (case 4), triangles to oscillating strips (case 5), and circles to branched patterns (case 6). Grayscale corresponds to different materials, from dark to light gray: polyethylene ($E=170\mathrm{MPa}$, $\nu =0.4$, blue online), polypropylene ($E=2.6\mathrm{GPa}$, $\nu =0.4$, red online), steel ($E=212\mathrm{GPa}$, $\nu =0.3$, pink online), and natural rubber ($E=1.3\mathrm{MPa}$, $\nu =0.5$, green online).