Stamping and Wrinkling of Elastic Plates

We study the peculiar wrinkling pattern of an elastic plate stamped into a spherical mold. We show that the wavelength of the wrinkles decreases with their amplitude, but reaches a minimum when the amplitude is of the order of the thickness of the plate. The force required for compressing the wrinkled plate presents a maximum independent of the thickness. A model is derived and verified experimentally for a simple one-dimensional case. This model is extended to the initial situation through an effective Young modulus representing the mechanical behavior of the wrinkled state. The theoretical predictions are shown to be in good agreement with the experiments. This approach provides a complement to the “tension field theory” developed for wrinkles with unconstrained amplitude.

Figure 1

Top: Experimental setup. A circular plate of radius $R$ is compressed between two rigid transparent hemispherical dies of radius $\rho$. The gap $\delta$ between the spheres is imposed. Bottom: Experimental observations of wrinkles formed when a plate is compressed between two hemispherical dies. $\delta$ is decreased from (b) to (d), while $R$ is maintained constant. (d) and (e) are characterized by the same $\delta$, but different values for $R$ ($\rho =58\mathrm{mm}$, $h=58\mu \mathrm{m}$, and $R=12.5/25\mathrm{mm}$).

Figure 2

Experimental setup: a plate of length $L$, thickness $h$, and unit width is subjected to symmetrical displacements at the ends $u_x(\pm L/2)=\pm \Delta /2$ and a maximal deflection $\delta$. We consider that $n$ similar wrinkles are formed (${\Delta }_{\mathrm{one}}=\Delta /n$) of wavelength $\lambda =L/n$ and amplitude $A$, with $A=\delta -h$.

Figure 3

Maximal number of wrinkles for polypropylene films of thicknesses ranging from $15\mu \mathrm{m}$ to $250\mu \mathrm{m}$. Open symbols, 1D experiments: strips of length 250 mm, width 25 mm with $\Delta /L=0.2\%$ and 0.3%. Filled symbols, 2D experiments: discs of radius $R$ ranging from 10 mm to 70 mm, radius of spheres $\rho =58$, 600 mm and $n_{\max }$ the number of wrinkles at the edge of the discs. For a comparison with 1D experiments, we take $L=2\pi R$ and $\Delta /L=R^2/8{\rho }^2$ [24]. The solid line corresponds to the best fit of the data: $n_{\max }=0.20(L/h)\sqrt{\Delta /L}$.

Figure 4

Evolution of the vertical force (per unit width) $f_z$ required to confine the plate as a function of the dimensionless amplitude $A/h$, defined as $f_z(A/h=0.94)=f_{z,\max }$. The solid line corresponds to Eq. (5b), and the dashed line to the inextensible approximation. Experiments with strips of length 250 mm, width 25 mm with $\Delta /L=0.07\%$ (circles) and 0.2% (squares). Inset: Raw force data for $h=90\mu \mathrm{m}$, solid line corresponds to $\Delta /L=0$, squares to $\Delta /L=0.2\%$.

Figure 5

Evolution of the vertical force $F_{z1}$ required to confine the plate as a function of the dimensionless amplitude $A/h$, defined as $F_{z1}(A/h=1.14)=F_{z1,\max }$. Experiments with discs of radius 25 mm, thicknesses $h=90\mu \mathrm{m}$ (squares), $150\mu \mathrm{m}$ (circles), and $250\mu \mathrm{m}$ (triangles), with $\rho =58\mathrm{mm}$. The solid line corresponds to the theoretical prediction. Inset: Raw force data, the solid thin line corresponds to stamping without plate, circles to $h=150\mu \mathrm{m}$.