Self-Similar Structures near Boundaries in Strained Systems

We study the buckling of thin elastic plates caused by residual strains concentrated near a free edge. This is a model for plant leaves and torn plastic sheet morphologies. We derive new governing equations explaining self-similar patterns reported earlier in experiments. We reveal the cascade mechanism, determine the bounds for its wavelengths, and predict a similarity factor of 3 in agreement with experiments. This is confirmed by numerical solutions with up to five generations of wrinkles.

Figure 1

An elastic plate buckles into a self-similar pattern due to residual strains near the edge (inset). Numerical solution for thickness $h=2.9\times {10}^{-5}$, width 1, metric perturbation $g(y)=1/(1+y/\ell )$, $\ell =0.008$, yielding wavelength $2\pi /k=0.54$.

Figure 2

Fourier components ${\zeta }^{[q]}(y)$ ($q=1,3,9,27$) of the deflection for a numerical self-similar solution with four generations (inset) and their collapse using transform (5). ${\zeta }^{[1]}(y)$ need not collapse away from the stretched edge (i.e., large $y$). Same parameters as in Fig. 1, except for $h=2.9\times {10}^{-4}$.

Figure 3

Numerical check of scaling law $\lambda \sim h^{2/5}{\ell }^{3/5}$: smallest wavelength $\lambda$ in the cascade versus plate thickness $h$ (left panel) and metric typical length $\ell$ (right panel) for the family of metric perturbations $g_{\ell }(y)=1/(1+y/\ell )$. Thick lines are best power-law fits, yielding exponents 0.38 and 0.62, respectively.

Figure 4

Total elastic energy of solutions with one to four generations of wrinkles for the one-parameter family of metrics $g_{\ell }(y)=1/(1+y/\ell )$. Unit of energy is the bending modulus [i.e., the prefactor of the bending energy in Eq. (2)]. Bifurcations occur while all branches remain close to embeddings: they are selected by minimization of bending energy.