Period fissioning and other instabilities of stressed elastic membranes

We study the shapes of elastic membranes under the simultaneous exertion of tensile and compressive forces when the translational symmetry along the tension direction is broken. We predict a multitude of morphological phases in various regimes of a two-dimensional parameter space $\left(ϵ,\nu \right)$ that defines the relevant mechanical and geometrical conditions. These parameters are, respectively, the ratio between compression and tension, and the wavelength contrast along the tension direction. The predicted patterns emerge through new transition and instability mechanisms and include several types of irregular and smooth cascades composed of wrinkles and sharp folds. In particular, the hierarchical morphology predicted under high tension and large wavelength contrast $\left(ϵ⪡1,\nu ⪢1\right)$, explains recent experimental observations on ultrathin membranes floating on liquid.

Figure 1

(Color online) Schematics of membrane geometry.

Figure 2

(Color online) A conjectured phase diagram for stressed membranes $\left(\tilde{\Delta }⪡1,\overline{L}⪢1\right)$. The axis correspond to the parameters $\left(ϵ,\nu \right)$. The irregular $\left(I_n\right)$, period-fissioning $\left(F_n\right)$, and mixed $\left(M_{n,k}\right)$ morphologies are described below.

Figure 3

(Color online) (a) Energies $U_n\left(ϵ,\nu \right)$ of $n$-strip ($ϵ⪡1$ and $n=1–3$) obtained by minimization over all $n$-strip shapes $\left({\vec{Q}}_n\right)$. Dimensionless convention [Eq. (6)] is used. [(b) and (c)] Computed patterns of 1-strip $\left(\nu =1\right)$ and 2-strip [$\nu =3$ with $\vec{Q}=\left(1,2,4\right)$].