Pressure and folding effects on the buckling of a freestanding compressed thin film

The combined effects of compressive stress, applied pressure, and edge folding of a freestanding thin film have been theoretically investigated on the buckle morphologies of the structure. In the framework of the Föppl–von Kármán theory of thin plates, the different buckle profiles have been analytically determined, and two buckling regimes have been identified for the film: one regime where the transition from upward to downward buckling is continuous, and one that is discontinuous (snap-through). The critical pressures characterizing the different regimes have then been determined, and an hysteresis cycle has been identified through the study of buckling versus pressure. The case in which the thin film is deposited on a substrate has also been discussed.

Figure 1

(Sketch not to scale) Thin film of thickness $h$ deposited on a semi-infinite substrate submitted to an in-plane stress ${\sigma }_{xx}^0={\sigma }_{yy}^0=-{\sigma }_0$. The free-standing part of the film of length $2b$ is submitted to an external applied pressure $p$. (a) Planar configuration. (b) Upward buckling of the free-standing film when the pressure is smaller than a critical value ${\tilde{p}}_c$. (c) Downward buckling when $\tilde{p}>{\tilde{p}}_c$.

Figure 2

Evolution of pressures ${\tilde{p}}_1$ and ${\tilde{p}}_2$ with respect to $\alpha$ for different values ${\alpha }_0=1$, 2, 3, and 4 and for $\theta =\pi /20$ and $\tilde{h}=0.1$. The pressures corresponding to buckle configurations in tension (compression) are shown by dotted lines (solid lines).

Figure 3

Buckle evolution under pressure for $\theta =\pi /20$ and $\tilde{h}=0.1$. Buckle profile $\tilde{w}(\tilde{x}-\tilde{u}\left(\tilde{x}\right))$ vs $\tilde{x}$ for (a), (c), (e) increasing pressure $\tilde{p}$ (from top profile to bottom) and (b), (d), (f) evolution of $\tilde{\delta }$ vs $\tilde{p}$ for (a), (b) ${\alpha }_0=1$; (c), (d) ${\alpha }_0=2$; and (e), (f) ${\alpha }_0=4$. Curves corresponding to pressures ${\tilde{p}}_1$ and ${\tilde{p}}_2$ (with ${\tilde{p}}_1<{\tilde{p}}_2$) are, respectively, blue (dark gray) and orange (light gray). Solid lines are used for buckled configurations in compression, and dotted lines for those in tension. Dashed lines correspond to buckled configurations at pressures in the range $[{\tilde{p}}_l,{\tilde{p}}_r]$.

Figure 4

Energy path between three buckle configurations A, B, and C in the range pressure $\tilde{p}\in ]{\tilde{p}}_l,{\tilde{p}}_r[$ for which the buckle can transit from the upward configuration A to the downward configuration B. The considered parameters are ${\alpha }_0=4$, $\theta =\pi /20$, and $\tilde{h}=0.1$.

Figure 5

Evolution of the maximum deflection $\tilde{\delta }$ with respect to the pressure $|\tilde{p}|$ for ${\alpha }_0=4$, $\theta =\pi /20$, and $\tilde{h}=0.1$. The filled arrows correspond to the path from upward to downward buckling, and the empty arrows correspond to the path from downward to upward buckling.

Figure 6

(a) $({\alpha }_0,|\tilde{p}\left|\right)$ diagram describing the thin-film behavior for $\tilde{h}=0.1$. The straight black solid line is given by $|{\tilde{p}}_0|$ variation with respect to ${\alpha }_0$ [Eq. (24)] and delimits the region C (buckles in a tensile stress state) from the region B (buckles in a compressive stress state), the buckle touching the plane $z=0$ by continuously increasing the pressure. The dash-dotted black curve, given by $|{\tilde{p}}_c|$ variation with respect to ${\alpha }_0$ [Eq. (26)], delimits the region B (continuous transition) from the regions D and A (abrupt upward to downward transition). The dotted black curve, given by $|{\tilde{p}}_c^0|$ variation with respect to ${\alpha }_0$ [Eq. (25)], delimits the region D, where the buckle touches the substrate after a snap-through for a pressure $|\tilde{p}|>|{\tilde{p}}_c|$, from the region A, where the buckle touches the substrate immediately after a snap-through at $|\tilde{p}|=|{\tilde{p}}_c|$. Dashed blue (dark gray) lines in the regions B, C, and D are the iso-$\theta$ curves indicating the pressure $\tilde{p}$ required for the buckles to touch the plane $z=0$ ($\tilde{\delta }=0$) vs ${\alpha }_0$. Dashed orange (light gray) lines give the critical pressure $|{\tilde{p}}_c|$ [Eq. (16)] at the abrupt transition in the regions A and D. (b) Evolution of $\tilde{\delta }$ with respect to $|\tilde{p}|$ for different values of the applied stress ${\alpha }_0=3.40$, 3.45, and 3.50. The hysteresis is displayed with dotted lines.