Compression-controlled dynamic buckling in thin soft sheets

We investigate experimentally the dynamic phase transition from compressed to buckled phases for thin sheets of rubber. We find that the rubber strips enter a highly compressed, metastable state when compression speed is high. During the compressed phase, higher modes grow, followed by mode coarsening. Mode growth is accompanied by an expansion of length while the system is still being compressed. We measure the forces and length of the sheet to confirm this, and we develop a mechanism for how modes grow and coarsen during dynamical buckling. The influence of crucial control parameters in the experiments, such as the material cross section and compression speed, on the buckling dynamics, are explained theoretically.

Figure 1

Demonstration of the key idea and experimental realisation of the buckling. (a) Demonstrates that while an eraser gets compresssed, a rubber band buckles easily. (b) A schematic representation of the experimental setup. (c) Image of a thread held between two clamps with the left clamp moving. The red dots show the detected points used to calculate the contour length $\ell \left(t\right)$ of the strip. Here $h(x,t)$ and $L\left(t\right)$ denote the height field of the strip and the distance between the clamps. The red points along the strip's length are representations of the central maxima of the strip's image along its vertical section. Connecting these points yields the contour along the length of the strip.

Figure 2

Spatiotemporal evolution of the mode structure during buckling. Panels (a) and (b) show data for a strip of acrylonitrile rubber (dimension: initial length ${\ell }_o=$ 86 mm, width $w_o=$ 5 mm, and thickness ${\zeta }_o=1\mathrm{mm}$) compressed at speed $v=1$ m/s. Panel (a) presents snapshots that capture the process of buckling. The top panel of (b) shows the spatiotemporal evolution of the height map $h(x,t)$ and the bottom panel shows the time evolution of the fractional amplitude $h_k/{\sum }_n{h}_n$ of modes obtained from the Fourier decomposition of $h(x,t)$. Here $h$ is taken to be positive in the direction of gravity. Panels (c) and (d) depict the corresponding results for the natural rubber (dimension: ${\ell }_o=80\mathrm{mm}$, $w_o=5\mathrm{mm}$, and ${\zeta }_o=0.2\mathrm{mm}$).

Figure 3

Structural response during buckling and evidence of negative compressibility. (a) The temporal variation of the measured force $\mathcal{F}$ (top panel), the contour length $\ell \left(t\right)$ (upper curve in the bottom panel) and distance between clamps $L$ (lower curve in the bottom panel). Here region I shows the subtle features near the instance of buckling, and region II shows the long-time evolution. (b) Variation of the total absolute curvature of the strip $\kappa$ as a function of time, for various values of the compression speed (marked against each curve). In these experiments, ${\ell }_o=86\mathrm{mm}$ and $L_o=78\mathrm{mm}$.

Figure 4

Observed scaling laws. Panels (a) and (b) show the variation of the jump length $\left(\delta \ell \right)$ and the maximum force sustained by the strip $\left({\mathcal{F}}_{\max }\right)$, respectively with the width of the sample $\left(w_o\right)$. Panels (c) and (d) show the variation of these parameters with the compression speed $v$. Here, ${\ell }_o\sim 8\mathrm{cm}$, thickness ${\zeta }_o=1\mathrm{mm}$ for (a)–(c) and ${\ell }_o\sim 15\mathrm{cm},{\zeta }_o=1\mathrm{mm}$ for (d). In panels (a) and (b), $v\approx 0.8$ m/s and in panels (c) and (d) $w_o=10\mathrm{mm}$. The corresponding numerical simulation results obtained by solving Eq. (3) are presented as red open circles with $I=w_o{\zeta }_o^3/12$ and $A=w_o{\zeta }_o$. The simulation parameters are same as experiments, with $\rho =1.1\mathrm{kg}/{\mathrm{m}}^3$ and $G=3$ MPa. The dash-dotted lines in the figures represent the various scaling laws which are obtained from Eqs. (6) and (7). These scaling laws are (i) $\delta \ell$ is independent of $w_o$ (ii) $\delta \ell \propto \left|v\right|{\ell }_o^2$ and (iii) $|{\mathcal{F}}_{\max }|\propto w_o\left|v\right|$.

Figure 5

Numerical results obtained using dynamical buckling equation. (a) Spatiotemporal variation of height, $h$ obtained numerically by solving Eq. (2). (b) The temporal evolution of mode fraction $h_k/{\sum }_n{h}_n$. The top and the bottom panels of (c) depict the temporal variations of $\mathcal{F}\left(t\right)$, and the contour length $\ell \left(t\right)$ of the strip, respectively. Here the parameters are the initial length, ${\ell }_0=86\mathrm{mm}$, thickness ${\zeta }_o=1\mathrm{mm}$, $w_o=10\mathrm{mm}$, $\rho =1.1\mathrm{kg}/{\mathrm{m}}^3$, and $G=3$ MPa.

Figure 6

Results from the three-spring model. (a) The boundary separating the buckled and compressed phases in the $(k_L/k_T,L/L_o)$ parameter space. The inset shows the three-spring model. (b) The contours of the potential $U(2\mathrm{\Delta }/L_o,L/L_w)$. Here, the white line represents the static buckling while colored lines show numerical solution for the dynamic buckling at various speeds. The speeds in units of m/s are marked beside each line. The parameters are $m=1\mathrm{g}$, and $k_L,k_T$ values are obtained from Eq. (5) using elasticity $G=3$ MPa, $w_o=5\mathrm{mm}$, ${\zeta }_o=1\mathrm{mm}$ and ${\ell }_o=L_o=100\mathrm{mm}$. These values are close to the experimental parameters as shown in Fig. 3. The white dotted arrows in panel (b) indicate the paths by which the springs decompress.