Confined buckling in thin sheets and its correlation to ripplocations: A deformation mechanism in layered solids

Recently, we have established that, when loaded in compression, edge-on, atomic layers in layered solid can fail by buckling. The resulting structure is termed a ripplocation. When more than one layer buckles, they outline standing waves with boundaries that we labeled ripplocation boundaries that are nearly fully recoverable. In this paper, we examine buckling of layers at the centimeter level to explore continuum buckling theory and its applicability to atomic layers. Specifically, we examine buckling by confining and cyclically loading thin steel sheets, edge-on, determining that increasing confining pressure, sheet thickness, and/or decreasing the number of layers increases the buckling load. Concomitantly, the resulting wavelengths and amplitudes are reduced. A nonlinear, folding mechanics model, which accounts for frictional bending and foundation energies, is adapted and verified on our experimental results. We also demonstrate that Coulombic friction between the layers can account for the energy dissipated per cycle. The predicted values of buckling nucleation loads and number of modes from the model are in good agreement—at low levels of confinement—with continuum and atomistic scale results. The wavelength estimates from the model correlate surprisingly well with the continuum buckling results; however, likely due to the complex mechanics at the lower length scales and limiting theoretical assumptions in the derivation, the accuracy decreases at the atomistic scale and at higher confining pressures.

Figure 1

(a) Experimental setup and coordinate system used. (b) Schematic showing typical loading/unloading protocol. In all cases, a cylindrical indenter was indented to a displacement corresponding to the maximum strain Y, unloaded to strain X, and reloaded to Y to yield a fully recoverable loop designated by hatched area. The process was then repeated for increasing values of unloading strains, resulting in fully recoverable nested loops sharing a single unloading trajectory (see Fig. 4). Before dismantling the setup, one last cycle (colored yellow) was carried out. (c) Plot of constraining loads as a function of indentation depth. (d) Actual constraining load as a function of time during cycling. Horizontal dashed red lines connect (c) and (d).

Figure 2

(a) Initial configuration where layers are confined by force $P_{N,0}$, (b) Configuration at maximum indentation $h_{2.5}$, showing vertically loaded four layers that buckled into four modes. (c) Definitions of $w$, wavelength λ = 2L, and amplitude $A_{\mathrm{RB}}$.

Figure 3

Pictures of steel decks at maximum indentations for runs (a) 120-0.13-400, (b) 60-0.13-400, (c) 30-0.13-400, (d) 30-0.13-1200, (e) 30-0.13-6500, (f) 160-0.05-400, and (g) 70-0.05-6500. The vertically stacked panels differ in the thicknesses of sheets used ($\mathrm{top}=0.13$; $\mathrm{bottom}=0.05$). Horizontal red arrows denote lateral confining loads, while the vertical ones denote indentation direction.

Figure 4

Indentation stress-strain curves as a function of (a) number of sheets per deck at $P_{N,0}$ 400 N and $t=0.13\mathrm{mm}$, (b) $P_{N,0}$ for a deck of 30 sheets, 0.13 mm thick, (c) $t$ at $P_{N,0}$ of 400 N and 60 and 160 sheets for the thick and thin sheets, respectively, and (d) same as (c) but at a $P_{N,0}$ of 6500 N with 30 and 70 sheets for the thick and thin sheets, respectively. Note that, in all cases, after the nested loops were obtained, the indenter load was fully removed and then reapplied to obtain a second full cycle. The yield stress of the steel is shown in all plots as a horizontal dashed line.

Figure 5

(a) ${\sigma }_{\mathrm{Bu}}$ as a function of decreasing number of sheets 120, 60, and 30 at $P_{N,0}$ of 400 N. (b) Same as (a) but with $P_{N,0}$ of 400, 1200, and 6500 N for the same total number of sheets (30). (c) ${\sigma }_{\mathrm{Bu}}$ as a function of $t$ and $P_{N,0}$ of 400 N (two left columns) and 6500 N (two right columns), while keeping total thickness of decks constant. (d) ${\lambda }_{\mathrm{RB}}$ for same variables as (a). (e) ${\lambda }_{\mathrm{RB}}$ for same variables as (b). (f) ${\lambda }_{\mathrm{RB}}$ for same variables as (c). (g) $A_{\mathrm{RB}}$ for same variables as (a). (h) $A_{\mathrm{RB}}$ for same variables as (b). (i) $A_{\mathrm{RB}}$ for same variables as (c). Horizontal arrows denote direction of increasing effective and/or direct constraining pressure.

Figure 6

Comparison of measured and calculated values of (a) buckling load and (b) ${\lambda }_{\mathrm{BR}}$ at the continuum and atomistic scales. The results of the paper of Edmund et al. [3] with a soft foam foundation are also plotted in green. The corresponding values for graphite ripplocations derived from DFT calculations at the atomistic scale are plotted as blue spheres/circles [11]. Here, $n$ stands for nano. Note this is not a log-log plot.

Figure 7

(a) $W_d$ vs ${\varepsilon }_{\mathrm{RB}}$ for various variables shown on panels, and (b) $W_d$ vs ($P_N\times {\varepsilon }_{\mathrm{BR}}$)/$A_{\mathrm{ind}}$ for all nested loops shown in Fig. 4. Slopes of lines ≈ μ(1 + ν). The two lines with higher slopes (circles) were obtained on decks with thinner sheets.

Figure 8

Comparison of energies dissipated during cycles 1 and 2 and the difference $W_{pl}$, which is a measure of plastic deformation during cycle 1 as a function of (a) sheets per deck (spd), (b) $P_{N,0}$, (c) sheet thickness at $P_{N,0}=400\mathrm{N}$, and (d) same as (c) but for $P_{N,0}=6500\mathrm{N}$. Horizontal arrows denote direction of increasing effective and/or direct constraining pressure.

Figure 9

(a) Picture of postcycling thin (top) and thick (bottom) sheets indented for $P_{N,0}=6500\mathrm{N}$. (b) Schematic of buckling mode responsible for linear regime in Fig. 4. (c) Same as (a) but showing full deck for $t=0.13\mathrm{mm}$, and (d) same as (c) but for $t=0.05\mathrm{mm}$.