Buckling-Fracture Transition and the Geometrical Charge of a Crack

We present a unifying approach that describes both surface bending and fracture in the same geometrical framework. An immediate outcome of this view is a prediction for a new mechanical transition: the buckling-fracture transition. Using responsive gel strips that are subjected to nonuniform osmotic stress, we show the existence of the transition: Thin plates do not fracture. Instead, they release energy via buckling, even at strains that can be orders of magnitude larger than the Griffith fracture criterion. The analysis of the system reveals the dependence of the transition on system’s parameters and agrees well with experimental results. Finally, we suggest a new description of a mode I crack as a line distribution of Gaussian curvature. It is thus exchangeable with extrinsic generation of curvature via buckling. The work opens the way for the study of mechanical problems within a single nonlinear framework. It suggests that fracture driven by internal stresses can be completely avoided by a proper geometrical design.

Figure 1

Schematics of the experimental system. (a) A polyacrylamide gel strip is immersed into a stratified fluid. The top layer of the fluid is 100% water. The bottom layer is 25% PEG4000 in water. In the transition region, the osmotic pressure gradient prescribes a non-Euclidean metric on the sheet. (b) An illustration of the reference strip widths that are induced by the PEG concentration profile. The parameter $\mathrm{\Delta }$ used in the text is defined as $\mathrm{\Delta }\equiv [(W-\tilde{W})/\tilde{W}]$. Regions of positive and negative reference Gaussian curvature are marked on the figure. (c) Scanning laser sheet illuminates the gel. A camera captures the signal from fluorescent powder within the sheet, allowing the construction of its 3D configuration (see Methods section in the Supplemental information file [32] for details).

Figure 2

Observation of a buckling-fracture transition. (a) Photos (top) and surface height measurements (bottom) of gel strips that are immersed into the PEG solution. Thin strips (left) buckle and do not support crack propagation. Thicker strips (right) hardly buckle. Instead, they are fractured by a straight or a wavy crack, or by several propagating cracks (extreme right). Strip thickness (left to right): 100, 750, 1500, 1750, and $2100\mu \mathrm{m}$. (b) Measurements of buckling-fracture transitions. Each vertical column of symbols indicates the number of propagating cracks along a single experiment. Black circles, no propagating cracks; red circles, a single propagating crack; red stars, two propagating cracks. The solid line is the theoretical transition line ${ϵ}_{\mathrm{FB}}\sim [(W\mathrm{\Gamma })/(E{h}^2)]$. Inset: schematics of the predicted phase diagram showing the buckling-dominated, the stretching-dominated, and the fracture-dominated regions. The curves separating these regions are ${ϵ}_{\mathrm{BS}}\sim (h^2/W^2)$, ${ϵ}_{\mathrm{FS}}\sim \sqrt{\mathrm{\Gamma }/\mathrm{EW}}$, and ${ϵ}_{\mathrm{FB}}\sim [(W\mathrm{\Gamma })/(E{h}^2)]$. The axes are as in the main figure.

Figure 3

Switching between fracturing and buckling. The crack opening (a) and average bending content (b) versus time plotted for propagating cracks in strips of thickness 1 mm (triangles), 0.85 mm (open squares), and 0.75 mm (circles). The 1 mm thick strip contained two propagating cracks and low bending content. Each crack arrest (indicated by arrows) was accompanied by an increase in the surface bending. The same process was observed in the 0.85 mm thick strip, which initially contained one propagating crack. No cracks propagated in the thinnest strip, which underwent the largest surface bending.

Figure 4

Curved kirigami and the geometric charge of a crack. Flat configurations with cuts (left), bent configurations with edges identified (middle), and maps of the Gaussian curvature (right). A triangular cut (a) leads to a disk with a missing angle $\theta$ for any loop that surrounds the tip (dotted lines) but not for loops that do not surround the tip (red circle). (b) The geometry of a cone. (c) The Gaussian curvature of such geometry is zero, except for a singularity of $+\theta$ at the origin. In a typical “crack opening” profile (d), the missing angle at the tip is $+\pi$ and it decreases to zero (for a finite opening crack) away from the tip. The bent configuration (e) as well as the Gaussian curvature map (f) show a positive point singularity followed by a line singularity (a cusp) of negative curvature integrated to a total of $-\pi$. Such a representation of a crack includes all the information for the nonlinear elastic problem.