Hierarchical Slice Patterns Inhibit Crack Propagation in Brittle Sheets

By introducing hierarchical patterns of load-parallel cuts into axially loaded brittle sheets, the resistance to propagation of mode-I cracks is very significantly enhanced. We demonstrate this effect by simulation of two-dimensional beam network models and experimentally by testing paper and polystyrene (PS) sheets that are sliced with a laser cutter to induce load-perpendicular hierarchical cut patterns. Samples endowed with nonhierarchical reference patterns of the same cut density and nonsliced sheets are considered for comparison. We demonstrate that hierarchical slicing can increase failure load, apparent fracture toughness, and work of fracture of notched paper and PS sheets by factors between 2 and 10.

Figure 1

Hierarchical patterning. (a) “Top-down construction” of a $n=$ 4-level-HBN structure: we start with a solid plate represented by a 2D FBN of size $L$; this is then divided by two load-parallel cuts into four lower-level modules—two groups of two modules loaded in series, connected by a remaining, system spanning cross-link (CL) connector of length $L$; then each of the four modules is again divided by two shorter cuts into four lower-level modules plus a module-spanning CL connector of length $L/2$, etc.; cuts are marked in blue, the remaining CL connectors are colored red. (b) Five-level HBN structure together with randomly permutated SHBN structure and reference RBN structure, illustrating the patterns of cuts and connectors for the different structures.

Figure 2

Simulation results. Simulation results for hierarchical (DHBN, SHBN) and nonhierarchical (RBN, FBN) structures of size $L=512$; (a) internal stress patterns at the peak stress, initial crack of length $a=100$ is marked in gray, the white line indicates the fracture path, the system is loaded in tension in $y$ direction; (b) representative stress-strain curves for different network types, crack length $a=100$, stresses and strains are given as multiples of the mean beam failure stress $⟨{\sigma }_B⟩$ (see Appendix pp1) and failure strain $⟨{\sigma }_B⟩/E$; (c) peak stress in units of $⟨{\sigma }_B⟩$ as a function of crack length for DHBN and RBN samples, all data are averaged over ten samples, the error bars indicate the corresponding standard deviation, red and blue lines represent fits according to Eqs. (1) and (2), respectively; (d) specific work of fracture for the same set of samples; the data points and error bars in (c),(d) each represent the average and standard deviation of 20 simulations.

Figure 3

Fracture experiments on paper sheets. The sample side length $L$ corresponds to 154 mm. (a) Fractured unnotched sample with SHBN-type hierarchical cut pattern, the blue line indicates the fracture surface; (b) stress-strain curves for DHBN- and RBN-type patterned and unpatterned sheets, all with a side notch of length $a=L/5=31\mathrm{mm}$; (c) peak stress as a function of notch length for DHBN- and RBN-patterned and unpatterned sheets, black and blue lines are fits according to Eqs. (1) and (2), respectively; inset: apparent fracture toughness relating peak stress to notch length via Eq. (1); (d) specific work of fracture as a function of notch length for the same set of samples as in (c); where error bars are given, the data points represent the average of two samples while the error bars indicate the larger and smaller value.

Figure 4

Fracture experiments on PS sheets. The sample side length $L$ corresponds to 160 mm. (a) Fractured samples with RBN cut pattern (left) and with SHBN-type cut pattern (right), initial notch length 4 mm, the fracture surfaces are traced in white; (b) stress-strain curves for sheets with SHBN patterns ($n=7$ hierarchical levels), RBN reference patterns and unpatterned sheets, all with a side notch of length $a=32\mathrm{mm}$; (c) peak stress as a function of notch length for SHBN- and RBN-patterned and unpatterned sheets, full lines are fits according to Eq. (1); (d) specific work of fracture as a function of notch length for the same set of samples as in (b); the data points and error bars in (b),(c) each represent the average and standard deviation of data from five samples.

Figure 5

Statistical size effects. Simulated stress-strain curves, $L=1024$, the label FB refers to a bundle of 1024 unconnected fibers of length $L$ that are loaded in parallel; top: Weibull exponent $\beta =4$, bottom: Weibull exponent $\beta =1.5$.

Figure 6

Generalization to 3D. (a) Three-dimensional generalization of a 2D HBN as depicted in Fig. 1, the depicted 3D sample consists of two $n=$ four-level hierarchical modules stacked on top of each other; (b) simulations of 3D networks of length $L=128$ containing cracks of different lengths, peak stress versus crack length, Weibull exponent $\beta =4$.