Forbidden Directions for the Fracture of Thin Anisotropic Sheets: An Analogy with the Wulff Plot

It is often postulated that quasistatic cracks propagate along the direction allowing fracture for the lowest load. Nevertheless, this statement is debated, in particular for anisotropic materials. We performed tearing experiments in anisotropic brittle thin sheets that validate this principle in the case of weak anisotropy. We also predict the existence of forbidden directions and facets in strongly anisotropic materials, through an analogy with the description of equilibrium shapes in crystals. However, we observe cracks that do not necessarily follow the easiest direction but can select a harder direction, which is only locally more advantageous than neighboring paths. These results challenge the traditional description of fracture propagation, and we suggest a modified, less restrictive criterion compatible with our experimental observations.

Figure 1

Trouser-test experiment. (a) Sample with a starting cut of orientation $\alpha$. The arrow indicates the sheet axis. (b) The fracture propagates for an applied force $F$ with a deflection angle $\theta -\alpha$. (c) Guided strip experiment (reinforcement in yellow).

Figure 2

Tearing of (a) weakly (material A) and (b) strongly anisotropic sheets (material B), where missing directions are evidenced. The direction of propagation $\theta$ (in colored thick lines) when tearing in direction $\alpha$ (thin and dashed lines) is represented in a rosace. Color codes indicate the amplitude of deflection $|\theta -\alpha |$.

Figure 3

Geometrical representation of the global criterion for fracture propagation [Eqs. (1, 2)]. The crack propagates at the first intersection of the parallel (black) lines (representing $G$) with the $G_c^{-1}$ curve (in red). (a) Isotropic case, the arrow indicates increasing force; (b) weakly anisotropic case; (c) singular anisotropy leading to facet; (d) anisotropy leading to “forbidden orientations” (grey sector).

Figure 4

Polar plots of the $G_c^{-1}$ curves for the experiments displayed in Fig. 2. Red dots correspond to the propagation of cracks at angle $\theta$ (diameter corresponds to the estimated error) and lines to the energy release-rate measured for each experiment ($\alpha$ is the pulling angle). Since the fracture energy follows the symmetry $G_c(\theta +\pi )=G_c(\theta )$, only half of the orientations were actually probed. In (a) weak anisotropy, material A, and (b) strong anisotropy, material B, energy release-rate $G(\theta )$ is normalized by typical value $G_0=2F_m/t$ (resp. $6.6\mathrm{kJ}/{\mathrm{m}}^2$ and $6.2\mathrm{kJ}/{\mathrm{m}}^2$), where $F_m$ is the maximal observed tearing force (resp. 0.165 and 0.156 N). In (b) the minimum measured tearing force was $0.07N$. (c) Geometrical construction [same data as in (b)], where the energy release-rate lines for $\alpha =30°$, ${\theta }_1=51°$, and ${\theta }_2=5°$ are presented in red and blue, respectively. The $G_c^{-1}$ curve was arbitrarily extended in light black line in the missing region where no measurement is available. The grey sector corresponds to orientations never observed even when fracture is guided (see Fig. 5).

Figure 5

Fracture guided along a forbidden direction [setup in Fig. 1c] in material B. A pair of adhesive tapes are placed 5 mm apart along the band of anisotropic polymer. Pulling is performed at $200\mathrm{mm}/\min$. (a) Sawtooth crack paths obtained when the crack is guided in a forbidden direction ($15°\le \alpha \le 35°$). When the direction is not forbidden ($\alpha =10°$, 40°), the crack path follows one of the boundaries. (b)–(c) Measured force as function of displacement for $\alpha =30°$, on a nonreinforced sample, in the most frequent case of propagation along $\theta =51°$ (b), and along $\theta =5°$ (c). (d) Force measured during zig-zag propagation ($\alpha =30°$) in a reinforced sample. The plateaus corresponding to propagation are consistent with measurements in (b) and (c).