Unified scenario for the morphology of crack paths in two-dimensional disordered solids

A combined experimental and numerical investigation of the roughness of intergranular cracks in two-dimensional disordered solids is presented. We focus on brittle materials for which the characteristic length scale of damage is much smaller than the grain size. Surprisingly, brittle cracks do not follow a persistent path with a roughness exponent $\zeta \approx 0.6–0.7$ as reported for a large range of materials. Instead, we show that they exhibit monoaffine scaling properties characterized by a roughness exponent $\zeta =0.50\pm 0.05$, which we explain theoretically from linear elastic fracture mechanics. Our findings support the description of the roughening process in two-dimensional brittle disordered solids by a random walk. Furthermore, they shed light on the failure mechanism at the origin of the persistent behavior with $\zeta \approx 0.6–0.7$ observed for fractures in other materials, suggesting a unified scenario for the geometry of crack paths in two-dimensional disordered solids.

Figure 1

Brittle intergranular crack profiles obtained from (a) the experimental fracture test of a thin sheet of expanded polystyrene, and (b) the large scale numerical simulation of the cohesive fracture of a random arrangement of polygonal elastic grains.

Figure 2

Setup of the double torsion test realized to investigate the crack path: (a) schematic representation of the sample and boundary conditions; (b) prenotched specimen and loading device.

Figure 3

Setup of the thin strip experiment realized to investigate the effect of the specimen size on the crack roughness geometry: (a) schematic representation of the sample and boundary conditions; (b) broken specimen and loading device.

Figure 4

Geometry and boundary conditions of the notched specimen employed in the simulations. For numerical convenience, the polycrystalline microstructure is used in the process zone only.

Figure 5

Crack path obtained with the sequential polycrystalline analysis applied to the specimen in Fig. 4. Only the process zone, with the embedded 3140-grain microstructure depicted in Fig. 1, is shown.

Figure 6

Logarithmic representation of the height correlation function of the experimental and computed crack profiles. At scales larger than the microstructural length $\ell$ (bead radius in the experiments and grain boundary length in the simulations), the crack roughness is self-affine with an exponent $\zeta \simeq 0.5$. The inset shows that the correlator of the crack local slopes decays exponentially fast over a few $\ell$, confirming the random walk behavior.

Figure 7

Statistical distribution $P_{\delta x}$ of height variations computed for different scales $\delta x$. After normalization by $\delta x^{\zeta }$ with $\zeta =0.5$, they collapse to a Gaussian distribution. The inset shows that the studied values of $\delta x$ belong to the self-affine domain $1\lesssim \delta x/\ell \lesssim 100$.

Figure 8

Multiaffine spectrum of the simulated fracture profiles. The main panel shows the correlation functions $\mathrm{\Delta }{h}_q\left(\delta x\right)$ of order $q$ in the range $0.5\le q\le 3$ as defined in the text. The inset shows the multiaffine spectrum $H\left(q\right)$ as obtained from the fit of the correlation functions $\mathrm{\Delta }{h}_q\sim \delta x^{H\left(q\right)}$. The constant value $H\left(q\right)\simeq 0.5$ is signature of monoaffine behavior.

Figure 9

Power spectrum of simulated fracture profiles in 2D consolidated granular solids and comparison with experimental fracture profiles in paper sheets (from Refs. [7, 45]). The power law behavior $S\left(k\right)\sim k^{-1-2\zeta }$ with $\zeta =0.55$ confirms the estimate of the roughness exponent obtained in the direct space in Fig. 7: it is significantly smaller than the roughness exponent $\zeta =0.7$ characterizing fractured paper sheets. The length scale $\ell$ used to normalize the wave number $k=2\pi /\lambda$ is taken as the grain size for the granular solid and a characteristic microstructural scale (like, e.g., the fiber length) $\ell \simeq 1\text{mm}$ for the paper sheet.

Figure 10

Logarithmic representation of the height correlation function of the experimental fracture profiles obtained from the thin strip geometry for various specimen widths $W$ (see Fig. 3). The crack profiles are self-affine with an exponent $\zeta \simeq 0.5$, up to some cutoff $\xi$ that depends on the sample size. The inset shows that the upper bound $\xi$ of the self-affine regime increases linearly with the sample width.

Figure 11

Correlation function of the fracture profiles as predicted by the path equation (5) solved numerically. For $\delta x>\ell$, the cracks follow a random walk with exponent $\zeta =0.49$ (red straight line). In the inset, the correlator of the slopes of the predicted fracture profiles is compared with the correlator of $\eta$ describing the effect of the material microstructure on the local shear conditions on the crack.

Figure 12

Sequential polycrystalline analysis of the specimen in Fig. 4 with a 90-grain microstructure embedded in the process zone: (a) discretized process zone with final crack path; (b), (c) subsimulation 1 and 2 with refined mesh in the active process zone—at least two elements along each grain boundary are provided outside the active process zone; (d) load-displacement curves corresponding to the simulations in (a)–(c), with point A the bottom, leftmost point of the specimen, as indicated in Fig. 4.

Figure 13

(a) Crack path obtained with the sequential polycrystalline analysis applied to the specimen in Fig. 4. (b), (c) First and last subsimulations in the sequential analysis. Only the process zone, with the embedded 3140-grain microstructure depicted in Fig. 1, is shown.

Figure 14

Load-displacement curves obtained through the proposed sequential polycrystalline analysis applied to the specimen in Fig. 4. The 3140-grain microstructure depicted in Fig. 1 is embedded in the process zone, and the corresponding crack path is reported in Fig. 13.

Figure 15

Nonsingular contribution $\mathrm{\Delta }{\sigma }_{xx}\left(x\right)={\sigma }_{xx}\left(x\right)-K_{\text{I}}/\sqrt{2\pi x}$ of the stress that applies parallel to the crack in the thin strip geometry considered in the simulations (see Fig. 4). The fit by an affine function $\mathrm{\Delta }{\sigma }_{xx}=T+A\sqrt{x}$ [see Eq. (B2)] provides the structural lengths ${\mathcal{L}}_1^{\text{sim}}$ and ${\mathcal{L}}_2^{\text{sim}}$ through Eq. (B1).