How Material Heterogeneity Creates Rough Fractures

Fractures are a critical process in how materials wear, weaken, and fail, whose unpredictable behavior can have dire consequences. While the behavior of smooth cracks in ideal materials is well understood, it is assumed that for real, heterogeneous systems, fracture propagation is complex, generating rough fracture surfaces that are highly sensitive to specific details of the medium. Here we show how fracture roughness and material heterogeneity are inextricably connected via a simple framework. Studying hydraulic fractures in brittle hydrogels that have been supplemented with microbeads or glycerol to create controlled material heterogeneity, we show that the morphology of the crack surface depends solely on one parameter: the probability to perturb the front above a critical size to produce a steplike instability. This probability scales linearly with the number density, and with heterogeneity size to the $5/2$ power. The ensuing behavior is universal and is captured by the 1D ballistic propagation and annihilation of steps along the singular fracture front.

Figure 1

(a) Image of typical fracture after crack propagation has stopped, showing the resultant penny-shaped crack (green square) in a hydrogel within our experimental apparatus. The fracture is broadly flat and smooth. (b) Microscopy image of a typical fracture surface segmented by many arched step lines. There is a clear boundary (green line) between the surface created by controlled hydraulic fracture (inside) and the surface created when separating the gel for imaging (outside). (c) Typical example of surface height map measured with confocal microscope showing that steps generate additional surface area and create topographic boundaries.

Figure 2

(a) Typical fracture surface with nucleations (red dots) and interactions (blue dots) indicated within the experimentally generated fracture area (black dashed line). (b) Typical analyzed fracture surface. Image processing is used to create a skeletonized map of step lines (green) for calculating ${\rho }_{\mathrm{SL}}$. (c) Step creations per unit area, $n_c$, for $1\mu \mathrm{m}$ silica beads as a function of volumetric bead density, $n_{\text{beads}}$. Error bars represent uncertainty in the measurement of nucleations. (d) ${\rho }_{\mathrm{SL}}$ as a function of $n_{\text{beads}}$ for bead sizes ranging from $1–40\mu \mathrm{m}$. Each appears to grow as roughly as $n_{\text{beads}}^{1/2}$. Note that the number of beads required to generate an equivalent ${\rho }_{\mathrm{SL}}$ decreases drastically with increasing bead size.

Figure 3

${\rho }_{\mathrm{SL}}/n_{\text{beads}}^2$ as a function of bead radius, $r$. Rescaling collapses all data within $1–40\mu \mathrm{m}$ onto a single line. The dashed line indicates a slope of $5/2$. The scaling regime breaks down sharply for $r$ $<1\mu \mathrm{m}$. Inset: ${\rho }_{\mathrm{SL}}$ as a function of $n_{\text{beads}}$, similar to Fig. 2, that includes the $r=750\mathrm{nm}$ data.

Figure 4

${\rho }_{\mathrm{SL}}$ as a function of glycerol volume fraction in hydrogel solvent, ${\varphi }_g$. Typical images of fracture surfaces for corresponding samples are shown. Each circle is 2.5 mm in diameter. The dashed red line indicates the solubility limit of PEGMEMA in glycerol-water solutions, ${\varphi }_g^c$. Inset: ${\rho }_{\mathrm{SL}}$ as a function of distance from the solubility limit.

Figure 5

Typical images of fracture surfaces in hydrogels without any added heterogeneity (black outline), with added microbeads which act as localized heterogeneities (blue outline), and with glycerol added to the solvent before polymerization (orange outline), which creates a more gradual background of inhomogeneity. When both glycerol and beads are used, creating complexity of multiple types and scales, significantly rougher surfaces (right) are generated that resemble those seen in natural materials. Fracture propagation proceeds approximately from the bottom to the top of each image. The white scale bars at the bottom left are 1 mm.