Hydraulic Fracture and Toughening of a Brittle Layer Bonded to a Hydrogel

Brittle materials propagate opening cracks under tension. When stress increases beyond a critical magnitude, then quasistatic crack propagation becomes unstable. In the presence of several precracks, a brittle material always propagates only the weakest crack, leading to catastrophic failure. Here, we show that all these features of brittle fracture are fundamentally modified when the material susceptible to cracking is bonded to a hydrogel, a common situation in biological tissues. In the presence of the hydrogel, the brittle material can fracture in compression and can hydraulically resist cracking in tension. Furthermore, the poroelastic coupling regularizes the crack dynamics and enhances material toughness by promoting multiple cracking.

Figure 1

Sketch of the model system at rest and under stretch (a). Different crack opening or closing scenarios (b), depending on strain rate $\dot{ϵ}$ and on the ratio of the shear modulus $G$ of the hydrogel to the effective Young’s modulus $\overline{E}$ of the brittle material. The color map represents the time rate of the solvent flux towards the crack cavity immediately after strain application, $\dot{\tilde{q}}(0)=\dot{q}(0)/(k\overline{E})=\dot{ϵ}\zeta /\gamma$. Its sign and magnitude express the tendency and rate of the crack opening or closing. The darker gray shade represents a stiffer brittle layer relative to the hydrogel. The labels in (b) describe the dominant driving force at the crack, hydraulic ($H$) or elastic ($E$); the phenomenology, fracture ($F$) or crack arrest ($A$); and the nature of the imposed deformation, tensile ($+$) or compressive ($-$). For instance, $HA+$ stands for hydraulic ($H$) arrest ($A$) under tension ($+$).

Figure 2

Opening and propagation dynamics of a single crack for different solvent diffusivities. Tensile cracking ($EF+$) is shown in (a),(b), $\zeta =0.8$, and compressive hydraulic fracturing ($HF-$) is shown in (c),(d), $\zeta =-19$. The plots depict crack length $\tilde{a}$ and pressure $\tilde{p}$ in the crack cavity computed by solving Eqs. (4) and (6), with ${\tilde{K}}_{\mathrm{Ic}}\approx 0.1$ and three different dimensionless diffusivities $\tilde{k}\approx 5500$ (red), 550 (blue), and 55 (green).

Figure 3

Dynamics of two competing cracks for different solvent diffusivities, in tension (a),(b), $\zeta =0.8$, and in compression (c),(d), $\zeta =-1$. The plots in (a),(c) show the relative crack length difference $\mathrm{\Delta }\tilde{a}$ for $\tilde{k}\approx 5500$ (red), 550 (blue), and 55 (green), with $|\dot{ϵ}|=0.01{\mathrm{s}}^{-1}$. The contour plots in (b),(d) show the longitudinal stress ${\tilde{\sigma }}_x$ in the brittle layer and the pressure ${\tilde{p}}_g$ in the hydrogel immediately prior to failure for the same values of $\tilde{k}$ as in (a),(c), decreasing from left to right. The scale bar is $5\mu \mathrm{m}$; the arrows in the hydrogel represent the solvent flux.