Morphogenesis and Propagation of Complex Cracks Induced by Thermal Shocks

We study the genesis and the selective propagation of complex crack networks induced by thermal shock or drying of brittle materials. We use a quasistatic gradient damage model to perform large-scale numerical simulations showing that the propagation of fully developed cracks follows Griffith criterion and depends only on the fracture toughness, while crack morphogenesis is driven by the material’s internal length. Our numerical simulations feature networks of parallel cracks and selective arrest in two dimensions and hexagonal columnar joints in three dimensions, without any hypotheses on cracks geometry, and are in good agreement with available experimental results.

Figure 1

Full-scale numerical simulation of a ceramic slab submitted to a thermal shock. (a) Damage field from the numerical simulation (blue, $\alpha =0$; red, $\alpha =1$). (b) Experimental results from Fig. 5(d) of Ref. 20. (c) Average crack spacing $d$ as a function of their depth $a$ for (a) and (b). The solid line is an approximate scaling law obtained in Ref. 13 by imposing a period doubling condition on a Griffith model. Here, $\ell =46\mu \mathrm{m}$ is the material internal length, ${\ell }_0=G_c/(E{\beta }^2\Delta T^2)=14\mu \mathrm{m}$ the Griffith length (loading parameter), $2L=9.8\mathrm{mm}$ the total depth of the slab. See Fig. 2 for the meaning of the distributed damage zone.

Figure 2

Crack nucleation. (a) Damage field near the shock surface before (left) and after (right) bifurcation time $t^{*}$ for ${\ell }_0/\ell =0.107$ showing the bifurcation of a horizontally homogeneous damaged band of depth $D^{*}$ toward a periodic solution with wavelength ${\lambda }^{*}$. (b) Wavelength, time, and damage penetration in numerical simulations for several intensities of the thermal shock ${\ell }_0/\ell$ (dots), compared to the semianalytical results from Ref. 23 (solid lines).

Figure 3

Three-dimensional version of the experiment from Fig. 1b showing the transition from two- to three-dimensional crack patterns. The simulations are performed on a subdomain of dimension $5\times 1\times 1\mathrm{mm}$ and temperature contrast (a) $380°\mathrm{C}$ (${\ell }_0=0.27\ell$), (b) $480°\mathrm{C}$ (${\ell }_0=0.17\ell$), (c) $580°\mathrm{C}$ (${\ell }_0=0.12\ell$), and (d) $680°\mathrm{C}$ (${\ell }_0=0.08\ell$).

Figure 4

Complex fracture pattern for ${\ell }_0=G_c/(E{\beta }^2\Delta T^2)=0.05\ell$ in a domain of size $150\ell \times 150\ell \times 20\ell$ color coded by distance from the bottom surface where the thermal shock is applied. The problem was discretized with approximately $4.4\times {10}^7$ linear finite elements in space (mesh size $h=\ell /5$) and 100 time steps. The computation was performed on 1536 cores of the NSF-XSEDE cluster Stampede at Texas Advanced Computing Center in 10 h.

Figure 5

Average size $d$ (square root of the average cross-sectional area) of the fracture-delimited cells as a function of the depth $a$ (distance to the exposed face) in cubic domains with edge length $L$ ranging from 2 mm to 2 m compared with the two-dimensional scaling law from Ref. 13 (solid line). All the simulations are with ${\ell }_0=14\mu \mathrm{m}$ and $\ell =L/40$. Inset: Top view of the crack patterns for (a) $L=2\mathrm{mm}$, (b) $L=63.2\mathrm{mm}$, and (c) $L=2\mathrm{m}$.