A complete phase-field fracture model for brittle materials subjected to thermal shocks

TL;DR

This work extends the complete phase-field fracture framework to thermo-mechanical loading by treating elasticity $E$, fracture energy $G_c$, and material strength as independent properties. The model incorporates a Drucker–Prager–type strength surface via external microforces $c_e$ in the phase-field evolution, enabling simultaneous prediction of crack nucleation and propagation under strong thermal gradients. Three canonical problems—progressive glass quenching, infrared-heated ceramic disks, and nuclear-fuel pellets under rapid power pulses—demonstrate the framework’s ability to capture both Griffith-like propagation and strength-governed nucleation, including pattern transitions and crack branching. The results highlight the practical potential of this unified approach for brittle materials in extreme environments and show how spatial strength perturbations can reproduce observed experimental scatter. Specifically, the intact-disk case requires perturbations to replicate single nucleation sites, and the fuel-pellet simulations show how load-induced variability can be captured by randomized strength fields.

Abstract

Brittle materials subjected to thermal shocks experience strong temperature gradients that in turn give rise to mechanical stresses that can be large enough to induce fracture. This work presents a complete model for phase-field fracture for coupled thermo-mechanical problems, wherein the bulk material properties, the material strength, and the fracture toughness are specified independently. The capabilities of the model are assessed across a wide span of scenarios in thermo-mechanical fracture, from the propagation of large pre-existing cracks to crack nucleation under spatially uniform states of stress. In particular, we revisit the controlled quenching of glass plates, and demonstrate how the model captures experimentally observed crack patterns across a range of thermal loads. Ceramic disks subjected to infrared radiation are also examined, with the model reproducing both straight cracks in notched specimens and branching in intact specimens. Finally, ceramic pellets subjected to rapid power pulses are examined, with the model explaining experimental transitions from intact to fractured pellets and the important role of material strength. The results demonstrate that the complete phase-field model unifies the treatment of distinct fracture scenarios under thermal shock, surpassing classical approaches and enabling more reliable prediction of brittle fracture in extreme environments.

Figures (22)

• Figure 1: Illustration of (a) the sharp crack model and (b) the general phase-field fracture model.
• Figure 2: Comparison between the experimental strength of graphite by Sato et al. sato_fracture_1987, the Drucker-Prager strength surface\ref{['eq: 2_pff/drucker-prager']} defined by $\sigma_{ts}=27$ MPa and $\sigma_{hs}=27.72$MPa, and the approximate strength surface\ref{['eq: 2_pff/approx_surface']}. The approximate strength surface (dashed lines) asymptotes to the Drucker-Prager surface as the regularization length $\ell$ decreases.
• Figure 3: (a) Schematic of the experimental setup kilic_prediction_2009 for the progressive quenching of glass plates and (b) a collection of experimental crack morphology photos from yuse_instabilities_1997sumi_thermally_2000, providing one example of straight crack propagation, one example of regular oscillatory propagation, and one example of chaotic propagation.
• Figure 4: Schematic of the simulated geometry. The evolution of the temperature field as a function of $V/D$ between the hot zone and cold front are as illustrated by the purple curves.
• Figure 5: Representative crack patterns obtained with $V=0.4$mm/s and selected temperature jumps $\Delta T$. (a) straight crack propagation, (b) regular oscillatory propagation, (c) snap-back and branching, followed by chaotic oscillatory propagation.