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

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.

💡 Research Summary

The paper presents a comprehensive phase‑field fracture framework specifically designed for brittle materials subjected to thermal shock. Unlike conventional variational phase‑field models that rely solely on Griffith‑type energy criteria and cannot predict crack nucleation, the authors adopt the “complete” phase‑field model in which three macroscopic material properties—elastic stiffness, fracture toughness (Gc), and material strength (compressive and tensile strengths)—are treated as independent inputs. Strength is introduced through a Drucker‑Prager failure surface, and its influence on fracture is incorporated as an external micro‑force term (ce) in the phase‑field evolution equation. This formulation enables the model to capture both energy‑driven crack propagation and stress‑driven crack nucleation within a single variational framework.

Thermo‑mechanical coupling is achieved by solving the heat conduction equation for temperature T, the linear momentum balance for displacement u, and the phase‑field equation for the damage variable v. Temperature affects the mechanical response only through thermal strain (αΔT), while the phase field does not alter thermal conductivity, preserving the standard heat flux formulation. The mechanical constitutive law uses a quadratic degradation function g(v)=v², and the stress is degraded accordingly. The governing equations are discretized with standard finite elements (quadrilateral in 2‑D, hexahedral in 3‑D) and a staggered solution scheme: at each time step the temperature field is updated first (implicit Euler), then the displacement and phase fields are solved iteratively using a primal‑dual active‑set method to enforce irreversibility (v̇≥0). Dynamic effects, when needed, are handled with the HHT‑α time integrator (α=‑0.3, β=0.4225, γ=0.8). To mitigate mesh‑size dependence of the effective fracture toughness, a correction factor δ is introduced that depends on the element size h, regularization length ℓ, and material parameters.

A key innovation is the inclusion of spatially heterogeneous strength fields. The authors generate random strength maps either through a translation model with prescribed correlation length or a simple mosaic of uncorrelated values, reflecting experimental observations of variability in uniaxial and biaxial strengths. This stochastic strength field directly influences the Drucker‑Prager surface and, consequently, the onset and path of cracking.

The model’s capabilities are demonstrated through three benchmark problems that span the spectrum from crack propagation to nucleation.

1. Quenching of Glass Plates – A pre‑existing crack in a glass plate is subjected to rapid cooling. Simulations reproduce the experimentally observed transition from straight crack extension at mild thermal loads to complex branching at higher ΔT. The results show that, even when Griffith energetics dominate, the material strength parameter still modulates crack speed and branching thresholds.

2. Infrared‑Heated Ceramic Disks – Both notched and intact ceramic disks are exposed to uniform infrared heating, creating nearly homogeneous thermal stresses. In the notched case the model predicts a straight crack that bisects the specimen, while in the intact case it captures the characteristic branching pattern. The ability to reproduce both outcomes without altering the model (only the geometry) underscores the importance of the stochastic strength field and the Drucker‑Prager surface.

3. Rapid Power Pulses in Nuclear Fuel Pellets – Pellets experience two levels of power pulses; low‑energy pulses cause no damage, whereas high‑energy pulses lead to a distribution of fractured fragments. Simulations reveal that the transition is governed by the competition between Gc‑driven crack growth and strength‑controlled nucleation. The model accurately predicts the fraction of fractured pellets and highlights that strength, rather than fracture energy alone, dictates failure under extreme thermo‑mechanical loading.

Across all cases, the complete phase‑field model unifies the treatment of distinct fracture mechanisms under thermal shock, surpassing classical sharp‑crack or regularized phase‑field approaches that lack independent strength representation. By allowing independent specification of elasticity, toughness, and strength, and by incorporating realistic spatial variability of strength, the framework delivers predictive accuracy for a wide range of industrially relevant scenarios, including glass tempering, ceramic processing, and nuclear fuel safety analysis. The implementation in the open‑source RACCOON code (built on MOOSE) ensures scalability to large three‑dimensional problems, making the approach a practical tool for engineers and researchers dealing with brittle fracture in extreme environments.