Thermal fracturing is a common phenomenon occurring during significant cooling or heating in diverse engineering problems and natural processes. Multiple thermal fractures caused by cooling may initiate, propagate, arrest, and reactivate under thermal stresses and inter-fracture stress interactions, leading to typical hierarchical fracture patterns. The reactivation, further propagation, and permanent arrest of initially arrested fractures have not been accounted for in existing analytical modeling. In this study, all the processes of thermal fracturing in an unconfined half-space plane are investigated using a plane strain model. A dimensionless solution is developed by solving the coupled elasticity equation and fracture-propagation criterion and performing stability analysis, with fractures discretized by the displacement discontinuity method, all in terms of dimensionless variables. A new stability criterion stricter than the classic one (without reactivation) is introduced to account for fracture reactivation and permanent arrest. The dimensionless solution, applied to ceramic quenching as an example, is validated using a two-dimensional, finite element method-based fracture model that automatically considers all processes of thermal fracturing. The solution includes two generic profiles of dimensionless fracture spacing and length scaled by the material’s properties (i.e., toughness, Young’s modulus, Poisson ratio, and linear thermal expansion coefficient) and the surface-cooling condition. At late time, the profile of fracture length ($l$) can be simplified by a scaling law of $\mathit{\text{l}}\propto \sqrt{t}$ with cooling time $t$. An algorithm is developed to produce the evolution of fracture pattern of interest, which is also validated by the numerical fracture model. Comparison of these solutions (with fracture reactivation) to those without reactivation indicates that fracture reactivation starts to affect fracture spacing and thus pattern after reaching a small fracture length and such an effect increases with fracture length and cooling time. These developed solutions are applicable to rapid and accurate prediction of evolution of fracture length, spacing, and pattern in thermal shock problems.

热破裂是在各种工程问题和自然过程中进行大量冷却或加热过程中发生的常见现象。由冷却引起的多个热断裂可在热应力和断裂间应力相互作用下引发，传播，阻止和重新活化，从而导致典型的分层断裂模式。在现有的分析模型中，还没有考虑到重新活化，进一步传播以及永久性阻止最初被捕的骨折。在这项研究中，使用平面应变模型研究了在无限制的半空间平面内的所有热裂过程。通过求解耦合弹性方程和裂缝传播准则并进行稳定性分析，开发了无量纲解决方案，并通过位移不连续性方法离散化了裂缝，所有这些都是无量纲变量。引入了比经典严格的新稳定性标准（无重新激活），以解决骨折重新激活和永久性停止的问题。例如，使用基于二维有限元方法的断裂模型（可自动考虑热断裂的所有过程）验证适用于陶瓷淬火的无量纲解决方案。该解决方案包括两个无因次断裂间距和长度的通用分布图，该分布由材料的特性（即韧性，杨氏模量，泊松比和线性热膨胀系数）和表面冷却条件决定。在后期，断口长度分布（引入了比经典严格的新稳定性标准（无重新激活），以解决骨折重新激活和永久性停止的问题。例如，使用基于二维有限元方法的断裂模型（可自动考虑热断裂的所有过程）验证适用于陶瓷淬火的无量纲解决方案。该解决方案包括两个无因次断裂间距和长度的通用分布图，该分布由材料的特性（即韧性，杨氏模量，泊松比和线性热膨胀系数）和表面冷却条件决定。在后期，断口长度分布（引入了比经典严格的新稳定性标准（无重新激活），以解决骨折重新激活和永久性停止的问题。例如，使用基于二维有限元方法的断裂模型（可自动考虑热断裂的所有过程）验证适用于陶瓷淬火的无量纲解决方案。该解决方案包括两个无因次断裂间距和长度的通用分布图，该分布由材料的特性（即韧性，杨氏模量，泊松比和线性热膨胀系数）和表面冷却条件决定。在后期，断口长度分布（基于有限元方法的断裂模型，可自动考虑热断裂的所有过程。该解决方案包括两个无因次断裂间距和长度的通用分布图，该分布由材料的特性（即韧性，杨氏模量，泊松比和线性热膨胀系数）和表面冷却条件决定。在后期，断口长度分布（基于有限元方法的断裂模型，可自动考虑热断裂的所有过程。该解决方案包括两个无因次断裂间距和长度的通用分布图，该分布由材料的特性（即韧性，杨氏模量，泊松比和线性热膨胀系数）和表面冷却条件决定。在后期，断口长度分布（$升$）可通过以下定标定律进行简化 $\mathit{\text{升}}\propto \sqrt{Ť}$ 随着冷却时间 $Ť$。开发了一种算法来产生感兴趣的裂缝模式的演化，该算法也已通过数值裂缝模型进行了验证。将这些解决方案（带有裂缝再活化）与没有重新活化的溶液进行比较表明，裂缝再活化开始影响裂缝间距，从而在达到较小裂缝长度后开始影响花纹，并且这种效果随裂缝长度和冷却时间的增加而增加。这些已开发的解决方案适用于快速，准确地预测热冲击问题中裂缝长度，间距和样式的演变。