标题： 费因克兰克：通过费曼-卡克杀死扩散的准脆性损伤连续体模型 显示英文标题

标题： FeynKrack: A continuum model for quasi-brittle damage through Feynman-Kac killed diffusion

摘要： 连续损伤力学（CDM）是用于模拟固体中裂纹扩展的流行框架。 CDM 使用一个损伤参数来定量评估人们通常称为“材料退化”的现象。 虽然这个参数有时具有物理意义，但其演变的数学方程通常与这些物理解释不一致。 有趣的是，CDM 中的退化可以被视为测度的变化，其中损伤变量表现为 Radon-Nikodym 导数。 我们采用这种观点，并使用概率测度值描述来表示准脆性损伤背后的随机微裂纹。 我们表明，底层密度的演变可以通过被杀死的扩散过程来描述，如 Feynman-Kac 理论中所示。 损伤的增长被解释为该测度在某个区域内的减少，这反过来通过相邻材料点之间力传递机制的损失来量化键的破坏。 值得注意的是，损伤的演变可以得到近似的显式解。 这带来了实质性的计算简便性，促进了大规模问题的快速而准确的模拟。 通过选择适当的杀死率，可以考虑损伤的不可逆性，从而消除通常在损伤相场建模中使用的经验性历史依赖路径。 我们的提议 FeynKrack（Feynman-Kac 裂纹传播器的简称）通过几个关于准脆性损伤的模拟得到了验证和展示，证明了其有效性。 它还为未来探索损伤的非平衡热力学方面提供了一条有前景的随机路径。 显示英文摘要

摘要： Continuum damage mechanics (CDM) is a popular framework for modelling crack propagation in solids. The CDM uses a damage parameter to quantitatively assess what one loosely calls `material degradation'. While this parameter is sometimes given a physical meaning, the mathematical equations for its evolution are generally not consistent with such physical interpretations. Curiously, degradation in the CDM may be viewed as a change of measures, wherein the damage variable appears as the Radon-Nikodym derivative. We adopt this point of view and use a probabilistic measure-valued description for the random microcracks underlying quasi-brittle damage. We show that the evolution of the underlying density may be described via killed diffusion as in the Feynman-Kac theory. Damage growth is then interpreted as the reduction in this measure over a region, which in turn quantifies the disruption of bonds through a loss of force-transmitting mechanisms between nearby material points. Remarkably, the evolution of damage admits an approximate closed-form solution. This brings forth substantive computational ease, facilitating fast yet accurate simulations of large dimensional problems. By selecting an appropriate killing rate, one accounts for the irreversibility of damage and thus eliminates the need for ad-hoc history-dependent routes typically employed, say, in phase field modelling of damage. Our proposal FeynKrack (a short form for Feynman-Kac crack propagator) is validated and demonstrated for its efficacy through several simulations on quasi-brittle damage. It also offers a promising stochastic route for future explorations of non-equilibrium thermodynamic aspects of damage.