标题： 基于累积分布函数的损伤定律：相容性、向Griffith断裂的Gamma收敛及适定性 显示英文标题

标题： CDF-Generated Damage Laws: Admissibility, Gamma-Convergence to Griffith Fracture, and Well-Posedness

摘要： 我们通过设定存储的能量密度为$\psi(\eta)=\int_{0}^{\eta}[1-F(s)]d s$，其中$F$包括指数分布、柯西分布、逻辑斯蒂分布、半正态分布、Gudermannian分布、超几何分布、根式分布、有理分布、分段分布和快速衰减累积分布函数（CDF），从而建立了一组标量软化定律。 我们证明了每一个这样的定律都生成一个单调的、有界的且耗散的退化映射，使得相关的超弹性材料在热力学上是可接受的。 直接在空间维度$d=2,3$上工作，我们建立了基于CDF的能量到尖锐界面Griffith泛函的紧致性和$\Gamma$-收敛性。 我们进一步通过构建同时满足稳定性和能量平衡的整体能量解，展示了速率无关的准静态演化的存在性。 这些分析结果为概率损伤公式与Griffith型断裂力学之间提供了一个严格的桥梁。 给出一个说明性的例子来展示当前损伤定律的有效性。 显示英文摘要

摘要： We formulate a family of scalar softening laws by setting the stored-energy density $\psi(\eta)=\int_{0}^{\eta}[1-F(s)]d s$, where $F$ ranges over exponential, Cauchy, logistic, half-normal, Gudermannian, hypergeometric, radical, rational, piece-wise, and rapid-decay cumulative-distribution functions (CDFs). We prove that every such law yields a degradation map that is monotone, bounded, and dissipative, rendering the associated hyperelastic material thermodynamically admissible. Working directly in spatial dimensions $d=2,3$, we establish compactness and $\Gamma$-convergence of the CDF-based energies to a sharp-interface Griffith functional. We further show the existence of rate-independent quasi-static evolutions by constructing global energetic solutions that satisfy both stability and energy balance. These analytical results provide a rigorous bridge between the probabilistic damage formulation and Griffith-type fracture mechanics. One illustrative example is presented to show the effectiveness of the current damage laws.