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

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.