Size effects on the mechanical behavior of single-layer graphene sheets with geometric defects: A probabilistic and machine learning assisted approach

This study investigates the mechanical behavior of defective single-layer graphene sheets, accounting explicitly for variability induced by structural defects and size effects. A comprehensive probabilistic framework is developed to model how defect type (single vacancy, double vacancy, Stone–Wales), defect density (1%–3%), and sheet size (4.82–55.24 nm) influence the elastic modulus and Poisson's ratio. The methodology combines a nonlinear stick-and-spring mechanical model, Monte Carlo simulations with random defect distributions, and machine learning-based sensitivity analysis (Gradient Boosting regression). Results reveal that defect type strongly governs stiffness degradation—double vacancies causing the largest reductions—while Stone–Wales defects enhance lateral deformation. The elastic moduli decrease linearly with defect density, and smaller sheets exhibit greater anisotropy and variability, whereas sheets larger than 35 nm approach isotropic behavior. Statistical modeling identifies lognormal and Weibull distributions as appropriate for quantifying property variability. The study makes four key contributions: (i) a novel probabilistic treatment of both stiffness and Poisson's ratio, providing new insights into variability and uncertainty; (ii) systematic quantification of size effects and their interaction with defect-induced variability; (iii) interpretable machine learning-based sensitivity analysis identifying dominant factors influencing mechanical behavior; and (iv) a modular, extensible modeling framework applicable to other 2D materials. The results advance the understanding of defect-driven variability in graphene mechanics and support the development of defect-aware design strategies for graphene-based materials.