Elucidating the complexity of stress interactions in polymer-coated granular materials via copula-based probabilistic dependence configurations

Coated granular materials, whether naturally occurring or synthetically produced in laboratories, offers substantial potential for various engineering applications. This study focuses on the use of dimethyldichlorosilane as a coating solution, inducing water repellency for granular materials, a property of interest in the development of advanced materials and structures. Although prior research suggests that established methods for natural granular materials analysis are generally applicable to coated materials, there remains an inherent stochasticity and probabilistic dependence in the properties of coated materials, influenced by variable extents of coating damage contingent on stress level. The deterministic empirical regression relationship alone is insufficient to represent the significant uncertainty evident in the experimental observations. In addressing these uncertainties, this study presents a probabilistic analysis approach, underpinned by copula theory, to define the probabilistic dependence structures of the coated materials. Compared to traditional measures of correlation, copula theory can reflect the various nonlinear probabilistic dependencies among multiple variables. The study utilizes lognormal probability density functions (PDFs) to assess the critical stress ratios for natural, thin-coated, and thick-coated materials. The results of natural and thin-coated materials indicate a congruity between the critical stress ratios derived from PDFs and those obtained through linear regression, implying the viability of the proposed probabilistic approach. Notably, for thick-coated granular material, significant uncertainties in the critical stress ratios emerge, correlating robustly with the imposed stress level. We thus propose a conditional PDF between the stress level and the corresponding critical stress ratios to better predict shearing behavior. To delineate the probabilistic dependence among the three key soil properties - initial state parameter, peak friction angle, and peak point dilatancy - multiple copula density functions are applied. The analysis highlights an increased variability and dispersion for thick-coated materials, attributable to the unpredictable nature of coating damage. Overall, the findings underscore the potential of probabilistic methodologies in the study of coated granular materials, leading to a more nuanced understanding of their behavior, particularly during shearing stages with different stress level. This ap