Microscopic Insights into the Critical State of Granular Materials with Different Fractal Dimensions Under Constant Stress Ratio Paths

Granular materials subjected to complex stress histories exhibit pronounced path dependence, multi-scale heterogeneity and scale-invariant characteristics, especially when particle breakage leads to gradation evolution with fractal features. Discrete element simulations are performed on granular assemblies with prescribed idealized fractal gradations under constant stress ratio loading–turning paths, while maintaining identical solid volume and initial relative density. The results show that, for a given gradation, both the peak strength envelope and the critical state line exhibit high consistency and are effectively independent of the examined stress paths, which are supported by high R2 values from regression. At the critical state, microstructural parameters together with energy measures consistently follow stable power–law relationships with mean effective stress. For different gradations, the critical stress ratio remains nearly unchanged, whereas peak strength increases with increasing fractal dimensions; although critical state points remain nearly collinear in the deviatoric stress (q) –mean effective stress (p) plane, the critical state line in void ratio (e)–p plane shifts downward as the particle size distribution becomes broader. The evolution of microstructural and energy-related power–law relationships with fractal dimension exhibits a clear saturation trend. This study demonstrates that, within the simulated framework, fractal gradation primarily governs the position of the critical state in e–p space without altering its fundamental path-independent nature, providing fundamental insights into the multi-scale mechanics of graded granular materials under complex loading.