Quantitatively descripting the gradation using equation has a great significance on the investigating of the optimal gradation and multi-scale effects for coarse-grained subgrade fillers. A new gradation equation for coarse-grained subgrade filler (CSF) was proposed according to the traditional fractal theory. Relationship between the shape of gradation curve and model parameters was analyzed, the curve shows sigmoid when \(- \sqrt { - 2\alpha } < \beta < \sqrt { - 2\alpha }\) , and the curve respects hyperbolic when \(\beta> \sqrt { - 2\alpha }\) . The values of model parameters when CSF is well-graded range were captured, i.e., β2 < 2.590α2 + 5.627α + 1.239 and α < 0. Applications of the new gradation equation in the scale effect and particle breakage were investigated and verified. Based on the traditional scaled methods, i.e., exclusion method, similar gradation method, equivalent substitution method and hybrid method, a new unified form of gradation scale formula was proposed. In addition, the new expressions for calculating breakage rate and disintegration ratio were established. Specifically, the particle breakage experiment for CSF using the large-scale triaxial test and the disintegration breakage experiment for red-bed soft rock under different dry–wet cycles were carried out to use and verify the correctness of these new expressions. Findings provides theoretical support for the study of gradation effects and multi-scale effects.
