Comparative Development of Engineering Predictive Models for the Deformation Modulus of Granular Materials Based on Triaxial Compression Data

Saule Sh. Kazhikenova, G.S. Shaikhova

Eng—Advances in Engineering, 2026

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DOI 10.3390/eng7100522
Data confidence 70 medium
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Abstract

Accurate prediction of the deformation modulus of granular materials is important for engineering analysis, but the coupled effects of material state, particle size, and frictional resistance are difficult to represent using a single empirical relationship. This study comparatively evaluates engineering predictive models using an experimental triaxial-compression database reported in a companion study accepted for publication, comprising 25 experimentally realized material states and 75 replicate deformation-modulus measurements. Five representative granular materials were used to establish the investigated property ranges. Bulk density ρ, equivalent particle diameter deq, and internal friction angle φ were used as quantitative predictors. A Protodyakonov multifactor formulation provided the engineering baseline, with R = 0.9029, R2 = 0.4481, and RMSE = 6.39 MPa. Among four predefined analytical formulations, the four-parameter multiplicative power-law Model 1 achieved R2 = 0.8769, RMSE = 3.02 MPa. The complete quadratic response-surface model provided the strongest calibration fit R2 = 0.9129, RMSE = 2.54 MPa, whereas sequential removal of the \(d_{eq}^2\) and φ2 terms produced an eight-coefficient reduced model with R2 = 0.9046, RMSE = 2.66 MPa. Leave-one-out cross-validation changed the model ranking: Model 1 provided the lowest cross-validated error \(R_{\mathrm{CV}}^2\) = 0.8304, RMSECV = 3.54 MPa, followed by the additive linear model 4.06 MPa, reduced model 4.22 MPa, and complete quadratic model 5.10 MPa. Nested validation of the complete model-selection procedure yielded \(R_{\mathrm{CV}}^2\) = 0.7263, and RMSE = 4.50 MPa. The results demonstrate that nonlinear and interaction terms improve calibration of the response surface, but greater model complexity does not necessarily improve prediction for a small, non-orthogonal database. The proposed comparison provides a reproducible basis for balancing calibration accuracy, model complexity, interpretability, and cross-validated predictive performance within the investigated material-state and loading domain.