Prediction of Young’s modulus of rock materials by multivariate regression analysis and neuro-fuzzy model

Abstract

Young’s modulus is one of the geomechanical properties used in the design phase of different rock engineering applications. Difficulties in sample preparation and the high cost of experimental equipment lead researchers to perform studies on the estimation of Young’s modulus. However, previous studies on this topic are often limited in terms of rock type and/or number of data. Therefore, a comprehensive database covering a wide variety of rock types is needed for reliable estimation of Young’s modulus. To address this deficiency, a large database including Schmidt rebound value, uniaxial compressive strength, and porosity was compiled from the literature to derive equations and models for Young’s modulus estimation. Multivariate regression analysis and adaptive-neuro-fuzzy inference system (ANFIS) were used to predict Young’s modulus of rock materials. The reliability of the derived multivariate regression equations was verified using F- and t-tests, and the equations were found to be statistically reliable. The prediction pperformance of multivariate regression analysis and neuro-fuzzy models was compared using root mean square error (RMSE) and mean absolute percentage error (MAPE). The ANFIS models yielded considerably lower absolute prediction errors than the regression models. Thus, the neuro‑fuzzy method provided significantly higher prediction accuracy than the multivariate regression approach. The results indicated that the neuro-fuzzy model constructed in this study using the uniaxial compressive strength (σ_c), Schmidt rebound value (R), and porosity (n) as input parameters yielded the best predictions of E when compared to those predicted in some previous studies.

Keywords

Young’s modulus / Schmidt rebound value / Porosity / Multivariate regression / Adaptive-neuro-fuzzy inference system

References

[1] Abdollahi A, Li D, Deng J, Amini A. An explainable artificial-intelligence-aided safety factor prediction of road embankments. Engineering Applications of Artificial Intelligence, 2024, 136 108854

[2] Armaghani D, Tonnizam Mohamad E, Momeni Eet al.. Prediction of the strength and elasticity modulus of granite through an expert artificial neural network. Arabian Journal of Geosciences, 2016, 9 48

[3] Aydin A, Basu A. The Schmidt hammer in rock material characterization. Engineering Geology, 2005, 81: 1-14

[4] Azeemuddin M. Pore collapse in weak rocks, 1995University of Oklahoma

[5] Barzegar R, Sattarpour M, Nikudel MRet al.. Comparative evaluation of artificial intelligence models for prediction of uniaxial compressive strength of travertine rocks, Case study: Azarshahr area, NW Iran. Modeling Earth Systems and Environment, 2016, 2 76

[6] Beiki M, Majdi A, Givshad A. Application of genetic programming to predict the uniaxial compressive strength and elastic modulus of carbonate rocks. International Journal of Rock Mechanics and Mining Sciences, 2013, 63: 159-169

[7] Briševac Z, Pollak D, Maričić A, Vlahek A. Modulus of elasticity for grain-supported carbonates—determination and estimation for preliminary engineering purposes. Applied Sciences, 2021, 11(13 6148

[8] Carmichael RS. Practical Handbook of Physical Properties of Rocks and Minerals, 1989CRC Press

[9] Deere D.U., Miller R.P. 1966. Engineering classification and index properties for intact rocks. Tech Rep Air Force Weapons Lab, New Mexico (No. AFNL-TR:65–116)

[10] Dehghan S, Sattari GH, Chelgani SC, Aliabadi MA. Prediction of uniaxial compressive strength and modulus of elasticity for Travertine samples using regression and artificial neural networks. Mining Science and Technology, 2010, 201): 41-46

[11] Demir AS, Dağdeviren U, Kurnaz TF, Erden C, Kökçam AH. An integrated SHAP-MCDM approach for slope stability prediction based on machine learning algorithms. Natural Hazards, 2025, 121: 21811-21836

[12] Diamantis K, Gartzos E, Migiros G. Study on uniaxial compressive strength, point load strength index, dynamic and physical properties of serpentinites from Central Greece: test results and empirical relations. Engineering Geology, 2009, 108: 199-207

[13] Dinçer I, Acar A, Çobanoğlu Iet al.. Correlation between Schmidt hardness, uniaxial compressive strength and Young’s modulus for andesites, basalts and tuffs. Bulletin of Engineering Geology and the Environment, 2004, 63: 141-148

[14] Fredrich J, Evans B, Wong T. Micromechanics of the brittle to plastic transition in Carrara marble. Journal of Geophysical Research, 1989, 94: 4129-4145

[15] Gokceoglu C, Zorlu K. A fuzzy model to predict the uniaxial compressive strength and the modulus of elasticity of a problematic rock. Engineering Applications of Artificial Intelligence, 2004, 17(1): 61-72

[16] Hussain J, Fu X, Chen Jet al.. Estimation of rock strength parameters from petrological contents using tree-based machine learning techniques. AI in Civil Engineering, 2025, 4 4

[17] ISRM. The Complete ISRM Suggested Methods for Rock Characterization, Testing and Monitoring: 1974–2006, 2007ISRM Turkish National Group

[18] Karakul, H. 2012. Investigation of relationships between strength parameters and P-wave velocity of rocks at different saturation conditions. Dept. of Geological Engineering, Hacettepe University, Ankara, Turkey (PhD Thesis, in Turkish, unpublished)

[19] Karakul H. Investigation of saturation effect on the relationship between compressive strength and Schmidt hammer rebound. Bulletin of Engineering Geology and the Environment, 2017, 76: 1143-1152

[20] Katz O, Reches Z, Roegiers JC. Evaluation of mechanical rock properties using a Schmidt Hammer. International Journal of Rock Mechanics and Mining Sciences, 2000, 37(4): 723-728

[21] Köken E. Assessment of deformation properties of coal measure sandstones through regression analyses and artificial neural networks. Archives of Mining Sciences, 2021, 66(4): 523-542

[22] Kováčik J. Correlation between Young's modulus and porosity in porous materials. Journal of Materials Science Letters, 1999, 18: 1007-1010

[23] Lashkaripour GR. Predicting mechanical properties of mudrock from index parameters. Bulletin of Engineering Geology and the Environment, 2002, 61: 73-77

[24] Liu N, Sun Y, Wang Jet al.. Estimation of static Young’s modulus of sandstone types: Effective machine learning and statistical models. Earth Science Informatics, 2024, 17: 4339-4359

[25] Lundberg S.M, Lee S.-I. 2017. A unified approach to interpreting model predictions. 31st Conference on Neural Information Processing Systems (NIPS 2017), 30.

[26] Mahdiabadi N, Khanlari G. Prediction of uniaxial compressive strength and modulus of elasticity in calcareous mudstones using neural networks, fuzzy systems, and regression analysis. Periodica Polytechnica. Civil Engineering, 2019, 63(1): 104-114

[27] Mahmoud AA, Elkatatny S, Al Shehri D. Application of machine learning in evaluation of the static Young’s modulus for sandstone formations. Sustainability, 2020, 12(5 1880

[28] Mishra DA, Basu A. Estimation of uniaxial compressive strength of rock materials by index tests using regression analysis and fuzzy inference system. Engineering Geology, 2013, 160: 54-68

[29] Okewale IA, Grobler H. Influence of fabric and mineralogy on the mechanics of dolomitic rocks. International Journal of Mining and Geo-Engineering, 2022, 56 2118

[30] Palchik V, Hatzor Y. The influence of porosity on tensile and compressive strength of porous chalks. Rock Mechanics and Rock Engineering, 2004, 37: 331-341

[31] Pappalardo G, Mineo S. Static elastic modulus of rocks predicted through regression models and Artificial Neural Network. Engineering Geology, 2022, 308 106829

[32] Phani KK, Niyogi S. Young’s modulus of porous brittle solids. Journal of Materials Science, 1987, 22: 257-263

[33] Polatligil M. Investigation of the effect of normalization methods on ANFIS success forestfire and diabets datasets. International Journal of Information Technology & Computer Science, 2022, 141): 1-8

[34] Sachpazis CI. Correlating Schmidt hardness with compressive strength and Young’s modulus of carbonate rocks. Bulletin of the International Association of Engineering Geology, 1990, 42: 75-83

[35] Singh BK, Verma K, Thoke AS. Investigations on impact of feature normalization techniques on classifier’s performance in breast tumor classification. International Journal of Computer Applications, 2015, 116(19): 11-15

[36] Tsesarsky, M. 1999. Stability of underground openings in jointed chalk rock-A case study from bell shaped caverns, Bet Guvrin National Park. MSc Thesis, Ben Gurion University of the Negev, Beer-Sheva, Israel.

[37] Vasconcelos G., Lourenço P.B., Alves C.S.A., Pamplona, J. 2007. Prediction of the mechanical properties of granites by ultrasonic pulse velocity and Schmidt hammer hardness. In: North American Masonry Conference. pp 981–991.

[38] Wang JC. Young's modulus of porous materials. Journal of Materials Science, 1984, 19: 801-808

[39] Wawersik WR, Fairhurst C. A study of brittle rock fracture in laboratory compression experiments. International Journal of Rock Mechanics and Mining Sciences, 1970, 7(5): 561-575

[40] Yagiz S. Predicting uniaxial compressive strength, modulus of elasticity and index properties of rocks using the Schmidt hammer. Bulletin of Engineering Geology and the Environment, 2009, 68: 55-63

[41] Yagiz S, Sezer EA, Gokceoglu C. Artificial neural networks and nonlinear regression techniques to assess the influence of slake durability cycles on the prediction of uniaxial compressive strength and modulus of elasticity for carbonate rocks. International Journal for Numerical and Analytical Methods in Geomechanics, 2012, 36(14): 1636-1650

[42] Yesiloglu-Gultekin N, Gokceoglu CA. Comparison among some non-linear prediction tools on indirect determination of uniaxial compressive strength and modulus of elasticity of basalt. Journal of Nondestructive Evaluation, 2022, 41 10

[43] Yesiloglu-Gultekin N, Sezer EA, Gokceoglu C, Bayhan H. An application of adaptive neuro fuzzy inference system for estimating the uniaxial compressive strength of certain granitic rocks from their mineral contents. Expert Systems with Applications, 2013, 40: 921-928

[44] Yilmaz I, Sendir H. Correlation of Schmidt hardness with unconfined compressive strength and Young’s modulus in gypsum from Sivas (Turkey). Engineering Geology, 2002, 66(3–4): 211-219

[45] Yilmaz I, Yuksek G. Prediction of the strength and elasticity modulus of gypsum using multiple regression, ANN, and ANFIS models. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(4): 803-810

[46] Zhang R, Huang J, Cheng Y, Zhang Y. Machine learning based prediction for bulk porosity and static elastic modulus of Yungang Grottoes sandstone. Heritage Science, 2024, 121 359

RIGHTS & PERMISSIONS

The Author(s)