Types of internal mechanical losses in rocks in infrasonic frequency range of treatment and their conformity with the Rayleigh model

The cyclic three-point bending tests of gabbro slab samples with the recording of the hysteresis loop parameters at the frequencies of 0.05–20 Hz revealed some special features of mechanical losses within this range. The change in internal mechanical losses in rocks in the infrasonic frequency range of cyclic treatment can be described by a sum of an inversely proportional function, characterizing viscous losses, and a constant value governed by losses due to dry friction. For the test frequency range, the revealed pattern of viscous losses conforms with the Rayleigh model which explains the increase in such losses at the decreased treatment frequency. The tests and calculations find out a characteristic frequency of change in the dominant mechanism of losses, within the range from 0.04 to 0.14 Hz depending on a specific sample. Transition between the frequency ranges is accompanied by the change in the orientation of the hysteresis loop, which is conditioned by physics of the processes. At the upper frequencies of the test range, the mechanism of dry friction dominates, when the displacements and loads have similar phases. At the lower frequencies of the test range, viscous losses prevail. In this case, the load phase falls behind of the governing displacement, which changes the orientation of the hysteresis loop. The finite element-based modeling in COMSOL Multiphysics confirmed that hypothesis. 

Voznesenskii A. S., Zadorozhnyy M. Yu., Salyukov V. S., Kutkin Ya. O. Types of internal mechanical losses in rocks in infrasonic frequency range of treatment and their conformity with the Rayleigh model. MIAB. Mining Inf. Anal. Bull. 2026;(4):30-42. [In Russ]. DOI: 10.25018/0236_1493_2026_4_0_30.

A.S. Voznesenskii¹, Dr. Sci. (Eng.), Professor, Professor, e-mail: asvoznesenskii@misis.ru, ORCID ID: 0000-0003-0926-1808,
M.Yu. Zadorozhnyy¹, Cand. Sci. (Eng.), Researcher, e-mail: zadorozhnyy.m.yu@mail.ru, ORCID: 0000-0001-8776-0595,
V.S. Salyukov¹, Graduate Student, e-mail: m1605021@edu.misis.ru, ORCID ID: 0009-0003-0343-7056,
Ya.O. Kutkin¹, Cand. Sci. (Eng.), Assistant Professor, e-mail: kutkin.yo@misis.ru, ORCID ID: 0000-0003-2644-3371,
¹ NUST MISIS, 119049, Moscow, Russia.

A.S. Voznesenskii, e-mail: asvoznesenskii@misis.ru.

1. Salukov V. S., Kutkin Ya. O., Voznesensky A. S. A method for experimental determination of the attenuation coefficient in rock samples of various types and genesis. Journal of Mining Sciences. 2025, no. 3, pp. 15—24. [In Russ]. DOI: 10.15372/ftprpi20250302.

2. Sagomonova V. A., Kislyakova V. I., Tyumeneva T. Yu., Bolshakov V. A. Influence of the composition of vibration absorbing materials on the mechanical loss factor. Proceedings of VIAM. 2015, no. 10, pp. 63—69. [In Russ].

3. Kopchenkov V. G. A method for assessing energy losses in elastomers under contact dynamic loading conditions. Izvestia of the Samara Scientific Center of the Russian Academy of Sciences. 2011, vol. 13, no. 4(3), pp. 1077—1079. [In Russ].

4. Mochugovskiy A. G., Mikhaylovskaya A. V., Zadorognyy M. Y., Golovin I. S. Effect of heat treatment on the grain size control, superplasticity, internal friction, and mechanical properties of zirconium bearing aluminum based alloy. Journal of Alloys and Compounds. 2021, vol. 856, article 157455. DOI: 10.1016/j.jallcom.2020.157455.

5. Del Río L., Nó M. L., Sota A., Perez Casero I., Gómez Cortés J. F., Pérez Cerrato M., Veiga A., Ruiz Larrea I., Ausejo S., Burgos N., San Juan J. M. Internal friction associated with ε martensite in shape memory steels produced by casting route and through additive manufacturing: influence of thermal cycling on the martensitic transformation. Journal of Alloys and Compounds. 2022, vol. 919, article 165806. DOI: 10.1016/j.jallcom.2022.165806.

6. Gavriljuk V. G., Haänninen H., Smouk S. Y. U., Tarasenko A. V., Ullakko K. Internal friction in hydrogen charged CrNi and CrNiMn austenitic stainless steels. Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science. 1996, vol. 27, no. 7, pp. 1815—1821. DOI: 10.1007/BF02651931.

7. Li Z., Li K., Qian C., Ji W., Cai Z., Liu Q. Mechanism of austenite mechanical stability enhancement in Cr4Mo4V bearing steel via pulsed magnetic field treatment: insights from internal friction behavior. Journal of Materials Science. 2025, vol. 60, no. 31, article 13556. DOI: 10.1007/s10853-025-11165-1.

8. Liu Y., Dai F. A review of experimental and theoretical research on the deformation and failure behavior of rocks subjected to cyclic loading. Journal of Rock Mechanics and Geotechnical Engineering. 2021, vol. 13, no. 5, pp. 1203—1230. DOI: 10.1016/j.jrmge.2021.03.012.

9. Müller T. M., Gurevich B., Lebedev M. Seismic wave attenuation and dispersion resulting from wave induced flow in porous rocks — A review. Geophysics. 2010, vol. 75, no. 5, pp. A147—A164. DOI: 10.1190/1.3463417.

10. Li T., Pei X., Wang D., Huang R., Tang H. Nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle. Engineering Fracture Mechanics. 2019, vol. 206, pp. 330—341. DOI: 10.1016/j.engfracmech.2018.12.010.

11. Zvereva A. S., Sobisevich A. L., Gabsatarova I. P. The quality factor of the geophysical medium in the eastern zone of the North Caucasus. Izvestiya, Physics of the Solid Earth. 2024, no. 1, pp. 1—17. [In Russ]. DOI: 10.31857/S0002333724010091.

12. Lebedev A. V., Ostrovsky L. A., Sutin A. M., Soustova I. A., Johnson P. A. Resonance acoustic spectroscopy at low quality factors. Acoustical Physics. 2003, vol. 49, no. 1, pp. 81—87. [In Russ].

13. Song S., Ren T., Dou L., Sun J., Yang X., Tan L. Fracture features of brittle coal under uniaxial and cyclic compression loads. International Journal of Coal Science and Technology. 2023, vol. 10, no. 1, pp. 9—22. DOI: 10.1007/s40789 023 00564 x.

14. Pang S., Stovas A., Xing H. Frequency dependent anisotropy in partially saturated porous rock with multiple sets of mesoscale fractures. Geophysical Journal International. 2021, vol. 227, no. 1, pp. 147—161. DOI: 10.1093/gji/ggab204.

15. Ran Q., Liang Y., Zou Q., Chen Z., Zhan J., Chen L., Wu Z., Ma T. Failure mechanisms of sandstone subjected to cyclic loading considering stress amplitude effects. International Journal of Coal Science and Technology. 2025, vol. 12, no. 68. DOI: 10.1007/s40789-025-00802-4.

16. Ahmadi Naghadeh R., Liu T., Vinck K., Jardine R. J., Kontoe S., Byrne B. W., McAdam R. A. A laboratory characterisation of the response of intact chalk to cyclic loading. Geotechnique. 2022, vol. 74, no. 6, pp. 527—539. DOI: 10.1680/jgeot.21.00198.

17. Bouchaala F., Ali M. Y., Matsushima J., Jouini M. S., Mohamed A. A. I., Nizamudin S. Experimental study of seismic wave attenuation in carbonate rocks. SPE Journal. 2024, vol. 29, no. 4, pp. 1933—1947. DOI: 10.2118/218406 PA.

18. Yuan W., Dong W., Zhang B., Huo J. Viscoelasticity induced fracture behavior of rock concrete interface after sustaining creep process. Cement and Concrete Composites. 2023, vol. 136, article 104901. DOI: 10.1016/j.cemconcomp.2022.104901.

19. Petrushin G. D., Petrushina A. G. Determination of the area of the mechanical hysteresis loop using mathematical models. Industrial Laboratory. Diagnostics of Materials. 2020, vol. 86, no. 5, pp. 59—64. [In Russ]. DOI: 10.26896/1028-6861-2020-86-5-59-64.

20. Mashinskii E. I. Amplitude dependent hysteresis of wave velocity in rocks in a wide frequency range: an experimental study.  Mining Science and Technology (Russia). 2021, vol. 6, no. 1, pp. 23—30. [In Russ]. DOI: 10.17073/2500-0632-2021-1-23-30.

21. Hou W., Ma D., Li Q., Zhang J., Liu Y., Zhou C. Mechanical and hydraulic properties of fault rocks under multi stage cyclic loading and unloading. International Journal of Coal Science and Technology. 2023, vol. 10, no. 54. DOI: 10.1007/s40789-023-00618-0.

22. Zou Q., Ning Y., Zhang B., Tian S., Jiang Z., An Y. Mechanical properties and failure characteristics of sandstone under ramp loading paths. Geomechanics and Geophysics for Geo Energy and Geo Resources. 2023, vol. 9, no. 1. DOI: 10.1007/s40948-023-00574-8.

23. Huang F., Mi J. L., Yang Y. H., Dong G. F., Zhang B., Liu X. C. Morphological characteristics of hysteretic curves of soil rock mixture under stepped axial cyclic loading. Rock and Soil Mechanics. 2024, vol. 45, no. 3, pp. 674—684. DOI: 10.16285/j.rsm.2023.1063.

24. Delle Piane C., Sarout J., Madonna C., Saenger E. H., Dewhurst D. N., Raven M. Frequency dependent seismic attenuation in shales: Experimental results and theoretical analysis. Geophysical Journal International. 2014, vol. 198, pp. 504—515. DOI: 10.1093/gji/ggu148.

25. Kogan S. Ya. Seysmicheskaya energiya i metody ee opredeleniya [Seismic energy and methods for its determination], Moscow, Nauka, 1975, 152 p.

26. Rayleigh B. The theory of sound, vol. 2. Reprinted: Dover, New York, 1945, 528 p.