Tunnel and Underground Space (터널과지하공간)

Korean Society for Rock Mechanics and Rock Engineering (한국암반공학회)
권호 표지
DOI QR코드

DOI QR Code

Directional Variation of Apparent Elastic Constants and Associated Constraints on Elastic Constants in Transversely Isotropic Rocks

횡등방성 암석에서 겉보기 탄성정수의 방향성 변화와 탄성정수 제약조건

  • Youn-Kyou Lee (Department of Coastal Construction Engineering, Kunsan National University)
  • 이연규 (군산대학교 건축.해양건설융합공학부)
  • Received : 2023.06.13
  • Accepted : 2023.06.20
  • Published : 2023.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

The anisotropic behavior of rocks is primarily attributed to the directional arrangement of rock-forming minerals and the distribution characteristics of microcracks. Notably, sedimentary and metamorphic rocks often exhibit distinct transverse isotropy in terms of their strength and deformation characteristics. Consequently, it is crucial to gain accurate insights into the deformation and failure characteristics of transversely isotropic rocks during rock mechanics design processes. The deformation of such rocks is described by five independent elastic constants, which are determined through laboratory testing. In this study, the characteristics of the directional variation of apparent elastic constants in transversely isotropic rocks were investigated using experimental data reported in the literature. To achieve this, the constitutive equation proposed by Mehrabadi & Cowin was introduced to calculate the apparent elastic constants more efficiently and systematically in a rotated Cartesian coordinate system. Four transversely isotropic rock types from the literature were selected, and the influence of changes in the orientation of the weak plane on the variations of the apparent elastic modulus, apparent shear modulus, and apparent Poisson's ratio was analyzed. Based on the investigation, a new constraint on the elastic constants has been proposed. If the proposed constraint is satisfied, the directional variation of the apparent elastic constants in transversely isotropic rocks aligns with intuitive predictions of their tendencies.

암석의 이방성 거동은 조암광물들의 방향성 배열과 미세균열의 분포 특성이 주된 원인이다. 특히, 퇴적암과 변성암은 횡등방성 강도 및 변형 특성을 또렷이 나타내는 경우가 많다. 그러므로 암석역학적 설계과정에서는 횡등방성 암석의 변형 및 파괴 특성을 정확하게 이해하는 것이 중요하다. 횡등방성 암석의 변형은 실내시험을 통해 측정할 수 있는 5개의 독립된 탄성정수를 이용하여 기술된다. 이 연구에서는 문헌에 보고된 실험자료를 이용하여 횡등방성 암석의 겉보기 탄성정수의 방향성 변화 특성을 분석하였다. 임의 방향으로 회전된 직각좌표계에서 겉보기 탄성정수 값을 효율적이고 체계적으로 계산하기 위해 Mehrabadi & Cowin의 구성방정식을 도입하였다. 문헌에 보고된 4개 횡등방성 암석을 선택하여 연약면의 방향 변화가 겉보기 탄성계수, 겉보기 전단 탄성계수, 겉보기 포아송비의 변화에 미치는 영향을 분석하였다. 이러한 분석 결과를 바탕으로 새로운 탄성정수 제약 조건식을 제안하였다. 제안한 제약 조건식을 만족하면 횡등방성 암석에서 겉보기 탄성정수의 방향성 변화는 직관적으로 예측되는 경향과 일치하게 된다.

Keywords

Acknowledgement

이 논문은 정부(과학기술정보통신부)의 재원으로 한국연구재단의 지원을 받아 수행된 연구임(No.2021R1F1A1048311).

References

  1. Adamus, F.P. and Kudela, I., 2019, On constraints imposed on a transversely isotropic elasticity tensor, arXiv preprint arXiv:1904.01707. 
  2. Alejano, L.R., Gonzalez-Fernandez, M.A., Estevez-Ventosa, X., and Song, F., 2021, Anisotropic deformability and strength of slate from NW-Spain, Int. J. Rock Mech. Min. Sci., 148, 104923. 
  3. Amadei, B., 1983, Rock anisotropy and the theory of stress measurements, Lecture notes in engineering series, Springer, New York. 
  4. Amadei, B., Rogers, J. D., and Goodman R. E., 1983, Elastic constants and tensile strength of anisotropic rocks, Proc. 5th ISRM Congr., Melbourne, A189-A196. 
  5. Amadei, B., Savage, W.Z., and Swolfs, H.S., 1987, Gravitational stresses in anisotropic rock masses, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 24(1), 5-14.  https://doi.org/10.1016/0148-9062(87)91227-7
  6. Aris, R., 1989, Vectors, tensors, and the basic equations of fluid mechanics, Dover Publications, New York. 
  7. Cho, J.W., Kim, H., Jeon, S., and Min, K.B., 2012, Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist, Int. J. Rock Mech. Min. Sci., 50, 158-169.  https://doi.org/10.1016/j.ijrmms.2011.12.004
  8. Chou, Y.C. and Chen, C.S., 2008, Determining elastic constants of transversely isotropic rocks using Brazilian test and iterative procedure, Int. J. Numer. Anal. Meth. Geomech., 32, 219-234.  https://doi.org/10.1002/nag.619
  9. Dambly, M.L.T., Nejati, M., Vogler, D., and Saar, M.O., 2019, On the direct measurement of shear moduli in transversely isotropic rocks using the uniaxial compression test, Int. J. Rock Mech. Min. Sci., 113, 220-240.  https://doi.org/10.1016/j.ijrmms.2018.10.025
  10. Exadaktylos, G.E., 2001, On the constraints and relations of elastic constants of transversely isotropic geomaterials, Int. J. Rock Mech. Min. Sci., 38(7), 941-956.  https://doi.org/10.1016/S1365-1609(01)00063-6
  11. Gurtin, M. E., 1981, An introduction to continuum mechanics, Academic Press, New York. 
  12. Hakala, M., Kuula, H., and Hudson, J.A., 2007, Estimating the transversely isotropic elastic intact rock properties for in situ stress measurement data reduction: A case study of the Olkiluoto mica gneiss, Finland, Int. J. Rock Mech. Min. Sci., 44, 14-46.  https://doi.org/10.1016/j.ijrmms.2006.04.003
  13. Jaeger, J.C., Cook, N.G.W., and Zimmerman, R.W., 2007, Fundamentals of rock mechanics (4th Ed.), Blackwell Publishing Ltd. 
  14. Jin, Z., Li, W., Jin, C., Hambleton, J. and Gusatis, G., 2018, Anisotropic elastic, strength, and fracture properties of Marcellus shale, Int. J. Rock Mech. Min. Sci., 109, 124-137.  https://doi.org/10.1016/j.ijrmms.2018.06.009
  15. Lekhnitskii, 1963, Theory of elasticity of an anisotropic elastic body, Holden-Day, Inc., San Francisco. 
  16. Mehrabadi, M. M., Cowin, S. C., and Jaric, J., 1995, Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions, Int. J. Solids and Struct., 32(3-4), 439-449.  https://doi.org/10.1016/0020-7683(94)00112-A
  17. Mehrabadi, M.M. and Cowin, S.C., 1990, Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl. Math., 43, 15-41. https://doi.org/10.1093/qjmam/43.1.15
  18. Pickering, D. J., 1970, Anisotropic elastic parameters for soil. Geotechnique, 20(3), 271-276.  https://doi.org/10.1680/geot.1970.20.3.271
  19. Voigt, W., 1928, Lehrbuch der Kristallphysik, Teubner, Leipzig.
  20. Yim, J, Hong, S., Lee, Y. and Min, K.B., 2022, A novel method to determine five elastic constants of a transversely isotropic rock using a single-orientation core by strip load test and strain inversion, Int. J. Rock Mech. Min. Sci., 154, 105115.