摘要
本文以顆粒流程式PFC2D(Particle Flow Code2D)模擬層狀岩石在巴西試驗(Brazilian test)下之張力強度(tensile strength)、破壞過程及破壞模態(failure modes)，並探討在材料層厚比與強度比變化對各巴西試驗之影響。進行層狀岩石模擬前，本文亦針對等向性岩石巴西試驗進行相關參數研究(包括：微觀參數敏感性分析、尺寸效應、位移速率與原生異向性等)。數值模擬結果顯示：層面傾角(θ)、材料強度比與層厚比皆會影響張力強度。隨著層面傾角增加，使張力強度遞減。隨材料強度比增加，在各傾角狀況下，張力強度皆有大幅遞增之趨勢。此外，改變材料層厚比所對應之張力強度則受到傾角所影響，於低傾角狀況下張力強度隨層厚比增加而遞增，而高傾角則不因層厚比影響張力強度。層狀岩石受力達尖峰強度前，僅在受力點附近有少數微裂隙生成，試體其餘處並無任何裂縫生成。過尖峰強度後，受力點附近之微裂隙迅速向試體中心處延伸。裂縫之發展亦受傾角影響甚鉅，形成不同的破壞模態，可分為四類：(1)穿層劈裂模態(Split across layer mode) (2)層間滑動模態(Sliding along layer mode) (3)混合模態(Mixed mode)與(4)層間劈裂模態(Split along layer mode)。最後，本文數值模擬結果亦與Cho et al. (2012)實驗結果進行比較，無論在強度、破壞模態與彈性常數部分結果相符。

This paper employs 2-D Particle Flow Code (PFC2D) to simulate layered rocks and focuses on the tensile strength, failure process, and failure modes under Brazilian test. Besides, this paper presents the effect of layer thickness ratio and strength ratio on tensile strength and failure modes. Before simulating layered rocks, this paper also performs the parametric studies (including the sensitive analysis of micro-parameters, size effect, displacement rate, and inherent anisotropy) of isotropic rocks under Brazilian test. Based on the numerical simulation results, the inclination angle (θ), layer thickness ratio, and strength ratio all have significant effects on Brazilian tensile strength. The tensile strength of layered rocks decrease with the increase of inclination angle, and the tensile strength would increase with the increase of strength ratio. The tensile strength of layered rock would also increase with the increase of layer thickness ratio when the low inclination angle; however, there is no significant effect when the layered rock with high inclination angle. During Brazilian test, the micro-crack of layered rock initiate at pre-peak. Besides, the micro-cracks are limited around the edge of specimen and propagate slowly until the peak is reached. After post-peak, the crack propagate rapidly and we observe four major failure modes in these numerical simulations:(1)Split across layers mode;(2)Sliding along layer mode;(3)Mixed mode;(4)Split along layer mode. In this paper, we also verify our simulation result to experimental results from Cho et al.(2012). The strength anisotropy, elastic constants and failure modes almost agree with experimental results.

1. 朱秀雯、鄭大偉、黃燦輝、薛文城與黃中杰，「無機聚合物修補水泥砂漿裂縫之力學機制探討」，第八屆海峽兩岸隧道與地下工程學術與技術研討會，台北 (2009)。
2. 李宏輝，「砂岩力學行為之微觀機制－以個別元素法探討」，博士論文，國立台灣大學土木研究所，臺北（2008）。
3. 徐書政，「異向性岩石張力強度之研究」，碩士論文，國立成功大學資源工程研究所，臺南（2000）。
4. 蕭永成，「異向性大理岩之力學性質研究」，碩士論文，國立臺北科技大學材料及資源工程研究所，臺北（2000）。
5. Amadei, B., “Rock Anisotropy and the Theory of Stress Measurements,” Springer-Verlag, New York (1983).
6. Amadei, B., “Important of anisotropy when estimating and measuring in situ stress in rock,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 33, pp. 293-325 (1996).
7. Chen, C.S., Pan, E., and Amadei, B., “Determination of deformability and tensile strength of anisotropic rock using Brazilian tests,” International Journal of Rock Mechanics and Mining Sciences, Vol.35, No. 1, pp. 43-61 (1998).
8. Cho, J.W., Kim, H., Jeon, S., and Min, K.B., “Deformation and strength anisotropy of Asan gneiss, Boryeong shale, and Yeoncheon schist,” International Journal of Rock Mechanics & Mining Sciences, Vol. 50, pp.158-169 (2012).
9. Cho, N., Martin, C.D., Sego, D.C., “A clumped particle model for rock,” International Journal of Rock Mechanics & Mining Sciences, Vol.44, pp. 997-1010 (2007).
10. Claesson, J., and Bohloli, B., “Brazilian test: stress field and tensile strength of anisotropic rocks using an analytical solution,” International Journal of Rock Mechanics & Mining Sciences, Vol. 39, pp. 991-1004 (2002).
11. Cundall, P.A. and Strack, O.D.L., “A discrete numerical model for granular assemblies,” Geotechnique, Vol. 29, pp. 47-65 (1979).
12. Dan, D.Q., Konietzky, H., and Herbst, M., “Brazilian tensile strength tests on some anisotropic rocks,” International Journal of Rock Mechanics and Mining Sciences, Vol. 58, pp. 1-7 (2013).
13. Debecker, B., and Vervoort, A., “Experimental observation of fracture patterns in layered slate,” Int J Fract, Vol. 159, Issue 1, pp. 51-62 (2009).
14. Fakhimi, A., “Application of slightly overlapped circular particles assembly in numerical simulation of rocks with high friction angles,” Engineering Geology, Vol. 74, Issue 1-2, pp. 129-138 (2004).
15. Fairhurst, C., “On the valisity of the Brazilian test for brittle materials,” International Journal of Rock Mechanics & Mining Sciences, Vol. 1, pp. 535-546 (1964).
16. Feng, X.T., Pan, P.Z., and Zhou, H., “Simulation of the rock microfracturing process under uniaxial compression using an elasto-plastic cellular automaton,” International Journal of Rock Mechanics & Mining Sciences, Vol. 43, Issue 7, pp. 1091-1108 (2006).
17. Fowell, F.J, “Suggested method for determining mode I fracture toughness using Cracked Chevron Notched Brazilian Disc (CCNBD) specimens,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 32, No. 1, pp. 57-64 (1995).
18. Ghazvinian, A., Sarfarazi, V., Schubert, W., Blumel, M., “A study of the failure mechanism of planar non-persistent open joints using PFC2D,” Rock Mechanics and Rock Engineering, Vol. 45, Issue 5, pp. 677-693 (2012).
19. Goodman, R.E., “Introduction to Rock Mechanics,” 2nd edn. John Wiley, Singapore (1989).
20. Hondros, G., “The evaluation of Poisson’s ratio and modulus of materials of a low tensile resistance by Brazilian test with particular reference to concrete,” Journal of Applied Science, Vol. 10, pp. 243-268 (1959).
21. ISRM, “Suggested methods for determining tensile strength of rock materials,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 15 No. 3, pp. 99-103 (1978).
22. Itasca Consulting Group Inc., PFC2D (Particle flow code in 2 dimensions), Verson 4.0 (manual), Itasca Consulting Group Inc., Minneapolis, MN:ICG (2008).
23. Lekhnitskii, S.G., “Anisotropic plates,” Gordon and Breach Scientific Publications, New York (1968).
24. Liao, J.J., Yang, M.T., and Hsieh, H.Y., “Direct tensile behavior of a transversely isotropic rock,” In J Rock Mech. Min. Sci.,Vol. 34, No. 5, pp. 837-849 (1997).
25. Liu, W.C., Tien, Y.M., and Juang, C.H., “Numerical simulation for layered rock under Brazilian test,” 46th U.S. Rock Mechanics / Geomechanics Symposium, Chicago, USA, 24–27 June 2012 .Paper No. 12-492 (2012).
26. Liu, W.C., Tien, Y.M., Juang, C.H., and Lin, J. S., “Numerical investigation of crack propagation and failure mechanism of layered rocks,” 47th U.S. Rock Mechanics / Geomechanics Symposium, San Francisco, USA, 23–26 June 2013 .Paper No. 13-673 (2013).
27. Ma, G.W., Wang, X.J., and Ren, F., “Numerical simulation of compressive failure of heterogeneous rock-like materials using SPH method,” International Journal of Rock Mechanics & Mining Sciences, Vol. 48, Issue 3, pp. 353-363 (2011).
28. Mahabadi, O.K., Grasselli, G., and Munjiza, A., “Numerical modelling of a Brazilian disc test of layered rocks using the combine dfinite-discrete element method,” Proceedings of the 3rd CANUS Rock Mechanics Symposium, Toronto (2009).
29. Margielewski, W., “Structural control and types of movements of rock mass in anistropic rocks: Case studies in the Polish Flysch Carpathians.” Geomorphology, Vol. 77, pp. 47-68 (2006).
30. Mellor, M., Hawkes, I., “Measurement of tensile strength by diametral compression of discs and annuli,” Engineering Geology, Vol. 5, Issue 3, pp. 173-225 (1971).
31. Nakashima, S., Taguchi, K., Moritoshi, A., and Shimizu, N., “Loading conditions in the Brazilian test simulation by DEM,” 47th U.S. Rock Mechanics / Geomechanics Symposium, San Fransico, USA, 23–26 June 2013 .Paper No. 13-515 (2013).
32. National Geoscience Database of IRAN, 取自http://www.ngdir.ir/geolab/GeoLabExp.asp?PExpCode=7275&PID=2677 (2012).
33. Park, B., and Min, K. B., “Discrete element modelong of transversely isotropic rock,” 47th U.S. Rock Mechanics / Geomechanics Symposium, San Fransico, USA, 23–26 June 2013 .Paper No. 13-490 (2013).
34. Potyondy, D.O., and Cundall, P.A., “A bonded-particle model for rock,” International Journal of Rock Mechanics & Mining Sciences, Vol. 41, pp. 1329–1346 (2004).
35. Stefanizzi, S., Barla, G., and Kaiser,P.K., “Numerical modeling of rock mechanics tests in layered media using a finite/discrete element approach,” International Association for Computer Methods and Advances in Geomechanics, India (2008).
36. Tavallali, A., and Vervoort, A., “Effect of layer orientation on the failure of layered sandstone under Brazilian test conditions,＂International Journal of Rock Mechanics & Mining Sciences, Vol. 47, Issue 2, pp. 313-322 (2010a).
37. Tavallali, A., and Vervoort, A., “Failure of layered sandstone under Brazilian Test conditions: effect of micro-scale parameters on macro-scale behaviour,＂ Rock Mech. Rock Eng. Vol. 43, pp. 641-653 (2010b).
38. Tien, Y.M., and Kuo, M.C., “A failure criterion for transversely isotropic rocks,” International Journal of Rock Mechanics and Mining Sciences, Vol. 38, No. 3, pp. 399-412 (2001).
39. Tien, Y.M., Kuo, M.C., and Juang, G.H., “An experimental investigation of the failure mechanism of simulated transversely isotropic rock,” In J Rock Mech. Min. Sci., Vol. 43, pp. 1163-1181 (2006).
40. Vutukuri, V.S., Lama, R.D., and Saluja, S. S., “Handbook on Mechanical Properties of Rocks,” Vol. I (1974).
41. Yanagidani, T., Sano, O, Terada, M., and Ito, I., “The observation of cracks propagating in diametrically-compressed rock disks,” Int J Rock Mech Min Sci Geomech Abstracts, Vol. 15, Issue 5, pp. 225-235 (1978).
42. Ye, J., Wu, F.Q., and Sun, J.Z., “Estimation of the tensile elastic modulus using Brazilian disc by applying diametrically opposed concentrated loads,” International Journal of Rock Mechanics & Mining Sciences, Vol. 46, Issue 3, pp. 568-576 (2009).
43. Yu, Y., Zhang, J., and Zhang, J., “A modified Brazilian disk tension test,” International Journal of Rock Mechanics & Mining Sciences, Vol. 46, pp. 421-425 (2009).
44. Zhu, W.C., and Tang, C.A., “Numerical simulation on of Brazilian disk rock failure under static and dynamic loading,” International Journal of Rock Mechanics & Mining Sciences, Vol. 43, Issue 2, pp. 236-252 (2006).
45. Zhong, X.P., and Wong, L. N. Y., “Loading rate effects on cracking behavior of flaw-contained specimens under uniaxial compression,” International Journal of Fracture, Vol. 180, pp. 93-110 (2013).