Abstract
The axial, crack and total volumetric strains, porosity, elastic constants, crack damage stresses, uniaxial compressive strengths, as well as fracture apertures and number of fracture traces in rock samples surface after compression were defined for different chalk, basalt, dolomite, granite, limestone and sandstone samples exhibiting negative total volumetric strain at failure. It is established that the total (summed) aperture of vertical fractures obtained on the lateral surface of rock sample is related to three characteristic strain parameters: axial strain at the onset of negative total volumetric strain, axial failure strain and negative total volumetric strain at failure. The relation is based on Descartes folium equation, where the length of the loop of folium is equal to axial strain coordinate at the onset of negative total volumetric strain. This relation shows that the total aperture increases according to power law with increasing difference between axial failure strain and axial strain at the onset of negative total volumetric strain. Simultaneously, an increase in this difference leads to an increase in the value of negative total volumetric strain at failure. It is found that a direct correlation between total aperture of fractures and negative total volumetric strain at failure is relatively weak. Nevertheless, total aperture of fractures tends to increase with increasing absolute value of negative total volumetric strain at failure. It is revealed that there is no connection between the number of fracture traces and negative total volumetric strain at failure.
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References
Besuelle P, Desrues J, Raynaud S (2000) Experimental characterization of the localization phenomenon inside a Vosges sandstone in a triaxial cell. Int J Rock Mech Min Sci 37:1223–1237
Bieniawski ZT (1967) Mechanism of brittle fracture of rock. Int J Rock Mech Min Sci 4(4):395–430
Brace WF (1978) Volume changes during fracture and frictional sliding: a review. PAGEOGH 116:603–614
Brace WF, Paulding B, Scholz C (1966) Dilatancy in the fracture of crystalline rocks. J Geoph Res 71(16):3939–3953
Brady BHG, Brown ET (1993) Rock mechanics for underground mining, 2nd edn. Chapman and Hall, London, p 571
Cai M, Kaiser PK, Tasaka Y, Maejima T, Morioka H, Minami M (2004) Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. Int J Rock Mech Min Sci 41(5):833–847
Coggan JS, Stead D, Howe JH, Faulks CI (2013) Mineralogical controls on the engineering behavior of hydrothermally altered granites under uniaxial compression. Engng Geol 160:89–102
Cundy H, Rollett A (1989) Mathematical Models. Tarquin Publications, Stradbroke, p 72
Eberhardt E, Stead D, Stimpson B (1999) Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. Int J Rock Mech Min Sci 36:361–380
Ghazvinian E, Perras M, Diederichs MS, Labrie D (2012) Formal approaches to definition damage thresholds in brittle rock granite and limestone. In: 46th Rock Mechanics/Geomechanics Symposium, 24–27 June, Chicago, USA, pp 966–974
Gray A (1997) Modern differential geometry of curves and surfaces with mathematica. CRC Press, Boca Raton, pp 77–82
Hatzor YH, Palchik V (1997) The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites. Int J Rock Mech Min Sci 34(5):805–816
Heo JS, Cho HK, Lee CI (2001) Measurement of acoustic emission and source location considering anisotropy of rock under triaxial compression. In: Sarkka P, Eloranta P (eds) Rock mechanics a challenge for society. Swets & Zeitlinger Lisse, Espoo Finland, pp 91–96
Hidalgo KP, Nordlund E (2012) Failure process analysis of spalling failure—comparison of laboratory test and numerical modelling data. Tunn Undergr Space Technol 32:66–77
Hidalgo KP, Nordlund E (2013) Comparison between stress and strain quantities of the failure—deformation process of Fennoscandian hard rocks using geological information. Rock Mech Rock Engng 46(1):45–51
Kolle JJ (1993) A model of dynamic confinement during drilling in pressurized boreholes. Int J Rock Mech Min Sci 30(7):1215–1218
Lawrence JD (1972) A catalog of special plane curves. Dover Publications, New York, pp 106–109
Martin CD (1993) Strength of massive Lac du Bonnet granite around underground openings. Ph.D. thesis, Department of Civil and Geological Engineering, University of Manitoba, Winnipeg
Martin CD, Chandler NA (1994) The progressive fracture of Lac du Bonnet Granite. Int J Rock Mech Min Sci 31(6):643–659
Nicksiar M, Martin CD (2012) Evaluation of methods for determining crack initiation in compressive tests on low-porosity rocks. Rock Mech Rock Engng 45(4):607–617
Palchik V (2010) Mechanical behavior of carbonate rocks at crack damage stress equal to uniaxial compressive strength. Rock Mech Rock Engng 43(4):497–503
Palchik V (2013) Is there link between the type of the volumetric strain curve and elastic constants, porosity, stress and strain characteristics? Rock Mech Rock Engng 46(2):315–326
Palchik V, Hatzor YH (2002) Crack damage stress as a composite function of porosity and elastic matrix stiffness in dolomites and limestones. Engng Geol 63(3–4):233–245
Palchik V, Hatzor YH (2004) The influence of porosity on tensile and compressive strength of porous chalks. Rock Mech Rock Engng 37(4):331–341
Paterson MS (1978) Experimental rock deformation-the Brittle field. Springer, New York, p 254
Pettitt WS, Young RP, Marsden JR (1998) Investigating the mechanics of microcrack damage induced under tru-triaxial unloading. In: Eurock 98, Society of Petroleum Engineering, pp SPE 47319
Schock RN, Heard HC, Stevens DR (1973) Stress–strain behavior of a granodiorite and two graywackes on compression to 20 kilobars. J Geophys Res 78:5922–5941
Simmons GF (2007) Calculus Gems: Brief Lives and Memorable Mathematics. McGraw-Hill, The Mathematical Association of America, New York, p 355
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I am grateful to the two anonymous reviewers for their critical and very useful comments.
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Palchik, V. Use of Descartes Folium Equation for Deriving a Relation between Total Aperture of Fractures after Uniaxial Compression and Strain Parameters of Different Rocks Exhibiting Negative Total Volumetric Strains. Rock Mech Rock Eng 47, 2075–2086 (2014). https://doi.org/10.1007/s00603-013-0510-8
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DOI: https://doi.org/10.1007/s00603-013-0510-8