STABLE CRACK PROPAGATION STUDIES IN MARBLE AND TRAVERTINE

Coulomb-Naiver Maximum Shear Stress Maximum Principal Stress Maximum Elastic Strain Constant Elastic Energy of Def. Shear Failure Constant Octehedral Shearing Str. Secondary Deviation Stress Change Original Griffith Statistical Failure Theory Fracture Toughness Griffith-modified Griffith-extended Coulomb Tresca Rankine St. Venant Beltrami Mohr Huber Von Mises Griffith Weibull Irwin Mc Clintock-Walsh Murrell 1773 1864 1869 1870 1885 1900 1904 1913 1921 1939 1960 1962 1963 Jaeger and Cook, 1979 Nadai, 1950 Nadai, 1950 Nadai, 1950 Nadai, 1950 Jaeger and Cook, 1979 Jaeger and Cook, 1979 Jaeger and Cook, 1979 Griffith, 1921, 1924 Weibull, 1952 Irwin, 1960 Mc Clintock-Walsh,1962 Murrell, 1963 Table 2. Empirical failure criterias developed for rock mass (Ozkan and Ozel, 2000). Criteria Proposer Date Reference Original Bieniawski Original Hoek-Brown Modified Bieniawski Ramamurthy Carter Modified Hoek-Brown Generalized Bieniawski Generalized Hoek-Brown Bieniawski Hoek-Brown Yudhbir and Bieniawsi Ramamurthy Carter et al

Hoek, Wood and Shah Kalamaras and Bieniawski

Hoek, Kaiser and Bawden 1974 1980 1983 1986 1991 1992 1993 1995 Bieniawski, 1974 Hoek-Brown, 1980 Yudhbir and Bieniawsi, 1983 Ramamurthy, 1986 Carter et al., 1991

Hoek, Wood and Shah, 1992 Kalamaras and Bieniawski,1993,1995

Hoek, Kaiser and Bawden, 1995

PREPARATION OF MODEL SAMPLES Two different model rock units (travertine and marble) were selected to be used in this study carried out for the purpose of the determination of crack propagation characteristics. The reason of selecting these materials is that they show more homogenous structure with respect to the other rock materials. A total of 25 samples were prepared for the aimed experimental study on selected natural model rock materials. The general appearance of the double cantilever beam (DCB) experimental samples prepared for the investigation of crack propagation and the number of prepared samples are shown in Table 6. Table 3. Research topics encountered in the literature (Ozkan, 2006). Studies related to the laboratory Studies related to the field Researcher Date L-1 L-2 L-3 L-4 F-1 F-2 Griffith Evans and Faber Huang and Sijing Kabayashi and Du Singh and Pahtan Lindquvist Koksal Marji Singh Singh and Sun Şantay Wittaker et al. Chan Hu Ersoy and Walker Kau Tan Marji Vasuduvan et al. Hatzor and Palchnik Hatzor et al. Buyurgan Eberhardt et al. Hirota et al. Zhoa Blair and Cook Hanson and Ingraftea Mishnaevsky Slowik et al. Wang and Xing Khan and Al-Shayea Van de Sten and Vervoot Chang et al. Shao ve Rudnicki Altındag Liu et all. Stephasson et al. Allabadi Mater Şener Başbay Backers et al. Ozkan et al. Gokay et al. 1921 1981 1985 1986 1988 1982 1989 1990 1990 1990 1990 1992 1993 1994 1995 1995 1996 1997 1997 1997 1997 1998 1998 1998 1998 1998 1998 1998 1998 1999 2000 2001 2002 2000 2000 2000 2001 2002 2002 2002 2002 2003 2003 2004 √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ L1: Crack propagation in laboratory L2: Rock texture effect in laboratory L3: Numerical modeling in laboratory L4: Study on artificial materials in laboratory F1: Crack propagation around underground openings F2: Numerical modeling in underground Table 4. Standart crack propagation tests for rock materials (Ozkan, 2006). Test name Short name References Double Cantilever Beam Chevron Bend Specimen Chevron-Notched Semicircular Bend Specimen Modified Ring Model Single Edge Cracked Brazilian Disc Parallel Misaligned Double Edge Cracked Brazilian Discs Collinear Double Edge Cracked Brazilian Discs DCB CBS SCB MR SECBD PMDECBD CDECBD ASTM, 1972 ISRM, 1988 ISRM, 1988 Thiercelin, et al. 1986 Singh and Pathan, 1988 ISRM, 1988 ISRM, 1988 Table 5. Sample types used in the previous studies (Ozkan and Kekec, 2005). Researcher Date of study Model material studied Griffith Evans and Faber Huang and Sijing Kabayashi and Du Singh and Pahtan Koksal Şantay Singh and Sun Whittaker et al. Hanson and Ingraffea Zhao Slowik et al. Wang and Xing Khan and Al-Shayea Van de Sten and Vervoot Chang et al. Şener Başbay Ozkan et al. Backers et al. Gokay et al. 1921 1981 1985 1986 1988 1989 1990 1990 1992 1998 1998 1998 1999 2000 2001 2002 2002 2002 2003 2003 2004 Glass material Ceramic material Basalt ve syenite Ceramic ve concrete Sandstone, siltstone, coal, basalt ve granite Ankara andesite Goynuk tufa Limestone Siltstone, sandstone, granite ve basalt Concrete materials Marble, calcite, dolomite, limestone and andstone Concrete materials Limestone Limestone Limestone, sandstone Granite and marble Ankara andesite Ankara andesite Andesite, travertine, marble types ve granite Limestone Marble types Table 6. Double cantilever beam sample sizes and numbers in order to observe crack propagation. Model Rock Unit L (mm) B (mm) 2h (mm) l (mm) a (mm) # Travertine-1 220 80 35 40 44/66/88/110/132 5 Travertine-2 220 80 35 40 44/66/88/110/132 5 Travertine-3 220 80 35 40 44/66/88/110/132 5 Marble-1 220 80 35 40 44/66/88/110/132 5 Marble-2 220 80 35 40 44/66/88/110/132 5

CRACK PROPAGATION TESTS A typical experimental mechanism of experiments conducted on double cantilever beam samples which were prepared by the model rock units selected for this study is shown in Figure 1. During the experiments, load value at the hydraulic pres, horizontal and vertical deformation values were measured in every 4 seconds and video recording of the experiment was done. In the beginning of the experiment, while the load (P) given by the hydraulic press was increasing, no fracture formation was observed. But, after it was reached to a specific load value, it was determined that the horizontal fracture on the material was beginning to growing with the decrease in the load (P) supplied by the hydraulic pres. After this instant, horizontal (HD) and vertical (VD) deformations continued to increase rapidly (Figure 2-6). After experiments, fracture length (k) that had growth on each sample and deviation angles of these fractures from the artificial fracture axis (β), horizontal and vertical deformations (HD and VD) and maximum load values that started fracture growing are all given in Table 7. It is seen in Table 7 that while artificially fracture sizes (a) were increasing, no systematic change was observed in the final fracture lengths (k) measured after the experiments. Same situation is valid for the horizontal (HD) and vertical (VD) deformation values. However, with the increase in the artificially fracture lengths (a), it was observed that the loads (P) causing fracture formation decreased systematically (Table 7). A typical graph showing this situation is presented in Figure 7 for the travertine 3 rock units.

Wedge loading unit

Crack propagated Figure 1. A view from a test executed on double cantilever beam type. 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 Load,P (kN) Vertical Deformation, Uv (mm) Stable crack propagation Sample : Travertine1-3 a = 132mm 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

0,1 0,2 0,3 0,4 0,5 0,6 0,7 Load,P (kN) Horizantal Deformation, Uh (mm) Stable crack propagation Sample : Travertine1-3 a = 132mm Figure 2. Stable crack propagation for travertine-1 rock type 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35

1 2 3 4 5 Load,P (kN) Vertical Deformation, Uv (mm) Stable crack propagation Sample : Travertine2-2 a = 132mm 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35

0,01 0,02 0,03 0,04 Load,P (kN) Horizantal Deformation, Uh (mm) Stable crack propagation Sample : Travertine2-2 a = 132mm Figure 3. Stable crack propagation for travertine-2 rock type 0 1 2 3 4 5 6

1 2 3 4 5 6 7 8 Load,P (kN) Vertical Deformation, Uv (mm) Stable crack propagation Sample : Travertine3-1 a = 132mm 0 1 2 3 4 5 6

0,2 0,4 0,6 0,8 1 1,2 Load,P (kN) Horizantal Deformation, Uh (mm) Stable crack propagation Sample : Travertine3-1 a = 132mm Figure 4. Stable crack propagation for travertine-3 rock type 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 Load,P (kN) Vertical Deformation, Uv (mm) Sample : Marble1-3 a = 88mm Stable crack propagation 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

0,05 0,1 0,15 0,2 0,25 0,3 0,35 Load,P (kN) Horizantal Deformation, Uh (mm) Stable crack propagation Sample : Marble1-3 a = 88mm Figure 5. Stable crack propagation for marble-1 rock type 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

1 2 3 4 5 6 7 8 9 10 11 Load,P (kN) Vertical Deformation, Uv (mm) Stable crack propagation Sample : Marble2-1 a = 44mm 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

0,05 0,1 0,15 0,2 0,25 0,3 0,35 Load,P (kN) Horizantal Deformation, Uh (mm) Stable crack propagation Sample : Marble2-1 a = 44mm Figure 6. Stable crack propagation for marble-2 rock type 0 0,5 1 1,5 2 2,5

0,5 1 1,5 2 2,5 3 Load,P (kN) Vertical Deformation, Uv (mm) a=44mm a=66mm a=88mm a=110 mm Stable crack Stable crack propagation Rock Type:Travertine -3 Figure 7. Stable crack propagation at different artificial crack length (a) for travertine-3 rock type.

FRACTURE TOUGHNESS Every material yields in a critical stress value (σc or σt). The stress density measured in this critical stress value for a material gives the fracture toughness value of that material and it is represented with Kc(MPa√m). This fracture toughness value is a characteristic of the material and it is a constant value which is determined depending on the used calculation method. The fracture toughness equation proposed for the model sample type Double Cantilever Beam (DCB) used in this study is given in Equation 1. For the samples subjected to the experiments, fracture toughness values were calculated using equation 1 and they are presented in Table 8. 2/332 Bh PaK I  [1] Where P is load (kN), a is artificial crack length (mm), B is the widely edge (mm) and h is thickness (mm). By using the experimental results given in Table 8, artificial fracture lengths (a) and the loads (P) which caused the fracture evolution were compared. It was observed that both parameters were related with each other in an exponential behavior (Figure 8). As seen in the figure 8, each rock unit showed an exponential character but they were placed in different levels. It is considered that this is resulted from the fact that materials have different texture structures that is they have different strengths. Also, experimental results were classified according to the artificial fracture lengths (a=44, 66, 88,110 and 132 mm) and then they were compared to the fracture toughness values (Kc). As seen in Figure 9, fracture toughness values are linearly increasing depending on the strength of the material. Table 7. Some results obtained from double cantilever beam test. Model Rock Unit Parameter of analysis Travertine 1 Travertine 2 Travertine 3 Marble 1 Marble 2 k1(mm) 141 21.5 14.2 12.8 54 β1(o) 40 60 85 80 45 HD1(mm) 0.25 0.045 0.48 0.01 0.3 VD1(mm) 5.45 9.36 5.32 6.24 9.76 Number 1 Artificial Crack a=44mm P1 (kN) 0.8 0.5 0.3 2 4.4 k2(mm) 16.1 28.2 24 61 96 β2(o) 95 70 17 44 33 HD2(mm) 0.11 0.01 2.49 2.28 2.54 VD2(mm) 7.6 5.63 2.16 8.82 8.12 Number 2 Artificial Crack a=66mm P2 (kN) 0.7 0.4 0.2 1.25 2.75 k3(mm) 41 30.8 14 20.6 80 β3(o) 18 65 80 70 49 HD3(mm) 2.95 0.35 1.0 0.29 5.85 VD3(mm) 1.84 2.4 7.18 6.86 3.80 Number 3 Artificial Crack a=88mm P3 (kN) 0.5 0.3 0.1 0.45 1.35 k4(mm) 39 37 18.2 8.5 28.5 β4(o) 61 11 47 91 85 HD4(mm) 0.05 3.25 0.04 0 0.375 VD4(mm) 8.12 2.73 1.71 5.8 6.02 Number 4 Artificial Crack a=110mm P4 (kN) 0.5 0.35 0.1 0.7 1 k5(mm) 33 34 46 41 34 β5(o) 70 73 17 33 80 HD5(mm) 0.62 0.01 2.75 5.95 0.58 VD5(mm) 6.05 3.84 7.78 14.75 5.85 Number 5 Artificial Crack a=132mm P5 (kN) 0.45 0.3 0.1 0.45 1 Fracture toughness as seen from the results presented in Figure 8 and 9 can change according to the artificial crack length used in DCB tests and strength of the material tested. It is determined that P load turns into a stable behavior by decreasing related with the increasing value of the artificial crack length (Figure 8). Authors suggest the artificial crack length (a) to be bigger than 100 mm for determining fracture toughness data from DCB tests. This situation can be determined from Figure 9. It can be seen from Figure 9 that Kc values obtained from the samples of 110 and 132 mm artificial crack length are closer to each other than the other ones (a=44, 66 and 88mm).

CONCLUSION Fracture toughness is taken into consideration in many different design studies such as drilling, blasting, excavation, block caving, mine and tunnel supports and slope stability. Experimental studies have been performed by the DCB test standard which has been developed in order to determine fracture toughness. It is suggested that the artificial crack length should be minimum 100 mm according to the results obtained from the performed tests. Under these circumstances, there will be no deviation in fracture toughness values and so design engineers will be able to use these data for their design studies. Table 8. Fracture toughness values for four rock units. Rock Unit a(mm) B(mm) 2h(mm) P(kN) Kc(MPa√m) Travertine -1 Travertine -1 Travertine -1 Travertine -1 Travertine -1 Travertine -2 Travertine -2 Travertine -2 Travertine -2 Travertine -2 Travertine -3 Travertine -3 Travertine -3 Travertine -3 Travertine -3 Marble -1 Marble -1 Marble -1 Marble -1 Marble -1 Marble -2 Marble -2 Marble -2 Marble -2 Marble -2 44 66 88 110 132 44 66 88 110 132 44 66 88 110 132 44 66 88 110 132 44 66 88 110 132 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35

4689 y = 1,0428e-0,0068x R2 = 0,8958 y = 0,576e-0,00x R? = 0,717 y = 0,454e-0,01x R? = 0,799 y = 3,3796e-0,0162x R2 = 0,7392 y = 8,5727e-0,0181x R2 = 0,9038 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

20 40 60 80 100 120 140 Load,P (kN) Artificial crack length, a (mm) Travertine -1 Travertine -2 Travertine -3 Marble -1 Marble -2 Figure 8. Artificial crack length vs. load behavior for marble rock units y = 0,823x R? = 1 y = 1,234x R? = 1 y = 1,646x R? = 1 y = 2,057x R? = 1 y = 2,469x R? = 1 0 2 4 6 8 10 12

1 2 3 4 5 Fracture toughness coeeficient, Kc (M Pa? m )

Load, P (kN) a=44mm a=66mm a=88mm a=110 mm a=132 mm Figure 9. Fracture toughness coefficient characterization based on load. ACKNOWLEDGEMENTS This study was supported by The Research Foundation of Selcuk University under Project No. BAP-07701058. REFERENCES

ASTM, E-399-72T, Amer. Soc.for Testing and Materials, 1972. Tentative method of test for plane strain fracture toughness of metallic materials, Annual Book of

Standards, Part 31.

ASTM, E-399-90, Amer. Soc.for Testing and Materials, 1991. Standard test method for plane strain fracture toughness of metallic materials, Annual Book of Standards, Part 31.

Bieniawski, Z.T., 1984. Rock Mechanics Design in Mining and Tunneling; Balkema, 272. ISRM, 1981. Rock Characterization Testing and Monitoring, Suggesting Methods, Oxford ISRM commission on testing methods, 1988. Suggested Method for Determinig Fracture toughness of Rocks, Int. J. Rock Mech. Min. Sci. And Geomech. Abstr., vol. 25, pp. 71-96 ISRM commission on testing methods, 1995. Suggested Method for Determinig ModI Fracture toughness Using Cracked Chevron Notched Brazilian Disc (CCNBD)

Specimens, Int. J. Rock Mech. Min. Sci. And Geomech. Abstr., vol. 32, pp. 57-64

Ozkan, I., Ozel, R., 2000. The rock failure criteria used in the stability analyses of the rock engineering structures, Proceedings of Vth National Rock Mechanics

Symposium, October 30-31, Isparta, Turkey, pp.129-137.

Ozkan, I. and Kekec,E. 2005. The first progress report on fracture mechanism, Selcuk

University, Mining Engineering Department, 63p.

Ozkan, I. 2006. The second progress report on fracture mechanism, Selcuk University, Mining Engineering Department, 48p.