Three independent parameters to describe conventional triaxial compressive strength of intact rocks

Transcription

1 Journal of Rok Mehanis and Geotehnial Engineering. 1, (): 了? Three independent parameters to desribe onventional triaxial ompressive strength of intat roks Mingqing You hool of Energy iene and Engineering, Henan Polytehni University, Jiaozuo, 1, China Reeived 8 Otober 1; reeived in revised form November 1; aepted 1 Deember 1 Abstrat: The strengths of 1 roks ited from literatures inrease in a nonlinear way with inreasing onfining pressure against the Coulomb. The riteria with power forms like the generalized Hoek-Brown are not available for desribing the strength properties in the whole test range for Indiana limestone, Yamaguhi and Vosges, of whih the differential stresses are approximately onstant at high onfining pressures. The exponential with three parameters fits the test data of those 1 roks well with a low misfit. The three parameters are independent of the uniaxial ompressive strength (UC), the initial inreasing rate of strength with onfining pressure, and the limitation of differential stress. Key words: strength riteria; parameters; exponential ; intat roks 1 Introdution The Coulomb has been widely applied to rok engineering [1 ], and it is onservative for rok engineering after negleting the effet of intermediate prinipal stress on rok failure []. The onventional triaxial ompression tests of ylindrial speimens are arried out in laboratories all over the world, and vast data of various roks have been presented. The ompressive strength of rok inreases in a nonlinear way with inreasing onfining pressure against the Coulomb, although the expresses a reasonable physial bakground: rok has fritions and ohesions. Therefore, numerous strength riteria have been proposed and studied. The onventional strength is express as f ( ) (1) where is the maximum prinipal stress, and is the minor prinipal stress. For the test data of ylindrial rok speimens, is the onfining pressure. The parameter is the peak axial stress in Doi: 1.7/P.J.1.1. Corresponding author. Tel: ; youmq9@yahoo.om.n upported by the National Natural iene Foundation of China (177) brittle failure and a strain at the stress-strain urve to produe a speified strain, usually.%.% of permanent strain in dutile failures [, ]. In the strength riteria, there are always materialdependent parameters, whih are determined by fitting the riteria with the test data [7, 8]. You [8] evaluated the auray of 1 riteria totally, among whih three riteria have one parameter, six riteria have two parameters and seven riteria have three parameters. Usually, the more parameters the has, the lower the misfit for test data is. Therefore, one question arises here immediately: how many parameters at least are needed in one to desribe the strength of roks? This paper demonstrates three independent parameters after analyzing the relationship between the strength and the onfining pressure of limestones, s and s, respetively. Test data of 1 roks Test data of 1 roks under the onventional triaxial ompression from other previous studies are presented in Table 1 [ 1]. The original test data of olnhofen limestone, Dunham dolomite and Yamaguhi are from Mogi []. The test data of Bunt and Jinping are from Gowd and Rummel [9] and

2 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): 1 Table 1 Conventional triaxial ompressive strengths of intat roks. MPa Indiana limestone (IL) [] olnhofen limestone (L) [] Tyndall limestone (TL) [1] Dunham dolomite (DD) [] Carrara (CM) [1] Nanyang (NM) [1] Yamaguhi (YM) [] Georgia (GM) [] Wang [1], respetively. The experiment of Vosges was arried out by Bésuelle et al. [11], but the test data in Table 1 are from Fang and Harrison [1]. The test of Nanyang was arried out by You [1]. Test data of other roks are digitized from figures in hwartz et al. [, 1 1]. Average magnitude of strengths of one rok under the same onfining pressure, if existed, is used in fitting the strength. Differential stresses of 1 roks, labeled with their aronyms (Table 1), are shown in Fig.1. The test data perfetly illustrate the effet of onfining pressure on the strengths of intat roks exept for a few ones as indiated with X and Y in Figs.1(b) and (), respetively. Tyndall limestone has the same UC with Indiana limestone, but a greater inreasing rate of strength with onfining pressure. While olnhofen limestone has the highest UC in four roks of limestone and dolomite, but its strength at onfining pressure over MPa is lower than that of Dunham dolomite, as shown in Fig.1(a). The differential stresses of olnhofen limestone and Indiana limestone beome onstant at a higher onfining pressure ( ) (MPa) 1 DD L TL IL Bunt (B) [9] Pottsville (P) [] Vosges (V) [11] Jinping (J) [1] (MPa) (a) s and dolomites ( ) (MPa) NM GM * 1 X YM CM * * Note: means that the strengths are in terms of unloading proess of onfining pressure while keeping axial stress onstant. 1 1 (MPa) (b) s.

3 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): ( ) (MPa) (MPa) () andstones. Fig.1 Conventional triaxial ompressive strengths of 1 roks and fitting solutions using the exponential. Nanyang has the highest inreasing rate of strength with onfining pressure, and Yamaguhi has the lowest one in four s as shown in Fig.1(b) although their UCs are only the intermediate. The UCs of Pottsville, Jinping and Bunt shown in Fig.1() are nearly the same, ranging from MPa to MPa, but their inreasing trends of strength with onfining pressures are totally different. Fitting solutions using exponential strength You [17] proposed an exponential equation with three material-dependent parameters to desribe the onventional triaxial ompressive strength, whih is the basis to onstrut a true triaxial strength for roks: ( K 1) ( ) exp () where is the UC from the, and is the limitation of differential stress when the onfining pressure inreases to infinite, K is the initial inreasing rate of strength, and the derivative of strength to the onfining pressure at = : d K () d The has been further studied by You et al. [1, 18]. The fitting solutions using the exponential on the least absolute derivative for 1 roks, labeled with their aronyms, are presented in Table, and the urves are plotted in Fig.1. The parameter in the best fitting solution is exatly equal to the real Y V P J B magnitude of UC for nine roks. However, for olnhofen limestone and Yamaguhi with UCs of 9 and 81 MPa, respetively, the values of are 99.8 and 8.7 MPa, respetively. If the parameter in the exponential, Eq.(), is set as the real UC for olnhofen limestone and Yamaguhi, the mean misfit of fitting solution only inreases by. and. MPa, respetively. Therefore, the magnitudes of parameter presented in Table for these two roks are also their real UCs. Rok Table Fitting solutions using the exponential. Tests Maximum (MPa) (MPa) (MPa) K A = / mf (MPa) IL L TL DD CM YM GM NM B P V J Note: A is the ratio of the maximum differential stress to UC, mf is the mean misfit. The UC of Jinping predited by the exponential is 8. MPa, whih is 1.8 MPa higher than the real magnitude of 1. MPa, and provides a huge ontribution of % to the total misfit of.8 MPa. Three frature strengths during unloading proess of a onfining pressure less than MPa are presented in Table 1. They are on the upper side of the fitting solution. It may be onluded that the real UC may ome from a flawed speimen and is negleted as an abnormal datum in the fitting. Therefore, the mean misfit using the exponential for the other data dereases from. to 1. MPa. The differential stresses of olnhofen limestone are and MPa under the onfining pressures of 19 and MPa, respetively, whih exatly math the fitting solutions but out of the sope of the absissa in Fig.1(a).

4 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): There are always satter within the test data of roks and the misfits from the fitting solutions. As shown in Fig.1, the exponential an desribe the relationship between strength and onfining pressure, in a global sense, with low misfits for brittle roks and dutile roks as well, and it also an redue few abnormal data with a huge deviation [1]. The parameters employed in the exponential for 1 roks are plotted in Fig.. There is no lear relationship between UC and initial inreasing (MPa) K (MPa) andstone 1 1 (MPa) 7 1 (a) Initial inreasing rate vs. UC. andstone (b) Differential stress limitation vs. initial inreasing rate (MPa) () Differential stress limitation vs. UC. Fig. Parameters employed in the exponential for 1 roks. K andstone rate K of strength, even for the same kind of roks. The is also independent of the initial inreasing rate K. As shown in Fig.(), the limitation of differential stress inreases overall with, and there is no lear relationship between and for 1 roks, but a linear equation looks like for four s, of whih the regression result is..9 () The pratial alulation shows that the mean misfit using Eq.() is 17 MPa for four s, and the largest one is MPa for Georgia. Therefore, two parameters and K are not enough to desribe aurately the onventional triaxial ompressive strength for four s, not to say the strength property of all roks. Dimensionless relationship between strength and onfining pressure The relationship among mehanial parameters may be expressed with a dimensionless mathematial equation to reveal the intrinsi harateristis. Clearly, the strength, differential stress and onfining pressure may be unified with the UC of roks. The test data of 1 roks listed in Table 1 are plotted in Fig.. Exept Tyndall limestone and Nanyang, the other roks were loaded up to a high onfining pressure ompared with their UCs. The exponential an also be written in a dimensionless form: ( K 1) A ( A 1)exp () A 1 where A is the asymptoti value of the differential stress with inreasing onfining pressure. The unified strengths of olnhofen limestone that has the highest atual UC are lower than those of other three roks. The dimensionless urves of strength with onfining pressure as shown in Fig.(a) are totally different from those of atual strength as shown in Fig.1(a). The UC of Dunham dolomite is times higher than that of Tyndall limestone; however, the relationship between strength and onfining pressure in dimensionless forms of the two roks are similar as shown in Fig.(a). As introdued by Carter et al. [1], the Tyndall limestone is partially dolomitized with intriate mottle.

5 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): ( ) / ( ) /. DD L TL. IL / (a) s and dolomite. NM GM CM YM The maximum onfining pressure is MPa for Nanyang with a UC of 8.1 MPa; meanwhile, it is 9 MPa for Georgia with a UC of. MPa. In a dimensionless sale plotted in Fig.(b), the test sope of Nanyang is muh smaller than that of Georgia. Although the strength seems to have nearly the same trend as onfining pressure for the two s, there are signifiantly different for the parameters K and A. For Nanyang, the test range of onfining pressure is relatively smaller at its UC, and for Georgia, the strengths are signifiantly sattered. The Vosges has the lowest UC and limitation of differential stress in four s as shown in Fig.1(); however, the dimensionless strength of Vosges is ompletely higher than that of Jinping under the same onfining pressure as shown in Fig.(). The omparison is not reliable yet between Vosges and Bunt for the lak of strength under low onfining pressures. The two parameters, K and A, in the dimensionless exponential Eq.() are independent for 1 roks and for the same kind of four roks as well, as shown in Fig.. / (b) s. () andstones. Fig. Dimensionless relations of strength with onfining pressure and the optimum fitting solutions using the exponential. The dimensionless strengths of Carrara are dealt with the same way as Yamaguhi as displayed in Fig.(b), although the tests of two s from different plaes were arried out by different researhers at different times. The fitting solutions using Eq.() for these two s, or the parameter K and A, are also nearly the same. ( ) / V P J B / / Fig. Parameters in the unified exponential for 1 roks. Clearly, three parameters,, K and or A, are needed to desribe the strength properties of 1 roks of limestones, s and s, using the exponential. Disussion K andstone The normal paraboli with one parameter, whih is usually different from the real magnitude of UC, may approximately desribe strength properties of numerous roks as illustrated by You et al. [8, 19]:

6 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): 1 () Eq.() is a universal funtion for roks with various parameters. As shown in Fig., differential stresses of Jinping, Dunham dolomite and Tyndall limestone are unified with parameter of 8., and. MPa, respetively, and the mean misfit using the normal paraboli for test data of three roks is.8 MPa, or about 8.% of the UC. The optimum fitting solutions for three roks using the Hoek-Brown with two parameters [] and the exponential are with mean misfit of. and.7, respetively: ( ) / / Fig. Dimensionless strengths of three roks and the fitting solutions using various riteria (7)...exp (8). The more parameters the has, the lower misfit for test data is. The normal paraboli provides an infinite inreasing rate in strength with the onfining pressure =, and is not available for strength at low onfining pressure. On the other hand, the fitting solution using the Hoek-Brown deviates from test data at high onfining pressures. Furthermore, the Hoek-Brown with any magnitude of parameter m an not properly desribe strengths of four roks as shown in Fig.. The inreasing rate of the strength with onfining pressure redues faster than the Hoek-Brown does, whih an be written as 1 m (9) Exponential Hoek-Brown Normal paraboli DD TL J ( ) / / Fig. Fitting solutions using the exponential for four roks and the Hoek-Brown with various magnitudes of parameter m. As shown in Fig.7, the fitting solutions using the Hoek-Brown, the generalized Hoek-Brown [1] and the exponential for Yamaguhi are with mean misfit of 11.,. and.1 MPa, respetively, i.e. 1. (1) (11) exp (1) ( ) (MPa) (MPa) Fig.7 Fitting solutions for Yamaguhi using three riteria. 1 1 B YM IL L The exponential is derived to be the best one. The generalized Hoek-Brown does not orrespond to the strength property at the highest onfining pressure. There are seven riteria with three parameters as 1 7 m 9 Exponential Hoek-Brown Hoek-Brown Generalized Hoek-Brown Exponential

7 Mingqing You / Journal of Rok Mehanis and Geotehnial Engineering. 1, (): disussed by You [8]. Exept the exponential, the others have various forms in terms of power funtion as the modifiation for the linear Coulomb. The differential stresses of Indiana limestone, olnhofen limestone and Vosges have beome onstant at a high onfining pressure, as shown in Figs.1(a) and (), therefore, the riteria with power form are not available for the strengths in the whole test range. The inreasing rate of the strength with onfining pressures for the Pottsville still keeps a high value at the highest onfining pressure, and the strength indiated with letter Y in Fig.1() is unusually high. These phenomena may result from the lamping effet at the ends of ylinder speimen. For the Bunt with a porosity of 1%, the speimens beome work-hardening without the development of marosopi singular shear fratures at onfining pressures higher than 1 MPa. The axial stress-strain urve of the exhibits a prominent bulging at a onfining pressure of MPa [9]. It may be onluded that the true failure strength of the ylinder speimen at high onfining pressures is lower than the peak stress during the ompression test, and then the exponential should provide lower misfits for Pottsville and Bunt than those presented in Table. Conlusions trength properties of some roks may be roughly desribed in a definite range by riteria with one or two parameters, suh as the normal paraboli and the Hoek-Brown. The exponential with three parameters fits the test data with low misfit for roks at brittle frature and dutile failure as well. The three parameters are independent of the UC, the initially inreasing rate of the strength with onfining pressures, and the limitation of differential stress. Referenes [1] Jaeger J C, Cook N G W, Zimmerman R W. Fundamentals of rok mehanis. New York: Wiley-Blakwell, 7. [] Brady B H G, Brown E T. Rok mehanis for underground mining. rd. Netherlands: pringer,. [] Hudson J A, Harrison J P. Engineering rok mehanis: part I, an introdution to the priniples. Oxford: Elsevier, [] Ministry of Water Resoures of the People s Republi of China. peifiations for rok tests in water onservany and hydroeletri engineering. Beijing: China Water Power Press, 1 (in Chinese). [] Mogi K. Experimental rok mehanis. London: Taylor and Franis, 7. [] hwartz A E. Failure of rok in the triaxial shear test. In: Proeedings of the th U ymposium on Rok Mehanis. Rolla, Missouri: [s. n.], 19: [7] Pariseau W G. Fitting failure riteria to laboratory strength tests. International Journal of Rok Mehanis and Mining ienes, 7, (): 7. [8] You M. Comparison of the auray of some onventional triaxial strength riteria for intat rok. International Journal of Rok Mehanis and Mining ienes, 1 (ubmitted). [9] Gowd T N, Rummel F. Effet of onfining pressure on the frature behavior of a porous rok. International Journal of Rok Mehanis and Mining ienes, 198, 7 (): 9. [1] Wang B, Zhu J B, Wu A, et al. Experimental validation of nonlinear strength property of rok under high geostress. Chinese Journal of Rok Mehanis and Engineering, 1, 9 (): 8 (in Chinese). [11] Bésuelle P, Desrues J, Raynaud. Experimental haraterisation of the loalization phenomenon inside a Vosges in a triaxial ell. International Journal of Rok Mehanis and Mining ienes,, 7 (8): [1] Fang Z, Harrison J P. A mehanial degradation index for rok. International Journal of Rok Mehanis and Mining ienes, 1, 8 (8): [1] You M. Mehanial harateristis of the exponential strength under onventional triaxial stresses. International Journal of Rok Mehanis and Mining ienes, 1, 7 (): 19. [1] Carter B J, Dunan E J, Lajtai E Z. Fitting strength riteria to intat rok. Geotehnial and Geologial Engineering, 1991, 9 (1): [1] Haimson B. True triaxial stresses and the brittle frature of rok. Pure and Applied Geophysis,, 1 (/): [1] Von Kármán T. Festigkeitsversuhe unter all seitigem Druk. Z Verein Deut Ingr, 1911, : [17] You M. True-triaxial strength riteria for rok. International Journal of Rok Mehanis and Mining ienes, 9, (1): [18] You M. Charateristis of exponential strength of rok in prinipal stress spae. Chinese Journal of Rok Mehanis and Engineering, 9, 8 (8): (in Chinese). [19] You M. tudy on mathematial equation and parameters of strength riteria for rok. Chinese Journal of Rok Mehanis and Engineering, 1, 9 (11): (in Chinese). [] Hoek E, Brown E T. Underground exavation of rok. London: The Institute of Mining and Metallurgy, 198. [1] Hoek E, Brown E T. Pratial estimates of rok mass strength. International Journal of Rok Mehanis and Mining ienes, 1997, (8):