EUROCK 2012 101 Mohr Coulomb

7/22/2019 EUROCK 2012 101 Mohr Coulomb

1/12

A modified Mohr-Coulomb failure criterion for intact granites

exposed to high temperature

H. Tian & M. Ziegler

Department of Geotechnical Engineering, RWTH Aachen University, Aachen,

Germany

T. Kempka

Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam,

Germany

N. -X. Xu

Department of Civil Engineering, China University of Geosciences (Beijing), Beijing,

China

Abstract:Rocks often experience high temperatures (several hundred degrees

Celsius) due to underground operations, such as deep geological disposal of

nuclear waste, geothermal heat extraction, CO2geological storage and underground

coal gasification as well as deep mining. Laboratory studies have shown that

mechanical properties such as compressive strength, tensile strength, elastic modulus,

etc. of rocks such as granite, marble and sandstone are temperature and temperature-

history dependent. Therefore, the conventional failure criteria may not be suitable

enough under high temperature conditions. In the present study, a thermo-mechanical

modified Mohr-Coulomb failure criterion is proposed based on the extensive review

and interpretation of mechanical properties of granites exposed to high temperatures.

The deduced criterion takes into consideration the effects of thermal damage and

confining conditions. The numerical study indicates that the proposed criterion

provides a higher quality depicting rock strength under high temperatures compared

with the conventional Mohr-Coulomb criterion. Moreover, according to analyses of

the behavior of other rock materials exposed to high temperatures, this criterion is also

suitable for other rocks.

Theme:Material Models.

Keywords: high-temperature rocks, failure criterion, thermal-mechanical modifiedMohr-Coulomb model, granite.

Rock Engineering and Technology for Sustainable Underground Construction

Eurock 2012 the 2012 ISRM International Symposium, 28-30 May 2012, Stockholm, Sweden.

BeFo and ISRM, 2012


2/12

Eurock 2012 Page 2

1 INTRODUCTION

Rock-mechanical engineering in high temperature environments is of universal

interests and a challenge to scientists and engineers of different disciplines. Rock mass

may undergo high temperatures (several hundred degrees Celsius) in modern projects,

such as deep underground nuclear waste disposal (Bergman 1980, Rutqvist et al.

2008), geothermal heat extraction (Zhao 2000, Zhao 2002), geological CO2storage

(Roddy & Younger 2010) and underground coal gasification (Burton et al. 2007,

Khnel et al. 1993), as well as deep mining (Zhou et al. 2005, He 2009). Under theaction of high temperature, the micro-structures of rocks change significantly

(Dwivedi et al. 2008), new micro-cracks are developed, and pre-existing ones are

extended/widened (Den'gina et al. 1994). Meanwhile, various physical and

mineralogical changes take place within these rocks. From a macroscopic point of

view, strength and deformation characteristics of rocks exposed to high temperature

are quite different from those at room temperature. Therefore, corresponding high-

temperature rock properties are key factors for successful implementation of different

rock engineering activities.

In the last few decades, special attention has been paid to mechanical, physical

and thermal properties of crystalline rocks such as granite (e.g. Dwivedi et al. 2008,

Heuze 1983) and marble (e.g. Ferrero & Marini 2001, Zhang et al. 2008), andsedimentary rocks such as sandstone (e.g. Qin et al. 2009, Wu et al. 2005) and

limestone (e.g. Zhang et al. 2006) during and after exposure to high temperatures. The

experimental research indicates: in general, rock mechanical properties such as elastic

modulus, compressive and tensile strength, cohesive strength and friction angle

decrease with increasing temperatures. Especially, from 500 C to 600 C onwards,

the mechanical indexes may be less than 50% of those at room temperature.

Therefore, a thermo-mechanical (TM) failure criterion is needed to depict the

mechanical phenomena of rocks.

In this paper, according to the latest published data, especially from Chinese

publications not considered in the English-speaking scientific community so far, the

review of the high-temperature mechanical properties of granites is updated firstly. It

was established that slow cooling of a preheated rock in the air does not significantly

affect its strength decrease acquired during the heating process (Dmitriyev et al.

1969). Thus, all the data reviewed here were obtained either under high temperature or

after slow cooling down conditions. Furthermore, as the Mohr-Coulomb criterion is

the most commonly used in practice, we propose a TM modified failure criterion,

based on the Mohr-Coulomb with the tension cut-off criterion. At last, after the

discussion of the mechanical behavior of other rocks exposed to high temperature, this

criterion is also suitable for other rocks.

2 THERMO-MECHANICAL PROPERTIES OF GRANITES

Heuze (1983) and Dwivedi et al. (2008) have given extensive reviews of mechanical,physical and thermal properties of granites exposed to high temperatures (below the

melting point). In this section, the review is updated with new data collected from

literature, especially from Chinese publications not considered in the English-speaking

scientific community so far, covering elastic modulus, compressive strength, cohesion

and friction angle, tensile strength as well as Poissons ratio. A normalized value is

defined as the ratio of the value at a testing temperature to the value at room

temperature. The granites reviewed along with their abbreviated names and references

are listed in Table 1.


3/12

Eurock 2012 Page 3

Table 1. The reviewed granites

Granites Abbreviated names References

British granites BG McLaren & Titchel (1981), Dwivedi et al. (2008)

Charcoal granites CcG Bauer & Johnson (1979)

Chinese granites CG Xu et al. (2011)

fine-grained granites FG Xu (2003)

Henan granites HG Qiu & Lin (2006)

Indian granites IG Dwivedi et al. (2008)

Japanese granites JG Inada et al. (1997)

Juyongguan granites JyG Jiang et al. (2000)

Luhui granites LG Xi & Zhao (2010)

Man-nari granites MnG Shimada & Liu (2000)

Remiremont granites RG Homand-Etienne & Houpert (1989)

Salisbury granites SbG Heuze (1983)

Senones granites SnG Homand-Etienne & Houpert (1989)

Stripa granites StG Swan (1978)

Suixian granites SxG Wang et al. (1989)

Westerly granites WG_0 Bauer & Johnson (1979)

Westerly granites WG_1 Wong (1982)

Westerly granites WG_2 Tullis & Yund (1977)

Westerly granites WG_3 Friedman et al. (1987)

Westerly granites WG_4 Bergman (1980), Dwivedi et al. (2008)

Woodbury granites WbG Clark (1966)

Yunnan granites YG Zhu et al. (2006)

2.1 Elastic modulus

Elastic modulus is temperature and pressure dependent. The variations of normalized

elastic modulus (E/E0) with increasing temperature, irrespective of pressure, are

plotted in Figure 1. At the atmospheric pressure and temperature up to 200 C,

granites show a mixed tendency with temperature, but from that onwards,E/E0

decreases with increasing temperature for the granites reviewed (Fig. 1). Under

confining pressure conditions, theE/E0-values almost decrease with increasing

temperature (Dwivedi et al. 2008).

Figure 1. Normalized elastic modulus vs. temperature under atmospheric pressure.


4/12

Eurock 2012 Page 4

2.2 Compressive strength

In general, the compressive strength of rocks is also temperature and pressure

dependent. Figure 2 shows the values of normalized unconfined compressive strength

(c/c0) as a function of temperature. Different trends are observed up to 200 C for

the granites reviewed, whereas a decrease trend is observed from that onwards.

Values of normalized ultimate compressive strength from tri-axial compressive

tests under different confining pressure conditions are plotted in Figure 3. A general

decrease trend with increasing temperature is observed, although mixed trends ofMnG and JyG appeared in the temperature range of 200 C 400 C.

Figure 2. Normalized unconfined compressive strength vs. temperature.

Figure 3. Normalized ultimate compressive strength vs. temperature under different confining pressures(in brackets).

2.3 Cohesive strength and angle of internal friction

Normalized cohesive strength (C/C0) and normalized friction angle (/0) are plotted

in Figs. 4 and 5, respectively. The C/C0-values decrease with increasing temperature

except for an increase observed for WG_1 at 150 C compared with that at room

temperature. /0always decreases with increasing temperature for the granites

reviewed.


5/12

Eurock 2012 Page 5

Figure 4. Normalized cohesive strength vs. temperature.

Figure 5. Normalized internal friction angle vs. temperature.

2.4 Tensile strength

Figure 6 plots the normalized tensile strength (t/t0) as a function of temperature. The

values of t/t0decrease with increasing temperature for the granites reviewed, except

for SnG where an increasing trend occurs at temperatures between 400 C and 500 C.

A faster decrease trend is observed in the temperature range of 400 600 C. Beyond

the temperature range, the rate becomes slow.

Figure 6. Normalized tensile strength vs. temperature (after Dwivedi et al. 2008).


6/12

Eurock 2012 Page 6

2.5 Poissons ratio

As shown in Figure 7, different trends of values of normalized Poissons ratio () for

the granites reviewed are observed under different confining pressure and

temperatures up to 200 C. However, the variation is very low in the range of 5%.

From 200 C onwards, a decreasing trend is observed for HG. Therefore, more tests

are needed to conclude the relation between normalized Poissons ratio and high

temperature (> 800 C).

Figure 7. Normalized ultimate compressive strength vs. temperature (the values in brackets areconfining pressures).

3 TM MOHR-COULOMB FAILURE CRITERION

Ohnaka (1995) proposed an empirical shear failure strength law of Westerly granites

in the brittle to brittle-plastic transition regimes, which is especially suitable for

predicting the shear failure strength under high pressure (a few hundred MPa) and

temperature (below the melting point). Hueckel et al. (1994) presented a framework ofconstitutive modeling of thermo-brittle-plastic behavior of granites and marbles using

at least ten constants. To better depict granite strength under common engineering

pressure range (below 100 MPa) and high temperature conditions, we propose a new

TM failure criterion based on the Mohr-coulomb criterion with a tension cut-off.

3.1 General

The TM criterion suggested in this paper is for isotropic rocks and does not take into

account the effect of intermediate principal stress. It is known that the Mohr-Coulomb

model predicts a tensile strength larger than the one observed in experiments (e.g.

Ottosen & Ristinmaa (2005)). This discrepancy can be mended to some extent by the

introduction of a tension cut-off criterion. Thus, TM tension cut-off criterion isproposed here to reflect the high-temperature tensile strength behavior. The general

form of the TM modified Mohr-Coulomb linear criterion can be expressed as follows:

TTTc sin)(cos2 3131 ++= , and (1)

01 = T

t , (2)

where 1 and 3are the principal stresses; cT

, Tand T

t are temperature-dependent

cohesive strength, friction angle and tensile strength, respectively, obtained from


7/12

Eurock 2012 Page 7

interpretation of high temperature tri-axial compression/extension and the Brazilian

tests. In this paper compression is considered to be positive.

In Figure 8, the criterion functions (1) and (2) at three temperature levels (T1, T2

and T3) are plotted in (1- 3) vs. (1+ 3) space. Apparently, the criterion contains

the influence of high temperature on the tensile and compressive strength; it can

therefore better describe the strength of granites at high temperatures.

Figure 8. TM modified Mohr-Coulomb model.

3.2 Relation with triaxial experiments

Theoretically, to obtain the parameters cT

, Tand T

t in the proposed criterion, a

series of high-temperature tri-axial compression/extension and/or tensile strength tests

(e.g. the Brazilian test), should be carried out. However, as depicted in Section 2, the

three parameters usually decrease with increasing temperature in a simplified linear or

bilinear way (Fig. 9). For a bilinear relation, the transition temperature is usually in the

range of 500 C 600 C.

Figure 9. Simplified relations between the parameters in the criterion with temperature (R.T. is short forroom temperature).

3.3 Case study

A TM finite element calculation was performed for a circular underground opening

excavated at a depth of 2000 m in a hydrostatic stress state (40MPa) on a granite


8/12

Eurock 2012 Page 8

material with the density of 2.7 g/cm3. The constitutive relation is thermo-elastic

perfectly plasticity with= 0.35, linear thermal expansion coefficient = 10-5/C and

temperature-dependent elastic modulusETwhereER.T.= 50 GPa. Two failure criteria

are employed. The first is the Mohr-Coulomb (MC) criterion with c= 40 MPa, = 40

and t= 8 MPa. The second is the proposed TM modified Mohr-Coulomb (TM-MC)

criterion, whereas its corresponding parameters at room temperature are equal to those

used in the MC criterion. The relations of temperature-dependent parameters with

temperature used in the study are based on the average values in Section 2 and plottedin Figure 10a.

A quarter symmetry model with r = 1 as well as its displacement, pressure and

temperature boundary conditions are shown in Figure 10b. 8-node quadrangle

elements are used. The temperature field is calculated by transient heat transfer.

a) b)

Figure 10. a) Simplified relations between the parameters with temperature, b) geometry andboundary conditions of the computational model.

Figure 11 plots the relations between normalized stresses with the distance ratio R/rwhere R is the distance from the center of the opening (Fig. 10b) to a point at the

positive horizontal direction of A, and the corresponding temperature distribution.

Since the TM-MC criterion considers the rock strength exposed to high temperature,

the results of the TM-MC are more reliable than the MC. It is seen that the maximum

radial stress (r) and tangential stress () of the Mohr-Coulomb criterion are larger

than those of the TM-MC criterion and the difference of in the two scenarios is

larger than that of r(as shown in the circled part of Fig. 11); the yield zone of the MC

is smaller than the TM-MC. Due to thermal expansion, the stress distributions in both

scenarios are different from those of pure mechanical calculation. Thus, we suggest

the TM failure criterion should be considered when rocks are exposed to high

temperature.

ET/E

R.T.

cT/c

R.T.

T/R.T.

/

..


9/12

Eurock 2012 Page 9

Figure 11. Stresses and temperature distributions along the positive horizontal direction from A.

4 DISCUSSION

4.1 Limitations

The following assumptions have been made in the proposed TM-MC criterion:

A rock is an isotropic material,

The temperature of this criterion is in the range of room temperature to high

temperature which is less than the rock melting point,

The confining pressure corresponds to common engineering pressures,

The linear Mohr-Coulomb with the tension cut-off is assumed to fit the strength

behavior of granites at each temperature level, The effect of the intermediate principal stress is neglected.

4.2 Suitability for other rocks

It is accepted that deformation and strength of rocks exposed to high temperatures are

different from those at normal temperature, and generally, a decreasing trend with

increasing temperature is observed in experiments. Various experiments on different

rock materials conducted at different high-temperature levels show that the Mohr-

Coulomb criterion fits the data obtained from uni-axial and conventional tri-axial tests.

In addition, under the conditions of high temperature (below the rock melting point)

and common engineering pressures, the failure mechanism of rocks is unchangedcompared to that at normal conditions (Chen 2005, Li 2008). Therefore, it is

experimentally and theoretically reasonable that the TM modified Mohr-Coulomb

criterion we proposed utilizes temperature-dependent cohesion, friction angle and

tensile strength to describe the effects of high temperature on strength. Thus, even

though the criterion is created based on the review of the high-temperature mechanical

properties of granites, it is also suitable for other rocks.

5 CONCLUSIONS

A TM modified Mohr-Coulomb (TM-MC) failure criterion is proposed on basis of

extensive review and analysis of the mechanical parameters of granites exposed to

r/v

T

/v


10/12

Eurock 2012 Page 10

high temperatures (below the melting point) and common engineering confining

pressures. To take account of the effects of high temperature on rock strength, the

proposed criterion contains three temperature-dependent parameters, cT,

Tand T

t .

The numerical study shows the yield zone of an underground opening exposed to high

temperature calculated by the TM-MC criterion is larger than that determined by the

MC. Thus, under high temperature conditions, a TM failure criterion should be

involved to depict temperature-dependent rock strength. Moreover, the proposed

criterion is also suitable for other rocks.

6 ACKNOWLEDGEMENTS

The first author appreciates the funding provided by the China Scholarship Council

(CSC).

7 REFERENCES

Bauer, S.J. & Johnson, B. 1979. Effects of slow uniform heating on the physicalproperties of the westerly and charcoal granites. In 20th U.S. symposium on rock

mechanics, Austin, Texas, June 46,1979, pp. 7-18. New York: ASCE.Bergman, M.S. 1980. Nuclear waste disposal. Subsurf. Space, 2: 791-1005.Burton, E., Friedmann, J. & Upadhye, R. 2007.Best practices in underground coal

gasification. Lawrence Livermore National Laboratory Report.Chen, L.J., Wu, Z., Qin, B.D. & Gu, H.T. 2005. Mechanical characteristics and

cracking mechanism of coal roof sandstone under high temperature.Journal ofChongqing University, 28(5): 123-126 (in Chinese).

Clark, J.S.P. (Ed.). 1966.Handbook of physical constant. New York: Geologicalsociety of America.

Den'gina, N.I., Kazak, V.N. & Pristash, V.V. 1994. Changes in rocks at hightemperatures.Journal of mining science, 29(5): 472-477.

Dmitriyev, A.P., Kuzyayev, L.S., Protasov, Y.I. & Yamsbcbikov, V.S. 1969.Physical

properties of rocks at high temperatures. Moscow: Nedra Press.Dwivedi, R.D., Goel, R.K., Prasad, V.V.R. & Sinha, A. 2008. Thermo-mechanical

properties of Indian and other granites.International Journal of Rock Mechanics& Mining Sciences, 45: 303-315.

Ferrero, A.M. & Marini, P. 2001. Experimental studies on the mechanical behaviourof two thermal cracked marbles.Rock mechanics and rock engineering, 34(1): 57-66.

Friedman, M., Bauer, S.J., Chester, F.M., Handin, J. & Hopkins, T. W. 1987.Mechanical properties of rocks at high temperatures and pressures. Center fortectonophysics Texas A&M University.

He, M.C. 2009. Application of HEMS cooling technology in deep mine heat hazardcontrol.Mining science and technology, 19: 269-275.

Heuze, F.E. 1983. High-temperature mechanical, physical and thermal properties ofgranitic rocks - a review.Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 20(1):3-10.

Homand-etienne, F. & Houpert, R. 1989. Thermally induced microcracking inGranites: characterization and analysis.Int. J. Rock Mech. Min. Sci. & Geomech.

Abstr., 26(2): 125-134.Hueckel, T., Peano, A. & Pellegrini, R. 1994. A thermo-plastic constitutive law for

brittle-plastic behavior of rocks at high temperatures.Pageoph, 143(1/2/3): 483-511.

Inada, Y., Kinoshita, N., Ebisawa, A. & Gomi, S. 1997. Strength and deformation

characteristics of rocks after undergoing thermal hysteresis of high and lowtemperatures.Int. J. Rock Mech. & Min. Sci, 34(3-4): Paper No. 140.


11/12

Eurock 2012 Page 11

Jiang, H.K., Zhang, L. & Zhou, Y.S. 2000. Characteristics of AE temporal sequencesin the process of deformation and failure of granite at high pressure and differenttemperatures.Earthquake, 20(3): 87-94 (in Chinese).

Khnel, R.A., Schmit, C.R., Eylands, K.E. & McCarthy, G. J. 1993. Comparison ofthe pyromeramorphism of clayey rocks during underground coal gasification andfiring of structural ceramics.Applied clay science, 8: 129-146.

Li, D.W., Zhu, Z.D., Jiang, Z.J. & Qu, W.P. 2008. Microstructural investigation of

mechanical characteristics of marbles under different temperatures.Journal ofHehai University (Natural Sciences), 36(3): 375-378 (in Chinese).McLaren, J.R. & Titchel, I. 1981.Physical properties of granites relevant to near field

conditions in a nuclear waste depository.Harwell: AERE.Ohnaka, M. 1995. A shear failure strength law of rock in the brittle-plastic transition

regime. Geophysical research letters, 22(1): 25-28.Ottosen, N. S. & Ristinmaa, M. 2005. The mechanics of constitutive modelling.

Elsevier.Qin, B.D., He, J. & Chen, L.J. 2009. Experimental research on mechanical properties

of limestone and sandstone under high temperature.Journal of geomechanics,15(3): 253-261(in Chinese).

Qiu, Y.P. & Lin, M.Y. 2006. Testing study on damage of granite samples after high

temperature.Rock and soil mechanics, 27(6): 1005-1010 (in Chinese).Roddy, D.J. & Younger, P. L. 2010. Underground coal gasification with CCS: a

pathway to decarbonising industry.Energy Environ. Sci., 3: 400-407.Rutqvist, J., Freifeld, B., Min, K.-B., Elsworth, D. & Tsang, Y. 2008. Analysis of

thermally induced changes in fractured rock permeability during 8 years ofheating and cooling at the Yucca Mountain Drift Scale Test.International Journalof Rock Mechanics & Mining Sciences, 45: 1373-1389.

Shimada, M., & Liu, J.L. 2000. Temperature dependence of strength of rocks underhigh confining pressure.Annuals of Disas.Prev.Res.Inst.KyotoUniv., 43(B-1): 75-84.

Swan, G. 1978. The mechanical properties of stripa granite.Dept. of Energy,Lawrence Berkeley Laboratory, Earth Sciences Division.

Tullis, J. & Yund, R.A. 1977. Experimental deformation of dry westerly granite.Journal of geophysical research, 82(36): 5705-5718.

Wang, J.T., Zhao, A.G. & Huang, M. C. 1989. Effect of high temperature on the

fracture toughness of granite. Chinese journal of geotechnical engineering, 11(6):113-119 (in Chinese).

Wong, T.F. 1982. Effects of temperature and pressure on failure and post-failurebehavior of Westerly granite.Mechanics of materials, 1: 3-17.

Wu, Z., Qin, B.D., Chen, H.J. & Luo, Y. J. 2005. Experimental study on mechanical

character of sandstone of the upper plank of coal bed under high temperature.Chinese Journal of Rock Mechanics and Engineering, 24(11): 1863-1867 (inChinese).

Xi, B.P. & Zhao, Y.S. 2010. Experimental study of thermophysico-mechanical

property of drilling surrounding rock in granite under high temperature and highpressure. Chinese Jounrnal of Rock Mechanics and Engineering, 29(6): 1245-1253 (in Chinese).

Xu, X.C. 2003. Study on the characteristics of thermal damage for granite.Rock andsoil mechanics, 24(Supp.): 188-192 (in Chinese).

Xu, X.L., Gao, F., Zhou, Q. & Chen, J. 2011. Energy analysis of rock deformation and

failure process after high temperature.Journal of Wuhan University ofTechnology, 33(1): 1671-4431 (in Chinese).

Zhang, L.Y., Mao, X.B., Sun, J.F. & Lu, A.H. 2008. Mechanical properties of marbleat high temperature.Journal of Chongqing jianzhu University, 30(6): 46-50 (inChinese).


12/12

Eurock 2012 Page 12

Zhang, L.Y., Mao, X.B., Yang, Y., Meng, M.M. & Du, S.J. 2006. Experiment studyon mechanical properties of limestone at high temperature.Journal of LiaoningTechnical University, 25(Suppl.): 121-123 (in Chinese).

Zhao, Y. S. 2000. The problem of rock mechanics in heat exploitation in hot dry rock.Inprocedings of6th academy conference of Chinese rock mechanics andengineering, Wuhan, Oct. 24-27, 2000, pp. 71-74. China academic journalelectronic publishing house (in Chinese).

Zhao, Y.S., Wang, R.F., Hu, Y.Q., Wan, Z.J. & Xie, Y.S. 2002. 3D numericalsimulation for coupled THM of rock matrix-fractured media in heat extraction inHDR Chinese Jounrnal of Rock Mechanics and Engineering, 21(12): 1751-1755(in Chinese).

Zhou, H.W., Xie, H.P. & Zuo, J. P. 2005. Developments in researches on mechanicalbehaviors of rocks under the confining of high ground pressure in the depths.Advances in Mechanics, 35(1): 91-99 (in Chinese).

Zhu, H.H., Yan, Z.G., Deng, T., Yao, J., Zeng, L.J. & Qiang, J. 2006. Testing study onmechanical properties of tuff, granite and breccia after high temperatures. ChineseJournal of Rock Mechanics and Engineering, 25(10): 1945-1950 (in Chinese).

Date post:	10-Feb-2018
Category:	Documents
Upload:	ricoo90e
View:	229 times
Download:	0 times

EUROCK 2012 101 Mohr Coulomb

Documents