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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
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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.
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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.
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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.
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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).
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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
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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
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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.
/
..
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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
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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).
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