Comparison of Post-Failure Strength of Micro-cracked Marble with Hoek-Brown Failure Criterion

Failure criteria of rock mass is the most important base for designing of surface and underground structures. However, behavior of jointed rock mass and its failure criteria are the controversial subjects of rock mechanics. Main reasons for this discussion are problems during or after the geotechnical application. However, some of the experimental and theoretical approaches are often preferred as they are practical, compatible with engineering considerations, and assist in decision-making process. On the other hand, the differentiation in the scale of the geosystem, which varies depending on the scale of geotechnical application, building process, and time, means that the failure conditions will also change. It is clear that the Mohr-Coulomb failure criterion, which is widely used in practice, cannot exactly represent discontinuous geo-environments (fractured rock) consisting of joint systems. Since the rock generally has a discontinuous character, it has been researched since the 1970s, and the Hoek-Brown failure criterion, put forth in the 1980s and modified many times until today, is widely accepted in application. Nevertheless, it is known that the empirical parameters used in this failure criteria proposed for different types of rocks are also open to discussion. In this paper, the results of the mechanical tests conducted on the previously-fissured model material, which is physically similar to rock mass are discussed. Marble samples whose grain boundaries were disturbed by cyclic thermal treatment were used as the model material. Post-failure curves of model material obtained from continuous failure state triaxial tests were compared with Hoek-Brown Failure Criteria. In conclusion, it was shown that the failure envelopes representing intergranular failure in the post-failure phase were similar and comparable to the Hoek-Brown Failure Criterion. However, it is found out that the post-failure strength in low confining stress may be lower than that of estimation by the Hoek-Brown criterion. Experimental studies have also shown that intergranular failure will develop among structural weaknesses in rock masses, and therefore the strength parameters commonly used in practice will depend on the size of geo-application.

In practice, strength criterions, commonly used in rock mechanics, are expressed by principal stresses. The classical Mohr-Coulomb failure envelope is defined by a linear line characterized by cohesion ( ) and angle of internal friction () (Equation 1).

= +
(1) When it is defined by principle stresses; Based on Equations 1 and 2, the ratio of compressive (c) to tensile (t) strength is as follows; If the angle of internal friction ( ) is taken as 45 and 60 degrees, the ratio between compressive and tensile strength will be found 5. 8 Where s, mb and are the rock mass material constants, and 1 , 3 and are the major and minor principal stresses, respectively; is the unconfined compressive strength of intact rock. The mb was introduced as Hoek-Brown constant for the rock mass, s and a are the constants which depend upon the characteristics of rock mass. The original mi value obtained as curve fitting parameters from triaxial testing of intact rock and mb is a reduced value of mi, which accounts for the strength reducing effects of rock mass. Hoek and Brown [5] re-examined the relationships between GSI, , and , and introduced D to account for near surface blast damage and stress relaxation. The last scaling relationships for these parameters were reported as: [ Where, the material constants for intact rock are donated by , = 1 and = 0.5 values are defined as material parameters corresponding to competent rock and D as the degree of disturbance to which the rock mass has been subjected to blast damage and stress relaxation. The GSI is a system of rock mass characterization that was developed, by [7] and [8], to correlate the failure curve to engineering geological investigations in the site. It was extended to cover folded and tectonically sheared rock masses in a series of papers by [9], [10]- [12], and [13].
Hoek and Brown have pointed out that above equations are valid for rock masses contain of interlocking angular elements in which the process is dominated by sliding and rotating without a real deal of intact rock failure, under confining stresses [14].
Hoek and Brown [6] have been taken into consideration of laboratory triaxial tests for more than 14 intact rocks. They have used the peak strengths ranging from 40 MPa to 580 MPa of rock samples. As a result, they have defined a non-linear criterion based on this review and the mi parameter was derived from best fit linear regression. The coefficient of determination (R 2 ) ranged from 0.68 to 0.99.
Zhao [15] has issued Mohr-Coulomb and Hoek-Brown linear regressions by a series of laboratory tests. He showed that the intact rock strength under dynamic loads can be better represented by the non-linear Hoek-Brown Failure Criterion. Ghazvinian et al [16] also suggested the non-linear Hoek-Brown provided a better fit than the linear Mohr-Coulomb. Pariseau [17] concluded the non-linear Hoek-Brown envelope gave a significantly better fit (low to high confining pressures) than Mohr-Coulomb envelope.
Eberhardt [18] noted that Hoek [19] recommends, where possible, the Hoek-Brown criterion be applied directly. However, given that many geotechnical design calculations are written for the Mohr-Coulomb Failure Criterion, it is often necessary to calculate equivalent rock mass cohesion, c, and friction angle, , values from the Hoek-Brown parameters.
Moreover, most practitioners have an intuitive feel for the physical meanings of cohesion and friction, which is not the case for , s and a. The quantitative conversion of Hoek-Brown to Mohr-Coulomb parameters is done by fitting an average linear relationship to the non-linear Hoek-Brown envelope for a range of minor principal stress values defined by t<3< ' 3max. Brown warns against applying programs that calculate equivalent Mohr-Coulomb parameters too automatically without thinking clearly about the range of effective normal stress that applies to the case being considered [20]. If high values of 3max are used, then the equivalent effective cohesion value may be too high and the equivalent effective friction angle too low.
This study takes as model material with the bonds between grains boundaries previously loosened by thermal expansion in consideration having similar structural properties of rock masses. The strength envelopes representing both failure and post-failure phases of test specimen were obtained from continuous failure state triaxial tests [21] and compared with Hoek-Brown Failure Criterion.
For this purpose, some of the cylindrical specimens of Carrara Marble with grain sizes varying between 95 and 150m was kept as an original while others were exposed to a number of thermal cycles each corresponding to 12 hours of heating and cooling durations. The number of heating-cooling cycles were designated as 0, 1, 2, 4, 8 and 16, refer to specimen categories. The original specimens correspond to specimen category 0 Pamukkale Univ Muh Bilim Derg, 26(8), [1365][1366][1367][1368][1369][1370][1371][1372]2020 (Special Issue of the National Symposium on Engineering Geology and Geotechnics 2019-ENGGEO'2019) Y. Mahmutoğlu, G. Şans were not exposed to thermal treatment. The maximum temperature (600 ºC) specimens were exposed to was determined with reference to differential thermal analysis results [2]. The tests were repeated on 2 specimens each corresponding to the same thermal cycles. The results of unconfined compression tests on testing material are given in Table 1 and the stress-strain curves demonstrating also post failure behaviour under uniaxial compression are also shown in Figure 2.   As it seen from Figure 2 and Table 1, both strength and modulus of elasticity are decreasing after thermal treatment. This was a consequence of the opening of micro-cracks during heating and cooling cycles because of the anisotropic thermal expansion of calcite grains constituting Carrara Marble (Figure 1).
The boundaries of calcite grains in Carrara Marble are irregular, partly curvilinear and planar. Therefore, the material used in experimental study resembles rock mass comprising single lithological type of interlocked blokes with unfilled and irregular discontinuities.
Continuous failure triaxial tests [21] and [27], [28] were repeated on two specimens representing each category to obtain peak and post-failure curves. During the tests, constant axial strain rate was applied as 1.5.10 -5 sec -1 and a servocontrolled electro-hydraulic pump maintained the confining pressure manually. As for axial loading, a stiff servo-controlled loading machine with an adjustable capacity of 60-3000 kN was used. The soft and pressure-sensitive elastic rubber membrane used in triaxial test was replaced in each test. The stages of tests are depicted with capital letters in Figure 3. At first stage, (stage A) constant confining pressure was kept as (3) 0.5 MPa until the beginning of failure. In stage B, it was gradually increased to 5 MPa in a way to linearly increase the axial stress (1) in stress-deformation diagram as parallel as to the pre-failure curve. In third stage (C), the specimen was exposed to failure continuously under maximum confining pressure of 5 MPa. In other word, the confining pressure was kept constant at its maximum value until the axial stress reached a residual value in the post failure stress-strain curves. At last stage of triaxial test (D), the confining pressure was regularly decreased down to zero. In other words, the envelopes correspond to continuous failure state and postfailure phases were recorded on principal stress plane (Figure 4).
In all tests, the initial confining stress was kept constant as 0.5 MPa. Finally, for each sample, both of the curves corresponding to pre and post failure strength were obtained as graphs on principal stresses plane (Figure 4b).
The tests demonstrated that the failure develops throughout surface similar cone and the calcite grains scattered into powder due to the intergranular failure ( Figure 5).

Strength envelope at continuous failure state
In stress-strain curves in Figure 4a, the first phase of the test until the point of failure is shown with thin and continuous lines. After this point, during in continuous failure state, confining pressure was regularly increased until 5 MPa, which is shown with dotted lines. Finally, the thick and dark lines indicate post-failure behaviours. Figure 4b shows the envelopes representing continuous failure and post-failure phases. Strength curves of failed specimens obtained during lowering the confining pressure is indicated by thick and dark lines in this figure.
As it shown in Figure 4, the results of triaxial tests generally conform to those obtained in uniaxial tests (Figure 2). The curves demonstrating continues failure states of each specimen are coherent with the level of disturbance, descending downwards by the number of thermal cycles. The peak strength envelops refer to failure (Figure 4b) are almost parallel and indicate a linearity. Therefore, failure envelope of intact rock having micro-fissures can also be explained by Mohr-Coulomb Failure Criteria.
The peak strength parameters (cP, ) can be calculated on the basis of Coulomb Failure Criteria (Eq. 8). Continuous failure envelopes appear to be roughly parallel with one another and therefore their gradient (m=1/3) can be taken as one same value for all. This demonstrates that, even though the bonds between grains were partly broken down through thermal treatment, there was no significant change in the angle of internal friction, but cohesion decreased by up to % 50.
Where * is the point where peak strength envelope intersects the major principal stress axis (1), and and shear strength parameters (Figure 3).

Strength envelope of post-failure phase
On the other hand, the envelopes indicated by dark thick lines in the lower part of Figure 4b After these comparisons, for all the equations representing residual strength curves of different sample categories, a high coefficient of determination (R 2 >0.99) was found. As is evident from Figure 4b, the higher the number of thermal cycles is, the more the loss of strength is and the strength curves corresponding to failed specimens generally descend downward. As a main result, it was found out that all of these post-failure strength curves can be defined by following equation; Where, the coefficients af and mf refer to the form of these curves, and cf corresponds to compressive strength of failed specimen.
The calculated values of af and mf as well as the ratio of cf/c * are presented in Table 2 and the correlation between cf and c * is also shown in Figure 6.
Likewise, best-fits denoting to the residual strength curves and corresponding failure envelope obtained for the same specimens by using Hoek-Brown Failure Criterion were compared as shown in Figure 7.  In other word, the comparisons demonstrated that the envelopes representing post-failure strength of tested samples do not match but residual strength curves bear similarity with Hoek-Brown Failure Criterion. In addition, estimated strength by this criterion is found to be lower in high confining pressure, but it is higher in low confining pressure (shallow depth) than those derived from continuous failure state triaxial tests.
Strength curves representing post-failure phase (Figure 7) and Equation 9 both clearly show the changes of residual shear strength parameters by confining pressure. Residual friction angles (r) and cohesions (cr) separately calculated by using Equation 8 and values of these parameters corresponding to the same interval of confining pressures are given in Table 3.  For the description of relationships between residual strength parameters (r and cr) and confining pressure, they were correlated in Figure 8 and Figure 9 respectively.
As shown in these correlations, it is obvious that both of these relationships can be described well by logarithmic functions. While internal friction angle decreases abruptly by increasing confining pressure, there is a considerable increase in residual cohesion. Although, equations describing the relations are similar for all specimen categories, however, highest coefficients of determination (R 2 >0.93 for internal friction angle and R 2 >0.94 for cohesion) are obtained for original specimen (Figure 8) Where Af and Bf empirical constants for residual internal friction angle and Ac and Bc for residual cohesion. The ranges of these parameter obtained for each specimen categories are shown on curves in Figure 8 and Figure 9.

Acknowledgement
Author of this paper, a visiting researcher in ETH Zurich Institute for Geotechnical Engineering, expresses his sincere thanks to the Head of the institute, Prof. K. Kovári for hosting the author and providing financial support to this research.