A Parabolic Failure Criterion for Transversely Isotropic Rock: Modification and Verification

Wang, Ding-Jian; Tang, Huiming; Shen, Peiwu; Cai, Yi

doi:https://doi.org/10.1155/2019/8052560

Mathematical Problems in Engineering

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Abstract Introduction Background Results Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8052560 | https://doi.org/10.1155/2019/8052560

A Parabolic Failure Criterion for Transversely Isotropic Rock: Modification and Verification

Ding-Jian Wang,^1,2Huiming Tang,¹Peiwu Shen,¹and Yi Cai³

Academic Editor: Michele Betti

Received31 Jul 2018

Revised25 Jan 2019

Accepted05 Feb 2019

Published04 Mar 2019

Abstract

It is of great significance to develop a failure criterion that can describe the orientation-dependent behavior of transversely isotropic rocks. This paper presents a simplified parabolic model that is successful in predicting the strengths of rocks under different confining pressures and bedding angles. The model is a modified version of the normal parabolic criterion for intact rocks. The two orientation-dependent parameters ( and ) in the model show trigonometric relationships with the bedding angle, and they can be readily determined through uniaxial and triaxial compression tests. The shape of the failure envelope is determined by , and only affects the level of rock strength. With application to 446 experimental data, the predicting results by the parabolic criterion are highly consistent with the experimental data, and the predictive capacity of the proposed criterion is better than those of the McLamore-Gray and Tien-Kuo criteria. Besides, the prediction errors for the high confining pressure condition and the bedding-sliding failure mode are smaller than those for the low confining pressure and the non-bedding-sliding failure. Moreover, the prediction error almost remains steady with the decrease of data set, indicating that the proposed criterion is of high precision even if the experimental data are limited.

1. Introduction

Determining the strength of rock is essential for engineering design and construction [1–3]. Rock strengths under different stress states are usually estimated by using a failure criterion with the assistant of triaxial compression tests. A powerful criterion is qualified to predict the rock strength with parameters easy to determine. Numerous failure criteria were proposed by many researchers over the past few decades, and some of them have been successfully applied to the engineering practice [4–12]. In recent years, several reports on ‘‘Suggested Methods for Failure Criteria’’ were organized by International Society of Rock Mechanics (ISRM) to standardize different failure criteria targeted towards the engineering practice [2]. These suggested methods include the Mohr-Coulomb [13], Hoek-Brown [14], 3D Hoek-Brown [15], Drucker-Prager [16], and Lade and modified Lade 3D [17] criteria.

Although they provided deep insight into the mechanical behaviors of rocks, these studies mainly focused on the intact rocks without any defects. However, natural rock masses consist of the intact rock matrix and multiple fractures, resulting in strong mechanical anisotropies [1, 18–20]. The transversely isotropic rocks, such as the stratified sedimentary rocks and foliated metamorphic rocks, are common rock types containing parallel bedding planes, and they present orientation-dependent behaviors due to the presence of bedding planes. To incorporate the effect of bedding planes into the mechanical behaviors, some failure criteria on the transversely isotropic rock were developed in previous studies. According to the assumptions and techniques, these criteria can be divided into three groups: mathematical continuous models, empirical continuous models, and “discontinuous weakness planes based” models [21]. In the first group, the transversely isotropic rock is assumed as continuous material, and strength anisotropy is quantitatively derived by using a mathematical technique along with the kind of material symmetries [22–24]. In the second group, models are generally extended from isotropic criteria with certain orientation-dependent parameters incorporated [25–29]. Contrasting to the former two groups, the third one basically assumes that the failure of an anisotropic rock is respectively due to the rupture of rock matrix (β approaches 0° or 90°) and bedding plane (β is located in the middle interval). Herein, orientation β is the angle between the major principal stress and the bedding plane. Therefore, distinct equations are adopted to describe the strengths of the two failure modes with different β values [30–33].

We proposed a modified parabolic criterion to predict the strength of the transversely isotropic rocks. The new model comprising orientation-dependent parameters is developed from the normal parabolic criterion for intact rock [11]. We also present the procedures for predicting rock strengths by using the proposed model. Then, the predicting results are compared to the experimental data with error variation evaluated. Finally, the parametric sensitivity and the predictive capability are analyzed.

2. Background

2.1. Rock Anisotropies

In terms of the formation mechanism, the anisotropies of rock masses are triggered by three factors those are protogenesis, tectonics, and exogenesis.

The protogenetic anisotropy formulates in the primary diagenesis process of rocks. The sedimentary rocks exhibit transversely isotropic behaviors because they contain multiple bedding planes, disconformity planes or weak intercalations. The intrusion-surrounding rock interfaces and condensation joints generate during the process of magmatic exhalation, and they lead to the protogenetic anisotropy of magmatic rocks. The anisotropy of metamorphic rocks originates from the inner smallish schistosities, gneissic schistosities or foliations. The anisotropic behavior of metamorphic rocks also depends on their protoliths.

The tectonic anisotropy is triggered by the tectonic movement that ruptures intact rocks into fissured masses containing multiple joints, faults, folds, intercalated sliding, cleavage, etc. Tectonic discontinuities have close correlations with the construction lines. It should be noted that the failures of rocks frequently happen in engineering due to these tectonic discontinuities.

The exogenic anisotropy is a common characteristic for rocks in the shallow strata. It formulates under the long-term actions of weathering, corrosion, and gravity. Although the exogenic discontinuities (e.g., weathering fractures, silted intercalations, and unloading cracks) are widely distributed in rock masses, they are of poor continuity and limited extension range as compared to tectonic discontinuities.

To deeply study the mechanical behaviors of heterogeneous rock, Singh et al. [34] classified the rock anisotropy as two types: the cleavage/planar anisotropy and bending plane anisotropy. The former one was further subdivided into the U-shape and undulatory anisotropies. This classification is based on three characteristics: the maximum and minimum strengths, strengths at various directions, and the shape of the anisotropic curve plotted for - β. Herein, is the uniaxial compressive strength of rock.

2.2. Existing Empirical Failure Criteria

In the past few decades, various failure criteria were developed by researchers to predict the strengths of the transversely isotropic rocks. For the sake of comparison with the forthcoming proposed model, two existing criteria with good predictive capability are briefly introduced below.

2.2.1. McLamore-Gray Criterion

McLamore and Gray [26] used the Mohr-Coulomb criterion with orientation-dependent cohesion and friction coefficient to describe the strengths of the anisotropic rocks. The modified criterion is expressed as follows:where and are respectively the friction angle and the cohesion at . where and , and are constants for the description of the friction coefficient variation for loading orientations and , respectively. is the loading orientation corresponding to the minimum cohesion. Similarly, and , and are constants for the description of the cohesion variation for the loading orientations and , respectively. is the loading orientation corresponding to the minimum friction. Finally, the two exponents and , named as the factors of anisotropy type, have typical values varying from 1 to 3 for plane anisotropy like cleavage and schistosity and from 5 to 6 for linear structures.

2.2.2. Tien-Kuo Criterion

Jaeger [30] proposed a physical-based criterion for anisotropic rocks with parallel discontinuities. The criterion is composed of two equations those are appropriate for the sliding failure along the discontinuity and the fracture failure of rock matrix, respectively. However, Jaeger’s criterion results in the same predicting strength values as β approaches 0° and 90°, which does not tally with the experimental results [33]. Tien and Kuo [32] developed a modified version of Jaeger’s criterion with the concept of elasticity theory incorporated. They followed Jaeger’s criterion for the description of the sliding failure, that isFor nonsliding failure, the elasticity theory and the strain-dominated criterion were combined to derive the strength equation that is expressed aswhere is the strength ratio expressed asand is the transversal anisotropy parameter. When the orientation angle β falls in the range of 60° ~90°, the value of becomes negligible, and hence (4) is rewritten asHence, can be computed using (6) along with three triaxial tests on specimens with at 0°, 90° and an angle in the range of 60° ~90°.

2.3. Approaches for Performance Evaluation

Three different error measurements are usually used to evaluate the performance of a proposed model - the regression R-square value (), discrepancy percentage or relative difference () and average absolute relative error percentage (AAREP), expressed as follows:where is the number of available experimental data, and , respectively, represent the maximum principal stress of experimental values and predicted results by the criterion, and is the average of the experimental data.

2.4. Database

Abundant experimental data including 358 triaxial and 88 uniaxial compression tests corresponding to 12 types of rocks are collected from published articles [35–37] to investigate the performance of the proposed criterion and characteristics parameters. As shown in Figure 1, all of the rocks are processed for cylindrical specimens with different bedding angles. Then uniaxial and triaxial compressive tests under different levels of confining pressure are conducted to obtain the failure strengths. Therefore, two factors affect the strength of anisotropic rocks and they should be taken into account for the failure criterion.

3. Parabolic Criterion for Transversely Isotropic Rock

3.1. Derivation of the Parabolic Criterion

The Mohr-Coulomb criterion is widely used to describe the mechanical behaviors of rock and soil. The criterion has a succinct form within which the relationship between shear and normal stresses is linear. However, experimental results indicate that the linear correlation is inaccurate especially under high normal stress condition. As a matter of fact, the increase ratio of the rock strength gradually decreases by increasing the normal stress, which means the gradient of the failure envelope tends to descend in higher stress condition. Concerning this characteristic, several parabolic criteria were proposed to fit the strength data of the uniaxial and triaxial compression tests on isotropic rocks [11, 12]. Since these criteria show high accuracy in previous studies, the parabola form will also be adopted and further improved to characterize the transversely isotropic rocks.

As shown in Figure 2, if the parabola is directly used to describe testing data in the shear strength - normal stress coordinate, the fitting equation can be written aswhere and are undetermined material parameters. When the Mohr circles under the uniaxial compressive and tensile conditions are tangent to the strength envelope, the expressions of , can be deduced:where is the ratio of the uniaxial compressive strength to the uniaxial tensile strength.

In the principal stress coordinate, (10) is rewritten asSince the uniaxial tensile strength has high discretization, (12) is not successful to predict the strengths under triaxial conditions. To make up for the accuracy, it can be transformed into a general formula with two fitting parameters, expressed asBased on the theoretical analysis and testing data analysis of the intact rocks, You [11] indicated that and are related to the uniaxial compressive strength, and (13) is further derived aswhere is the uniaxial compressive strength of the isotropic rock. The explicit expression of (14) isEquation (15) was called the normal parabolic criterion by You [11]. The fitting results of ten rocks showed that “it is suitable for some rocks in compression stress state”. Besides, there is only one undetermined parameter in the criterion, which makes it more convenient for engineering application.

To evaluate the strength of transversely isotropic rocks, the normal parabolic criterion is improved as follows: where is the uniaxial compressive strength of transversely isotropic rocks, and is the confining coefficient determining the contribution of the confining pressure to the strength. Note that the contribution of the confining pressure changes when the bedding angle decreases/increases, indicating that confining coefficient also depends on the bedding angle. The proposed criterion has a very concise form with only two undetermined parameters. Since the values of and are both orientation-dependent, more efforts should be made to determine them.

3.2. Data Fitting

The proposed failure criterion (Equation (16)) describes a parabolic relationship between the shear and normal stresses at the critical failure state. Two parameters ( and ) in the criterion need to be further determined from experimental data. The value of and its variation with can be directly acquired from uniaxial compression tests, but the value of should be calculated through compression data for specimens with various confining pressures and orientations. We take five sets of triaxial and uniaxial compression data from the database mentioned in Section 2.4 for example to calculate the parameter . These experimental data are respectively collected from the tests on dry phyllite, dry orthoquartzite, dry slate [35], chlorite schist, and biotite schist [36].

Figure 3 illustrates the comparison of the fitting envelopes using the experimental data, and Table 1 shows the values of as well as R² obtained from the fitting results. It can be learned that the proposed failure criterion performs well in fitting experimental data for transversely isotropic rocks since the value of R² in each data set is greater than 0.95. The good fitting capacity verifies the effectiveness of the parabolic failure criterion in characterizing the strengths of anysotropic rocks. The values of fall in the range from 1.0 to 2.0. For the same lithology, varies with the bedding angle, but the fluctuation is slight. We will use the fitting results to derive the relationship between and β in the following section.

(a)

(b)

(c)

(d)

(e)

3.3. Anisotropy of Parameters

Two parameters in the proposed parabolic criterion show apparent anisotropy, and they are both dependent on the orientation of the bedding plane. The expressions of and can be summarized/derived by using the collected data in the listed references above [35, 36].

The most popular equation for the uniaxial compressive strength is the trigonometric expression that was proposed by Jaeger [30] as follows:where is the bedding angle for which the uniaxial compressive strength is minimum, and and are constants obtained by data fitting.

Equation (17) is also used in the present study to describe the anisotropy of uniaxial compressive strength. Examples are taken from the experimental data for five different rocks shown in the former section. Comparison of the testing values and predictive results is illustrated in Figure 4, and the values of the fitting parameters are shown in Table 2. It can be observed that (17) has good fitting effects for various anisotropic rocks with R² >0.90. The values of parameters A and B positively determine the average level and the fluctuation range of the uniaxial compressive strength, respectively. The value of θ is usually located between 30° and 45° according to the fitting results.

Values of the other anisotropic parameter show regular fluctuations referred to Table 1. After analyzing the change rule of , the following expression is suggested:where is the value of when β=0°, , , and are constants to express the correlation between and . Parameters and determine the fluctuation range of and parameter determines the fluctuation period.

The comparison of the fitting curves by (18) with the testing data is illustrated in Figure 5 and the values of the derived parameters are shown in Table 3. It is obvious that the suggested equation can well reflect the fluctuation of and quite consistent with the testing data. However, it should be noted that the specimens with more than four different bedding angles are required for uniaxial and triaxial compression tests since there are four undetermined constants in (18).

3.4. Procedures of Using the Parabolic Criterion for Prediction

As presented above, a parabolic criterion for transversely isotropic rocks is proposed on the basis of the normal parabolic criterion. There are two orientation-dependent parameters in the parabolic criterion: and , both of which can be quantitatively evaluated by certain equations. To intuitively understand the application of the parabolic criterion, the procedures are illustrated in Figure 6 and described below.

Step 1. Collect the uniaxial and triaxial compression data under various confining pressures and bedding angles. Note that at least 4 different bedding angles should be involved, and the specimens for each bedding angle should have more than 3 levels of confining pressure.

Step 2. Use (16) to fit the experimental data. The and values under various conditions are derived through the fitting processes.

Step 3. Fit and by (17) and (18), respectively, to determine the parameters in both equations. Note that the fitting data of is obtained from Step 2, rather than collected from experimental data.

Step 4. Predict the values by (16)~(18) with the use of the evaluated parameters in Step 3. The and AAREP values can be taken as the index to examine the precision of the model.

4. Results

4.1. Compared with Experimental Data and Existing Criteria

In the present study, 446 uniaxial and triaxial compression data for 12 different rock types (see Section 2.4) are used to evaluate the predictive capacity of failure criteria. Besides, the accuracy of the proposed parabolic criterion is compared with those of the McLamore-Gray criterion (1) and Tien-Kuo criterion (Equations (3) and (4)) with the use of the same database. The comparison of the experiments and predictions by the three criteria are respectively illustrated in Figure 7.

(a)

(b)

(c)

The R² value of the predicting results by the proposed criterion is 0.981, which is larger than those by McLamore-Gray criterion (R² =0.968) and by Tien-Kuo criterion (R² =0.961). The AAREP values for predictions by the three criteria are respectively 11.18%, 12.09%, and 11.34%. Therefore, the predicting results of the parabolic criterion are highly consistent with the experimental data, and the predictive capacity is superior to those of the existing criteria.

4.2. Error Variation

It should be noted that the accuracy of the predicting results varies with the confining pressure and bedding angle. Therefore, the influences of the two factors on the predictive capacity are investigated in this section. The errors between predictions and experiments are calculated with the use of the database above and then grouped by confining pressure and bedding angle as shown in Figures 8(a) and 8(b).

(a)

(b)

For different confining pressure levels, the values of R² and AAREP range from 0.965 to 0.987 and from 5.6% to 15.4%, respectively. With the increase of confining pressure, the R² value has a fluctuation of increase-decrease-increase, but the AAREP value shows a monotonic decrease trend. Therefore, it can be learned that the error for high confining pressure level is smaller than that for low level.

For different bedding angles, the values of R² and AAREP range from 0.974 to 0.984 and from 7.5% to 19.8%, respectively. They both show certain fluctuations with the increase of bedding angle, but their variation trends are the reverse. The minimum R² value and maximum AAREP value are respectively located in [15°, 30°) and [30°, 45°) which are close to the friction angle of the bedding plane. It is noted that the specimen with bedding angle between 15° and 45° mostly ruptures as the bedding-sliding mode. Therefore, the prediction error for this failure mode is larger than that for non-bedding-sliding mode.

5. Discussion

5.1. Parametric Sensitivity

There are two critical parameters in the proposed criterion (Equation. (16)), and they determine the level of the rock strength and the shape of the envelope. The influences of the two parameters on the failure envelope are illustrated in Figure 9 and described as follows.

(a)

(b)

On one hand, is set as a constant value of 30 MPa, and changes from 1.0 to 2.0 incremented by 0.2 to reveal the influence of on strength (see Figure 9(a)). It can be observed that all the envelopes radiate from the same vertex. With the increase of , the envelope gradually moves up, and its slope increases accordingly. An interval of 35 MPa is marked on the coordinate for comparison. The interval shifts from the bottom right to top left, indicating that the differences between adjacent envelopes grow with the confining pressure as well as the value of .

On the other hand, is kept constant (1.5 for example) to investigate the change of failure envelope subject to (see Figure 9(b)). When changes from 30 MPa to 60 MPa, the envelope gradually moves up and changes from 258.7 to 336.8 MPa when =50 MPa. Different from Figure 9(a), the envelopes with varied are approximately parallel, which indicates that has few impacts on the shape of envelope. Therefore, the shape of the failure envelope is determined by , and only affects the level of rock strength.

5.2. Capability with Limited Experimental Data

A superior failure criterion is of good capability for predicting rock strength, even if the experimental data are limited. In the proposed parabolic model, parameters can be determined through the uniaxial and triaxial data under only four different bedding angles. To examine the predictive capability of the model, we evaluate the prediction errors of the dry phyllite [35] with the use of various sets of data. The procedures are as follows:(1)Choose the data of specimens with different bedding angles. At least four sets of data should be used.(2)Evaluate the parameters in the and equations in turn, and calculate the relative R² and AAREP values.(3)Predict the values under various bedding angles and confining pressures by using the determined parameters above. Then, compare the predictions with the experimental data to obtain the R² and AAREP values.

The comparisons of the fitting curves of and , using different sets of data, are illustrated in Figure 10, and the relative evaluation errors are presented in Table 4. It is apparent that the fitting curves coincide with the experimental data no matter how many sets of data are used. Although the set of data gradually decreases from 7 to 4, the prediction errors of remain at a low degree of increase: the R² value decreases from 0.926 to 0.911, and the AAREP value increases from 9.09% to 10.22%. By decreasing the set of data, on the other hand, the prediction errors of firstly show a sharp decrease and then remain steady. Hence, the prediction errors of both and are at low level, even if the experimental data are limited.

(a)

(b)

Based on the above-evaluated and , the σ₁ values are predicted and compared with the experimental results, as shown in Figure 11. The scatters in Figure 11(a) are densely attached to the center line, indicating that the predictions are approximate to the testing values. Although the set of data is changed, the R² values are greater than 0.96, and the AAREP values range from 11% to 12%, which verifies the strong capability of the proposed model using limited data.

(a)

(b)

6. Conclusion

A parabolic failure criterion is developed to predict the orientation-dependent strength of the transversely isotropic rock. The proposed model is a modified version of the normal parabolic criterion for isotropic rocks by incorporating two orientation-dependent parameters: and . It is further found that both of the two parameters show trigonometric relationships with β. The results are as follows:

The shape of the failure envelope is determined by , and only affects the level of rock strength. With the increase of , all the envelopes radiate from the same vertex and their slopes increase accordingly. With the change of , the envelopes are approximately parallel.

With application to 446 experimental data, the predicting results by the parabolic criterion are highly consistent with the experimental values, and the predictive capacity of the proposed criterion is better than those of the McLamore-Gray and Tien-Kuo criteria.

The prediction error for the high confining pressure condition is smaller than that for the low confining pressure. Besides, the prediction error for the bedding-sliding failure mode is larger than that for the non-bedding-sliding failure.

It is verified that the proposed criterion is of high precision even if the experimental data are limited. The prediction error remains steady when the set of data is decreased.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the National Key R&D Program of China (Grant No. 2017YFC1501305) and the Key Program of National Science Foundation of China (Grant No. 41230637). This financial support is appreciated.

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