Numerical Analysis of the Shape Effect on the Mechanical Behaviors of Rocks in True 3D Compression Test

Abstract

Rock strength parameters are usually indispensable for rock engineering design. Under the experimental testing conditions, the shape effect could significantly affect the measured results. Previous findings from uniaxial tests reveal that the measured rock strength gradually decreases with the increase in the slenderness ratio, which is mainly ascribed to the end effect. However, it is still unclear how strong the influence of the end effect on the shape effect is in true 3D tests. In the present study, rock mechanical behaviors in response to the variation in slenderness ratio are detailly examined in true 3D tests. The calculated results show that rock strength progressively decreases as the slenderness ratio increases. But the rock strength in true 3D tests still deviates far from the actual even though the slenderness ratio goes beyond 2, which is mainly caused by the interface friction along the two extra ${\sigma }_2$-normal specimen faces. It is interesting that the slenderness ratio increases lead to an increase in the measured stiffness as well. The calculated results suggest that symmetry in the experimentally defined typical arc-shaped curves ${\sigma }_1\left({\sigma }_2\right)$ at constant ${\sigma }_3$ are neglected most likely due to the stronger end effects in true 3D tests, and the accurate rock strength parameters are not obtained only through using slender specimens.

1. Introduction

Reliable mechanical properties of rocks are always required for most engineering designs such as underground mining, civil tunneling and geophysical applications. Currently, rock laboratory testing is regarded as the main approach to investigating this topic. However, it has been proven that the material properties can be strongly affected by the testing conditions including specimen geometry (e.g., size effect and shape effect) [1,2,3,4,5,6,7,8,9], contact conditions (e.g., the end effect) [10,11,12,13,14,15,16,17,18,19] and loading conditions (e.g., stress path, strain rate, confining pressure) [20,21,22]. Among these factors, specimen geometry has a significant influence on the experimental measured results, which has been extensively studied under uniaxial compression. Generally, the specimen geometry effect comprises two categories, namely shape effect and size effect. The shape effect depicts the influence of slenderness ratios of a specimen on strength properties; the size effect is related to the impact of absolute size of a specimen where the slenderness ratio remains unchanged. The slenderness ratio is defined as the ratio of length to diameter (L/D) for a cylindrical specimen or height to width (H/W) for a square prism. In the present study, the shape effect is the main subject of interest.

To date, the shape effect has been investigated both experimentally and numerically within the framework of the uniaxial experimental setup. The reported results showed that the shape effect could not only affect the stress–strain curves of rocks but also the failure modes of rocks [4]. Well known is that rock strength decreases with the increase of slenderness ratio. To obtain reliable results, most researchers concluded that the slenderness ratio should be set to at least 2.0–2.5, which was also suggested by the ISRM (the International Society of Rock Mechanics) [23] and ASTM (the American Society for Testing and Materials) [24]. It is worth noting that the abovementioned research works on the shape effects are limited to the uniaxial compression state. In fact, buried rock masses in situ are mostly subjected to a general stress state (${\sigma }_3<{\sigma }_2<{\sigma }_1$) rather than only to the uniaxial compression state. The true 3D experimental results have revealed that the mechanical response of rocks depends on both ${\sigma }_2$ and ${\sigma }_3$ [25,26,27,28,29,30,31,32,33,34,35,36,37]. However, due to the different specimen shapes, the ${\sigma }_2$ effects obtained by different laboratories vary as usual. At present, the preferential slenderness ratio for true 3D compression was always a reference to that for uniaxial compression. Up to now, it has not yet been unified. Obviously, clarifying the shape effect in true 3D tests is helpful for a better understanding of the ${\sigma }_2$ effects. Nevertheless, most research studies were focused on the shape effect in uniaxial compression and few provided details on that in true 3D compression.

It is generally accepted that the shape effect is positively correlated with the end effect [2,6,8]. In the laboratory tests, due to the presence of the elastic misfit between the steel platen and specimen, extra shear stresses at the interface are always generated when the load is imposed on the specimen, which corresponds to an additional local “confining pressure” effect near the specimen’s ends and could further perturb both the stress and strain fields within the specimen. Figure 1 shows the end confined zones where the local “confining pressure” exists for different slenderness ratios under uniaxial compression [38]. Clearly, the end confined zones increase with the decrease in slenderness ratio, which makes the stress field within the specimen more heterogeneous and eventually results in the increase of rock strength. The end effect in true 3D tests is more significant than in the uniaxial test for the reason that two extra faces (${\sigma }_2$-normal), or even four faces (${\sigma }_2$-normal and ${\sigma }_3$-normal), are in contact with the steel platens. Unfortunately, it is also not yet fully understood how strong the end effect is in true 3D tests. So far, research on this subject is very limited. In addition, the impact of the end effect on the shape effect in true 3D tests was hardly mentioned.

In this paper, the idea is that accurate evaluation of rock deformation and strength under true 3D compression forms the basis of the constitutive model of the rock. The artificial failure errors caused by the shape and end effects may result in a less accurate failure model and then mislead further predictions. The novelty of this paper is to study the shape effect and the impact of the end friction on the shape effect under true 3D compression, which is hardly mentioned in previous research. Aiming at evaluating the shape and end effects contributing to experimental observations, true 3D simulations are conducted in the finite-difference 3D dynamic time-matching explicit code (Flac 3D). We first examined the effects of specimen shape on the rock strength properties with different friction coefficients at various ${\sigma }_2$ and ${\sigma }_3$ levels, and a preferential slenderness ratio under true 3D compression was recommended. Then, we compared the results with those in uniaxial tests, showing the contribution of the end friction at the ${\sigma }_2$-normal interface. The calculated results indicate that the shape and end effects on the mechanical properties of rocks are significant in true 3D tests.

2. Material and Methods

Some research studies show that the end effect can be largely reduced with effective anti-friction measures, but it cannot be completely excluded in rock laboratory tests. The difficulties in laboratory tests can be overcome by numerical simulation where the end friction could be easily set to zero. Hence, numerical simulation is preferential for studying the impact of the end effect. The designed modeling setup (Figure 2) briefly comprises four steel loading platens (Young’s modulus $E_{pl}$ = 200 GPa, Poisson’s ratio ${\nu }_{pl}$ = 0.3) and the cuboid model of the specimen (elastic–plastic properties described below), which is close to the real experimental conditions. The model dimensions are 50 × 50 × 100 mm³ with a standard slenderness (H/W = 2), as shown in Figure 2. Other specimen models used can be obtained by changing the H/W ratio, where W is left constant. In this paper, we have adopted five values of H/W ratio: 1, 1.5, 2, 2.5, 3. Each dimension is listed in

Table 1. Model dimension for each slenderness ratio.

Slenderness Ratio	Dimension (mm)
1	50 × 50 × 50
1.5	50 × 50 × 75
2	50 × 50 × 100
2.5	50 × 50 × 125
3	50 × 50 × 150

. The thickness of the steel loading platen is half of the specimen width, 25 mm. The specimen and platens are separated by the frictional interfaces with zero cohesion and the friction angles $\varphi$. The end frictions exist only in the directions of the ${\sigma }_1$-normal and ${\sigma }_2$-normal faces of the specimen, namely ${\varphi }_1$ and ${\varphi }_2$ (or friction coefficients ${\mu }_1$ and ${\mu }_2$). ${\sigma }_3$ is directly imposed to the ${\sigma }_3$-normal faces of the specimen simulating the oil pressure in reality, and the corresponding ${\varphi }_3$ is therefore zero. Gaps exist around the vicinity of the specimen ends normal to ${\sigma }_1$, which allows the specimen to be freely deformed in the ${\sigma }_1$ direction. The shear and normal stiffness of the interfaces are set to [39]:

$$k_n=k_s=10\max \left[\frac{4G+3K}{3\mathsf{\Delta }z}\right]$$

(1)

where $\mathsf{\Delta }z$ is the grid size, $\mathrm{m}$; “max” means that the maximum value of all zones adjacent to the interface is adopted.

During the calculation, the boundary conditions are shown in Figure 3a. Arrows in Figure 3a indicate the direction of the principal stresses. The thick orange arrow represents that the principal stresses are imposed by steel platens, and the thin blue arrows mean that the principal stress is applied by hydraulic oil. A common loading path (Figure 3b) is applied as follows: (1) all the three principal stresses (${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$) are simultaneously raised to the preset ${\sigma }_3$ value at constant stress incremental rate, which corresponds to a hydrostatical state; (2) thereafter, ${\sigma }_1$ and ${\sigma }_2$ increase at the same stress incremental rate while ${\sigma }_3$ is held constant until the predesigned ${\sigma }_2$ is reached; (3) ${\sigma }_1$ is raised by strain-control mode ($V_z=-2\times {10}^{-9}\mathrm{m}/\mathrm{s}$) until the specimen model fails.

The previous work [15,16,17,18] has shown that the end effect could largely enhance rock strength, and it changes with various stress levels. Besides, the experimentally defined arc-shaped curves ${\sigma }_1\left({\sigma }_2\right)$ at constant ${\sigma }_3$ are more or less uplifted by the end effect when ${\sigma }_2$ varies from ${\sigma }_3$ to ${\sigma }_1$. That is, 3D empirical failure criteria developed from the experimental data comprise two parts, namely the real/intrinsic ${\sigma }_2$ effect and the platen-induced ${\sigma }_2$ effect. In this work, our aim was to find the specific friction angle purely contributing to the discrepancy in ${\sigma }_1$ and thus a ${\sigma }_2$-independent rock failure criterion is preferable and suitable in the numerical simulations. The classical Mohr–Coulomb criterion, which ignores the ${\sigma }_2$ effect, is used to define the rock strength parameters. Non-associated flow rule is applied in the simulations, and the dilatancy angle is arbitrarily set to half of $\phi$ at each ${\sigma }_3$. In addition, strain-softening behavior is also considered in view of the experimental post-peak observations in rocks. Note that, generally, inherent mesh size is commonly regarded as an obstacle for simulating strain-softening behaviors and is not yet satisfactorily unsettled. To minimize the numerical error, the mesh shape (hexahedron) and size (relatively fine) are set to be the same for each case, and thus the numerical results are within the same order of errors. The mechanical properties as well as strain-softening behaviors of Beishan granite used for different slenderness ratio cases are the same as those used in [40], as summarized in

Table 2. Parameters for Beishan granite.

Properties	Young’s Modulus GPa	Poisson’s Ratio -	Cohesion MPa	Friction Angle °	Dilatancy Angle °	Tension MPa
Specimen	38	0.22	19	58	29	7

. Strain-softening behavior is defined by degradation of the rock’s cohesion as a function of plastic strain, as listed in

Table 3. Strain-softening parameters for Beishan granite.

Cohesion MPa	Plastic Strain -
19	0
16	0.008
9.5	0.035
1	0.090

.

3. Results

3.1. The Influence of the End Effect on the Shape Effect

Mesh size in numerical simulation often affects the results strongly, so it is necessary to conduct some runs to examine it. In this study, the grid resolution ($N_x$) is only indicated by the number of zones in the $x$-direction. Figure 4 shows the calculated ${\sigma }_1$(${\epsilon }_1$) curves for different $N_x$ during the loading process with H/W = 2 under ${\sigma }_2={\sigma }_3=$ 5 MPa. ${\sigma }_1$ is calculated as the value that the total vertical force on the upper platen is divided by its horizontal area. The strain ${\epsilon }_1$ is defined as the value that the vertical displacement of the upper model face is divided by the initial model height. It illustrates in Figure 4 that the strength at failure does not practically depend on the mesh size, and therefore, the errors among different $N_x$ can be neglected. Therefore, to save calculation time, the smallest size ($N_x=20,20\times 20\times 40$ numerical zones in the $x, y, z$ directions, respectively) is adopted for the following simulations. In addition, one can obviously see the post-failure stress reduction with the increase in $N_x$, but as mentioned above, it is not a subject of interest in this paper.

The reported experimental results show that with effective anti-friction measures, the end friction can be reduced to less than 0.018 (or $\varphi =$ 1$°$). In this paper, four values of the interface friction angle are adopted, namely $\varphi =$ 0$°$, $\varphi =$ 1$°$, $\varphi =$ 5$°$, and $\varphi =$ 10$°$, respectively corresponding to $\mu =$ 0, $\mu =$ 0.018, $\mu =$ 0.087 and $\mu =$ 0.173 (arbitrarily set). If not indicated, the end frictions of all the interfaces are the same and expressed as $\varphi$. Figure 5 shows the variations in the calculated apparent strength in response to the variation in the slenderness ratio for different end frictions at ${\sigma }_2={\sigma }_3=$ 5 MPa. The apparent strength is normalized against the specimen strength for $\varphi =$ 0$°$. It is seen that the slenderness ratio has no significant impacts on the apparent strength for $\varphi =$ 0$°$; however, for $\varphi \ne$ 0$°$, the apparent strength is significantly influenced by the slenderness ratio, showing that the larger the end friction, the larger the apparent strength for all the slenderness ratios. Besides, as the slenderness ratio rises, the curves of $\varphi \ne$ 0$°$ gradually descend and approach the curve of $\varphi =$ 0$°$. For example, for $\varphi =$ 5$°$, the calculated apparent strength decreases gradually from 1.13 (H/W = 1) to 1.01 (H/W = 3). It seems that the slenderness ratio does not result in any further significant variation in the apparent strength values when it goes beyond 2, which agrees well with previous results from uniaxial compression. The calculated results confirm that the end friction is the main cause of the variation in rock strength in response to the variation in slenderness ratio.

3.2. The Influence of ${\sigma }_2$ on the Shape Effect

In true 3D tests, the specimen is usually subjected to a large-range ${\sigma }_2$ (much larger than ${\sigma }_3$), which varies from ${\sigma }_2={\sigma }_3$ to ${\sigma }_2={\sigma }_1$. Thus, the effects of ${\sigma }_2$ on the apparent strength in response to the variation in slenderness ratio for $\varphi =$ 5$°$ at ${\sigma }_3=$ 5 MPa are examined in Figure 6. The apparent strength is normalized against the strength of the specimen for $\varphi =$ 0$°$ at ${\sigma }_2={\sigma }_3=$ 5 MPa. Figure 6 demonstrates that all the curves exhibit the same variation trend as that in Figure 5, but a larger ${\sigma }_2$ leads to a larger apparent strength for all the slenderness ratios, showing that the curves extend further away from that of ${\sigma }_2=$ 5 MPa. For instance, for H/W = 3, the apparent strength at ${\sigma }_2=$ 5 MPa, 20 MPa, 50 MPa, 100 MPa are about 1.01, 1.06, 1.13, 1.25, respectively. We can see that at high ${\sigma }_2$ levels, even with a slenderness ratio of 3, the apparent strength still deviates far from the actual ($\varphi =$ 0$°$). Moreover, the stress–strain curves corresponding to the points in Figure 6 are also given in Figure 7. It is shown that not only the failure stresses but the stiffnesses (proportional to the slopes) are also strongly dependent on the slenderness ratio in the whole loading process, particularly at this stage before the pre-designed ${\sigma }_2$ is reached. The stiffness grows progressively with the slenderness ratio increasing. As ${\sigma }_2$ increases, both rock strength and stiffness increase, which indicates that the shape effect becomes more significant. With these observations, we can reasonably infer that the interface friction (${\varphi }_2$) along the ${\sigma }_2$-normal faces plays a dominant role on account that it is proportional to ${\sigma }_2$ values, which further brings about large deviations of stress distribution within the specimen from the uniform. Take the slenderness ratio of 2 as an example, the ${\sigma }_3$ contours are shown in Figure 8 when the strains in Figure 7 reach 0.51$\%$ for each case (${\sigma }_3=$ 5 MPa, 50 MPa, 100 MPa). As ${\sigma }_2$ increases, the stress distribution within the specimen becomes more heterogeneous: the maximum and minimum ${\sigma }_3$ values are, respectively, 4.27 MPa and 31.4 MPa for ${\sigma }_2=$ 5 MPa (the average/nominal value ${\sigma }_3^n$ is 5.99 MPa), 5.02 MPa and 35.1 MPa for ${\sigma }_2=$ 50 MPa (${\sigma }_3^n$ is 8.4 MPa); 4.99 MPa and 37.2 MPa for ${\sigma }_2=$ 100 MPa (${\sigma }_3^n$ is 9.28 MPa). Obviously, each ${\sigma }_3^n$ is larger than the applied value (5 MPa), and the discrepancy increases with the increase in ${\sigma }_2$, which indicates the degree of stress disturbance caused by the end friction.

Next, we adopted three cases for comparing the relative contribution of the interface friction along the ${\sigma }_1$- and ${\sigma }_2$-normal faces (${\varphi }_1$, ${\varphi }_2$): (1) ${\varphi }_1={\varphi }_2=0$; (2) ${\varphi }_1=5°$, ${\varphi }_2=0$, i.e., without the interface friction along the ${\sigma }_2$-normal faces (similar with the conventional tests); (3) ${\varphi }_1={\varphi }_2=5°$ (true 3D tests). The comparisons of the apparent strength for different ${\varphi }_1$ and ${\varphi }_2$ at different ${\sigma }_2$ levels are plotted in Figure 9. It is seen from Figure 9 that the strong dependence of the apparent strength on slenderness ratio is different for three cases, but their variation trends are similar. In the conventional triaxial state (${\sigma }_2={\sigma }_3=$ 5 MPa), both curves for $\varphi \ne$ 0$°$ almost coincide for each slenderness ratio, which indicates that the ${\sigma }_1$-normal end effect plays a major role in the increment of apparent strength. In true 3D tests, due to the presence of the friction along the ${\sigma }_2$-normal specimen faces, both curves for $\varphi \ne$ 0$°$ separate clearly and extend further as ${\sigma }_2$ increases. This indicates that as the slenderness ratio rises, the ${\sigma }_1$-normal end effect progressively degrades, whereas the ${\sigma }_2$-normal end effect is always significant. On the whole, the ${\sigma }_1$-normal end effect is dominant for low slenderness ratios, but for high slenderness ratios the increment in apparent strength is primarily due to the ${\sigma }_2$-normal end effect. Additionally, Figure 9 also depicts that the relative contribution of the ${\sigma }_2$-normal end effect ascends with ${\sigma }_2$ increasing for each slenderness ratio. Take the slenderness ratio of 2 as an example, its stress–strain curves at ${\sigma }_3=$ 5 MPa, ${\sigma }_2=$ 100 MPa are shown in Figure 10. It is seen that the dependence of both failure strength and stiffness on ${\sigma }_2$ at $\varphi \ne$ 0$°$ is mainly attributed to the ${\sigma }_2$-normal end effect (${\varphi }_2$). Figure 11 shows ${\sigma }_1$ and ${\sigma }_3$ contours when the strains in Figure 10 reach 0.45$\%$ for each case. Note that at $\varphi =$ 0$°$ (Figure 11a,b), the stress distribution is not strictly uniform due to the existence of the gap (Figure 2) between the platens and specimen, but the stress field is uniformly distributed in the other parts of the specimen. ${\sigma }_3^n$ for Figure 11b,d,f are 5.03 MPa, 7.98 MPa and 9.95 MPa, respectively. This directly indicates that stress distributions within the specimen under true 3D tests (Figure 11e,f) are more heterogenous than that under “conventional” tests (${\varphi }_1=5°$, ${\varphi }_2=0$) (Figure 11c,d).

3.3. The Influence of ${\sigma }_3$ on the Shape Effect

Figure 12 shows the variations in apparent strength in response to the variation in slenderness ratio for two ${\sigma }_3$ values, 20 MPa and 50 MPa. The two curves for $\varphi \ne$ 0$°$ are becoming closer with the increase in ${\sigma }_3$, which indicates that a higher ${\sigma }_3$ could provide a stronger restricting effect on the interface friction along the ${\sigma }_2$-normal specimen faces.

4. Discussion

4.1. The Reason for Shape Effect

The numerical results confirm that the shape effect is mainly due to the interface friction between the specimen and platens. The interface friction provides a horizontal restricting effect (along the interface in ${\sigma }_1$-normal direction) near the vicinity of the specimen end, which can be visualized in Figure 1 under uniaxial compression. The elements in these highly confined zones have higher failure stress because of the increased ${\sigma }_3$/confinement, in which local plasticization and higher strain occur. On another aspect, for a constant slenderness ratio, those zones will be broadened if the end friction increases (i.e., boundary constraint is strengthened). Those confined zones influenced by the end friction gradually decrease with the increase in the slenderness ratio (Figure 1). Apparently, true 3D compression cases are similar, but additional vertical restricting effects exist along the ${\sigma }_2$-normal interface. Figure 13 compares the variations in ${\sigma }_3$ contour (${\epsilon }_1=0.45\%$) in response to the variation in the slenderness ratio at ${\sigma }_3=$ 5 MPa, ${\sigma }_2=$ 100 MPa for “conventional” and true 3D compressions. The confined zones in Figure 13 are the ones in which ${\sigma }_3$ is greater than 5 MPa. Obviously, more confined zones which are lumped near the vertical (top and bottom) and horizontal (in ${\sigma }_2$ direction) specimen ends forming cone shapes are influenced by the end friction under true 3D compression. Taking the top confined zones as an example, for “conventional” compression, its lower edge is about 12 mm away from the specimen’s top end, whereas 15 mm for true 3D compression. As for the horizontal confined zones, they extend to the core, and the final edge is approximately 8 mm away from the specimen’s end in the ${\sigma }_2$ direction. The increased ${\sigma }_3$ values in these confined elements control the enhancement of the overall strength of the specimen, which is the main reason for the higher apparent strength under true 3D compression in Figure 9 and Figure 12.

Figure 14 presents the percentage of the confined zones to the total number of the elements in each specimen in Figure 13. On the whole, due to the lateral confined zones in the ${\sigma }_2$ direction, the percentage of the confined zones under true 3D compression for each slenderness ratio is about 5$\%$ larger than that under “conventional” compression. Note that the confined zones and their percentage progressively decrease as the slenderness ratio increases. The specimen with H/W = 1 has the highest proportion of the confined zones in its volume as well as the highest confinement (${\sigma }_3^n$) compared to the other cases. In contrast, the lowest percentage and confinement are for the specimen with H/W = 3. This is why the slenderness effect is more significant when H/W is small. In essence, the slenderness effect is mainly caused by the change in the distribution of the confined zones induced by the end constraint. The distribution of the confined zones is largely impacted by ${\sigma }_2$ and ${\sigma }_3$, which leads to variations in ${\sigma }_1$ as shown in Figure 9 and Figure 12.

4.2. End Effect for Rock Strength

In “conventional” compression tests, the apparent strength is approximately equal to the actual for a high slenderness specimen, which is not the case for true 3D tests. Especially at high ${\sigma }_2$ levels (Figure 9 and Figure 12), the big discrepancy between the actual and apparent strength still exists even for a high slenderness specimen, which implies that rock strength obtained from experimental tests always deviates far from the actual in true 3D tests. That is, reliable measured results cannot be obtained only by using the specimen with a high slenderness ratio. Therefore, the end effect could not be ignored in true 3D tests. The ascending–descending trend in ${\sigma }_1$ with the ${\sigma }_2$ increase might be partly uplifted. As often observed from experimental tests, this trend is non-symmetrical, showing that the rock strength (${\sigma }_1$, at failure) under axisymmetric extension (AE) is larger than that under axisymmetric compression (AC). This phenomenon is also supported by some results from conventional triaxial tests. However, some exceptions are noticed in the experimental results of Bentheim, Coconino [32] and Zigong sandstones [37], which show that ${\sigma }_1$ at failure are almost identical for both AC and AE conditions at different ${\sigma }_3$ levels. These tests are all performed with strict specimen preparation and effective anti-friction measures (the end friction is $\le$0.02 for Bentheim and Coconino sandstones [15], and $\le$0.006 for Zigong sandstone [16], respectively). Interestingly, the phenomena (${\sigma }^{AE}>{\sigma }^{AC}$ and ${\sigma }^{AE}\approx {\sigma }^{AC}$) appears at the same time when Fan et al. 2017 [33] tested the influence of the end effect on the experimental results (Figure 15a). The results from [33] show that the end effect is the main reason for the difference, which implies that if the end friction was quite small or close to zero, ideally in the laboratory, the actual intrinsic shape of the envelope in ${\sigma }_1\left({\sigma }_2\right)$ could be symmetrical. This is also in good agreement with Mogi’s criterion [41].

Mogi’s criterion, as suggested by the ISRM [42], could provide excellent fittings to most experimental data from true 3D tests. It states that failure occurs when the distortional energy reaches the critical value (which monotonically increases with ${\sigma }_{m,2}$) on the failure planes. It is formulated by

$${\tau }_{oct}=f\left({\sigma }_{m,2}\right)$$

(2)

where ${\tau }_{oct}=\sqrt{s_{ij}{s}_{ij}}/3$ is the octahedral shear stress, $s_{ij}$ is the stress deviator tensor ($i,j=\mathrm{1,2},3$); ${\sigma }_{m,2}=({\sigma }_1+{\sigma }_3)/2$; $f\left({\sigma }_{m,2}\right)$ can be any monotonic increasing function, but only linear and power-law ones are recommended by the ISRM [42]. For convenience in analysis, Equation (2) is rearranged by Jimenez and Ma 2013 [43] using a linear function, and the new form reads:

$${\sigma }_1=\frac{a}{\frac{1}{3}\sqrt{2({\beta }^2-\beta +1)}-\frac{b}{2}}+\frac{\frac{1}{3}\sqrt{2({\beta }^2-\beta +1)}+\frac{b}{2}}{\frac{1}{3}\sqrt{2({\beta }^2-\beta +1)}-\frac{b}{2}}{\sigma }_3$$

(3)

where $\beta =({\sigma }_2-{\sigma }_3)/({\sigma }_1-{\sigma }_3)$ is the stress state factor; $a,b$ are the material constant. This envelope is symmetrical with respect to $\beta =1/2$, which can be attained equally well by other types of the function. That is, Mogi’s criterion implies that for a given ${\sigma }_3$ value, ${\sigma }_1$ at failure for both AC and AE tests are identical. Indeed, this is contradictory with most true 3D tests. Since the data available are scarce and limited, it is still hard to point out whether the intrinsic envelop in the form of ${\sigma }_1\left({\sigma }_2\right)$ at a certain ${\sigma }_3$ is symmetrical or not. However, the actual intrinsic shape of the envelope in ${\sigma }_1\left({\sigma }_2\right)$ is strongly influenced by the end effect.

4.3. Comparison with the Available Experimental Data

Recently, some true 3D experimental tests were performed to examine the influence of the shape effect on the measured rock property [35]. Although rock strength in [35] is tested under true 3D unloading conditions, it still can provide valuable guidelines for true 3D loading conditions. As the calculated numerical results, the study [35] also reveals a strong influence of specimen shape on the measured rock strength (Figure 15b). This influence is enhanced when the deviatoric stress state varies from AC to AE. Compared with the case of H/W = 2, the rock strength increment for H/W = 0.5 increases approximately from 31$\%$ at ${\sigma }_2$ = 15 MPa to 65$\%$ at ${\sigma }_2$ = 120 MPa; for H/W = 1, the strength increment increases from about 6$\%$ at ${\sigma }_2$ = 15 MPa to 11$\%$ at ${\sigma }_2$ = 120 MPa. This indicates that ${\sigma }_2$ dependence on ${\sigma }_1$ decreases with the increase of the slenderness ratio. Moreover, the envelops (dot lines in Figure 15b) seems to be always symmetrical for each slenderness ratio under true 3D unloading conditions. These features are consistent with the above numerical results and analysis. But more true 3D experimental tests are needed to further confirm the results obtained in this study.

5. Conclusions

This paper numerically studied the influence of shape effect (slenderness ratio of the rock specimen) in true 3D tests on various kinds of rock. The primary aim was to verify the existing overestimated strengths obtained from experimental mechanical tests due to the shape effect. The shape effect was examined under different ${\sigma }_2$ and ${\sigma }_3$ levels, respectively. The following conclusions are drawn:

(1)

The shape effect could significantly affect both the measured rock strength and stiffness. The apparent strength decreases sharply with the increase in slenderness ratio and remains almost unchanged when the slenderness ratio extends beyond 2. For a high slenderness ratio ($\ge$2), the shape effect is still significant in true 3D tests, which is not the case for uniaxial compression tests. The shape effect could be enhanced by ${\sigma }_2$, whereas restricted by ${\sigma }_3$.

(2)

The shape effect is mainly ascribed to the end effect, which is caused by the mismatch in the elastic parameters between steel platen and rocks. Stress distribution within the specimen becomes not uniform due to the end effect. The stress heterogeneity grows practically as the slenderness ratio decreases. This heterogeneity could be mostly reflected by the highly confined zones near the vicinity of specimen faces in contact with platens.

(3)

The actual intrinsic shape of the envelope in ${\sigma }_1\left({\sigma }_2\right)$ at each ${\sigma }_3$ is partly uplifted due to the end effect. It seems that the intrinsic shape should be symmetrical, but more relevant research studies are needed to confirm this point.