A novel obtaining method and mesoscopic mechanism of pseudo-shear strength parameter evolution of sandstone

The pre-peak microstructure evolution of the material is crucial in determining the strength and failure scheme of the material, but is not captured by the existing shear strength parameters models which focus on post-peak stage information. Novel concepts of pseudo-cohesion and pseudo-internal friction angle that account for both the pre-peak and post-peak stage information were proposed in this study, based on the experimentally observed response of the rock mechanics parameters to an external load (the maximum principal stress). These two parameters were analytically derived as functions of maximum principal strain, with coefficients to be determined (i.e., by applying the Mohr–Coulomb criterion with these parameters to experimental data). The dependence of pseudo-cohesion and pseudo-internal friction angle on the axial strain and confining pressure was investigated. The evolution of the strength parameters was found to relate to the propagation of micro-crack developed in the loading process, as revealed by PFC2D simulations reproducing experiments. The results show that (1) analytical model characterizing the pseudo-cohesion and internal friction angle as the quadratic function of axial strain accurately reflects the mechanical response characteristics of the specimen; (2) before the residual stage, the pseudo-cohesion first increases and then decreases with increasing strain, while the pseudo-internal friction angle increases continuously; (3) for the sandstone specimens studied, the pseudo-cohesion at the peak stress increases by a gradually decreasing rate with increasing confining pressure, while the pseudo-internal friction angle at the peak stress decreases by a decreasing rate with the increasing confining pressure; (4) CPM well characterized the mechanical properties of rock under multiple confining pressures, as the total stress–strain curves and failure modes of numerical specimens reproduced the experimental results. The evolution of rock pseudo-shear strength parameters is closely related to the external load and the generation, propagation, and penetration process of rock cracks. The rock pseudo-shear strength parameters are gradually mobilized with the increasing load, and the propagating of cracks owing to the decrease of mobilizable maximum cohesion and the increase of mobilizable maximum internal friction angle.


Introduction
The shear strength parameters-cohesion and internal friction angle of materials (i.e., rocks) are the macroscopic manifestation of the total bonding and friction strength in between the material particles which describes the inherent properties of the material (Bai et al. 2020;Chen et al. 2019;Comanici and Barsanescu 2016;Farajzadehha et al. 2020;Lin et al. 2019;Shi et al. 2020aShi et al. , 2022Yang et al. 2020). The bond strength and the friction state between particles change in response to external loading and internal damage development, which leads to a time-dependent character of cohesion and internal friction angle in the loading process. In geological engineering, the shear strength parameters are instructive information for engineering calculations, support design, etc. (Wu et al. 2019). Overestimated strength parameters lead to overestimated safety factor of the support which increases the risk of failure of the support; on the other hand, underestimated strength parameters lead to excessive support and significantly increases the cost. Therefore, it is important to obtain the time-evolution of cohesion and internal friction angle of materials.
The evolutions of rock cohesion and internal friction have been studied for centuries Li et al. 2015;Lu et al. 2010;Xu et al. 2015;Zhang et al. 2015a). In general, two traditional research methods were adopted by researchers to obtain rock cohesion and internal friction, i.e., the Mohr Strength Envelope Method (MSEM) and the Yield Surface Fitting Method (YSFM). The MSEM is the most often used method, in which the cohesion and internal friction angle are, respectively, the intercept of the envelope tangent on the vertical axis and the angle between the envelope tangent and the horizontal axis in the τ-σ coordinate system (Edelbro 2010;Hajiabdolmajid et al. 2002;Martin and Chandler 1994;Rafiei Renani and Martin 2018;Shi et al. 2020a;Zhang et al. 2015a). Albeit its prevalence, the YSEM has a few drawbacks, i.e., the cyclic loading and unloading procedure used in the experiment is time consuming and laborious Shen et al. 2020;Shi et al. 2020a). Unlike the MSEM, the YSFM combines the rock stress-strain curve under monotonic loading and strength criterion to obtain the evolution of rock strength parameters (Lu et al. 2010;Joseph 2000). In YSFM, the functional relationship between the shear parameters and the damage variables is first assumed with undetermined coefficients; the functional relationship is then introduced into a yield criterion to establish an analytical model of the shear parameters; finally, the undetermined coefficients are obtained by taking points on the yield stage of the measured stress-strain curve. Shear parameters obtained with both methods focus on the material strength at peak or the post-peak yield stage. The pre-peak evolution of bonding and cohesion was not captured.
Latest studies suggested that the pre-peak evolution of microstructure and the cognition of macro parameters greatly affect the failure of rock material (Hajiabdolmajid et al. 2002;Zhao et al. 2005). Hajiabdolmajid et al. (2002) believe that the cohesive strength plays a major role in the early failure stage of brittle rock; and the micro-cracks grow gradually as the loading increases. The generation and coalescence of the tensile cracks destroy the total cohesive strength component of the rock, which also results in the mobilization of the friction strength component. For limited specimens, the MSEM can also be used to obtain the pre-peak evolution of shear strength parameters. The completely decomposed granite is among the few special specimens, whose pre-peak strength parameter curve was found to clearly separated with the confining pressure (Zhao et al. 2005); the corresponding stress-strain curves of the granite specimen and the result obtained are shown in Fig. 1. The cohesion was observed to significantly reduce in the pre-peak stage, but was not captured by existing research involving only post-peak shear strength parameters. The lack of understanding of pre-peak process is obviously not conducive to the accurate identification of engineering disaster risks. For most rock specimens, the pre-peak strength parameter curve entangles with the confining pressure curve, which limits the applicability of MSEM. For the YSFM, the fitting equation to characterize the pre-peak evolutions of shear strength parameters has not been developed yet. The YSFM under a single confining pressure is the most promising method to obtain the evolution of the whole loading process for most rock specimens, but requires the establishment of a suitable analytical model.
To fully understand and predict the failure process of the structure, it is necessary to study the internal mechanism of the evolution of shear strength parameters. Existing researches stay mostly in the subjective description stage. Process-based numerical models such as Particle Flow Code 2D (PFC 2D ) have been applied to describe the microevolution of material under loading and to reveal the mesomechanism of shear strength parameter (Shi et al. 2020a). However, the particle unit used in such studies is mostly spherical, which is difficult to reflect the friction caused by inlay and meshing effect between rock particles. Due to the limitation of the method for obtaining shear strength parameters, the revelation of the meso-mechanism for the evolution of rock shear strength parameters is also only limited in the post-peak stage.
In this study, new concepts of pseudo-cohesion and pseudo-internal friction angle that account for both prepeak and post-peak stage were proposed based on the response property of the rock mechanics parameters to an external load. These two parameters were analytically derived as functions of maximum principal strain, with coefficients to be determined by implementing these two parameters into the Mohr-Coulomb (M-C) criterion with experimental data. The proposed method was verified through a comparison between the Fast Lagrangian Analysis of Continua 3D (FLAC 3D ) simulation and the experiment results. Finally, a Clumped-Particle Model (CPM) which can effectively reflect the inlay and meshing effect between particles was established by PFC 2D to simulate the bearing failure process of rock, and reveal the meso-mechanism of the evolution of pseudo-shear strength parameters combined with the micro-crack propagation process.

Cohesion weakening friction strengthening model (CWFS)
In the CWFS model proposed by Hajiabdolmajid et al. (2002), the rock stress-strain curve can be roughly divided into three stages, namely, the pre-peak elastic and elastoplastic stage (OB), the post-peak softening stage (BC), and the residual strength stage (i.e., after the point C in Fig. 2). Prior to the generation of micro-cracks, only the cohesive strength responds to the external load (OA); with the development of micro-cracks, the friction strength increases, and bears the external load together with the cohesive strength; in the and C are the stress points, while A′, B′, and C′ are strain points corresponding to A, B, and C, respectively.) softening stage (BC), the build-up of shear band diminishes the continuity of the specimen and the cohesive strength component; in the residual strength stage (after point C in Fig. 2a), the friction strength is the dominant component of the axial stress. It can be seen from Fig. 2b that the initial value of the cohesive strength component mobilized is 0 (point O); as the strain increases, it increases with a gradually decreasing rate until reaches the peak value at around the point B′; After point B′, as the strain increases, the mobilized cohesive strength gradually decreases and reaches a steady state at point C′, at which it is relatively small, as shown in Fig. 2b. Similarly, it can be known from the mobilized friction strength curve in Fig. 2c that the appearance of mobilized friction strength is accompanied with the generation of micro-cracks at the point A′ (Hajiabdolmajid et al. 2003;Shi et al. 2019). With the increase of strain, the increasing process of the mobilized friction strength can be roughly divided into the fast-growing section A'B' and the slow-increasing section B′C; after the point C′, the mobilized friction strength also remains unchanged.
According to the CWFS model, the active rock cohesion and internal friction responding to the change of axial strain (actual mobilized components) are defined as pseudocohesion (c v ) and pseudo-internal friction angle (φ v ) here, to distinguish the peak shear parameters at limit equilibrium state (cohesion c and internal friction angle φ) in the M-C criterion. Note that natural rock carries many cracks, as a result of hundred-million-years sedimentation (Asadizadeh et al. 2017;Diederichs et al. 2004). Therefore, the mobilization of the internal friction angle of rock should be considered from the beginning of the load (i.e., see the dotted line in Fig. 2c as suggested by Zhao et al. (2005)).

New evolution model of rock pseudo-cohesion and pseudo-internal friction angle
The shear strength of cohesionless soil is proportional to the normal stress of the shear surface, in the form as following (He et al. 2006;Rahimi and Nygaard 2008), where τ is the shear strength of the soil, σ is the normal stress exerted on the soil, σtanφ is collectively called the friction strength. For cohesionless soil, the shear strength is essentially the friction resistance produced by the sliding friction between soil particles and the inlay between the concave and convex surfaces of the soil particles, which mainly depends on the roughness, size, and assembly of the soil particles. The friction between soil particles is reflected by the angle of internal friction φ. For cohesive soil, Coulomb added a cohesion term onto the friction resistance to represent its shear strength (He et al. 2006): where the cohesion (c) is independent of the normal pressure. Equations 1 and 2 are collectively referred to as the Coulomb formula. The strength parameters rock cohesion (c) and internal friction angle (φ) in Eq. 2 can be determined by the σ-τ coordinate system (Martin and Chandler 1994), i.e., the intercept of the common tangent of Mohr's circle on the τ-axis and the angle with the σ-axis. These two parameters represent the inherent mechanical property of rock (Zhang et al. 2015a), and play a role in the whole process of rock loading and failure .
The M-C criterion was established by extending and perfecting Coulomb formula (Labuz and Zang 2012;Liu et al. 2018). The criterion is widely used in rock mechanics and plasticity theory (Martin and Chandler 1994;Zhang et al. 2009) and given by, where σ 1 and σ 3 are the maximum and minimum principal stress, respectively. The whole loading process of servo control, as used in this paper, is quasi-static. Equation 3 is suitable for the peak point and post-peak stage of the specimen, where cohesion c and internal friction angle φ are defined as the peak shear parameter, i.e., describing the full strength of the cohesion and internal friction between rock particles at limit equilibrium state, as used in the M-C criterion. In the pre-peak stage, the specimen has not yet yielded. Therefore, the following relationship should be satisfied, By definition (Cohesion weakening-friction strengthening model (CWFS), the pseudo-cohesion and pseudo-internal friction angle should satisfy the following formula both before and after the peak (Zhao et al. 2005), According to the CWFS model and the research by Zhao et al. (2005), before the residual stage, the dependence of the pseudo-cohesion and pseudo-internal friction angle on the axial strain (ε 1 ) can be characterized as second-degree polynomial functions, i.e., where, H 1-6 are coefficients undetermined.
Substituting Eq. 6 into Eq. 5 leads to, (2) = c+ tan Once the stress-strain curve is obtained, the undetermined coefficients (H 1-6 ) can be obtained by fitting characteristic points on the curve, and then the variation of pseudo-cohesion and pseudo-internal friction angle can be ascertained by putting the coefficient into Eq. 6. Note that the selected characteristic points should reflect the trend of the curve and should be more than 6 (i.e., great than the number of unknown coefficients). In combination with the M-C strength criterion, it can be known that the σ 1 should be selected as the sum of the deviatoric stress and the confining pressure. This conversion relationship is used in the calculation and verification of the following texts.

Complete stress-strain curves of sandstone under triaxial compression
The MTS815 testing machine of China University of Mining and Technology was used for the triaxial compression experiment of turquoise sandstone, as shown in Fig. 3. The specimens were taken from a deep coal mine in China, with the average dry density, porosity, and water content of 2700 kg/m 3 , 7.8%, and 1.8%, respectively. The mineral contained albite (65%), quartz (16.9%), anorthoclase (12.3%), laumontite (4.4%), and santafeite (1.4%), as shown by the detection with the X-ray diffraction method. Besides, the cylinder rock samples are in a standard size of 50 mm × 100 mm and without visible joints or cracks. The experiments were carried out by a displacement control method at a rate of 0.002 mm/s, and the applied confining pressures are 5 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa. The representative complete stress-strain curves of sandstones under various confining pressures are presented in Fig. 4. Note that the strain of the specimen is calculated based on the loading displacement, with a zero value corresponding to the loading state of hydrostatic equilibrium, of which the axial stress of the specimen equals to the confining pressure.

Analysis example of sandstone strength parameters
The analytical model of the pseudo-cohesion and pseudointernal friction angle of sandstone has been proposed in "New evolution model of rock pseudo-cohesion and pseudointernal friction angle". According to the model, the complete stress-strain curve of rock sample with a confining pressure of 10 MPa in Fig. 4 is selected as an example to obtain the parameters. First, select 14 characteristic points (should be more than 6) that reflect the trend of the curve before the residual phase, as shown in Fig. 5. σ 1 and ε 1 corresponding to each characteristic point are listed in Table 1.
According to Fig. 6, when the axial strain is 0, the pseudo-cohesion and pseudo-internal friction angle have not been mobilized. The pseudo-cohesion-strain curve seems have a symmetrical bell shape. With increasing strain, the continuously increases with increasing strain at a gradually decreasing rate. Its peak value of about 56° is corresponded to the strain of about 1.0%, see point φ v, b . At the peak strain (about 0.77%), the pseudo-cohesion of the rock has been reduced to about 23 MPa, see point c v, b in Fig. 6, while the pseudo-internal friction angle is about 47°, see point φ v , a in Fig. 6. In the residual stage, the axial stress does not change with the strain. The strength parameters are considered remaining unchanged. Thus, it can be determined that the pseudo-cohesion is constantly 0 (see the dotted line after c v, c in Fig. 6), and the pseudo-internal friction angle is constantly 56° (see the dashed line after φ v, b in Fig. 6) in the residual phase. The pseudo-shear strength parameters are supported by the study of Zhao et al. (2005) on completely weathered granite. Using the traditional molar strength envelope method with axial strain as the internal variable, Zhao et al. obtained the evolution law of internal friction angle (Fig. 1b) which shows similar pattern as the pseudo-strength parameters presented here. In general, the evolution of cohesion and internal friction angle obtained by this study is consistent with that of Zhao et al. (2005). The fact that Zhao et al. (2005) were able to obtain the evolution of pre-peak cohesion and internal friction angle using the molar strength   envelope method is because the cementation between completely weathered granite particles is very weak, and the stress-strain curve of the completely weathered granite has been significantly separated before the peak. However, general rock specimens do not share this mechanical property, which limits the applicability of the method of Zhao et al. (2005). Comparative reference value of shear parameter evolution can also be found in literature, see Fig. 6b. For instance, Li et al. (2019) established the rock stress-strain constitutive model considering the crack propagation mechanism and applied it to simplify the stress-strain curve of granite. On this basis, the evolution of cohesion and internal friction angle was solved by taking the axial strain corresponding to the crack initiation strength as the zero point of parameter evolution. It can be seen from the results that both cohesion and internal friction angle show an increasing trend before the peak, and cohesion has decreased to some extent before the peak, while the internal friction angle has not reached the peak. Above characteristics are consistent with the results shown in Fig. 6a. Additional support rose from the work of Liu et al. (2018) who found that the internal friction angles corresponding to blue sandstone and red sandstone at the crack initiation strength stage have reached 18.36° and 21.16°, respectively (about half the internal friction angles at the peak stress). They believe that the internal friction angle of rock is smaller than that of rock fracture strength stage because the friction strength is not fully developed before fracture initiation, and conclude that the internal friction angle increases gradually with the accumulation of damage through overall analysis.
For the evolution of post-peak cohesion, it is widely recognized that there exists a decreasing trend with the increase of internal variables, which is consistent with the analytical results of this study. Contradictory recognitions were found for the evolution of internal friction angle after the peak, which can be mainly divided into the following three aspects: (1) researchers represented by Martin and Chandler (1994) believed that internal friction angle decreases with a decreasing rate after the peak, which is also the mainstream view at present; (2) based on extensive test data, researchers represented by Zhang et al. (2009) andLu et al. (2010) believed that the internal friction angle remains constant after the peak; (3) researchers represented by Taylor (Terzaghi et al. 1996), Hajiabdolmajid et al. (2002;2003), Edelbro (2010) and Zhao et al. (2005) believed that the internal friction angle increases after the peak, which has attracted more and more attention from the academic community. For example, Liu et al. (2018) gave the conclusion that the internal friction angle increases gradually with the accumulation of specimen damage. Zhang et al. (2015a) believed that an important reason for the above-mentioned divergence was the difference in the evolution variables of shear strength parameters. In addition, rock types and even the specific trend of loading curve may affect the evolution of internal friction angle. The specific reasons why the pseudo-internal friction angle presents the evolution in Fig. 6a will be analyzed in detail in "Meso-mechanism analysis of sandstone strength parameter evolution under the same confining pressure" based on the research of Liu et al. (2018). The above views are listed here to avoid readers' misunderstanding of the results in Fig. 6a.

Result analysis of sandstone strength parameters
Following the procedure in "Application of new evolution model for sandstone strength parameters", the unknown coefficients of other stress-strain curves in Fig. 4 were obtained and listed in Table 2.
Substituting the coefficients in Table 2 into Eq. 6 and considering different confining pressures, the variation of the pseudo-cohesion and the pseudo-internal friction angle with axial strain can be obtained, as shown in Fig. 7.
Under different confining pressures, the pseudo-cohesion first increases, then reduces, and finally remains unchanged (see Fig. 7a). For the initial loading point (the strain is 0) and the residual phase, the corresponding pseudo-cohesion is 0. The peak value of the pseudo-cohesion increases with the confining pressures, so does the pseudo-cohesion at the peak strain. In the pre-peak stage, the pseudo-cohesion at the peak stress is significantly lower than its maximum value, i.e., 24.51%, 30.65%, 33.09%, 38.16%, and 37.90% lower under the confining pressures of 5 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa, respectively. In general, with increasing confining pressure, the decrease range of pseudocohesion before peak shows an increasing trend. In addition, considering that damage may have occurred before loading to the strain corresponding to the pseudo-cohesion peak, that is, the pseudo-cohesion peak may be less than the maximum cohesion that can be invoked before loading, the reduction range of the pseudo-cohesion before the peak calculated above may be smaller than the actual situation, which means that the cohesion value calculated by the traditional MSEM may be less than 60% of the inherent cohesion of the rock. The pseudo-internal friction angle of sandstone increases continuously with the strain until the residual phase (Fig. 7b). With increasing confining pressure, the peak value of pseudo-internal friction angle reduces, so does the pseudo-internal friction angle at the peak strain. At the peak stress, the pseudo-internal friction angle under different confining pressures is in the rising stage. It can be calculated that the rising ranges of the pseudo-internal friction angle after the peak are 9.23%, 18.03%, 16.76%, 16.56%, and 18.28%, respectively, under the confining pressures of 5 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa.
Considering that the strength parameters at the peak stress determine the limit load of the rock, it can be stated that it is the core content of rock strength parameter research. Therefore, the pseudo-cohesion and pseudo-internal friction angle at the peak strength should be analyzed with the confining pressure. The pseudo-cohesion and the pseudointernal friction angle at peak stress of different confining pressures are obtained from Fig. 7, and listed in Table 3, as shown in Fig. 8.
With increasing confining pressure, the pseudo-cohesion at the peak stress increases with a decreasing rate. When the confining pressure increases from 5 to 20 MPa, the pseudo-cohesion at the peak stress increases by about 6.08 MPa (from 17.28 MPa to about 23.36 MPa, see Fig. 8), and when the confining pressure increases from 20 to 40 MPa, the pseudo-cohesion only increases by   With the increase of confining pressure, the pseudointernal friction angle at the peak stress decreases by a decreasing rate. As the confining pressure increased from 5 to 40 MPa, the pseudo-internal friction angle reduced from about 51.44° to 39.44°. With confining pressure increasing from 5 to 20 MPa, the pseudo-internal friction angle reduces from 51.44° to 43.09°, i.e., a reduction of 8.35°, while with confining pressure increasing from 20 to 40 MPa, the pseudo-internal friction angle reduces from 43.09° to 39.44°, i.e., a reduction of 3.65°.

Verification of new evolution model for sandstone strength parameters
To verify the effectiveness of new evolution model for sandstone strength parameters, in this section, the triaxial compression test simulation will be carried out using FLAC 3D with the corresponding relationship between the sandstone pseudo-shear strength parameters and the axial strain obtained by the analysis in "Result analysis of sandstone strength parameters". The total stress-strain curve obtained by the test and the numerical simulation will be compared.

FLAC 3D numerical model, constraints, and model parameters
FLAC 3D is based on explicit Lagrange fast algorithm for calculation using macroscopic mechanical parameters, and is widely used to analyze the reliability of shear strength parameter acquisition method. For the three-dimensional model used for verification simulation, two sections along the central axis and perpendicular to the central axis are selected to intuitively express the model information, by which the specimen size and the boundary conditions can be also better labeled, as shown in Fig. 9. The numerical model is with the size of 50 mm 100 mm, which is same as that of the laboratory specimen. What's more, the model is with the displacement constraint applied at the bottom, the confining pressure applied laterally, and the axial stress applied on the top. The applied confining pressures are 5 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa, respectively; and the loading process is controlled by displacement with the rate of 0.002 mm/s. By rewriting the fish function, the evolution of rock pseudo-cohesion and pseudo-internal friction angle obtained in "Result analysis of sandstone strength parameters" is introduced into FLAC 3D . The rock strength parameters are updated for each strain increment following the relationships shown in Fig. 7. Other parameters are introduced in Table 4. The model density and tensile strength of the laboratory specimen were used, while the elastic modulus and the Poisson's ratio were both determined from the experimental curves (see Fig. 4). The curves of axial stress with lateral strain were collected by ring extensometer in Fig. 3 (not present here).

Comparison of simulated and experimental curves
The simulated curves agree well with the experimental curves (Fig. 10), indicating a successful representation of material strength by the new evolution model of pseudostrength parameters. Under different confining pressures, although all the calculation models are set at the same elastic modulus, the pre-peak curves do not completely overlap (especially for the nonlinear deformation stage before the peak). Rather, it is closer to the analytical template-test curves, which further demonstrates that the analytical results can truly reflect the force-deformation characteristics of the sandstone.
In the post-peak stage, the simulation curves under different confining pressures oscillate to various extent. This is because the quality solution range of internal friction angle is 0°-45° for FLAC 3D (Itasca Consulting Group Inc 2014); any value outside of this interval causes the oscillation of axial stress curve. The strain corresponding to the internal friction angle of 45° increases with the confining pressures (see Fig. 7), which explains the phenomenon that as the  confining pressure increases, the starting point of the oscillation shifts from the peak strength (see the 5 MPa simulation curve in Fig. 10) to the residual stage (see the 40 MPa simulation curve in Fig. 10).

Mesoscopic mechanism for the evolution of pseudo-strength parameters
To throughly understand the internal mechanism of rock pseudo-shear strength parameter evolution, PFC 2D will be used in this section to reproduce the loading failure process of sandstone specimens under different confining pressures, and analyze the crack distribution and propagation during the failure process.

Micro-bond model
The PFC 2D software used in this paper has been successfully applied to simulate the meso-damage process of rock and soil, where the geotechnical media are characterized through particles and bonds (Yang et al. 2014). The particle is represented by rigid disk body with normal and tangential stiffness, whereas the bonds between particles include contact bond and parallel bond (see Fig. 11). Contact bond reflects the normal and tangential action, but cannot capture the moment between particles (see Fig. 11a). Parallel bond can be regarded as a set of springs distributed in a rectangular area centered on the contact point, which transmit both the force and moment (see Fig. 11b). Both these two bonds are included in the bonded-particle model (BPM) which is used in this paper to simulate the rock material (Cho et al. 2017).

Numerical specimen
Numerical specimen is established based on the CPM. Unlike the ball particle model (BPM), the particle rotation of CPM is constrained by surrounding particles, which effectively solve the inherent problem of low internal friction angle of BPM (Li et al. 2016). In addition, the clump is subjected to the force as a whole, so that the force within the clump can be ignored to improve the calculation efficiency (Potyondy and Cundall 2004) Table 4. The CPM model consisting of about 6000 clumps used in this paper has the same size (in two dimensions), boundary conditions, and loading settings as the FLAC model (see "FLAC 3D numerical model, constraints, and model parameters"), as shown in Fig. 12.

Determination procedure of micro-parameters of sandstone
The numerical simulation should first determine the model parameters, and the basic element of the PFC 2D software is the particle and the bond between particles. So, the input model parameters need to reflect the physical and mechanical properties of the particles and the bonding, which cannot be directly obtained by experiment. Researchers usually select the full stress-strain curve of the rock obtained by laboratory test and use the failure mode as a reference to determine the mesoscopic parameters of the model with the "trial and error" method (Castro-Filgueira et al. 2017). The parameters of model in Fig. 12 were determined in the same way, and are listed in Table 5.

Comparison of the simulation and experimental results
The simulated stress-strain curve was compared with the experimental results (Fig. 13). Under different confining pressures, all simulation curves agree well with the experimental curves. The final failure modes of sandstone obtained by simulation and experiment are compared and shown in Fig. 14. There are multiple shear bands inside the simulated and experimental specimens after loading failure, especially under confining pressures of 30 MPa and 40 MPa. Both the simulated and experimental specimens are failed in the conjugate shear planes, which indicates a same failure mode for both cases.
Regarding the stress-strain curve, the simulated peak stress and failure mode match well with the experiment results. It is concluded that the PFC 2D numerical model and its mesoscopic parameters successfully represent the mechanical properties of real sandstone specimens.

Propagation of rock crack
Detailed analysis of the crack propagation of specimen during the whole loading process is present in Fig. 15. During the whole loading process for the specimens, the growth rate of both the tensile and shear cracks follows a S-shaped curve, i.e., first increases and then decreases. With increasing confining pressure, the growth rate of tensile cracks  decreases, whereas the growth rate of shear cracks increases. For cracks number at peak stress points and at the final failure state, higher confining pressure leads to less tensile cracks but more shear cracks (see Fig. 15b). Note that the tensile cracks are generated at an axial strain of about 0.13% (point I, see Fig. 15a), and the strain corresponding to shearcrack generation decreases with increasing confining pressures. In particular, the corresponding strain of the specimen under the confining pressure of 5 MPa and 10 MPa is about 0.22% (point I1, see Fig. 15b); for specimens subjected to confining pressures of 20 MPa, 30 MPa, and 40 MPa, shear cracks emerge when the strain reaches about 0.09% (point I2, see Fig. 15b). For specimens whose shear crack generated under the same strain, their shear-crack growth curves with axial strain are highly overlapped, as shown in Fig. 15b.
The crack distribution at typical stress points is analyzed and shown in Fig. 16. The cracks are randomly distributed for low stress, i.e., see crack distribution diagram corresponding to point a in Fig. 16. At stress points b and c, localized crack distribution emerges, and cracks of multi type interact with each other (i.e., intersection and connection), and nuclear cracks gradually appear. Onward the peak stress point, the nucleated cracks gradually develop and coalesce, and the potential sliding shear plane gradually form (from point c to d in Fig. 16). In this stage, the specimens can be regarded as a continuous medium with visible cracks. In the residual deformation stage (point e), the sliding shear plane fully develops, which destroys the continuous property of the specimen, and leaves external load to be absorbed by the interaction of multiple blocks.

Meso-mechanism analysis of sandstone strength parameter evolution under the same confining pressure
In this section, the specimen under a confining pressure of 10 MPa is selected to study the meso-mechanism of rock strength parameter evolution (see Fig. 17). The relationship between strength parameter evolution and crack propagation of sandstone is divided into seven stages for analysis according to the axial stress curve.
(1) Stage O-A: there is no micro-crack generated inside the model and the rock strength parameters are not fully mobilized at this stage, so the changes of pseudo-cohesion and pseudo-internal friction angle both depend on the axial load. This is a combined result of no prefabri-  Li et al. 2021;Rao et al. 2021;Xu et al. 2022). As the compressive load increases, the tensile cracks generated first inside the model where the tensile strength of bonds is relatively small (Nezhad et al. 2018;Shi et al. 2020b;Yang et al. 2014). Tensile cracks cause the instantaneous disappearance of the tensile strength (electrostatic attraction or cementation) between particles (Cho et al. 2017;Zhao et al. 2018), which leads to the decrease of the maximum cohesion that can be mobilized. The damage of the specimen is relatively small at this stage, and the pseudo-cohesion of the rock is in the increasing stage of the bell curve (from c v, A to c v, B ). After the generation of cracks, the friction between the particles and the inlaying effect of the uneven face increases, which leads to the increase of the maximum internal friction angle that can be mobilized. Therefore, the generation of cracks makes it possible to respond to the increased axial load, so the pseudo-internal friction angle increases from φ v, A to φ v, B .
(3) Stage B-C: the continuous increasing of axial load leads to increasing deviatoric stress of the rock, generating shear cracks (Yang et al. 2014). The micro-destruction (mostly tensile cracks) distributes randomly and dispersedly (see the crack distribution of point C in Fig. 17). The number of tensile cracks is about 2000, i.e., one quarter of the final number of tensile cracks (about 8000), indicating that the cohesion that can be mobilized has been significantly reduced (Zhao et al. 2005). The pseudo-cohesion reaches peak value at point C, so the cohesion is fully mobilized for the first time at point C. In this stage, the greater the number of micro-cracks, the greater the inherent internal friction angle. Correspondingly, the pseudo-internal friction angle increases from φ v, B to φ v, C . (4) Stage C-D: after point C, the specimen gradually enters the stage of nonlinear deformation, and the number of micro-crack events under unit strain increases significantly (see AE curve); local concentration of crack emerges with interactions like crossing, connection, etc., between multi-type cracks (see crack distribution of point D in Fig. 17) (Zhang et al. 2007(Zhang et al. , 2015b, which damages the specimen's continuity and leads to continuous reduction of inherent cohesion. Thus, the pseudo-cohesion is reduced from c v, C to c v, D (since the pseudo-cohesion has been completely mobilized at point C, the pseudo-cohesion c v after C is equal to the maximum cohesion that the specimen can provide in real time). The interlock effect inside the specimen arises from the interaction of the cracks, which moves the pseudo-internal friction angle from φ v, C to φ v, D . (5) Stage D-E: the specimen is in the strain softening stage with sharply increasing cracks (see growth curves of tensile and shear crack in Fig. 17). Nucleated cracks gradually connect, which initiates the potential slip plane (see the crack distribution of point E in Fig. 17). The maximum cohesion that the rock can be provided is further reduced (the pseudo-cohesion is gradually reduced from c v, D to c v, E , see Fig. 17). The specimen can still be considered as a continuous medium although with visible cracks (Zhang et al. 2007). The interlock resistance inside the specimen increases with increasing crack, but decreases with the formation of the shear band. These two processes coexist at this stage, and the amount of growth is greater than the amount of failure (but the failure rate of the interlock resistance is gradually increased). Consequently the pseudo-internal friction angle increases from φ v, D to φ v, E , as shown in Fig. 17. The reduction of axial stress indicates that both the cohesion c E and internal friction angle φ v, E have been fully mobilized, so after point C, the pseudo-friction angle φ v, max is equal to the maximum internal friction angle in real time. (6) Stage E-F: during this stage, the axial stress is rapidly reduced with the gradual formation of the sliding shear plane, and the specimen cannot be regarded as a continuous medium. The continuity of the specimen is gradually lost as the breaking down of a single specimen into plural blocks (see Fig. 17 for the rock failure pattern at point F). The pseudo-cohesion continuously reduces to 0, and the interlock resistance is negligible. In this process, the friction force at the shear plane, as the main component of the axial stress, increases with the formation of the conjugate shear plane. Corre-spondingly, the pseudo-internal friction angle gradually increases from φ v, E to φ v, F , as can be seen from Fig. 17. (7) Stage F: the specimen is in the residual deformation stage. In this stage, the external load is absorbed by the interaction of multiple blocks, and there will be no damage in discrete rock blocks. It can be seen from Fig. 17 that the axial stress, the pseudo-cohesion and pseudo-internal friction angle all remain unchanged.
To summarize, the evolution of rock pseudo-shear strength parameters is closely related to the whole process of rock loading and the generation, propagation, and penetration of internal cracks. Before the residual stage, the pseudo-cohesion of rock first increases at a decreasing rate and then decreases at a gradually increasing rate. Both the initial and final value of the pseudo-cohesion are 0, the maximum value corresponds to the initiation and nucleation of cracks inside the specimen as well. The pseudo-internal friction angle, whose initial value is 0, increases at a gradually decreasing rate as the specimen loading and crack growing.

Conclusion
In this paper, the pseudo-shear strength parameters were proposed to describe both the pre-and post-peak response of cohesion and internal friction angle to external load. An analytical model for the evolution of rock pseudo-shear strength parameters was established, by characterizing the pseudoshear strength parameters in M-C equation as a quadratic function of axial strain. Combined with the full stress-strain curves of hard brittle sandstone under consecutively increasing confining pressures (i.e., from 5 to 40 MPa), the evolution of pseudo-shear strength parameters of sandstone in the whole loading and failure process was obtained. The influence of confining pressure on the evolution of pseudo-shear strength parameters was analyzed, and the reliability of the analytical model was verified against sandstone specimen. Finally, the compression, loading, and failure process of the sandstone specimen was simulated using PFC 2D , considering multiple confining pressures. The meso-mechanism for the evolution of pseudo-shear strength parameters was revealed in combination with a detailed analysis on the simulated micro-crack propagation. The main conclusions are as following: (1) An analytical model for the evolution of rock pseudoshear strength parameters was established based on M-C strength criterion. The evolution of pseudocohesion and pseudo-internal friction angle during the whole process of loading failure was obtained combined with the full stress-strain curve of sandstone triaxial compression. The analytical model is straightforward to application, and is proved to well describe the stress and deformation characteristics of sandstone.
(2) The evolution of pseudo-shear strength parameters of the sandstone specimens studied in this paper is summarized: before the residual stage, the pseudo-cohesion first increases at a decreasing rate and then decreases at an increasing rate, with both initial and final values to be 0; the pseudo-internal friction angle, initially to be 0, increases at a decreasing rate. In particular, the pseudocohesion has decreased significantly in the pre-peak stage. The decrease range of pseudo-cohesion before the peak was more than 38% under the confining pressure of 30 MPa. At the peak stress, the pseudo-internal friction angle is still in the growth stage, and the postpeak rise of the pseudo-internal friction angle was more than 18% under the confining pressure of 40 MPa. The evolution of pseudo-shear strength parameters obtained qualitatively agrees with results obtained by the MSEM in the literature of Zhao et al. (2005).
(3) For the sandstone specimens studied, the pseudo-cohesion at the peak stress increases by a gradually decreasing rate with increasing confining pressure in the range of 5-40 MPa. When the confining pressure increases from 5 to 20 MPa, the pseudo-cohesion increases by 6.08 MPa, whereas increases from 20 to 40 MPa, the pseudo-cohesion increases by only 1.36 MPa. The pseudo-internal friction angle at the peak stress decreases by a decreasing rate with the increasing confining pressure. When the confining pressure increases from 5 to 20 MPa, the pseudo-internal friction angle decreases by 8.35°, and when the confining pressure increases from 20 to 40 MPa, the pseudo-internal friction angle decreases by only 3.65°. (4) CPM well characterized the mechanical properties of rock under multiple confining pressures, as the total stress-strain curves and failure modes of numerical specimens reproduced the experimental results. The evolution of rock pseudo-shear strength parameters is closely related to the external load and the generation, propagation, and penetration process of rock cracks. It is concluded that the rock pseudo-shear strength parameters are gradually mobilized with the increasing load, and the propagation of cracks owing to the decrease of mobilizable maximum cohesion and the increase of mobilizable maximum internal friction angle. In particular, the maximum value of pseudo-cohesion occurs at the beginning of crack aggregation and nucleation in the specimen.
The analytical model established has also been successfully applied in the analytical acquisition of mudstone and Ordovician limestone (Shi 2021). Furthermore, the analytical model is suitable for the specimens with main shear failure rather than splitting failure in which the post-peak curve of the specimen decreases relatively gently. For future research, we will not only expand the adaptability of the analytical model, but also improve the method to reflect the crack propagation of rock.