1. College of Resources and Environment Engineering, Guizhou University, Guiyang 550025, China; 2. Key Laboratory of Karst Geologic Resource and Environment of the Ministry of Education,
Abstract: The cohesion weakening and friction strengthening (CWFS) model for rock reveals the strength components mobilization process during progressive brittle failure process of rock, which is very helpful in understanding mechanical properties of rock. However, the used incremental cyclic loading−unloading compression test for the determination of strength components is very complicated, which limits the application of CWFS model. In this paper, incremental cyclic loading−unloading compression test was firstly carried out to study the evolution of deformation and the strength properties of Beishan granite after various temperatures treated under different confining pressures. We found the axial and lateral unloading modulus are closely related to the applied stress and damage state of rock. Based on these findings, we can accurately determine the plastic strain during the entire failure process using conventional tri-axial compression test data. Furthermore, a strength component (cohesive and frictional strength) determination method was developed using conventional triaxial compression test. Using this method, we analyzed the variation of strength mobilization and deformation properties of Beishan granite after various temperatures treated. At last, a non-simultaneous strength mobilization model for thermally treated granite was obtained and verified by numerical simulation, which demonstrated the effectiveness of the proposed strength determination method. Key words: strength components mobilization; secant unloading modulus; Beishan granite; thermally treated; post-failure Cite this article as: CHEN Shi-wan, LIANG Feng, ZUO Shuang-ying, WU Dao-yong. Evolution of deformation property and strength component mobilization for thermally treated Beishan granite under compression [J]. Journal of Central South University, 2021, 28(1): 219−234. DOI: https://doi.org/10.1007/s11771-021-4598-9.
1 Introduction
Brittle rock under compression often goes through five distinct stages: 1) natural crack closure stage, 2) crack initiation and elastic stage, 3) stable crack growth stage, 4) unstable crack growth stage, and 5) post-failure stage [1, 2]. The development of
cracks in rock will result in nonlinear deformation behavior such as strain hardening, strain softening and stiffness degradation, which brings in challenging task to describe the deformation and strength property of rock. Strain-softening model has been widely used in studying the non-linear mechanical behavior of rock [3−5].
Many researchers studied the post-failure
Foundation item: Project(41902301) supported by the National Natural Science Foundation of China; Project(20201Y185) supported by the Science and Technology Foundation of Guizhou Province, China; Project(Z018023) supported by the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering, IRSM, CAS; Project (201822) supported by the Foundation for Young Talents of Guizhou University, China; Project(2017-5402) supported by the Mountain Geohazard Prevention R&D Center of Guizhou Province, China
mechanical properties of rock to deepen the understanding of strain softening behavior, especially for post-peak regime [6−10]. The degradation of stiffness of rock was taken into account in the strain-softening model. HAN et al [11] proposed a simple numerical procedure of strain-softening model with elasto-plastic coupling. UNTEREGGER et al [12] developed a damage plasticity model for describing the nonlinear mechanical behavior (strain hardening, softening and the degradation of stiffness). DAI et al [13] studied the evolution of secant modulus for rock under loading and confining pressure unloading process. In addition, many studies showed temperature could significantly change rock mechanical properties, such as post-failure behavior [10], dynamic strength [14], strength and brittleness [15], and creep behavior [16]. Rock strength is composed of cohesive strength and frictional strength. MARTIN [17] studied the evolution of cohesive strength and frictional strength of Lac du Bonnet granite using incremental cyclic loading-unloading test. HAJIABDOLMAJID [18] proposed a cohesion weakening and friction strengthening (CWFS) model for brittle rock. He found cohesive strength decreases rapidly with increasing damage once the applied stress exceeded crack damage stress, meanwhile the frictional strength is mobilized to sustain the increasing load. This model was proved to be effective in modeling brittle failure of rock under compression. RAFIEI et al [2] further studied the strength evolution of granite and limestone, and found cohesion degradation was determined by accumulated damage and not sensitive to confining stress, while friction mobilization depended on both damage level and confining pressure. LI et al [19] studied strength property of Beishan granite using incremental cyclic loading−unloading compression test under 0.2−40 MPa confining pressure. They used a new method to calculate cohesion and friction angle. Curves of plastic strain versus crack damage stress at each loading cycle were firstly identified for samples under different confining pressure. Then the strength corresponding to the same plastic strain for samples under different confining pressures were used to calculate cohesion and friction angle based on Morh-Coulomb criterion. Using this method, the obtained cohesion and friction angle are applicable to samples under
various confining pressure condition. In previous studies, the incremental cyclic loading−unloading compression test is essential to analyze the mobilization of strength components during compression [20, 21]. But the incremental cyclic loading−unloading test is very complicated and time consuming compared to conventional triaxial compression test. Easier analysis method is worth developing to study the evolution of strength and deformation properties of rock. In this study, we developed a new method to determine the mobilization of strength components for conventional triaxial compression test [22]. We firstly carried out incremental cyclic loading− unloading compression test and conventional triaxial compression test on Beishan granite samples after different temperatures treated. Then a method for strength components identification was proposed for conventional triaxial compression test based on accurate identification of plastic strain and Morh-Columb criterion. Using this method, the strength mobilization and deformation properties of Beishan granite after various temperatures treated are analyzed. At last a non-simultaneous strength mobilization model is obtained and verified using numerical simulation, which demonstrates the effectiveness of the proposed strength determination method. 2 Experimental work 2.1 Samples preparation Granite samples were collected from Jijicao, Beishan Area, Gansu Province, China. The natural density of Beishan granite is about 2.61 g/cm3. Petrophysical analysis with X-ray diffraction shows that the main mineral components of Beishan granite are 34.09% quartz, 60.59% feldspar, and 5.32% biotite (mass fraction). As shown in Figure 1,
Figure 1 Beishan granite samples
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cylindrical granite samples with 50 mm in diameter and 100 mm in length were prepared according to IRSM suggested method [23]. Granite samples were heated to 100, 300, 500, 600 and 800 °C respectively at 4 °C/min using a tube-type high temperature furnace GR.TF80/11, and temperature was kept for 2 h, then cooled down gradually at the furnace. 2.2 Mechanical test on thermally treated granite To study the deformation and strength properties of Beishan granite, incremental cyclic loading−unloading compression test and conventional triaxial compression test were carried out. An electro-hydraulic servo-controlled rock mechanics testing system (MTS 815.03) was used to carry out mechanical experiment. 2.2.1 Group A: Incremental cyclic loading−
unloading compression test 1) To study the evolution of deformation property of granite under different confining pressure, incremental cyclic loading−unloading compression test was carried out under different confining pressures respectively (0, 5, 10, 20 MPa). 2) To study the influence of thermal damage on mechanical property of Beishan granite after different temperatures treated (100, 300, 500, 800 °C), incremental cyclic loading−unloading compression test was carried out on granite samples after treated at different temperatures (confining pressure is 5 MPa). For incremental cyclic loading−unloading compression test, the confining pressure was firstly applied to the target pressure. Then the axial stress was applied to the predetermined stress under the axial displacement control mode with the rate of 0.002 mm/s. Once the axial stress reached the target stress, the stress was unloaded with rate of −0.002 mm/s until the deviatoric stress was nearby zero (about 0.2 MPa). Samples were loaded and unloaded to residual strength to obtain complete pre-peak and post-peak stress−strain curve. As shown in Table 1, incremental cyclic loading− unloading compression tests were carried out under different pressures on Beishan granite after treated at various temperatures. 2.2.2 Group B: Conventional triaxial compression
test We also carried out conventional triaxial compression test on Beishan granite after treated at
Table 1 Summary of incremental cyclic test and
conventional triaxial compression tests
Sample Treated temperature/°C σ3/MPa σp/MPa
M-U-100 100 0 103.78
M-5-100 100 5 201.98
M-10-100 100 10 264.40
M-20-100 100 20 345.83
M-5-300 300 5 194.62
M-5-500 500 5 189.50
M-5-800 800 5 105.62
Note: To make the sample name more meaningful, samples were renamed compared to Ref. [22]. The first word M means multi-cycle, the second word ‘U’ means uniaxial compression, 5 ,10, 20 represent confining pressure. The last word (100, 300, 500, 800) represents the treated temperature. Samples regarded as 100 °C treated were not specially thermal treated at 100 °C, but used for oven drying method at 100 °C for porosity measurement. Although many studies showed granite after 100 °C treated was almost unchanged compared to untreated granite. In this study they were regarded as 100 °C treated for rigorism.
various temperatures (100, 300, 500, 600, 800 °C). The confining pressures were 5, 10, 20 and 30 MPa. During conventional triaxial compression test, the axial stress was applied by the axial displacement controlled mode with the strain rate of 0.002 mm/s. To study the residual strength and post-failure deformation property, samples were loaded to residual strength and unloaded at the rate of −0.002 mm/s. Finally, samples were reloaded at the rate of 0.002 mm/s. 3 Results 3.1 Results of mechanical test 3.1.1 Stress−strain curves of incremental cyclic
loading−unloading compression test The incremental cyclic loading−unloading stress−strain curves of granite samples under different confining pressures are shown in Figure 2. For uniaxial compression test, post-failure stress− strain curve was missed due to brittle failure. However, for granite under confining pressure, post-failure stress−strain curves were continuously recorded. The incremental cyclic loading−unloading stress−strain curves of granite after treated at different temperatures (σ3=5 MPa) are shown in Figure 3. The deformation characteristics are similar for granite samples after treated at 100 and 300 °C. Before yield stress (stress lower than crack damage stress, σcd), the irreversible strain (plastic
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Figure 2 Incremental cyclic loading−unloading stress−strain curves of Beishan granite under various confining
pressures: (a) 100 °C treated, σ3=0 MPa; (b) 100 °C treated, σ3=5 MPa; (c) 100 °C treated, σ3=10 MPa; (d) 100 °C
treated, σ3=20 MPa
Figure 3 Incremental cyclic loading− unloading stress−strain curves of Beishan granite after treated at different temperatures: (a) 300 °C treated, σ3=5 MPa; (b) 500 °C treated, σ3=5 MPa; (c) 800 °C treated, σ3= 5 MPa
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strain) is very small. However, for granite sample after treated at 500 °C, large irreversible plastic strain appears after the first loading−unloading cycle, the plastic strain also appears in the following loading−unloading cycle. Considerable irreversible plastic strain appears after the first loading−unloading cycle for granite sample after treated at 800 °C, and the peak strength decreases significantly. The crack closure stage is more obvious and longer when the temperature is higher than 500 °C. Overall, the irreversible plastic strain increases and the strength decreases with increasing temperature from 500 °C upward. 3.1.2 Evolution of unloading secant modulus Axial unloading secant modulus (Et
a) and lateral unloading secant modulus (Et
l) are determined using the method as illustrated in Figure 4. Axial and lateral unloading secant modulus of granite under different confining pressures are calculated and shown in Figure 5. For granite samples under different confining pressures
Figure 4 Diagram of unloading secant modulus
and after different temperature treated, Eta increases
with increasing applied axial stress during pre-failure loading process, and decreases with the decrease of strength during post-failure loading process. It is worth noting that the axial unloading secant modulus during post-failure process is lower than that during pre-failure process at the same stress level. For granite treated at temperature
Figure 5 Axial and lateral unloading secant modulus for granite after treatment: (a, b) Under different confining
pressures, treated at 100 °C; (c, d) After different temperatures treated, σ3=5 MPa
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higher than 300 °C, Et
a is degraded by high temperature. Et
l decreases with the increase of applied axial stress in pre-failure stage, however, during post-failure process, Et
l almost remains constant. Figure 6(a) shows axial unloading secant modulus under different confining pressures. Et
a is linearly correlated with the applied axial stress for granite during pre-yielding stage and post-failure stage respectively. Take granite under σ3=5 MPa as an example, the evolution of Et
a can be divided into three parts: before crack damage stress, Et
a linearly increases with the increase of applied axial stress until yield stress (crack damage stress, σcd); then as the applied axial stress increases from crack damage stress to peak stress, Et
a is almost a constant; in the post-failure stage, the secant modulus Et
a decreases linearly with decreasing strength. The slope of the linear-fitting line of axial unloading secant modulus with axial stress in pre-yield stage is similar to that in post failure stage, and the difference between them is almost a constant, characterizing as a constant reduction in Et
a due to mechanical damage. Figure 6(b) shows the evolution of lateral unloading secant modulus, Et
l, during compression test. The lateral unloading secant modulus decreases linearly with applied axial stress before yield, indicating the lateral strain increment is enlarged with the increase of axial stress. The lateral unloading secant modulus finally stabilizes at a constant value after crack damage stress, which is close to the average value of axial unloading secant modulus. Figure 7 shows the axial and lateral unloading secant modulus of granite after treatment. The evolution of unloading secant modulus with the applied axial stress is similar. Et
a is linearly correlated with the applied stress (or post-failure strength) at pre-failure stage and post-failure stage respectively. But the axial unloading secant modulus was obviously weakening with the increase of treated temperatures. For granite after treated at 300 and 500 °C, evolution trend of Et
l is
similar with that after treated at 100 °C. But for granite after treated at 800 °C, Et
l does not linearly decrease but slightly increases with the applied stress. This is attributed to severely thermally damaged at high temperature.
3.2 Results of conventional triaxial compression test on thermally treated samples
3.2.1 Stress−strain curves Stress−strain curves of Beishan granite after treated at different temperatures are shown in Figure 8. For granite samples after treated at lower than 500 °C, the strength and deformation behaviors of granite samples are similar. Stress drops rapidly at the peak stress, characterized as typical brittles failure. For granite samples after treated at 600 and 800 °C, the initial strain is large (corresponding to crack closure stage). Granite samples become more ductile. Interestingly, the lateral strain of granite after treated at higher than 500 °C treated increases significantly at low stress, like dilation deformation. However, samples are not immediately followed by unstable crack growth but followed by linear deformation. This could contribute to higher porosity of thermal treated granite [24], which cause pore/crack adjustment at initial loading stage. The axial unloading secant modulus at residual strength increases with increasing confining pressure. The re-loading stress−strain curve is similar with the initial loading curves, characterizing as initial crack closure deformation, then the elastic deformation [22]. 3.2.2 Strength parameters Table 2 summarizes strength parameters of Beishan granite samples calculated by Morh- Coulomb criterion after treatment. Strength parameters, cohesion (c) and internal friction angle (ϕ) are similar after treated at 100, 300 and 500 °C, suggesting that the strength of Beishan granite is not significantly influenced by temperature when treated within 500 °C. However, cohesion decreases significantly after treated at 600 °C. For granite samples after treated at 800 °C, cohesion significantly decreases to 2.74 MPa, which dedicates the cohesive strength has been depleted essentially. The internal friction angle (ϕ) decreases slightly from 600 °C upward treated. 4 Discussion 4.1 A strength components determination
method The non-simultaneous mobilization of cohesion weakening and friction strength (CWFS) model is shown effective in describing the deformation and strength behaviors under various
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Figure 6 Evolution of axial (a) and lateral (b) unloading secant modulus during entire compression under different
confining pressures
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Figure 7 Evolution of axial and lateral unloading secant modulus during entire compressive failure of granite after
various temperatures treated
confining pressure conditions. However, its application is limited for the complicated damage- controlled test (incremental cyclic loading− unloading test). Some researchers tried to determine the strength components mobilization using conventional triaxial compression test data [25−27]. They used a constant unloading modulus (usually the secant modulus at peak strength) to determine plastic strains for samples under different confining
pressures. Then the strength components were calculated using Morh-Columb criterion at the corresponding plastic strain for test results under various confining pressure. So to determine plastic strain accurately is crucial for this method. Figure 9 shows the comparison of measured plastic strain and calculated strain using a constant unloading modulus. We find the calculated plastic strain using constant unloading modulus, especially the axial
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Table 2 Strength parameters for Beishan granite after
different temperatures treatment
Parameter/°C c/MPa ϕ/(°)
100 23.31 52.59
300 23.62 53.64
500 23.71 53.35
600 13.78 52.20
800 2.74 50.25
plastic strain differs remarkably with the measured axial plastic strain. Therefore, non-negligible error would be introduced to the plastic strain identification using an unloading secant modulus at peak strength, which would result in unacceptable error for strength components determination [22]. According to incremental cyclic loading− unloading test results, typical axial and lateral unloading secant modulus are shown in Figure 10.
Figure 8 Stress−strain curves of Beishan granite after treatment at various temperatures [22]: (a) 100 °C; (b) 300 °C; (c) 500 °C; (d) 600 °C; (e) 800 °C
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Figure 9 Illustration of error of plastic strain
identification (modified from Ref. [22])
As analyzed above, the evolution of unloading secant axial modulus (Ea
t) with the applied axial stress can be divided into three parts (Figures 10(a) and (b)). The axial unloading secant modulus (Ea
t) is similar to axial loading secant modulus. The slope k can be determined using Ea
t at σcd and Eat
at σr. Eat at σr needs an unloading cycle at residual
stress, which is easy to conduct in conventional triaxial compression test. The lateral unloading secant modulus, shown in Figure 10(c), can be simplified as Figure 10(d). El
t at residual stress can be determined by the unloading cycle at residual stress, and El
t at crack damage stress is almost equal to that at residual stress for conventional triaxial compression test. Overall, the evolution of secant modulus during compression test shows that secant modulus can be determined by the applied stress and damage state. Secant modulus can be expressed as [22]: E(t)=k(ωi)σ+b(ωi) (1) where ωi represents the damage state of rock at three damage stages: ω1 for pre-yielding stage, ω2 for pre-yielding to peak strength, ω3 for post-failure stage. These damage stages determine the slope k and constant b. From the analysis above, axial and lateral plastic strain are negligible before crack damage
Figure 10 Variation of deformation parameters with applied stress and damage state (modified from Ref. [22]):
stress, so it only needs to determine k and b for yield stage and post-failure stage. Ea
t almost keeps consistent from crack damage stress to peak stress. Crack damage stress and its corresponding strain, residual stress and its corresponding plastic strain can be used to determine k and b for post-failure stage, and Ea
t at crack damage stress can be used to determine b for pre-yielded stage. For lateral unloading secant modulus from crack damage stress to post-failure stage, El
t can be determined using residual stress, corresponding strain and lateral plastic strain. As shown in Figure 11(a), the axial and lateral plastic strain of samples under various
Figure 11 Developed method for strength components
identification (modified from Ref. [22]): (a) Diagram of
axial and lateral plastic strain identification;
(b) Strength−plastic strain curves; (c) Cohesion and
friction angle calculation
confining pressure can be accurately determined using Eq. (1). Subsequently, the plastic strain−strength curves were determined as shown in Figure 11(b). The Morh-Coulomb criterion can be applied to calculate cohesion and internal friction angle using strength values corresponding to the same plastic strain of samples under various confining pressures according to Figure 11(c).
4.2 Strength parameters determination using proposed method
We determine strength components mobilization of granite after treated at different temperatures using the proposed method as following. 4.2.1 Plastic strain identification According to the proposed method, the axial and the lateral plastic strain are firstly calculated as shown in Figure 12. The calculated plastic strain is
Figure 12 Comparison of measured plastic strain with
calculated plastic strain using proposed method:
(a) 300 °C treated, σ3=20 MPa; (b) 500 °C treated, σ3=
20 MPa
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consistent with the real plastic strain, indicating the plastic strain determination method could precisely determine the axial and lateral residual plastic strain. 4.2.2 σ−γp curves Then the evolution of strength with plastic strain (σ−γp curve after crack damage stress) is determined as shown in Figure 13 [22]. 4.2.3 c and determination Finally, strength parameters, cohesion c and internal friction angle corresponding to the same plastic strain of granite samples under different temperatures, are calculated using Morh-Coulomb criterion by following equations [22]:
s 3K P 1 = + (2)
21 sin πtan
1 sin 4 2K
(3)
cos π
2 2 tan1 sin 4 2
P c c
(4)
where σ1s is the applied stress or strength with the same plastic strain under different confining
pressure, σ3, K and P are functions of cohesion c, and internal friction angle . The evolution of cohesion and internal friction angle for granite after treated at various temperatures is shown in Figures 14(a) and (b). For granite after 100−500 °C treated, cohesion increases firstly with plastic strain from crack damage stress to peak strength, then decreases with plastic strain until residual strength, finally stabilizes at residual strength. The internal friction angle is firstly strengthened before peak strength, then decreases with plastic strain until residual strength, finally stabilizes during residual strength stage. The cohesion of granite after treated at 500 °C is slightly weakened compared to that after treated at lower temperatures. For granite samples after treated at 600 °C, cohesion decreases remarkably. This is attributed to the α−β phase transition of quartz around 573 °C, which causes sharp thermal expansion of quartz (8% volume increase), inducing large thermal stress to cause marked thermal damage on granite [10, 28, 29]. The residual cohesion of granite after treated at 100−600 °C
Figure 13 σ−γp curves for granite after different temperatures treatment (γp=1/2(ε1
is similar. The influence of temperature on friction angle is limited compared to cohesive strength. According to the calculation method described in Refs. [21, 22], dilation angle of granite (σ3= 10 MPa) after thermally treated during compression
is also calculated as shown in Figure 14(c). The evolution of dilation angle of granite after treated at 100−500 °C shows similar trend, the dilation angle increases rapidly from crack damage stress to peak stress, then decreases with the development of plastic strain during post failure stage. Dilatancy is weakened after treated at 500 °C compared to granite after treated at lower temperature. Dilatancy is further weakened after treated at 600 °C. This is also contributed to higher thermal damage caused by thermal expansion of quartz by α−β transition at around 573 °C. 4.3 Verification of determined model According to the evolution of cohesion and internal friction angle determined by the proposed method, a strength mobilization model [22] is derived as shown in Figure 15. To verify the applicability of the proposed simplified model, a code was developed in FLAC3D. Model parameters for different temperatures treated granite are listed in Table 3. Figure 16 shows the comparison of the
pc3 are the plastic strain at peak stress, the initial
residual stress and the end residual stress; ψini, ψp, ψres are dilation angle at initial stage; cini, ϕini, cp, ϕp, cres, ϕres are cohesion and friction angle at initial stage; E and μ are elastic modulus and Poisson ratio.
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Figure 16 Comparison of experimental and numerical
results for triaxial compression tests: (a) Treated at
100 °C; (b) Treated at 500 °C
experimental results with numerical calculation results. It can be seen that the axial and lateral stress−strain behaviors are well captured by the proposed model for all confining pressure. The deformation behaviors (strain hardening after crack damage stress, dilatancy, stress drop at peak stress) and strength properties (peak strength and residual strength) are all well captured by the model developed from the proposed method. Overall, the strain-softening model with strength parameters identified by the proposed method is capable of predicting key features of strain-hardening, dilatancy and strain-softening behavior. We should note that the in-situ stress (σ1>σ2>σ3) is more complex than laboratory triaxial compression test condition (σ1>σ2=σ3). Many researches showed the intermediate principal stress does influence mechanical behaviors of rock material in many aspects [30−33], such as failure
modes, strength and deformation properties of intact rock, jointed rock [30] and veined rock [31]. In addition, engineering construction will induce stress-redistribution. Mechanical behavior of granite under more complex stress condition (complex in-situ stress condition and change of principal stress) could be more complicated, but more valuable for real engineering, which is worth further study. 5 Conclusions In this study, we carried out incremental cyclic loading−unloading compression tests and conventional triaxial compression tests on Beishan granite after treated at various temperatures under different confining pressures. Conclusions are as follows: 1) The axial and lateral unloading secant modulus is piecewise linear with applied stress which determined by damage state. Based on this finding, a method to accurately identify plastic strain is proposed for conventional triaxial compression tests. 2) Based on accurate plastic strain determination, a method to determine strength components mobilization using conventional triaxial compression test is proposed. Using this method, the strength mobilization process of Beishan granite after various temperatures treated is analyzed. 3) Cohesion firstly increases from crack damage stress to peak stress, then decreases rapidly with increasing damage. Meanwhile, the internal friction angle firstly increases then decreases slightly. Strength parameters are similar for granite after treated at 100 and 300 °C, while the cohesion decreases after treated higher than 500 °C treated. The internal friction angle is not sensitive to the treated temperature compared to cohesion. 4) The dilation angle increases rapidly from crack damage stress to peak stress, then decreases with the development of plastic strain during post failure stage for granite after treated at 100−500 °C. The dilatancy of granite is weakened after treated higher than 500 °C. At last, the capability of the strain-softening model determined by the proposed method in capturing the mechanical behavior of rock is demonstrated using numerical simulation.
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Contributors CHEN Shi-wan provided the concept, conducted experimental work and edited the draft of manuscript. LIANG Feng, ZUO Shuang-ying and WU Dao-yong edited the draft of manuscript, replied to reviewers’ comments and revised the final version. Conflict of interest CHEN Shi-wan, LIANG Feng, ZUO Shuang- ying and WU Dao-yong declare that they have no conflict of interest.
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