Crack propagation of a specimen shaped into 150 × 75 × 40 mm plates was observed using a digital image correlation technique (Rubino et al., 2015; G.; Gao et al., 2017; F.; Huang et al., 2020).

For mortar samples, the mortar was poured into a special mold, demolded, and cured for 28 days to obtain jointed specimens. Before casting, a 25 × 40 × 2 mm steel sheet and a 6 mm diameter steel bar were placed in the center of the mold and drawn out during demolding to form prefabricated joints and bolt holes, respectively (Figure 1). The angles of the joint with respect to bolts were 45° and 75°, respectively.

According to the size similarity constant C l , the Mg/Mn alloy bar was machined into a 4 mm diameter bolt, with the thread machined on the bolt surface to simulate the rough surface of a prototype rock bolt. In addition, a 0.5 mm deep groove was machined on the bolt surface of the bolt, which was used to lay an optical fiber to measure the bolt strain. Epoxy resin was used for providing adhesion between the optical fiber and bolt groove and also serving as the anchoring agent between the bolt and mortar.

The specimen production process and size are shown in Figure 1.

The test machine was an RLW-3000 hydraulic servo testing machine. In these experiments, loading was applied in the displacement-controlled mode, with the loading rate of 0.2 mm/s.

The AE sensor with a resonant frequency of 55 kHz, an operating frequency of 30–100 kHz, and a peak sensitivity of 75 dB, was adopted. Four sensors were arranged on both sides of the sample.

The noncontact full-field strain measurement system consisted of two parts: artificial speckle and CCD camera. In order to ensure the measurement accuracy, a fixed focus lens installed on the CCD camera was placed in front of the specimen, which ensured that the aperture and focal length were stable during the test. Artificial speckles were drawn into points with different shapes, and each of them occupied 5–10 pixels in the digital image which is beneficial to be accurately recognized. An artificial speckle pattern was produced on the front of the model, and its layout is shown in Figure 2. In addition, the light in the laboratory was also required to be extremely stable.

The 0.5 mm diameter optical fiber was embedded into the groove of the simulated bolt using epoxy resin to ensure their synchronous deformation while loading. In this regard, the deformation of the bolt can be measured.

The stress–strain curves of specimens with the joint angles being 75° and 45° under different anchorage conditions are given in Figure 3 and Supplementary Table S3. From these results, after bolting, the UCS of 75° and 45° jointed specimens increased by 16.63 and 21.54%, respectively. The peak strains of 75° and 45° jointed specimens increased by 11.11 and 10.81%, respectively. However, the elastic module of jointed samples hardly changed after the bolt was installed.

The maximum principle strain under uniaxial compression actually developed along the edge of the moving zone, which might have led to the formation of wing cracks and anti-wing cracks (Figure 4). The maximum principle strain fields of jointed and bolted jointed specimens under uniaxial compression were similar to those reported by L.N.Y. Wong and Einstein (2009b). The angles of the maximum principal strain direction of jointed and bolted jointed specimens are listed in Supplementary Table S4. Because the bolt did not pass through the wing crack propagation direction, the development of the maximum principal strain in the wing crack direction was not significantly affected. Specifically, after anchoring, the angle of the maximum principal strain of 75° and 45° jointed specimens shrunk by 0.08 and 3.21% in the development direction of the wing crack, respectively. In contrast, the propagation direction of the anti-wing crack passed through the anchorage zone, which made the maximum principal strain development obstructed and shrunk closer to the anchorage zone. The angle of the maximum principal strain of 75° and 45° jointed specimens shrunk by 13.46 and 10.30% in the development direction of the anti-wing crack, respectively.

AE signals generated during the rock failure process are used to interpret the damage evolution. The crack radius can be calculated using the Brune model (J. Zhou et al., 2018; Brune 1970; J. Zhou et al., 2021) as r = K v s 2 π f c , (2) where K is the Brune constant, the value of which is generally 2.34, v s is the shear wave velocity, and f 0 is the corner frequency.

The crack radius distribution of the specimens with the joint angles of 75° and 45° is shown in Figure 5. Clearly, the radius of most cracks was between 1.5 and 3.0 mm whether the jointed specimens were anchored or not. However, the crack propagation of jointed specimens was still restrained by the bolt, which was embodied in the following aspects. First, the average crack radius of jointed specimens was larger than that of bolted jointed specimens. Second, cracks with a radius close to 9.0 mm occurred during the failure process of jointed specimens, although their number was few. In contrast, cracks with a radius over 6.0 mm were difficult to be produced during the failure process of bolted jointed specimens.

The severity of the damage is reflected by the AE magnitude. The magnitude during the failure process of jointed and bolted jointed specimens was calculated by the method described according to Liu et al. (2021) (Figure 6). After anchorage, large-magnitude AE events during the compaction stage of jointed specimens became reduced. The number of AE events with a magnitude close to 8 in bolted jointed specimens was clearly less than that in jointed specimens near the instability stage. With bolt installation, the average magnitude of AE events during the failure of jointed rock specimens becomes low. This suggests that the anchorage effect reduced the severity of damage and improved the anti-disturbance capacity of jointed specimens.

The RA value (rising time/maximum amplitude) and average frequency (AF) of AE signals can be used to determine the tensile or shear damage modes (Soulioti et al., 2009). When RA − AF > 0 , the signal is predominated in the shear damage mode, and it is predominated by tensile damage when RA − AF < 0 (Du et al., 2020; Y.-Q.; Wang et al., 2021). Tensile damage appeared to be the main mode of both jointed and bolted jointed specimens during uniaxial loading, while the shear mode appeared in large numbers near the instability stage (Figure 7), which indicated that the anchorage effect did not change the damage mode of jointed rock masses under uniaxial compression. The difference was that the proportion of AE signals produced in the shear mode in bolted jointed specimens was smaller than those in jointed specimens before the failure of specimens. More specifically, the proportion of AE signals predominated in shear damage before instability in 75° jointed specimens was 15.49%, while that of 75° bolted jointed specimens was reduced to 12.50%. The proportion of 45° bolted jointed samples decreased from 10.87 to 9.59% after the specimen was bolted. This phenomenon was attributed to the inhibition effects of the bolt in the formation and slip of shear zones, which led to decreased AE signals driven by the shear mode in bolted jointed samples. Therefore, it was considered that the anchorage effect inhibited shear damage in jointed specimens.

Under uniaxial compressive loading, the joint tip is always prone to cracking, which is due to the dislocation along the joint plane. After the jointed specimen is anchored, the cracking at the joint tip is alleviated. This is because dislocation is restrained by the bolt. As a cost, the interface between the bolt and jointed specimen becomes a location easy to crack.

Through distributed optical fiber measurements, the strain distribution along the whole length of the bolt was obtained (Figure 8). Bolt deformation in bolted jointed specimens was mainly tensile. The Mg/Mn alloy bolt is an elastic homogenous material, with an elastic modulus being 45 GPa, such that the strain in the bolt was always in tension during the tests. The tensile stress in the middle of the bolt was the highest, and the tensile stress was the lowest at both ends.

However, the highest strain was not located at the midpoint of the bolt. In specimen bolted with a 45° joint, the peak strain was 0.138%, and its location was 4 mm away from the bolt midpoint, while the peak strain point of 75° bolted jointed specimens was 12 mm away from the midpoint, and the strain was 0.137%, which almost equaled the value of the anchorage specimen with a 45° joint. This suggested that the reinforced range of a bolt was greater in a 75° jointed specimen than in a 45° jointed specimen.

The damage constitutive relationship of rocks under various stress states showed that damage to a rock was closely related to its stress state (Figure 9). When the rock stress state met the maximum tensile stress criterion or Mohr–Coulomb criterion, rock damage began to occur (Zhu et al., 2010). Under any stress condition, the maximum tensile stress criterion was preferred. The maximum tensile stress criterion and Mohr–Coulomb criterion were expressed as F 1 = σ 1 − f t 0 , (3) F 2 = ( cos ⁡ θ − 1 3 sin ⁡ θ ⁡ sin ⁡ ϕ ) J 2 + σ m ⁡ sin ⁡ ϕ − 1 − sin ⁡ ϕ 2 f c 0 , (4) where σ 1 is the major principle stress, f t 0 and f c 0 are the rock tensile strength and uniaxial compressive strength, respectively, and ϕ is the internal friction angle. The mean stress was σ m = ( σ 1 + σ 2 + σ 3 ) / 3 , where σ 2 and σ 3 are the intermediate and minor principle stresses, respectively. For lode angle θ = 1 3 sin − 1 ( − 3 3 J 3 2 J 2 3 / 2 ) , ( − π 6 ≤ θ ≤ π 6 ) are defined. J 2 and J 3 are the second and third principle invariants of the stress deviator, respectively,where D is the damage variable determined by (G.-l. Zhou et al., 2020; Y.-Q. Wang et al., 2021), = 0 ∼ 1 . The detailed formula is expressed in Eq. 4 as D = { 0                             F 1 < 0   a n d   F 2 < 0   1 − ε t 0 ε 1 F 1 = 0   a n d   d F 1 > 0 1 − ε c 0 ε 3 F 2 = 0   a n d   d F 2 > 0 . (5)

These damage models, given as equations. (2)–(4), were implemented into FLAC^3D to simulate the damage and failure of rocks under various loading conditions.

In the nonlinear shear-sliding (query) model (M. Huang, Zhou, and Ou 2014), the shear stress and shear displacement relationship of the anchoring interface was τ = τ r + a ⁡ exp ( − b s ) − ( a + τ r ) exp ( − 2 b s ) , (6) where τ r ≥ 0 , a ≥ 0 , and b > 0 , with τ being the shear stress of the anchoring interface, s being the bolt slip, and τ r being the residual shear strength of the anchoring interface. Parameters a and b are determined by the peak shear strength τ u and peak shear displacements s u and τ r .

Suppose the residual shear strength τ r was expressed as τ r = γ τ u , (7) where γ is the ratio of the residual shear strength to peak shear strength, 0 < γ < 1 . Parameters a and b were expressed as a = 2 τ u ( 1 − γ + 1 − γ ) , (8) b = 1 s u [ l n ( 1 + γ 2 ( 1 − γ + 1 − γ ) ) + ln ( 2 ) ] . (9)

The relationship between the anchoring interface shear stress and shear displacement is shown in Figure 9.

Referring to the experimental scheme, boundary conditions of the numerical model were set. Both ends of the specimen were steel plates, and the plates and bolts were assumed to be homogenous materials.

Because rock heterogeneity was considered in numerical simulations, the damage of rock elements in a specimen should be different under the same input of Weibull distribution parameters of UCS and elastic modulus, but the failure pattern observed here was indeed similar (Zhu and Tang 2004). The mechanical parameters of the specimen are listed in Supplementary Table S3.