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摘要: 為揭示巴西圓盤起裂模式的變化規律及其破裂演化過程，運用連續介質彈塑性分析開展巴西圓盤劈裂二維及三維數值模擬研究。通過開展二維模擬研究，探究壓拉比及加載角對試樣起裂破壞模式的影響；通過三維模擬研究，探究圓盤試樣三維破裂面的形成及擴展過程。二維數值模擬結果表明，接觸加載角及壓拉比越大，巴西圓盤試樣越容易發生中心起裂；端部起裂由剪切破壞引起，而劈裂裂紋進一步擴展則由張拉破壞驅動。三維數值模擬結果表明，初始起裂點位于三維圓盤端面，隨加載角增大其逐漸向端面圓心移動；當圓盤發生端面中心起裂時，三維破裂面以弧形邊界向試樣內部發散擴展。無論圓盤試樣發生中心起裂還是端部起裂，由于三維效應巴西劈裂試驗可能都會低估巖石的抗拉強度。Abstract: The Brazilian splitting test is widely used to determine the tensile strength of rocks and rock-like materials due to its easy sample preparation and an easier compressive test setup as an indirect testing method compared with performing a direct uniaxial tensile test. However, the accuracy of this method has also been criticized for a long time in the literature since its introduction. This paper carried out two-dimensional (2D)/three-dimensional (3D) numerical simulations of the Brazilian tensile test using a continuum elastoplastic analysis to reveal the variation of fracture modes of the Brazilian disk and its fracture evolution process. The effect of compression-tension ratios and contact loading angles on the fracture modes of the disk specimens was studied through 2D simulations. Through 3D simulations, the initiation and expansion processes of the 3D fracture under different loading angles were explored. The simulated results of failure modes, stress distributions, and calculated tensile strengths were analyzed. The 2D numerical results show that the larger the contact loading angle and the compression–tension ratio, the more likely the Brazilian disk specimens crack first at the disk center. The fracture initiation under the loading rims is caused by shear failure, but further propagation of the split fracture is driven by tension failure. The 3D numerical simulation results show that the crack initiation point is always located on the end face of the disk and gradually moves to the center from the loading ends as the loading angle increases. When the central tensile cracking appears, the 3D fracture expanded toward the inside of the specimen with an arc boundary. Regardless of whether the disk specimen starts to fracture initially at the disk center or the loading points, the Brazilian tensile test may underestimate the tensile strength of rocks due to the 3D effect.
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Key words:
- Brazilian test /
- numerical simulation /
- compression–tension ratio /
- loading angle /
- 3D fracture
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圖 1 巴西圓盤弧形均布載荷加載示意圖
Figure 1. A disk subjected to diametrically distributed loads in the Brazilian test
圖 3 不同壓拉比時巴西圓盤試樣的破壞過程。 （a）均勻試樣λ=20；（b） 均勻試樣λ=10；（c）非均勻試樣λ=20；（d）非均勻試樣λ=10；（e）非均勻試樣λ=8
Figure 3. Failure processes of the numerical disks with different compression–tension ratios (λ) under the loading angle of 9.18°: (a) homogeneous disk λ = 20; (b) homogeneous disk λ = 10; (c) heterogeneous disk λ = 20; (d) heterogeneous disk λ = 10; (e) heterogeneous disk λ = 8
圖 5 不同加載角時巴西圓盤試樣三維破裂面的起裂及擴展過程。 （a）2α=29.7°；（b）2α=25.7°；（c）2α=14.4°；（d）2α=4.4°
Figure 5. Fracture initiation and propagation processes of the 3D Brazilian disks with different contact loading angles: (a) 2α = 29.7°; (b) 2α = 25.7°; (c) 2α = 14.4°; (d) 2α = 4.4°
圖 6 不同加載角時二維巴西圓盤受壓直徑上應力分布的數值模擬結果與理論計算結果對比示意圖。 （a） 2α=4.58°；（b）2α=6.88°；（c）2α=9.18°；（d）2α=11.48°；（e）2α=13.78°；（f）2α=16.10°
Figure 6. Comparison of normalized stresses along the compressed diameter between the numerical results and Hondros’ solutions with different contact loading angles: (a) 2α = 4.58°; (b) 2α = 6.88°; (c) 2α = 9.18°; (d) 2α = 11.48°; (e) 2α = 13.78°; (f) 2α = 16.10°
圖 7 不同加載角下三維巴西圓盤受壓直徑上應力分布的數值模擬結果與理論計算結果對比示意圖。（a）2α=4.4°；（b）2α=14.4°；（c）2α=25.7°；（d）2α=29.7°
Figure 7. Comparison of normalized stresses along the compressed diameter between the numerical results and Hondros’ solutions with different contact loading angles: (a) 2α = 4.4°; (b) 2α = 14.4°; (c) 2α = 25.7°; (d) 2α = 29.7°
圖 8 不同加載角下三維巴西圓盤試樣中心剖面及端面受壓直徑上切向應力分布示意圖。（a）中心剖面（Y=0.0125）切向應力；（b）端面（Y=0.025）切向應力
Figure 8. Normalized tangential stresses along the compressed diameter of the surface and middle section of 3D disks with different contact loading angles: (a) middle section (Y=0.0125); (b) surface (Y=0.025)
圖 9 不同加載角時三維巴西圓盤試樣軸向受壓平面上（X=0）切向應力分布云圖。（a）2α=29.7°；（b）2α=25.7°；（c）2α=14.4°；（d）2α=7.6°；（e）2α=4.4°
Figure 9. Contour plots of normalized tangential stresses (σxx) on the compressed middle section plane (X = 0) of the 3D disks with different contact loading angles: (a) 2α = 29.7°; (b) 2α = 25.7°; (c) 2α = 14.4°; (d) 2α = 7.6°; (e) 2α = 4.4°
圖 10 不同泊松比時最大切向拉應力點及最大切向拉應變點位置隨接觸加載角的變化情況。（a）最大切向拉應力點；（b）最大切向拉應變點
Figure 10. Change in the position of the maximum tangential tensile stress point and strain point with the contact loading angles under different Poisson’s ratios: (a) the maximum tangential tensile stress point; (b) the maximum tangential tensile strain point
圖 11 最大Griffith等效拉應力位置隨加載角度的變化情況
Figure 11. Change in the position of the maximum Griffith equivalent tensile stresses with the contact loading angles
表 1 數值模型參數
Table 1. Material properties of the numerical model
Elastic modulus/
GPaPoisson’s ratio Compressive strength/
MPaTensile strength/
MPaCohesion/
MPa60 0.25 200 8 38.39 Friction angle/
(°)Residue cohesion/
MPaResidue tensile strength/
MPaPlastic shear strain limit Plastic tension strain limit 48 1.92 0.4 5 × 10?4 2 × 10?4 
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表 2 不同壓拉比及加載角巴西試樣的破壞模式
Table 2. Failure modes of the numerical Brazilian disks with different compression?tension ratios (λ) and contact loading angles (2α)
λ Failure modes 2α = 16.10° 2α = 13.78° 2α = 11.48° 2α = 9.18° 2α = 6.88° 2α = 4.58° 20 
18 
16 
14 
12 
10 
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表 3 不同加載角及壓拉比下二維巴西圓盤試樣抗拉強度計算結果
Table 3. Calculated tensile strength of 2D numerical Brazilian disks with different contact loading angles and compression–tension ratios
λ Real strength/MPa Calculated tensile strength/MPa 2α = 16.10° 2α = 13.78° 2α = 11.48° 2α = 9.18° 2α = 6.88° 2α = 4.58° 20 10.0 10.3 (2.6%) 10.2 (2.2%) 10.2 (1.7%) 10.1 (1.4%) 10.1 (1.2%) 8.7 (13.3%) 18 11.1 11.4 (2.5%) 11.3 (2.0%) 11.3 (1.6%) 11.3 (1.3%) 10.7 (3.3%) 8.6 (22.6%) 16 12.5 12.8 (2.5%) 12.7 (1.9%) 12.7 (1.5%) 12.1 (3.4%) 11.3 (9.7%) 8.7 (30.8%) 14 14.3 14.6 (2.2%) 14.5 (1.6%) 14.5 (1.2%) 11.9 (16.7%) 11.6 (19.0%) 8.7 (39.5%) 12 16.7 16.8 (1.0%) 16.3 (2.5%) 14.7 (12.1%) 11.7 (29.9%) 10.7 (35.6%) 8.6 (48.3%) 10 20.0 17.0 (14.8%) 16.6 (17.0%) 14.6 (26.8%) 12.9 (35.7%) 10.7 (46.7%) 8.6 (56.9%) Note: The data in parentheses are relative errors.
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表 4 不同加載角下三維巴西圓盤試樣抗拉強度計算結果
Table 4. Calculated tensile strength of the 3D numerical Brazilian disks with different loading angles
Loading angles/
(°)Peak loads/
NTensile strength/
MPaRelative error/
%Distance/
mmFailure modes 4.4 5427 2.76 65.5 25 Crack initiation at loading point 7.6 8329 4.22 47.3 20.5 Crack initiation off central point 10.9 10100 5.08 36.5 18.5 Crack initiation off central point 14.4 11784 5.88 26.5 17.5 Crack initiation off central point 17.9 13237 6.52 18.5 14.5 Crack initiation off central point 21.7 14132 6.86 14.3 12.5 Crack initiation off central point 25.7 14925 7.10 11.3 8.5 Crack initiation off central point 29.7 15441 7.17 10.4 0 Crack initiation at central point 33.7 15868 7.16 10.4 0 Crack initiation at central point
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