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Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading

Initiation and Fracture Characteristics of Different Width Cracks of Concretes under... applied sciences Article Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading 1 1 , 1 1 1 , 2 , 3 Lizhou Wu , Jianting Zhou * , Jun Yang , Jingzhou Xin , Hong Zhang * and Bu Li State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, China; lzwu@cqjtu.edu.cn (L.W.); yangjun@cqjtu.edu.cn (J.Y.); xinjz@cqjtu.edu.cn (J.X.) China Southwest Geotechnical Investigation & Design Institute Co., Ltd., Chengdu 610052, China; lbcdut@foxmail.com College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu 610059, China * Correspondence: jtzhou@cqjtu.edu.cn (J.Z.); zhanghong@cqjtu.edu.cn (H.Z.) Abstract: A stress concentration at a crack tip may cause fracture initiation even under low-stress con- ditions. The maximum axial stress theory meets the challenges of explaining the fracture propagation of a non-closed fracture of cracked concretes under compressional loading. Uniaxial loading tests of single-crack concrete specimens were carried out and a numerical simulation of fracture propagation under uniaxial compression was performed. The radial shear stress criterion for a mode-II fracture is proposed to examine the stress intensity factor (SIF) of the pre-crack specimens under compressional loading. When the maximum radial shear stress at the crack tip is larger than the maximum axial tensile stress, and the maximum dimensionless SIFs can satisfy f /f > 1 and f /f rq max q max rq max q max p p > K /K ( f = K /s pa and f = K /s pa are maximum dimensionless mode-I IIC IC qmax Ie y rqmax IIe y and mode-II SIFs, respectively), the crack will extend along the direction of the maximum radial shear stress. The influence of the single-crack angle and width on the mechanical properties of the specimens was examined. The experimental and numerical results indicate that the existence of cracks can considerably weaken the strength of the specimen. The distribution and width of the Citation: Wu, L.; Zhou, J.; Yang, J.; cracks had a significant effect on the specimen strength. The strength of the concrete specimen initially Xin, J.; Zhang, H.; Li, B. Initiation and Fracture Characteristics of Different decreased and then increased with increasing fracture angle. The failure mechanism and rupture Width Cracks of Concretes under angle of pre-crack brittle material while considering crack width will be discovered. Compressional Loading. Appl. Sci. 2022, 12, 4803. https://doi.org/ Keywords: crack; concrete material; uniaxial compression; strength; mixed-mode fracture criterion 10.3390/app12104803 Academic Editor: Dario De Domenico 1. Introduction Received: 1 April 2022 Brittle materials such as concrete or rock usually contain macroscopic cracks or defects Accepted: 1 May 2022 that develop because of complex environmental conditions, which has a significant effect on Published: 10 May 2022 the properties of brittle materials [1,2]. These defects destroy the material integrity, weaken Publisher’s Note: MDPI stays neutral the mechanical properties of the materials, and modify the stress distribution. Moreover, a with regard to jurisdictional claims in stress concentration can be generated at the crack tip, which can influence the failure mode published maps and institutional affil- of brittle materials. Therefore, to study the deformation and failure characteristics of brittle iations. materials, many theoretical and numerical studies show that internal cracks in concrete or rock play a significant role in determining the deformation pattern, the strength of the material, and the fracture mode, although crack initiations for pre-existing defects have a long history [3–14]. Copyright: © 2022 by the authors. Many brittle materials including rock and concrete show high elastic modulus and Licensee MDPI, Basel, Switzerland. strength. Brittle materials are considered to be linearly elastic. When studying cracks in This article is an open access article brittle materials, the stress intensity factor (SIF) can be employed to describe the stress state distributed under the terms and at the crack tip using the fracture mechanics method [15–19]. Three important fracture conditions of the Creative Commons initiation criteria are commonly employed to analyze the crack propagation mechanism of Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ brittle materials: the maximum tangential stress, the maximum energy release rate, and the 4.0/). minimum energy density criterion [20–24]. The F-criterion, which is a modified version Appl. Sci. 2022, 12, 4803. https://doi.org/10.3390/app12104803 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 4803 2 of 14 of the energy release rate criterion, may also be employed to study the fracture behavior of quasi-brittle materials [25,26]. These criteria are based on the assumption of a mode-I fracture. However, the fracture extension of compression shear cracks while considering mixed-mode I/II fracture has rarely been investigated [27]. Many experiments were carried out to examine the crack initiation, propagation path, and eventual coalescence of cracks in samples made of various materials, including artificial materials, under compressive loading [3,8,28–41]. In experimental studies, the brittle material is often made into specimens with embedded cracks in the laboratory and uniaxial compression tests are carried out on the rock or concrete samples [10,42,43]. Brazilian disk testing of rock and concrete samples is commonly used to analyze the tensile strength, fracture toughness, and mixed-mode fracture process in uncracked and pre-cracked samples under compressive loading [15,44–55]. A number of numerical approaches are used to examine fracture and crack propagation and crack growth in brittle materials; these include the finite element method (FEM), the boundary element method, and the discrete element method [6,24,33,40]. SIF, energy- release rate (G), crack propagation, fracturing time, static tensile, and normal-distributed stresses were computed to describe the fracture initiation and propagation in brittle material samples [16]. The objective of the study presented in this paper was to carry out uniaxial compres- sion tests and numerical analyses of specimens containing a crack and to examine suitable fracture criteria to explain the initiation angle of cracked concretes. The crack growth and the stress and strain characteristics of the concrete specimen for different crack widths and crack angles were analyzed based on the maximum circumferential stress theory and radial shear stress criterion. The relationship between the stress threshold of crack propagation with pre-crack angle and width is discussed, the influence of the crack width on the fracture propagation mode and fracture initiation is examined, and the fracture propagation of the non-closed cracks is discussed. Based on the experimental results, the methods suitable for the concrete fracture propagation incorporating crack width will be discussed. 2. Materials and Methods 2.1. Specimen Preparation The samples consisted of concrete material composed of cement, gypsum, quartz sand, water, a water-reducing agent, and a waterproofing agent, the details of which are listed in Table 1. The elastic modulus is 1.35 GPa and Poisson’s ratio is 0.21, which is obtained by uniaxial compression tests. Table 1. The ratio of concrete material. Quartz Sand Water-Reducing Water-Proofing Cement Gypsum Water Agent Agent 0.60 mm 0.30 mm 0.15 mm 0.075 mm 33.60% 8.06% 12.33% 13.41% 13.60% 0.96% 16.89% 0.86% 0.29% To replicate the crack in the sample, a thin steel sheet was embedded in the concrete material. Cuboid specimens with a central through-crack (L W H = 50 50 100 mm) were prepared for the compression-shear tests. In the study of the effect of crack angles on the strength and deformation, the fracture angle difference of the materials is generally set to 15 [40]. We applied various crack angles: 15 , 30 , 45 , 60 , and 75 , and a crack width of 0.5 or 3 mm (Figure 1). The thin steel sheet was first set in the sample mold, and then the concrete material was poured into the mold. The sample was vibrated to compact the material and left to set for 12 h at a temperature of 20 C. Then, the sheet steel was removed from the mold. The smoothness of the samples and cracks were checked. Finally, to provide proper curing humidity, the samples were cured in water for 28 days at room temperature. Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 15 Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 15 the material and left to set for 12 h at a temperature of 20 °C. Then, the sheet steel was the material and left to set for 12 h at a temperature of 20 °C. Then, the sheet steel was removed from the mold. The smoothness of the samples and cracks were checked. Finally, removed from the mold. The smoothness of the samples and cracks were checked. Finally, Appl. Sci. 2022, 12, 4803 3 of 14 to provide proper curing humidity, the samples were cured in water for 28 days at room to provide proper curing humidity, the samples were cured in water for 28 days at room temperature. temperature. (a) (b) (c) (a) (b) (c) Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample with with 3.0 mm width single hollow crack; (c) samples with different crack angles. with 3.0 mm width single hollow crack; (c) samples with different crack angles. 3.0 mm width single hollow crack; (c) samples with different crack angles. 2.2. Compression Shear Fracture Test of Concrete Samples 2.2. 2.2. Co Compr mpression ession Shear Shear Fractur Fracture Tes e Test tof of Co Concr ncrete ete Samples Samples Each brittle concrete specimen was placed in the loading apparatus and loaded under Each Eachbrittle brittleconcr concrete spec ete specimen imen w was as plac placed ed in in the the loading loadingapparatus apparatus an and d loaded loadedunder under uniaxial compression (Figure 2). The tests were carried out on the MTS 815 testing ma- uniaxial uniaxial com compr pression ession (Fig (Figur ure e 22). The ). The te tests sts were were c carried arried out out on the on the MTS MTS 815 te 815 testing sting m ma- a- chine, which showed stable and reliable performance. The crack widths and angles of the chine, which showed stable and reliable performance. The crack widths and angles of the chine, which showed stable and reliable performance. The crack widths and angles of the specimens are listed in Table 2. specimens are listed in Table 2. specimens are listed in Table 2. (a) (b) (a) (b) Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single frac- Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single frac- Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single fracture. ture. The applied compressive stress is represented by ; β is the angle between the crack axis The applied compressive stress is represented by s ; b is σ the angle between the crack axis and the ture. The applied compressive stress is represented by y ; β is the angle between the crack axis direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen height. and the direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen and the direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen height. height. Table 2. Crack parameters for specimens in uniaxial compression tests. Table 2. Crack parameters for specimens in uniaxial compression tests. Table 2. Crack parameters for specimens in uniaxial compression tests. Cases Crack Width d (mm) Crack Length a (mm) Crack Angle b ( ) Cases Crack Width d (mm) Crack Length a (mm) Crack Angle β (°) Cases Crack Width d (mm) Crack Length a (mm) Crack Angle β (°) 1 0.5 10 15 1 0.5 10 15 1 0.5 10 15 2 0.5 10 30 2 0.5 10 30 3 0.5 10 45 2 0.5 10 30 4 0.5 10 60 3 0.5 10 45 3 0.5 10 45 5 0.5 10 75 4 0.5 10 60 4 0.5 10 60 6 3 10 15 5 0.5 10 75 5 0.5 10 75 7 3 10 30 8 3 10 45 9 3 10 60 10 3 10 75 Note: b is the angle between the crack and loading direction. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 15 6 3 10 15 7 3 10 30 8 3 10 45 9 3 10 60 10 3 10 75 Appl. Sci. 2022, 12, 4803 4 of 14 Note: β is the angle between the crack and loading direction. The width of the cracks for specimens 1–5 was 0.5 mm and 3 mm for specimens 6– 10. A small steel sheet, 0.5 and 3 mm thick, was inserted into each sample to fill the crack The width of the cracks for specimens 1–5 was 0.5 mm and 3 mm for specimens 6–10. (Figure 1b,c). The loading was controlled by changing the displacement, with a loading A small steel sheet, 0.5 and 3 mm thick, was inserted into each sample to fill the crack rate of 0.5 mm/min. The crack propagation of the specimen was captured by a high-reso- (Figure 1b,c). The loading was controlled by changing the displacement, with a loading rate lution camera during the tests. of 0.5 mm/min. The crack propagation of the specimen was captured by a high-resolution camera during the tests. 2.3. Analysis of the Experimental Results 2.3. Analysis of the Experimental Results Figure 3 shows the stress–strain curves of the specimens under uniaxial loading. The uniaxialFigur comp er3es shows sion pthe roce str ssess–strain of the sincurves gle-crac of k c the onspecimens crete specim under en ca uniaxial n be divlioading. ded into The uniaxial compression process of the single-crack concrete specimen can be divided into four stages: (1) The nonlinear compaction stage where mostly internal micro-cracks are four stages: (1) The nonlinear compaction stage where mostly internal micro-cracks are present in the specimen. The strain is small, and the stress remains unchanged, resulting present in the specimen. The strain is small, and the stress remains unchanged, resulting in a relatively smooth curve. (2) The linear elastic deformation stage, with no obvious in a relatively smooth curve. (2) The linear elastic deformation stage, with no obvious crack development. (3) The stable crack expansion stage. The slope of the curve gradually crack development. (3) The stable crack expansion stage. The slope of the curve gradually decreases as wing cracks form and develop rapidly with increasing loading. When the decreases as wing cracks form and develop rapidly with increasing loading. When the slope of the curve is close to zero, secondary cracks often occur in the periphery of the slope of the curve is close to zero, secondary cracks often occur in the periphery of the wing wing cracks. (4) The unstable crack expansion stage. At this stage, the stress reaches its cracks. (4) The unstable crack expansion stage. At this stage, the stress reaches its limit limit and the material loses its stability; the wing cracks expand rapidly, reaching the spec- and the material loses its stability; the wing cracks expand rapidly, reaching the specimen imen surface, then the curve falls suddenly, indicating quasi-brittle failure. surface, then the curve falls suddenly, indicating quasi-brittle failure. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 15 (a) (b) Figure 3. Stress–strain curves of concrete specimens with different crack angles. (a) Crack width = Figure 3. Stress–strain curves of concrete specimens with different crack angles. (a) Crack width = 0.5 mm; (b) Crack width = 3.0 mm. 0.5 mm; (b) Crack width = 3.0 mm. It can be seen in Table 3 and Figure 3 that the stress–strain curves and the peak strength under uniaxial compression vary with the pre-load crack angle β. The presence of cracks weakens the peak strength of the samples from 10.19% (case 1) to 41.09% (case 7). The fracture angle and width play a significant role in the mechanical properties of the concrete samples. As the crack angle increases, the strength of the specimen first decreases and then increases. Under the same single fracture angle, the strength of the fracture spec- imens including 0.5 mm wide crack is higher than that of the specimens with 3.0 mm wide crack. Table 3. Peak strength and crack initiation angle in uniaxial compression tests. Crack Width d Peak Strength Cases Crack Angle β (°) Initiation Angle (°) (mm) (MPa) 1 0.5 15 22.03 10 2 0.5 30 19.60 54 3 0.5 45 17.18 55 4 0.5 60 20.30 - 5 0.5 75 21.97 67 7 3.0 15 14.80 11 8 3.0 30 14.45 39 9 3.0 45 16.17 41 10 3.0 60 16.40 - 11 3.0 75 18.36 - Figure 4 depicts the failure modes of the crack tip for the specimens with different pre-load crack angles. The fracture angle of the single non-closed crack for the specimens decreases and then increases gradually under normal conditions, for example, the rupture angle is smaller when β = 15° and 45°. Appl. Sci. 2022, 12, 4803 5 of 14 It can be seen in Table 3 and Figure 3 that the stress–strain curves and the peak strength under uniaxial compression vary with the pre-load crack angle b. The presence of cracks weakens the peak strength of the samples from 10.19% (case 1) to 41.09% (case 7). The fracture angle and width play a significant role in the mechanical properties of the concrete samples. As the crack angle increases, the strength of the specimen first decreases and then increases. Under the same single fracture angle, the strength of the fracture specimens including 0.5 mm wide crack is higher than that of the specimens with 3.0 mm wide crack. Table 3. Peak strength and crack initiation angle in uniaxial compression tests. Crack Width d Peak Strength Cases Crack Angle b ( ) Initiation Angle ( ) (mm) (MPa) 1 0.5 15 22.03 10 2 0.5 30 19.60 54 3 0.5 45 17.18 55 4 0.5 60 20.30 - 5 0.5 75 21.97 67 7 3.0 15 14.80 11 8 3.0 30 14.45 39 9 3.0 45 16.17 41 10 3.0 60 16.40 - 11 3.0 75 18.36 - Figure 4 depicts the failure modes of the crack tip for the specimens with different pre-load crack angles. The fracture angle of the single non-closed crack for the specimens Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 15 decreases and then increases gradually under normal conditions, for example, the rupture angle is smaller when b = 15 and 45 . (a) (b) (c) (d) (e) Figure 4. Failure mode of jointed concrete with a single crack (crack width = 3.0 mm). (a) β = 15°, (b) Figure 4. Failure mode of jointed concrete with a single crack (crack width = 3.0 mm). (a) b = 15 , β = 30°, (c) β = 45°, (d) β = 60°, (e) β = 75° Note: T represents tensile crack and S represents shear crack. (b) b = 30 , (c) b = 45 , (d) b = 60 , (e) b = 75 Note: T represents tensile crack and S represents shear crack. 3. Fracture Criterion of Brittle Material 3. Fracture Criterion of Brittle Material 3.1. The Maximum Circumferential Stress Theory 3.1. The Maximum Circumferential Stress Theory There are stress components of mixed-mode I-II at the crack tip in the polar coordi- There are stress components of mixed-mode I-II at the crack tip in the polar coordinates. nates. According to the linear elastic theory, the stress components can be expressed as According to the linear elastic theory, the stress components can be expressed as [56]: [56]: 1 1 qθθ q σ=− Kcos33 cosθ+Ksin cosθ−1 s = p K cos (3() cos q) + K sin (() 3 cos q 1) , (1 (1) ) r ⅠⅡ r I II  22πr 2 2 2 2pr 1 θ σ=+ cos K13 cosθ− Ksinθ , () (2) θ ⅠⅡ  22πr 1 θ τ=+ cos K sinθ Kc31 osθ− () , (3) rθ ⅠⅡ  22πr where σrr, σθθ, and τrθ are the radial stress, circumferential tensile stress, and shear stress at the crack tip, respectively; r is the distance from the crack tip; θ is the angle that the surface deviates from the original crack tip direction (counterclockwise is positive). The ∞ ∞ expressions Ka =σπ and Ka =τπ are the mixed-mode I and mode II I II stress intensity factors of an infinite plate with a central crack; where a is half the length ∞ ∞ of crack; and σ , τ are the far-field tensile stress and the far-field shear stress, respectively. The crack propagation direction at the crack tip is very close to that of the maximum circumferential stress [20]. According to the maximum circumferential stress theory, a mixed-mode I-II fracture can transform into an equivalent pure mode-I fracture, where the equivalent mode-I stress intensity factor (KIe) and initiation angle (θI0) are expressed as: 1 θ Ⅰ 0 KK = cos  1+− cosθθ 3K sin  (4) () , Ⅰe Ⅰ Ⅰ00 Ⅱ Ⅰ  11 −+8(KK) ⅡⅠ θ =2arctan (5) Ⅰ0 4(KK ) ⅡⅠ when KIe reaches the mode-I fracture toughness (KIC), the crack begins to expand. For a closed crack, the mode I stress intensity factor is KI = 0. Thus, the initial propagation di- rection is θI0 = 70.5°, where KIe reaches its maximum value regardless of the crack angle. Appl. Sci. 2022, 12, 4803 6 of 14 1 q s = p cos [K (1 + cos q) 3K sin q], (2) q I II 2 2pr 1 q t = p cos [K sin q + K (3 cos q 1)], (3) rq I II 2 2pr where s , s , and t are the radial stress, circumferential tensile stress, and shear stress rr qq rq at the crack tip, respectively; r is the distance from the crack tip; q is the angle that the surface deviates from the original crack tip direction (counterclockwise is positive). The p p ¥ ¥ expressions K = s pa and K = t pa are the mixed-mode I and mode II stress I II intensity factors of an infinite plate with a central crack; where a is half the length of crack; ¥ ¥ and s , t are the far-field tensile stress and the far-field shear stress, respectively. The crack propagation direction at the crack tip is very close to that of the maximum circumferential stress [20]. According to the maximum circumferential stress theory, a mixed-mode I-II fracture can transform into an equivalent pure mode-I fracture, where the equivalent mode-I stress intensity factor (K ) and initiation angle (q ) are expressed as: Ie I0 1 q I0 K = cos [K (1 + cos q ) 3K sin q ], (4) Ie I I0 II I0 2 2 1 1 + 8(K /K ) II I q = 2arctan (5) I0 4(K /K ) II I when K reaches the mode-I fracture toughness (K ), the crack begins to expand. For Ie IC a closed crack, the mode I stress intensity factor is K = 0. Thus, the initial propagation direction is q = 70.5 , where K reaches its maximum value regardless of the crack angle. I0 Ie 3.2. The Radial Shear Stress Criterion The absolute value of the maximum shear stress at the crack tip should satisfy the following equation [55]: 2 2 ¶t ¶ t ¶ t rq rq rq = 0, < 0 or > 0,jt (q = q )j , (6) rq II0 max 2 2 ¶q ¶q ¶q q is given by: II0 p p 2 + A cos(a /3) 3 sin(a /3) 0 0 q = 2arctan , (7) II0 3k 2 3 where A = 4 + 42(K /K ) , B =4(K /K ), k = 2(K /K ), a = arccos(T), T = (4A3k B)/(2 A ), II I II I 0 II I 0 0 and (a 2(0, p), 1 < T < 1). These are the coefficients related to K and K . 0 II I When the crack expansion follows the mode-II fracture pattern, the fracturing depends on the mode-II fracture toughness (K ). Thus, we can transform this crack into an IIC equivalent pure mode-II crack and the equivalent stress intensity factor can be expressed as: 1 q II0 K = cos [K sin q + K (3 cos q 1)]. (8) IIe I II0 II II0 2 2 4. Numerical Analysis and Failure Mechanism of a Single-Crack Specimen 4.1. Numerical Model and Parameter Analysis The stress field and stress intensity factor of a simulated single-crack concrete specimen were analyzed using the finite element program ABAQUS. The extended FEM was used to simulate the fracture propagation of a single-crack concrete sample. The two-dimensional computational model of the concrete sample (Figure 5) consists of 3040 grid elements. The mechanical parameters of the simulated material are listed in Table 4. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 15 3.2. The Radial Shear Stress Criterion The absolute value of the maximum shear stress at the crack tip should satisfy the following equation [55]: ∂ τ ∂∂ττ rθ rr θθ >= 0, τθ θ =< 0, 0 () (6) rθ Ⅱ0 2 2 max ∂∂ θθ ∂θ or , θII0 is given by: −+2c A os(αα 3)− 3sin( 3) () (7) θ =2arctan , Ⅱ0 3k where A = 4 + 42(KII/KI) , B = −4(KII/KI), k0 = 2(KII/KI), α0 = arccos(T), T = (−4A−3k0B)/(2√𝐴 ), and (α0∈(0, π), −1 < T < 1). These are the coefficients related to KII and KI. When the crack expansion follows the mode-II fracture pattern, the fracturing de- pends on the mode-II fracture toughness (KIIC). Thus, we can transform this crack into an equivalent pure mode-II crack and the equivalent stress intensity factor can be expressed as: 1 θ Ⅱ0 KK = cos  sinθθ+−K 3 cos 1  . () (8) Ⅱe Ⅰ Ⅱ0 Ⅱ Ⅱ0  4. Numerical Analysis and Failure Mechanism of a Single-Crack Specimen Appl. Sci. 2022, 12, 4803 7 of 14 4.1. Numerical Model and Parameter Analysis The stress field and stress intensity factor of a simulated single-crack concrete speci- men were analyzed using the finite element program ABAQUS. The extended FEM was Table 4. Mechanical parameters used in numerical modeling. used to simulate the fracture propagation of a single-crack concrete sample. Elastic Modulus Poisson’s Ratio Material Friction Coefficient u The two-dimensional computational model of the concrete sample (Figure 5) consists E (GPa) v of 3040 grid elements. The mechanical parameters of the simulated material are listed in Concrete 1.354 0.12 0.21 Table 4. (a) (b) Figure 5. Numerical model of single-crack concrete sample with b = 45 and d = 3.0 mm. Figure 5. Numerical model of single-crack concrete sample with β = 45° and d = 3.0 mm. (a) Numer- (a) Numerical model of single-crack sample; (b) Stress distribution of single-crack. ical model of single-crack sample; (b) Stress distribution of single-crack. Figure 5 shows the stress nephogram of specimen No. 9 under normal conditions. Table 5 lists the changes in the maximum circumferential tensile stress and radial shear stress at the crack tip for the specimens. The initiation angles obtained in the experimental tests and numerical simulations are shown in Table 6. Figure 6 compares the stress–strain relationships between the numerical and experimental results. The numerical solution shows a linear relationship before the peak strength. However, the stress–strain from the experiments is nonlinear and experiences compaction at the early loading stage. The nu- merical method can simplify complex stress–strain relations of cracked concretes. Figure 7 shows the computational results of the fracture propagation of the single-crack concrete sample. The SIF at the crack tip changes the pre-crack angle. The SIF at the crack tip becomes high and then low with the pre-crack angle. The SIF arrives at the maximum value when the pre-crack angle is 45 . Appl. Sci. 2022, 12, 4803 8 of 14 Table 5. Numerical results of stress near the crack tip. Maximum The Direction of The Direction of Crack Maximum Radial Crack Circumferential the Maximum the Maximum Cases Width Shear Stress at Angle b ( ) Stress at Crack Tip Circumferential Radial Shear Stress (mm) Crack Tip (MPa) (MPa) Stress ( ) ( ) 1 15 10.25 13.5 8.12 0 2 30 13.0 46.5 10.44 0 3 45 14.97 61 11.98 0 0.5 4 60 12.74 66 9.44 0 5 75 11.73 72 9.44 0 6 15 13.13 14 8.53 10 7 30 15.42 46 14.67 2.5 8 3.0 45 12.33 81 17.75 7.5 9 60 7.74 102 17.07 12 10 75 1.55 124 14.68 41 Table 6. Initiation angles derived from experimental tests and numerical simulation. The Direction of the The Initiation Angles The Initiation Angles The Direction of the Crack Angle b Maximum Cases of Experimental Work by Numerical Maximum Radial ( ) Circumferential ( ) Simulation ( ) Shear Stress ( ) Stress ( ) 1 15 11 13 13 0 2 30 47 47 47 0 3 45 68 63 63 0 Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 15 4 60 73 66 65 0 5 75 69 70 73 0 Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). (a) (b) (c) (d) (e) Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where the single crack thickness is 0.5 mm. (a) β = 15°; (b) β = 30°; (c) β = 45°; (d) β = 60°; (e) β = 75°. Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 15 Appl. Sci. 2022, 12, 4803 9 of 14 4.2. Failure Mechanism of a Closed Pre-Crack Concrete Sample The mode-I fracture toughness of brittle material is lower than the mode-II frac- ture toughness. The maximum dimensionless mode-I and mode-II stress intensity fac- p p tors are defined as f = K /s pa and f = K /s pa, respectively. When qmax Ie rqmax IIe y y f / f < 1 or f / f < K /K , the crack will expand according to the max max max max IIC IC rq q rq q mode-I fracture pattern. If f / f > 1 and f / f > K /K the fracture rq max q max rq max q max IIC IC will follow a mode-II fracture pattern. For a 0.5 mm wide crack, K = 0, the dimensionless stress intensity factor is shown in Figure 8, where f / f < 1. Thus, mode-I fracture rq max q max occurred, and the angle of f was 70.5 based on the maximum circumferential stress qmax theory. Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). (a) (b) (c) (d) (e) Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where the single crack thickness is 0.5 mm. (a) β = 15°; (b) β = 30°; (c) β = 45°; (d) β = 60°; (e) β = 75°. the single crack thickness is 0.5 mm. (a) b = 15 ; (b) b = 30 ; (c) b = 45 ; (d) b = 60 ; (e) b = 75 . Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 15 4.2. Failure Mechanism of a Closed Pre-Crack Concrete Sample The mode-I fracture toughness of brittle material is lower than the mode-II fracture toughness. The maximum dimensionless mode-I and mode-II stress intensity factors are fK = σπa fK = σπa defined as and , respectively. When θmax Ie y rθmax II e y ff < 1 or f fK < K , the crack will expand according to the rθθ max max rθθ max max ⅡⅠ C C mode-I fracture pattern. If ff > 1 and f fK > K the fracture rθθ max max rθθ max max ⅡⅠ C C will follow a mode-II fracture pattern. For a 0.5 mm wide crack, KI = 0, the dimensionless stress intensity factor is shown in Figure 8, where . Thus, mode-I fracture ff < 1 rθθ max max occurred, and the angle of was 70.5° based on the maximum circumferential stress Appl. Sci. 2022, 12, 4803 θmax 10 of 14 theory. Figure 8. Distribution curve of dimensionless intensity factor for a closed crack (f denotes dimension- Figure 8. Distribution curve of dimensionless intensity factor for a closed crack (f denotes dimen- less SIF). sionless SIF). The maximum circumferential stress at the crack tip is greater than the maximum radial The maximum circumferential stress at the crack tip is greater than the maximum shear stress for specimens 1–5 (Table 5). The crack propagation angles for the specimens radial shear stress for specimens 1–5 (Table 5). The crack propagation angles for the spec- obtained from the numerical and experimental results are close to those obtained from the imens obtained from the numerical and experimental results are close to those obtained maximum circumferential stress (Table 6). from the maximum circumferential stress (Table 6). 4.3. Failure Mechanism of a Non-Closed Single-Crack Concrete Sample 4.3. Failure Mechanism of a Non-Closed Single-Crack Concrete Sample The mode-I stress intensity factor for the non-closed crack is negative when the crack The mode-I stress intensity factor for the non-closed crack is negative when the crack is under compression. The dimensionless SIF at the crack tip achieved from maximum is under compression. The dimensionless SIF at the crack tip achieved from maximum circumferential tensile stress will be less than that from the radial shearing stress when circumferential tensile stress will be less than that from the radial shearing stress when the pre-crack angle is small (Figure 9). Thus, mode-II fracture in a non-closed crack will the pre-crack angle is small (Figure 9). Thus, mode-II fracture in a non-closed crack will Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 15 occur only if f / f > 1 and f / f > K /K . Therefore, the mode domain occur only if max max and max max . Therefore, the mode domain ff rq q > 1 f rqfK q> K IIC IC rθθ max max rθθ max max ⅡⅠ C C can be divided into two regions—the mode-I and mode-II fracture regions—as shown can be divided into two regions—the mode-I and mode-II fracture regions—as shown in in Figure 10. Figure 10. Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity factors. factors. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. A mode-I fracture will occur in a sample with a non-closed crack when the crack angle is less than 45° (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the crack rupture angles of the specimens with small pre-crack angles do not agree with the initiation angles obtained by the maximum circumferential stress theory. However, as β approaches 45°, the experimental fracture initiation angles are closer to those obtained by the maximum circumferential stress theory. For crack angles greater than 30° (specimens 9–11), there are considerable differences between the rupture angles of the non-closed and closed cracks because the stress patterns are different for the two conditions. The initiation angles of the non-closed cracks are close to the results obtained by the radial shear stress criterion. Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 15 Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity Appl. Sci. 2022, 12, 4803 11 of 14 factors. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. A mode-I fracture will occur in a sample with a non-closed crack when the crack angle A mode-I fracture will occur in a sample with a non-closed crack when the crack is less than 45 (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the crack angle is less than 45° (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the rupture angles of the specimens with small pre-crack angles do not agree with the initiation angles obtained by the maximum circumferential stress theory. However, as b approaches crack rupture angles of the specimens with small pre-crack angles do not agree with the 45 , the experimental fracture initiation angles are closer to those obtained by the maximum initiation angles obtained by the maximum circumferential stress theory. However, as β circumferential stress theory. For crack angles greater than 30 (specimens 9–11), there are approaches 45°, the experimental fracture initiation angles are closer to those obtained by considerable differences between the rupture angles of the non-closed and closed cracks Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 15 the maximum circumferential stress theory. For crack angles greater than 30° (specimens because the stress patterns are different for the two conditions. The initiation angles of the 9–11), there are considerable differences between the rupture angles of the non-closed and non-closed cracks are close to the results obtained by the radial shear stress criterion. closed cracks because the stress patterns are different for the two conditions. The initiation angles of the non-closed cracks are close to the results obtained by the radial shear stress criterion. Figure 11. Comparison between theoretical and experimental initiation angles. Figure 11. Comparison between theoretical and experimental initiation angles. It can be seen in Table 5 that under the same crack width, as the crack angle increases, It can be seen in Table 5 that under the same crack width, as the crack angle increases, the maximum circumferential stress and the maximum radial shear stress at the crack tip the maximum circumferential stress and the maximum radial shear stress at the crack tip first increase and then decrease. For closed cracks (crack width 0.5 mm), the maximum first increase and then decrease. For closed cracks (crack width 0.5 mm), the maximum circumferential stress at the crack tip is greater than the radial shear stress at the same crack circumferential stress at the crack tip is greater than the  radial shear stress at the same angle and the maximum of both stresses appears at 45 (case 3). In the case of non-closed crack angle and the maximum of both stresses appears at 45° (case 3). In the case of non- cracks, the maximum radial shear stress is greater than the maximum circumferential tensile closed cracks, the maximum radial shear stress is greater than the maximum circumferen- stress (cases 8, 9, and 10) after the crack angle is greater than 45 . When the crack angle is tial tensile stress (cases 8, 9, and 10) after the crack angle is greater than 45°. When the 75 , the maximum circumferential tensile stress is only 1.55 MPa. There is good agreement crack angle is 75°, the maximum circumferential tensile stress is only 1.55 MPa. There is good agreement between the rupture angles derived from the numerical and experi- mental tests and those calculated by the maximum circumferential stress theory. How- ever, for specimens 9–11, the maximum circumferential tensile stress is smaller than the maximum radial shear stress even though ff > 1 and f fK > K rθθ max max rθθ max max ⅡC ⅠC . The rupture angles of the experimental results are close to those of the maximum radial stress for a mode-II fracture. The results presented above indicate that the maximum circumferential stress theory has some limitations in interpreting the fracture propagation of the non-closed cracks un- der compressional loading. The initiation angles approach those of the radial shear stress criterion if ff > 1 and f fK > K . rθθ max max rθθ max max ⅡⅠ C C Appl. Sci. 2022, 12, 4803 12 of 14 between the rupture angles derived from the numerical and experimental tests and those calculated by the maximum circumferential stress theory. However, for specimens 9–11, the maximum circumferential tensile stress is smaller than the maximum radial shear stress even though f / f > 1 and f / f > K /K . The rupture angles of the rq max q max rq max q max IIC IC experimental results are close to those of the maximum radial stress for a mode-II fracture. The results presented above indicate that the maximum circumferential stress theory has some limitations in interpreting the fracture propagation of the non-closed cracks under compressional loading. The initiation angles approach those of the radial shear stress criterion if f / f > 1 and f / f > K /K . max max max max rq q rq q IIC IC 5. Conclusions We performed uniaxial compression tests on concrete specimens containing cracks at various angles and widths. The results were compared with theoretical and numerical simulation results; based on the analysis of the results, we conclude the following. (1) The strength of the concrete sample decreases initially and then increases with increas- ing crack angle. For the same crack angle, the greater the crack width, the higher the strength of the non-closed crack sample. The crack width has a significant effect on the initiation angle. (2) With the maximum circumferential stress theory, it is difficult to depict the fracture propagation of non-closed cracks under compression. When K < 0, a non-closed crack under uniaxial compression will have a mode-I stress intensity factor. The circumferential compressive stress created by the mode-I stress intensity factor will restrain the circumferential tensile stress caused by the mode-II stress intensity factor. If f / f > 1 and f / f > K /K , the rupture angle will be close to rq max q max rq max q max IIC IC the direction of the maximum radial shear stress for a non-closed crack. Author Contributions: Conceptualization, J.Z. and L.W.; methodology, L.W.; software, J.Z. and L.W.; validation, J.Z. and L.W.; formal analysis, J.Z., B.L. and L.W.; writing—original draft preparation, J.Z. and L.W.; writing—review and editing, J.Z. and L.W.; visualization, J.Y.; supervision, J.X. and H.Z.; funding acquisition, L.W. 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Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading

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applied sciences Article Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading 1 1 , 1 1 1 , 2 , 3 Lizhou Wu , Jianting Zhou * , Jun Yang , Jingzhou Xin , Hong Zhang * and Bu Li State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, China; lzwu@cqjtu.edu.cn (L.W.); yangjun@cqjtu.edu.cn (J.Y.); xinjz@cqjtu.edu.cn (J.X.) China Southwest Geotechnical Investigation & Design Institute Co., Ltd., Chengdu 610052, China; lbcdut@foxmail.com College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu 610059, China * Correspondence: jtzhou@cqjtu.edu.cn (J.Z.); zhanghong@cqjtu.edu.cn (H.Z.) Abstract: A stress concentration at a crack tip may cause fracture initiation even under low-stress con- ditions. The maximum axial stress theory meets the challenges of explaining the fracture propagation of a non-closed fracture of cracked concretes under compressional loading. Uniaxial loading tests of single-crack concrete specimens were carried out and a numerical simulation of fracture propagation under uniaxial compression was performed. The radial shear stress criterion for a mode-II fracture is proposed to examine the stress intensity factor (SIF) of the pre-crack specimens under compressional loading. When the maximum radial shear stress at the crack tip is larger than the maximum axial tensile stress, and the maximum dimensionless SIFs can satisfy f /f > 1 and f /f rq max q max rq max q max p p > K /K ( f = K /s pa and f = K /s pa are maximum dimensionless mode-I IIC IC qmax Ie y rqmax IIe y and mode-II SIFs, respectively), the crack will extend along the direction of the maximum radial shear stress. The influence of the single-crack angle and width on the mechanical properties of the specimens was examined. The experimental and numerical results indicate that the existence of cracks can considerably weaken the strength of the specimen. The distribution and width of the Citation: Wu, L.; Zhou, J.; Yang, J.; cracks had a significant effect on the specimen strength. The strength of the concrete specimen initially Xin, J.; Zhang, H.; Li, B. Initiation and Fracture Characteristics of Different decreased and then increased with increasing fracture angle. The failure mechanism and rupture Width Cracks of Concretes under angle of pre-crack brittle material while considering crack width will be discovered. Compressional Loading. Appl. Sci. 2022, 12, 4803. https://doi.org/ Keywords: crack; concrete material; uniaxial compression; strength; mixed-mode fracture criterion 10.3390/app12104803 Academic Editor: Dario De Domenico 1. Introduction Received: 1 April 2022 Brittle materials such as concrete or rock usually contain macroscopic cracks or defects Accepted: 1 May 2022 that develop because of complex environmental conditions, which has a significant effect on Published: 10 May 2022 the properties of brittle materials [1,2]. These defects destroy the material integrity, weaken Publisher’s Note: MDPI stays neutral the mechanical properties of the materials, and modify the stress distribution. Moreover, a with regard to jurisdictional claims in stress concentration can be generated at the crack tip, which can influence the failure mode published maps and institutional affil- of brittle materials. Therefore, to study the deformation and failure characteristics of brittle iations. materials, many theoretical and numerical studies show that internal cracks in concrete or rock play a significant role in determining the deformation pattern, the strength of the material, and the fracture mode, although crack initiations for pre-existing defects have a long history [3–14]. Copyright: © 2022 by the authors. Many brittle materials including rock and concrete show high elastic modulus and Licensee MDPI, Basel, Switzerland. strength. Brittle materials are considered to be linearly elastic. When studying cracks in This article is an open access article brittle materials, the stress intensity factor (SIF) can be employed to describe the stress state distributed under the terms and at the crack tip using the fracture mechanics method [15–19]. Three important fracture conditions of the Creative Commons initiation criteria are commonly employed to analyze the crack propagation mechanism of Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ brittle materials: the maximum tangential stress, the maximum energy release rate, and the 4.0/). minimum energy density criterion [20–24]. The F-criterion, which is a modified version Appl. Sci. 2022, 12, 4803. https://doi.org/10.3390/app12104803 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 4803 2 of 14 of the energy release rate criterion, may also be employed to study the fracture behavior of quasi-brittle materials [25,26]. These criteria are based on the assumption of a mode-I fracture. However, the fracture extension of compression shear cracks while considering mixed-mode I/II fracture has rarely been investigated [27]. Many experiments were carried out to examine the crack initiation, propagation path, and eventual coalescence of cracks in samples made of various materials, including artificial materials, under compressive loading [3,8,28–41]. In experimental studies, the brittle material is often made into specimens with embedded cracks in the laboratory and uniaxial compression tests are carried out on the rock or concrete samples [10,42,43]. Brazilian disk testing of rock and concrete samples is commonly used to analyze the tensile strength, fracture toughness, and mixed-mode fracture process in uncracked and pre-cracked samples under compressive loading [15,44–55]. A number of numerical approaches are used to examine fracture and crack propagation and crack growth in brittle materials; these include the finite element method (FEM), the boundary element method, and the discrete element method [6,24,33,40]. SIF, energy- release rate (G), crack propagation, fracturing time, static tensile, and normal-distributed stresses were computed to describe the fracture initiation and propagation in brittle material samples [16]. The objective of the study presented in this paper was to carry out uniaxial compres- sion tests and numerical analyses of specimens containing a crack and to examine suitable fracture criteria to explain the initiation angle of cracked concretes. The crack growth and the stress and strain characteristics of the concrete specimen for different crack widths and crack angles were analyzed based on the maximum circumferential stress theory and radial shear stress criterion. The relationship between the stress threshold of crack propagation with pre-crack angle and width is discussed, the influence of the crack width on the fracture propagation mode and fracture initiation is examined, and the fracture propagation of the non-closed cracks is discussed. Based on the experimental results, the methods suitable for the concrete fracture propagation incorporating crack width will be discussed. 2. Materials and Methods 2.1. Specimen Preparation The samples consisted of concrete material composed of cement, gypsum, quartz sand, water, a water-reducing agent, and a waterproofing agent, the details of which are listed in Table 1. The elastic modulus is 1.35 GPa and Poisson’s ratio is 0.21, which is obtained by uniaxial compression tests. Table 1. The ratio of concrete material. Quartz Sand Water-Reducing Water-Proofing Cement Gypsum Water Agent Agent 0.60 mm 0.30 mm 0.15 mm 0.075 mm 33.60% 8.06% 12.33% 13.41% 13.60% 0.96% 16.89% 0.86% 0.29% To replicate the crack in the sample, a thin steel sheet was embedded in the concrete material. Cuboid specimens with a central through-crack (L W H = 50 50 100 mm) were prepared for the compression-shear tests. In the study of the effect of crack angles on the strength and deformation, the fracture angle difference of the materials is generally set to 15 [40]. We applied various crack angles: 15 , 30 , 45 , 60 , and 75 , and a crack width of 0.5 or 3 mm (Figure 1). The thin steel sheet was first set in the sample mold, and then the concrete material was poured into the mold. The sample was vibrated to compact the material and left to set for 12 h at a temperature of 20 C. Then, the sheet steel was removed from the mold. The smoothness of the samples and cracks were checked. Finally, to provide proper curing humidity, the samples were cured in water for 28 days at room temperature. Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 15 Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 15 the material and left to set for 12 h at a temperature of 20 °C. Then, the sheet steel was the material and left to set for 12 h at a temperature of 20 °C. Then, the sheet steel was removed from the mold. The smoothness of the samples and cracks were checked. Finally, removed from the mold. The smoothness of the samples and cracks were checked. Finally, Appl. Sci. 2022, 12, 4803 3 of 14 to provide proper curing humidity, the samples were cured in water for 28 days at room to provide proper curing humidity, the samples were cured in water for 28 days at room temperature. temperature. (a) (b) (c) (a) (b) (c) Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack Figure 1. Concrete specimen with a single fracture. (a) Test sample with 0.5 mm width single crack (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample (left), and the 0.5 mm thick steel sheet is inserted into the cracks to fill the cracks; (b) Test sample with with 3.0 mm width single hollow crack; (c) samples with different crack angles. with 3.0 mm width single hollow crack; (c) samples with different crack angles. 3.0 mm width single hollow crack; (c) samples with different crack angles. 2.2. Compression Shear Fracture Test of Concrete Samples 2.2. 2.2. Co Compr mpression ession Shear Shear Fractur Fracture Tes e Test tof of Co Concr ncrete ete Samples Samples Each brittle concrete specimen was placed in the loading apparatus and loaded under Each Eachbrittle brittleconcr concrete spec ete specimen imen w was as plac placed ed in in the the loading loadingapparatus apparatus an and d loaded loadedunder under uniaxial compression (Figure 2). The tests were carried out on the MTS 815 testing ma- uniaxial uniaxial com compr pression ession (Fig (Figur ure e 22). The ). The te tests sts were were c carried arried out out on the on the MTS MTS 815 te 815 testing sting m ma- a- chine, which showed stable and reliable performance. The crack widths and angles of the chine, which showed stable and reliable performance. The crack widths and angles of the chine, which showed stable and reliable performance. The crack widths and angles of the specimens are listed in Table 2. specimens are listed in Table 2. specimens are listed in Table 2. (a) (b) (a) (b) Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single frac- Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single frac- Figure 2. (a) Test machine; (b) Diagram of the loading of the concrete specimen with a single fracture. ture. The applied compressive stress is represented by ; β is the angle between the crack axis The applied compressive stress is represented by s ; b is σ the angle between the crack axis and the ture. The applied compressive stress is represented by y ; β is the angle between the crack axis direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen height. and the direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen and the direction of loading; 2a is the crack length; L is the specimen width, and H is the specimen height. height. Table 2. Crack parameters for specimens in uniaxial compression tests. Table 2. Crack parameters for specimens in uniaxial compression tests. Table 2. Crack parameters for specimens in uniaxial compression tests. Cases Crack Width d (mm) Crack Length a (mm) Crack Angle b ( ) Cases Crack Width d (mm) Crack Length a (mm) Crack Angle β (°) Cases Crack Width d (mm) Crack Length a (mm) Crack Angle β (°) 1 0.5 10 15 1 0.5 10 15 1 0.5 10 15 2 0.5 10 30 2 0.5 10 30 3 0.5 10 45 2 0.5 10 30 4 0.5 10 60 3 0.5 10 45 3 0.5 10 45 5 0.5 10 75 4 0.5 10 60 4 0.5 10 60 6 3 10 15 5 0.5 10 75 5 0.5 10 75 7 3 10 30 8 3 10 45 9 3 10 60 10 3 10 75 Note: b is the angle between the crack and loading direction. Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 15 6 3 10 15 7 3 10 30 8 3 10 45 9 3 10 60 10 3 10 75 Appl. Sci. 2022, 12, 4803 4 of 14 Note: β is the angle between the crack and loading direction. The width of the cracks for specimens 1–5 was 0.5 mm and 3 mm for specimens 6– 10. A small steel sheet, 0.5 and 3 mm thick, was inserted into each sample to fill the crack The width of the cracks for specimens 1–5 was 0.5 mm and 3 mm for specimens 6–10. (Figure 1b,c). The loading was controlled by changing the displacement, with a loading A small steel sheet, 0.5 and 3 mm thick, was inserted into each sample to fill the crack rate of 0.5 mm/min. The crack propagation of the specimen was captured by a high-reso- (Figure 1b,c). The loading was controlled by changing the displacement, with a loading rate lution camera during the tests. of 0.5 mm/min. The crack propagation of the specimen was captured by a high-resolution camera during the tests. 2.3. Analysis of the Experimental Results 2.3. Analysis of the Experimental Results Figure 3 shows the stress–strain curves of the specimens under uniaxial loading. The uniaxialFigur comp er3es shows sion pthe roce str ssess–strain of the sincurves gle-crac of k c the onspecimens crete specim under en ca uniaxial n be divlioading. ded into The uniaxial compression process of the single-crack concrete specimen can be divided into four stages: (1) The nonlinear compaction stage where mostly internal micro-cracks are four stages: (1) The nonlinear compaction stage where mostly internal micro-cracks are present in the specimen. The strain is small, and the stress remains unchanged, resulting present in the specimen. The strain is small, and the stress remains unchanged, resulting in a relatively smooth curve. (2) The linear elastic deformation stage, with no obvious in a relatively smooth curve. (2) The linear elastic deformation stage, with no obvious crack development. (3) The stable crack expansion stage. The slope of the curve gradually crack development. (3) The stable crack expansion stage. The slope of the curve gradually decreases as wing cracks form and develop rapidly with increasing loading. When the decreases as wing cracks form and develop rapidly with increasing loading. When the slope of the curve is close to zero, secondary cracks often occur in the periphery of the slope of the curve is close to zero, secondary cracks often occur in the periphery of the wing wing cracks. (4) The unstable crack expansion stage. At this stage, the stress reaches its cracks. (4) The unstable crack expansion stage. At this stage, the stress reaches its limit limit and the material loses its stability; the wing cracks expand rapidly, reaching the spec- and the material loses its stability; the wing cracks expand rapidly, reaching the specimen imen surface, then the curve falls suddenly, indicating quasi-brittle failure. surface, then the curve falls suddenly, indicating quasi-brittle failure. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 15 (a) (b) Figure 3. Stress–strain curves of concrete specimens with different crack angles. (a) Crack width = Figure 3. Stress–strain curves of concrete specimens with different crack angles. (a) Crack width = 0.5 mm; (b) Crack width = 3.0 mm. 0.5 mm; (b) Crack width = 3.0 mm. It can be seen in Table 3 and Figure 3 that the stress–strain curves and the peak strength under uniaxial compression vary with the pre-load crack angle β. The presence of cracks weakens the peak strength of the samples from 10.19% (case 1) to 41.09% (case 7). The fracture angle and width play a significant role in the mechanical properties of the concrete samples. As the crack angle increases, the strength of the specimen first decreases and then increases. Under the same single fracture angle, the strength of the fracture spec- imens including 0.5 mm wide crack is higher than that of the specimens with 3.0 mm wide crack. Table 3. Peak strength and crack initiation angle in uniaxial compression tests. Crack Width d Peak Strength Cases Crack Angle β (°) Initiation Angle (°) (mm) (MPa) 1 0.5 15 22.03 10 2 0.5 30 19.60 54 3 0.5 45 17.18 55 4 0.5 60 20.30 - 5 0.5 75 21.97 67 7 3.0 15 14.80 11 8 3.0 30 14.45 39 9 3.0 45 16.17 41 10 3.0 60 16.40 - 11 3.0 75 18.36 - Figure 4 depicts the failure modes of the crack tip for the specimens with different pre-load crack angles. The fracture angle of the single non-closed crack for the specimens decreases and then increases gradually under normal conditions, for example, the rupture angle is smaller when β = 15° and 45°. Appl. Sci. 2022, 12, 4803 5 of 14 It can be seen in Table 3 and Figure 3 that the stress–strain curves and the peak strength under uniaxial compression vary with the pre-load crack angle b. The presence of cracks weakens the peak strength of the samples from 10.19% (case 1) to 41.09% (case 7). The fracture angle and width play a significant role in the mechanical properties of the concrete samples. As the crack angle increases, the strength of the specimen first decreases and then increases. Under the same single fracture angle, the strength of the fracture specimens including 0.5 mm wide crack is higher than that of the specimens with 3.0 mm wide crack. Table 3. Peak strength and crack initiation angle in uniaxial compression tests. Crack Width d Peak Strength Cases Crack Angle b ( ) Initiation Angle ( ) (mm) (MPa) 1 0.5 15 22.03 10 2 0.5 30 19.60 54 3 0.5 45 17.18 55 4 0.5 60 20.30 - 5 0.5 75 21.97 67 7 3.0 15 14.80 11 8 3.0 30 14.45 39 9 3.0 45 16.17 41 10 3.0 60 16.40 - 11 3.0 75 18.36 - Figure 4 depicts the failure modes of the crack tip for the specimens with different pre-load crack angles. The fracture angle of the single non-closed crack for the specimens Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 15 decreases and then increases gradually under normal conditions, for example, the rupture angle is smaller when b = 15 and 45 . (a) (b) (c) (d) (e) Figure 4. Failure mode of jointed concrete with a single crack (crack width = 3.0 mm). (a) β = 15°, (b) Figure 4. Failure mode of jointed concrete with a single crack (crack width = 3.0 mm). (a) b = 15 , β = 30°, (c) β = 45°, (d) β = 60°, (e) β = 75° Note: T represents tensile crack and S represents shear crack. (b) b = 30 , (c) b = 45 , (d) b = 60 , (e) b = 75 Note: T represents tensile crack and S represents shear crack. 3. Fracture Criterion of Brittle Material 3. Fracture Criterion of Brittle Material 3.1. The Maximum Circumferential Stress Theory 3.1. The Maximum Circumferential Stress Theory There are stress components of mixed-mode I-II at the crack tip in the polar coordi- There are stress components of mixed-mode I-II at the crack tip in the polar coordinates. nates. According to the linear elastic theory, the stress components can be expressed as According to the linear elastic theory, the stress components can be expressed as [56]: [56]: 1 1 qθθ q σ=− Kcos33 cosθ+Ksin cosθ−1 s = p K cos (3() cos q) + K sin (() 3 cos q 1) , (1 (1) ) r ⅠⅡ r I II  22πr 2 2 2 2pr 1 θ σ=+ cos K13 cosθ− Ksinθ , () (2) θ ⅠⅡ  22πr 1 θ τ=+ cos K sinθ Kc31 osθ− () , (3) rθ ⅠⅡ  22πr where σrr, σθθ, and τrθ are the radial stress, circumferential tensile stress, and shear stress at the crack tip, respectively; r is the distance from the crack tip; θ is the angle that the surface deviates from the original crack tip direction (counterclockwise is positive). The ∞ ∞ expressions Ka =σπ and Ka =τπ are the mixed-mode I and mode II I II stress intensity factors of an infinite plate with a central crack; where a is half the length ∞ ∞ of crack; and σ , τ are the far-field tensile stress and the far-field shear stress, respectively. The crack propagation direction at the crack tip is very close to that of the maximum circumferential stress [20]. According to the maximum circumferential stress theory, a mixed-mode I-II fracture can transform into an equivalent pure mode-I fracture, where the equivalent mode-I stress intensity factor (KIe) and initiation angle (θI0) are expressed as: 1 θ Ⅰ 0 KK = cos  1+− cosθθ 3K sin  (4) () , Ⅰe Ⅰ Ⅰ00 Ⅱ Ⅰ  11 −+8(KK) ⅡⅠ θ =2arctan (5) Ⅰ0 4(KK ) ⅡⅠ when KIe reaches the mode-I fracture toughness (KIC), the crack begins to expand. For a closed crack, the mode I stress intensity factor is KI = 0. Thus, the initial propagation di- rection is θI0 = 70.5°, where KIe reaches its maximum value regardless of the crack angle. Appl. Sci. 2022, 12, 4803 6 of 14 1 q s = p cos [K (1 + cos q) 3K sin q], (2) q I II 2 2pr 1 q t = p cos [K sin q + K (3 cos q 1)], (3) rq I II 2 2pr where s , s , and t are the radial stress, circumferential tensile stress, and shear stress rr qq rq at the crack tip, respectively; r is the distance from the crack tip; q is the angle that the surface deviates from the original crack tip direction (counterclockwise is positive). The p p ¥ ¥ expressions K = s pa and K = t pa are the mixed-mode I and mode II stress I II intensity factors of an infinite plate with a central crack; where a is half the length of crack; ¥ ¥ and s , t are the far-field tensile stress and the far-field shear stress, respectively. The crack propagation direction at the crack tip is very close to that of the maximum circumferential stress [20]. According to the maximum circumferential stress theory, a mixed-mode I-II fracture can transform into an equivalent pure mode-I fracture, where the equivalent mode-I stress intensity factor (K ) and initiation angle (q ) are expressed as: Ie I0 1 q I0 K = cos [K (1 + cos q ) 3K sin q ], (4) Ie I I0 II I0 2 2 1 1 + 8(K /K ) II I q = 2arctan (5) I0 4(K /K ) II I when K reaches the mode-I fracture toughness (K ), the crack begins to expand. For Ie IC a closed crack, the mode I stress intensity factor is K = 0. Thus, the initial propagation direction is q = 70.5 , where K reaches its maximum value regardless of the crack angle. I0 Ie 3.2. The Radial Shear Stress Criterion The absolute value of the maximum shear stress at the crack tip should satisfy the following equation [55]: 2 2 ¶t ¶ t ¶ t rq rq rq = 0, < 0 or > 0,jt (q = q )j , (6) rq II0 max 2 2 ¶q ¶q ¶q q is given by: II0 p p 2 + A cos(a /3) 3 sin(a /3) 0 0 q = 2arctan , (7) II0 3k 2 3 where A = 4 + 42(K /K ) , B =4(K /K ), k = 2(K /K ), a = arccos(T), T = (4A3k B)/(2 A ), II I II I 0 II I 0 0 and (a 2(0, p), 1 < T < 1). These are the coefficients related to K and K . 0 II I When the crack expansion follows the mode-II fracture pattern, the fracturing depends on the mode-II fracture toughness (K ). Thus, we can transform this crack into an IIC equivalent pure mode-II crack and the equivalent stress intensity factor can be expressed as: 1 q II0 K = cos [K sin q + K (3 cos q 1)]. (8) IIe I II0 II II0 2 2 4. Numerical Analysis and Failure Mechanism of a Single-Crack Specimen 4.1. Numerical Model and Parameter Analysis The stress field and stress intensity factor of a simulated single-crack concrete specimen were analyzed using the finite element program ABAQUS. The extended FEM was used to simulate the fracture propagation of a single-crack concrete sample. The two-dimensional computational model of the concrete sample (Figure 5) consists of 3040 grid elements. The mechanical parameters of the simulated material are listed in Table 4. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 15 3.2. The Radial Shear Stress Criterion The absolute value of the maximum shear stress at the crack tip should satisfy the following equation [55]: ∂ τ ∂∂ττ rθ rr θθ >= 0, τθ θ =< 0, 0 () (6) rθ Ⅱ0 2 2 max ∂∂ θθ ∂θ or , θII0 is given by: −+2c A os(αα 3)− 3sin( 3) () (7) θ =2arctan , Ⅱ0 3k where A = 4 + 42(KII/KI) , B = −4(KII/KI), k0 = 2(KII/KI), α0 = arccos(T), T = (−4A−3k0B)/(2√𝐴 ), and (α0∈(0, π), −1 < T < 1). These are the coefficients related to KII and KI. When the crack expansion follows the mode-II fracture pattern, the fracturing de- pends on the mode-II fracture toughness (KIIC). Thus, we can transform this crack into an equivalent pure mode-II crack and the equivalent stress intensity factor can be expressed as: 1 θ Ⅱ0 KK = cos  sinθθ+−K 3 cos 1  . () (8) Ⅱe Ⅰ Ⅱ0 Ⅱ Ⅱ0  4. Numerical Analysis and Failure Mechanism of a Single-Crack Specimen Appl. Sci. 2022, 12, 4803 7 of 14 4.1. Numerical Model and Parameter Analysis The stress field and stress intensity factor of a simulated single-crack concrete speci- men were analyzed using the finite element program ABAQUS. The extended FEM was Table 4. Mechanical parameters used in numerical modeling. used to simulate the fracture propagation of a single-crack concrete sample. Elastic Modulus Poisson’s Ratio Material Friction Coefficient u The two-dimensional computational model of the concrete sample (Figure 5) consists E (GPa) v of 3040 grid elements. The mechanical parameters of the simulated material are listed in Concrete 1.354 0.12 0.21 Table 4. (a) (b) Figure 5. Numerical model of single-crack concrete sample with b = 45 and d = 3.0 mm. Figure 5. Numerical model of single-crack concrete sample with β = 45° and d = 3.0 mm. (a) Numer- (a) Numerical model of single-crack sample; (b) Stress distribution of single-crack. ical model of single-crack sample; (b) Stress distribution of single-crack. Figure 5 shows the stress nephogram of specimen No. 9 under normal conditions. Table 5 lists the changes in the maximum circumferential tensile stress and radial shear stress at the crack tip for the specimens. The initiation angles obtained in the experimental tests and numerical simulations are shown in Table 6. Figure 6 compares the stress–strain relationships between the numerical and experimental results. The numerical solution shows a linear relationship before the peak strength. However, the stress–strain from the experiments is nonlinear and experiences compaction at the early loading stage. The nu- merical method can simplify complex stress–strain relations of cracked concretes. Figure 7 shows the computational results of the fracture propagation of the single-crack concrete sample. The SIF at the crack tip changes the pre-crack angle. The SIF at the crack tip becomes high and then low with the pre-crack angle. The SIF arrives at the maximum value when the pre-crack angle is 45 . Appl. Sci. 2022, 12, 4803 8 of 14 Table 5. Numerical results of stress near the crack tip. Maximum The Direction of The Direction of Crack Maximum Radial Crack Circumferential the Maximum the Maximum Cases Width Shear Stress at Angle b ( ) Stress at Crack Tip Circumferential Radial Shear Stress (mm) Crack Tip (MPa) (MPa) Stress ( ) ( ) 1 15 10.25 13.5 8.12 0 2 30 13.0 46.5 10.44 0 3 45 14.97 61 11.98 0 0.5 4 60 12.74 66 9.44 0 5 75 11.73 72 9.44 0 6 15 13.13 14 8.53 10 7 30 15.42 46 14.67 2.5 8 3.0 45 12.33 81 17.75 7.5 9 60 7.74 102 17.07 12 10 75 1.55 124 14.68 41 Table 6. Initiation angles derived from experimental tests and numerical simulation. The Direction of the The Initiation Angles The Initiation Angles The Direction of the Crack Angle b Maximum Cases of Experimental Work by Numerical Maximum Radial ( ) Circumferential ( ) Simulation ( ) Shear Stress ( ) Stress ( ) 1 15 11 13 13 0 2 30 47 47 47 0 3 45 68 63 63 0 Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 15 4 60 73 66 65 0 5 75 69 70 73 0 Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). (a) (b) (c) (d) (e) Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where the single crack thickness is 0.5 mm. (a) β = 15°; (b) β = 30°; (c) β = 45°; (d) β = 60°; (e) β = 75°. Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 15 Appl. Sci. 2022, 12, 4803 9 of 14 4.2. Failure Mechanism of a Closed Pre-Crack Concrete Sample The mode-I fracture toughness of brittle material is lower than the mode-II frac- ture toughness. The maximum dimensionless mode-I and mode-II stress intensity fac- p p tors are defined as f = K /s pa and f = K /s pa, respectively. When qmax Ie rqmax IIe y y f / f < 1 or f / f < K /K , the crack will expand according to the max max max max IIC IC rq q rq q mode-I fracture pattern. If f / f > 1 and f / f > K /K the fracture rq max q max rq max q max IIC IC will follow a mode-II fracture pattern. For a 0.5 mm wide crack, K = 0, the dimensionless stress intensity factor is shown in Figure 8, where f / f < 1. Thus, mode-I fracture rq max q max occurred, and the angle of f was 70.5 based on the maximum circumferential stress qmax theory. Figure 6. Stress–strain curves under numerical and experimental results (crack width = 3.0 mm). (a) (b) (c) (d) (e) Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where Figure 7. Fracture propagation numerical simulation of single joint-fissured concrete sample where the single crack thickness is 0.5 mm. (a) β = 15°; (b) β = 30°; (c) β = 45°; (d) β = 60°; (e) β = 75°. the single crack thickness is 0.5 mm. (a) b = 15 ; (b) b = 30 ; (c) b = 45 ; (d) b = 60 ; (e) b = 75 . Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 15 4.2. Failure Mechanism of a Closed Pre-Crack Concrete Sample The mode-I fracture toughness of brittle material is lower than the mode-II fracture toughness. The maximum dimensionless mode-I and mode-II stress intensity factors are fK = σπa fK = σπa defined as and , respectively. When θmax Ie y rθmax II e y ff < 1 or f fK < K , the crack will expand according to the rθθ max max rθθ max max ⅡⅠ C C mode-I fracture pattern. If ff > 1 and f fK > K the fracture rθθ max max rθθ max max ⅡⅠ C C will follow a mode-II fracture pattern. For a 0.5 mm wide crack, KI = 0, the dimensionless stress intensity factor is shown in Figure 8, where . Thus, mode-I fracture ff < 1 rθθ max max occurred, and the angle of was 70.5° based on the maximum circumferential stress Appl. Sci. 2022, 12, 4803 θmax 10 of 14 theory. Figure 8. Distribution curve of dimensionless intensity factor for a closed crack (f denotes dimension- Figure 8. Distribution curve of dimensionless intensity factor for a closed crack (f denotes dimen- less SIF). sionless SIF). The maximum circumferential stress at the crack tip is greater than the maximum radial The maximum circumferential stress at the crack tip is greater than the maximum shear stress for specimens 1–5 (Table 5). The crack propagation angles for the specimens radial shear stress for specimens 1–5 (Table 5). The crack propagation angles for the spec- obtained from the numerical and experimental results are close to those obtained from the imens obtained from the numerical and experimental results are close to those obtained maximum circumferential stress (Table 6). from the maximum circumferential stress (Table 6). 4.3. Failure Mechanism of a Non-Closed Single-Crack Concrete Sample 4.3. Failure Mechanism of a Non-Closed Single-Crack Concrete Sample The mode-I stress intensity factor for the non-closed crack is negative when the crack The mode-I stress intensity factor for the non-closed crack is negative when the crack is under compression. The dimensionless SIF at the crack tip achieved from maximum is under compression. The dimensionless SIF at the crack tip achieved from maximum circumferential tensile stress will be less than that from the radial shearing stress when circumferential tensile stress will be less than that from the radial shearing stress when the pre-crack angle is small (Figure 9). Thus, mode-II fracture in a non-closed crack will the pre-crack angle is small (Figure 9). Thus, mode-II fracture in a non-closed crack will Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 15 occur only if f / f > 1 and f / f > K /K . Therefore, the mode domain occur only if max max and max max . Therefore, the mode domain ff rq q > 1 f rqfK q> K IIC IC rθθ max max rθθ max max ⅡⅠ C C can be divided into two regions—the mode-I and mode-II fracture regions—as shown can be divided into two regions—the mode-I and mode-II fracture regions—as shown in in Figure 10. Figure 10. Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity factors. factors. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. A mode-I fracture will occur in a sample with a non-closed crack when the crack angle is less than 45° (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the crack rupture angles of the specimens with small pre-crack angles do not agree with the initiation angles obtained by the maximum circumferential stress theory. However, as β approaches 45°, the experimental fracture initiation angles are closer to those obtained by the maximum circumferential stress theory. For crack angles greater than 30° (specimens 9–11), there are considerable differences between the rupture angles of the non-closed and closed cracks because the stress patterns are different for the two conditions. The initiation angles of the non-closed cracks are close to the results obtained by the radial shear stress criterion. Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 15 Figure 9. Relationship between the maximum mode-I and mode-II dimensionless stress intensity Appl. Sci. 2022, 12, 4803 11 of 14 factors. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. Figure 10. Mode-I and mode-II fracture regions for hollow cracks. A mode-I fracture will occur in a sample with a non-closed crack when the crack angle A mode-I fracture will occur in a sample with a non-closed crack when the crack is less than 45 (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the crack angle is less than 45° (Figure 11). Figure 11 and Table 3 (cases 6 and 7) indicate that the rupture angles of the specimens with small pre-crack angles do not agree with the initiation angles obtained by the maximum circumferential stress theory. However, as b approaches crack rupture angles of the specimens with small pre-crack angles do not agree with the 45 , the experimental fracture initiation angles are closer to those obtained by the maximum initiation angles obtained by the maximum circumferential stress theory. However, as β circumferential stress theory. For crack angles greater than 30 (specimens 9–11), there are approaches 45°, the experimental fracture initiation angles are closer to those obtained by considerable differences between the rupture angles of the non-closed and closed cracks Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 15 the maximum circumferential stress theory. For crack angles greater than 30° (specimens because the stress patterns are different for the two conditions. The initiation angles of the 9–11), there are considerable differences between the rupture angles of the non-closed and non-closed cracks are close to the results obtained by the radial shear stress criterion. closed cracks because the stress patterns are different for the two conditions. The initiation angles of the non-closed cracks are close to the results obtained by the radial shear stress criterion. Figure 11. Comparison between theoretical and experimental initiation angles. Figure 11. Comparison between theoretical and experimental initiation angles. It can be seen in Table 5 that under the same crack width, as the crack angle increases, It can be seen in Table 5 that under the same crack width, as the crack angle increases, the maximum circumferential stress and the maximum radial shear stress at the crack tip the maximum circumferential stress and the maximum radial shear stress at the crack tip first increase and then decrease. For closed cracks (crack width 0.5 mm), the maximum first increase and then decrease. For closed cracks (crack width 0.5 mm), the maximum circumferential stress at the crack tip is greater than the radial shear stress at the same crack circumferential stress at the crack tip is greater than the  radial shear stress at the same angle and the maximum of both stresses appears at 45 (case 3). In the case of non-closed crack angle and the maximum of both stresses appears at 45° (case 3). In the case of non- cracks, the maximum radial shear stress is greater than the maximum circumferential tensile closed cracks, the maximum radial shear stress is greater than the maximum circumferen- stress (cases 8, 9, and 10) after the crack angle is greater than 45 . When the crack angle is tial tensile stress (cases 8, 9, and 10) after the crack angle is greater than 45°. When the 75 , the maximum circumferential tensile stress is only 1.55 MPa. There is good agreement crack angle is 75°, the maximum circumferential tensile stress is only 1.55 MPa. There is good agreement between the rupture angles derived from the numerical and experi- mental tests and those calculated by the maximum circumferential stress theory. How- ever, for specimens 9–11, the maximum circumferential tensile stress is smaller than the maximum radial shear stress even though ff > 1 and f fK > K rθθ max max rθθ max max ⅡC ⅠC . The rupture angles of the experimental results are close to those of the maximum radial stress for a mode-II fracture. The results presented above indicate that the maximum circumferential stress theory has some limitations in interpreting the fracture propagation of the non-closed cracks un- der compressional loading. The initiation angles approach those of the radial shear stress criterion if ff > 1 and f fK > K . rθθ max max rθθ max max ⅡⅠ C C Appl. Sci. 2022, 12, 4803 12 of 14 between the rupture angles derived from the numerical and experimental tests and those calculated by the maximum circumferential stress theory. However, for specimens 9–11, the maximum circumferential tensile stress is smaller than the maximum radial shear stress even though f / f > 1 and f / f > K /K . The rupture angles of the rq max q max rq max q max IIC IC experimental results are close to those of the maximum radial stress for a mode-II fracture. The results presented above indicate that the maximum circumferential stress theory has some limitations in interpreting the fracture propagation of the non-closed cracks under compressional loading. The initiation angles approach those of the radial shear stress criterion if f / f > 1 and f / f > K /K . max max max max rq q rq q IIC IC 5. Conclusions We performed uniaxial compression tests on concrete specimens containing cracks at various angles and widths. The results were compared with theoretical and numerical simulation results; based on the analysis of the results, we conclude the following. (1) The strength of the concrete sample decreases initially and then increases with increas- ing crack angle. For the same crack angle, the greater the crack width, the higher the strength of the non-closed crack sample. The crack width has a significant effect on the initiation angle. (2) With the maximum circumferential stress theory, it is difficult to depict the fracture propagation of non-closed cracks under compression. When K < 0, a non-closed crack under uniaxial compression will have a mode-I stress intensity factor. The circumferential compressive stress created by the mode-I stress intensity factor will restrain the circumferential tensile stress caused by the mode-II stress intensity factor. If f / f > 1 and f / f > K /K , the rupture angle will be close to rq max q max rq max q max IIC IC the direction of the maximum radial shear stress for a non-closed crack. Author Contributions: Conceptualization, J.Z. and L.W.; methodology, L.W.; software, J.Z. and L.W.; validation, J.Z. and L.W.; formal analysis, J.Z., B.L. and L.W.; writing—original draft preparation, J.Z. and L.W.; writing—review and editing, J.Z. and L.W.; visualization, J.Y.; supervision, J.X. and H.Z.; funding acquisition, L.W. 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Journal

Applied SciencesMultidisciplinary Digital Publishing Institute

Published: May 10, 2022

Keywords: crack; concrete material; uniaxial compression; strength; mixed-mode fracture criterion

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