Investigating the Fracture Process and Tensile Mechanical Behaviours of Brittle Materials under Concentrated and Distributed Boundary Conditions

Abstract

In this study, concrete was selected to investigate the real-time splitting tensile mechanical and fracture behaviours of brittle materials using the Brazilian test under concentrated and distributed boundary conditions. The digital image correlation (DIC) method was adopted to evaluate the tensile strength and failure process in Brazilian tests using a high-resolution camera. The DIC results showed that the position of the crack initiation randomly occurred at the centre of a disc and at the boundary in Brazilian tests with concentrated loads (BTC). Comparatively, the crack initiated at the centre of discs in most Brazilian tests with distributed loads (BTD), as validated by the DIC results. Our results indicated that the average nominal tensile strength of specimens cracking at the boundary was larger than that of specimens cracking at the centre in the BTC, suggesting that the nominal tensile strength measured by the Brazilian test was related to the failure process of the discs. Moreover, this study demonstrated that the tensile strength measured by the Brazilian test was dependent on the boundary conditions. The equation recommended by the ISRM (International Society for Rock Mechanics) might slightly overestimate the tensile strength of brittle materials based on the pure elastic theory and hyperbolic-distributed loading conditions at the boundaries.

1. Introduction

Mechanical strength is deemed as a significant feature for brittle materials such as concrete, rock, and ceramic [1,2,3,4]. Numerical research [5,6,7] has suggested that the instability of structures composed of these brittle materials generally arises from a tensile failure. It is thus of vital importance to accurately quantify the tensile mechanical behaviours of brittle materials [8].

Measuring the tensile strength of brittle materials is mainly concentrated on direct and indirect tensile-testing methods [9,10,11]. In the direct tensile test, a bone-shaped specimen is subjected to a uniaxial tensile load directly [12,13,14,15]. Theoretically, the direct tensile method aims to measure the uniaxial tensile strength of materials. However, the local stress concentration and eccentricity are generally inevitable, leading to unexpected crushing and an apparent dispersion of the tensile strength [16,17,18].

The Brazilian test is the most widely used indirect tensile method which translates a compressive load to a local tensile stress concentrated at the centre of the disc specimen [19,20,21,22,23]. In a Brazilian test, a cylindrical sample is submitted to a compressive force along its diameter through diametrical compression between two flat platens, thus inducing tensile stresses perpendicular to the direction of compressive force. Currently, the Brazilian test has been recommended as the acknowledged approach to measure the tensile strength of brittle materials by the ISRM [24], ASME(American Society of Mechanical Engineers) [25], RILEM (International Union of Laboratories and Experts in Construction Materials, Systems and Structures) [26], and Chinese standards [27].

Generally, the Brazilian test supposes an elastic disc in a 2D-plane stress state and deduces the maximum tensile stress located at the centre of discs using the elastic theory [28]. Considering failure criteria based on the maximum tensile stress, the tensile strength is determined by the peak force which causes the failure [29]. To obtain reliable and scientific experimental outputs, the Brazilian test requires two preconditions as follows: (1) the crack initiates at the centre of the disc; (2) the in-plane stress/strain fields follow the elastic theory. However, it is hard to satisfy the condition for the initiation of cracks at the centre of specimens, due to incorrect failure patterns caused by the concentrated force near the loading area. The tiny contact area between platens and cylinder promotes the development of stress concentration. This indicates that a large enough contracting stress can result in contacting areas before the initiation of the central crack.

To avoid shear failure by contacting, studies have adapted the Brazilian test to reduce the stress concentration caused by linear loads. These adaptions involve changing the contact geometry between the loading platens and specimens (such as curved loading platens) [30,31], flat loading platens with strips [32,33,34,35], and flattened Brazilian discs [36,37,38,39]. However, changing the contact configurations means the transition of boundary conditions from concentrated loads to distributed loads. Therefore, setting concentrated boundary conditions may bring about a great inaccuracy in the measured tensile strength of materials with distributed boundary conditions.

Previous works have conducted numerical, analytical/semianalytical, and experimental investigations on elastic solutions of tensile strength considering different boundary conditions. However, their research focus largely remains at the theoretical level, which is in the absence of real-time experimental investigations. Consequently, it is necessary to discuss the real-time failure process of cracks before evaluating the tensile strength of brittle materials for Brazilian tests with different boundary conditions. The combination of a high-speed camera and digital image correlation (DIC) techniques [40,41,42] provides an alternative method to obtain the real-time in-plane strain fields on the surface of specimens and to identify the first initiation of cracks by virtue of the strain concentration [43,44].

In this work, we selected concrete as the targeting brittle material considering the wide use of concrete in civil engineering. The Brazilian tests were conducted on Φ75 mm concrete discs with concentrated-load and distributed-load boundary conditions. Similar to the ordinary Brazilian test, the disc was directly compressed by two flat rigid platens, which resulted in a concentrated-load boundary. For this reason, the Brazilian discs were loaded with distributed loads and connected using two curved steel jaws to depict the distributed-load boundary. In addition, a high-resolution camera was utilized to record the deformation and failure of the specimens. To synchronously collect the images and forcing, a synchronous data acquisition system was employed. Moreover, the strain evolution on the surface of the specimens was analysed by the DIC method to examine the effects of the stress state, crack initiation, as well as crack propagation on the tensile strength of concrete.

In Section 2, basic elastic theories and the experimental methodology used in the Brazilian test including the tensile strength, fracture criterion, and strain field are explained in detail. In the Brazilian test, the material of the Brazilian disc was regarded as homogeneous, isotropic, and linearly elastic before the initiation of cracks in brittle materials. In Section 3, we report the key results of the experiment. In Section 4, the failure process of specimens was examined using the DIC method and the tensile strength based on the elastic solutions is evaluated.

2. Methodology

2.1. Basic Theories

The basic theories of all Brazilian tests can be simplified into a plane stress model pertaining to the elasticity theory. In this model, the tensile stress field is developed by the compression acting at geometric boundaries of discs. The compression is loaded with two parallel plates which are flattened and rigid. Generally, the contact is assumed to be rigid, where the disc and plates in contact do not deform, or their deformation is negligible. However, significant deformation occurs at the loading point of concrete discs. The distribution of stress at the boundaries can be more accurately solved by Hertz’s solution. As a result, the theoretical basis of the Brazilian test can be described as a plane stress model where diametrical concentrated compressive forces act at diametrically opposite points of a cylindrical disk with D and L represented as its diameter and thickness, respectively. However, the results of the Brazilian test have shown evidence that the crack begins near to the loading points due to the stress singularity at the boundaries. To improve the stress singularity, the distributed loads can be applied at the boundaries of discs by increasing the contact area between the rigid plate and disc. One common method is to use a pair of steel-loading jaws with a radius 1.5 times that of the disc sample to exert a load on the Brazilian disc.

Under the assumption that cracks initiate at the centre of discs, the stress state along the central line in the BTC with rigid plates can be described by

$$\left\{\begin{array}{cc}{\sigma }_x=& \frac{-2P}{\pi DL}\\ \\ {\sigma }_y=& \frac{2P}{\pi DL}\left(\frac{4D^2}{D^2-4y^2}-1\right)\\ \\ {\tau }_{xy}=& 0\end{array}\right.$$

(1)

where ${\sigma }_x$ describes the normal stress which is perpendicular to the loading direction, ${\sigma }_y$ is the normal stress along the loading direction, ${\tau }_{xy}=$ is the shear stress, P is the force loaded at the boundary, D is the diameter of the disc specimen, and L is the thickness of the disc specimen.

Accordingly, the theoretical tension strength of rigid plates loaded at the centre of the disc according to the Griffith criterion can be described by

$${\sigma }_T=\frac{-2P}{\pi DL}$$

(2)

The Brazilian test with rigid jaws also assumes that a crack initiates at the centre of the disc. The stress state along the central line for the Brazilian test with rigid jaws can be described by

$$\begin{array}{c}\left\{\begin{array}{c}{\sigma }_x=\frac{-2P}{\pi DL\theta }\left[\frac{\left(1-{\rho }^2\right)sin2\theta }{{\rho }^4-2{\rho }^{2}cos2\theta +1}-{tan}^{-1}\left(\frac{1+{\rho }^2}{1-{\rho }^2}tan\theta \right)\right]\\ \\ {\sigma }_y=\frac{2P}{\pi DL\theta }\left[\frac{-\left(1-{\rho }^2\right)sin2\theta }{{\rho }^4-2{\rho }^{2}cos2\theta +1}-{tan}^{-1}\left(\frac{1+{\rho }^2}{1-{\rho }^2}tan\theta \right)\right]\end{array}\right.\end{array}$$

(3)

where $\theta$ is the contact angle. The ISRM suggests two steel-loading jaws designed to contact a disc-shaped sample at diametrically opposed surfaces over an arc of contact of approx 10${}^{\circ }$ at failure; $\rho =R_0/R_1$, where $R_0$ is the radius of the disc specimen, and $R_1$ is the radius of the jaws.

According to the “Griffith criterion”, the tension strength at the centre of disc can be described by

$${\sigma }_T=\frac{-2P}{\pi DL}\frac{sin\theta {cos}^2\theta }{\theta }$$

(4)

Table 1. The calculation of stress and tensile strength in BTC and BTD tests with concentrated and distributed loading boundary conditions.

Boundary Conditions	Loading Types	Stress field in Central Line	Tensile Strength at the Center of the Disc	Source
		$\begin{array}{c}{\sigma }_x=\frac{-2P}{\pi DL}\hfill \\ {\sigma }_y=\frac{2P}{\pi DL}\left(\frac{4D^2}{D^2-4y^2}-1\right)\hfill \\ {\sigma }_{xy}=0\hfill \end{array}$	$\begin{array}{c}{\sigma }_T=\frac{-2P}{\pi DL}\hfill \end{array}$	[25,28]
		$\begin{array}{c}{\sigma }_x=\frac{-2P}{\pi DL\theta }\left[\frac{\left(1-{\rho }^2\right)sin2\theta }{{\rho }^4-2{\rho }^{2}cos2\theta +1}-\right.\hfill \\ \left.{tan}^{-1}\left(\frac{1+{\rho }^2}{1-{\rho }^2}tan\theta \right)\right]\hfill \\ {\sigma }_y=\frac{2P}{\pi DL\theta }\left[\frac{-\left(1-{\rho }^2\right)sin2\theta }{{\rho }^4-2{\rho }^{2}cos2\theta +1}-\right.\hfill \\ \left.{tan}^{-1}\left(\frac{1+{\rho }^2}{1-{\rho }^2}tan\theta \right)\right]\hfill \\ {\sigma }_{xy}=0,\rho =r/R\hfill \\ P=\int qd\theta \hfill \end{array}$	$\begin{array}{c}{\sigma }_T=\frac{-2P}{\pi DL}\frac{sin\theta {cos}^2\theta }{\theta }\hfill \end{array}$	[26,30,31,38]

summarises the stress distribution along the central line of the disc and the corresponding tensile strength at the disc centre according to the Griffith cracking criterion for Brazilian tests with concentrated boundary conditions (BTC) and distributed boundary conditions (BTD). Figure 1 shows the distribution of tensile and compressive stress along the central line of discs with different boundary conditions. Obviously, the stress distribution remains the same at the centre of the disc but different at the periphery for the two boundary conditions. The distributed loading boundary could significantly improve the stress singularity at the boundaries, as shown in Figure 1b.

2.2. Experimental Program

2.2.1. Material and Specimen

To obtain the tension’s constitutive parameters of concrete, the Brazilian test was conducted on cylindrical discs. Concrete is one kind of multiple-phase material that includes cement, fine aggregates, coarse aggregates, and bubbles. Hence, the geometrical dimension is expected to be large enough to ensure that the average mechanical properties of the multiple-phase materials are obtained. The specimen dimension is generally required to be 5 times larger than the maximum diameter of coarse aggregates for ordinary concrete. In this work, the selected concrete discs should be greater than 60 mm in diameter, given that the maximum diameter of coarse aggregates was about 12 mm. Finally, cylindrical discs with a diameter of 74 mm, a length of 37 mm, and a flatness tolerance of 0.075 mm at both surfaces of specimens were built.

The ordinary Portland cement (P.I. 42.5) with a 28-day compression strength of 42.5 MPa was used. The coarse aggregates were rounded in shape with a size ranging from 12 mm down to 5 mm. The fine aggregates consisted of river sand with a specific gravity of 2.6. The ratio of water to cement was set as 0.5.

The mixes were prepared and cured under laboratory conditions. All experimental samples were then cast simultaneously using the same concrete batches. Afterward, these samples were cured for 28 days under the same ambient conditions (20 ± 2 °C and 95% relative humidity). The quasi-static compressive mechanical properties of concrete are summarised in

Table 2. Quasi-static compressive mechanical properties of concrete.

Density	Compressive Strength	Passion’s Ratio	Compressive Young’s Modulus
GPa	kg/m3	MPa
32	2430	60	0.21

.

2.2.2. Setups and Data Acquisition

Figure 2 shows specific setups of Brazilian tests for evaluating the tensile behaviours of concrete. A hydraulic servomaterial testing machine (MTS-180-50) with a constant compressive velocity of 0.01 mm/s was utilized to load a concrete disc (diameter: Φ75 mm) under the quasi-static condition. As shown in Figure 3, the Brazilian test was then performed under BTC and BTD boundary conditions, with the disc loaded by two steel plates in BTC, and the disc loaded by two steel-arc jaws with a diameter of 1.5 times larger than that of the disc in BTD tests, respectively (see Figure 3a).

To capture the initiation of the tensile crack, a Photron FASTCAM SA.5 camera was adopted to record the surface deformation in the specimens. We then identified the initiation of crack patterns using a DIC technique by measuring the real-time strain concentration. Figure 2c presents the random distribution of grey levels by spraying speckles on the surface of specimens. The relative motion of the specimens could be tracked by comparing the grey-level distributions of images before and after deformations. In general, the development of cracks is highly correlated with strain concentration. The location of cracks can thus be determined by the strain concentration.

In this paper, a camera equipped with a 50 mm lens was used to acquire the planar images at the resolution of 1024 × 1024 pixel² during deformation. The images were captured. The parameters settings of the DIC analysis windows are given in

Table 3. The parameters of the analysis windows in this study.

Parameters	Setting
Subset	15 × 15 pixel2
Step	10 pixel
Magnification factor	55 mm/pixel
Strain filter size	Gaussian (5)
Matching criterion	Normalized squared differences
Interpolation	Optimized 8-tap interpolation
Shape function	Affine

.

3. Application and Results

3.1. Loads and Loading-Rate

Figure 4 plots the typical loading response (P) for both BTC and BTD tests under a quasi-static condition. Clearly, the load–time curves increased linearly from the initial time for the BTC. However, for the BTD tests, the load of P firstly grew at a slow loading rate and then went up rapidly after the densification of the contacting interface. Since the later stage of the load curve increased almost linearly with the increase of time in the BTC, the increasing slope of the linear curve could be regarded as a constant loading rate. Therefore, the loading rate $\dot{\gamma }$ could be defined as:

$$\dot{\gamma }=\frac{dP}{dt}$$

(5)

Thus, as shown in Figure 4, the average loading rate $\dot{\gamma }$ for the BTD and BTC was calculated as about 0.512 kN/s and 0.598 kN/s, respectively. Although the results of $\dot{\gamma }$ of the BTD tests was larger than that of the BTC, the loading rates were approximately the same. Hence, both BTC and BTD were considered to have the same loading rate. According to Figure 4, the maximum loads of the BTD tests were apparently larger than those of the BTC even though the loading rate of the BTC was slightly lower than that of the BTD. It could also be observed that three load–time curves of the BTD did not coincide perfectly before the peak force, but the maximum loads were approximately the same (a maximum value of 17.15 kN and a minimum value of 16.13 kN). However, a larger difference in maximum loads, which were 16.90 kN and 14.53 kN, was observed in the results of the BTC. Thus, the tensile strength of the Brazilian test could be expected to be dependent on the boundary conditions.

3.2. Crack-Pattern Profiles

Figure 5 and Figure 6 show the typical crack patterns for both BTC and BTD, respectively. It should be noted that these tests were conducted under quasi-static conditions. In Figure 5, it can be clearly seen that a crack branching existed near the loading points at the boundaries in the BTC, although the fracture approximately passed through the centre of the discs. Therefore, it was inferred that the local concentrated load led to the shear failure of concrete at the boundaries. Meanwhile, through an observation of the crack-pattern profiles of the BTD in Figure 6, it was found that the cracks at the centre of discs in the BTD were significantly finer and straighter compared with the BTC. The crack branching was located at the loading points in the BTD, while several fine cracks occurred at the contacting boundary.

4. Discussion

Despite the apparent differences in the crack profiles observed in the BTC and BTD (as described in Section 3), the strength of the two tests was approximately equal to that calculated by Equations (2) and (4), correspondingly. Generally, to obtain the real tension strength of brittle materials, two preconditions are required as the Griffith criterion in the linear elastic theory. One is that the tensile strength of materials is inferred upon assuming that cracking initiates at the centre of the discs. The other is that the stress or the strain field should be restricted to match the theoretical resolution as described by Equations (1) and (3). However, it is difficult to determine whether the cracks initiate at the centre of discs in spite of the final fracture found along the central line. In view of this, an accurate and comprehensive recording and analysis the evolution of failures in BTC and BTD were carried out with the aid of a camera and DIC technology.

4.1. Evolution of Failure

4.1.1. Characteristics of Fracture Profiles

To examine the initial location of cracks, the DIC method was applied to analyse the deformation fields on the surface of discs by calculating the relative motion of speckles between deformed images and the reference image recorded by the camera during the loading process. Figure 7 shows the initial location of cracks on the surface of discs in the BTC and BTD by observing the map of the horizontal displacement U and the tension strain ${\epsilon }_x$ (c and f) at the occurrence of the first crack. The cracks initiated at the centre of discs were observed in both BTC and BTD as the map of ${\epsilon }_x$ shows in Figure 7c,f. However, it was found that the cracks were not straight enough at the centre of the discs where a crack turn occurred for the multiple phases of concrete in the BTC and BTD. Additionally, the DIC method can be adopted to determine the initiation and propagation of cracks using maps of the horizontal displacement, U, and tension strain, ${\epsilon }_x$.

4.1.2. Fractures of BTC

Figure 8 shows the load–displacement curves and the corresponding maps of U in BTC tests. The initiation and propagation of cracks in the discs could be determined by observing the development of the horizontal displacement resulting from the DIC. In the BTC-01 test, the distribution of U was approximately symmetrical along the vertical central line when BTC-01-a approached the peak force, as shown in Figure 8c. With a careful observation, the crack initiated from the upper loading point at the boundary and then propagated from the upper boundary to the centre of the disc as shown in Figure 8d. At the same time, another crack was also observed at the downside of the disc centre near the bottom steel plate at the time of BTC-01-b. Then, the crack propagated through the disc and the two fracture patterns extended oppositely, as shown in Figure 8e. According to Figure 5a, the final fracture profiles of the BTC-01 test also revealed the serious damage of the sample near the loading point. Consequently, it was determined that fractures did not initiate at the centre of the disc but at the boundary in the BTC-01 test. By contrast, it was found that the crack formed at the centre of the disc in the BTC-03 test as shown in Figure 8f–h. Based on the above, it can be concluded that the initiating location of cracks is uncertain without recording and analysing the fracture process in Brazilian tests with the central loads at the boundaries.

Interestingly, in the BTC-03 test, although the load had already passed its peak and started to decrease to zero quickly, the crack had not yet crossed through the central line at the time of BTC-03b. This proved that the propagating speed of the crack initiating at the centre of the disc was significantly lower than that of the crack at the boundary. Figure 9 shows the horizontal strains of ${\epsilon }_x$ at the centre of the disc and near the boundaries. In the BTC-01 test, the ${\epsilon }_x$ of $P_{up}$ increased more than that of $P_0$ and $P_{down}$ as shown in Figure 9a, indicating that the concrete connecting to the upper steel plate was compressed and crushed first, which led to the crack initiating at the upper boundary. However, in the BTC-03 test, the ${\epsilon }_x$ of $P_0$ increased sharply first, which meant the crack initiated at the centre of the disc as shown in Figure 9b. There was a clear gap of about 3 s between the history curves of $P_0$ and $P_{up}$ at the time of the crack forming, with a sharp increase in the strain (see Figure 9b). However, the cross-through time of the crack in the BTC-01 test was quite shorter than 0.2 s. These results demonstrated that the propagating speed of the crack at the centre of the disc was lower than that at the boundary of the disc in the BTC.

4.1.3. Fractures of BTD

The development of fracture was described by the DIC method in the BTD as shown in Figure 10. In Figure 10c, it can be seen that the crack initiated at the centre of the disc at the time of BTD-01-a with the load reaching the peak, indicating the initial crack was located at the centre of the sample in the BTD-01 test. The results of the BTD-02 test also demonstrated a crack initiated at the centre of the disc, even though the crack slightly deviated from the vertical centre line as shown in Figure 10f. Similar results were obtained in other BTD, which contributed to the conclusion that the crack initiated at the centre of discs in most BTD. It is noteworthy that the load reached the maximum when the central crack was observed correspondingly, which meant that the maximum load can be adopted to calculate the tensile strength of brittle materials in BTD. Indeed, this conclusion is of great significance for the evaluation of the brittle materials’ tensile strength using indirect loading methods.

4.2. Tensile Strength

A Brazilian test is generally considered to be valid on the premise that the fracture straightly passed through the centre of the disc based on the final crack-pattern profile results. Consequently, the tensile strength was calculated in this study using Equations (2) and (4) (see Table 1) under the assumption of a central breaking of the discs. Meanwhile, the real-time initiation of cracks was examined by analysing the results of the DIC method (see Section 4.1).

4.2.1. Tensile Strength of BTC Tests

The crack initiation position and corresponding tensile strength are summarised in

Table 4. Statistics of the maximum of load, crack initiation position, and the tensile strength in BTC.

Number	Loading Rate/kN/s	Maximum Load/kN	Crack Initiation	Tensile Strength/MPa
03	0.478	14.53	Center	3.291
04	0.513	15.12		3.424
07	0.509	15.45		3.499
Average	0.500	15.03		3.405
01	0.515	16.25	Boundary	3.680
02	0.544	16.90		3.490
05	0.480	15.41		3.691
06	0.476	16.30		3.506
08	0.514	15.48		3.827
Average	0.506	16.07		3.639

, after carefully checking all BTC using the DIC method. Obviously, only three specimens appeared with a crack initiation at the centre of the disc and five specimens at the boundary. Interestingly, it was observed that the average nominal tensile strength of specimens cracking at the boundary was larger than that of specimens cracking at the centre. An appropriate explanation for this might be that the tensile strength was related to the failure process of discs. As can be seen from Figure 11a, the strain concentration formed at the upper loading point in the BTC-01 test was exactly calculating the vertical compressing strain field by the DIC method. The significant compressing strain concentration resulted in a change of boundary conditions from a concentrated force to a flatten distributed pressure as shown in Figure 11b,c. However, the boundary of the specimens cracking at the centre approximately maintained the state of concentrated force boundary as shown in Figure 11d–f. The nominal tensile strength of a crack at the boundary was thus larger than that of a crack at the centre.

4.2.2. Tensile Strength of BTD Tests

With the validation of the DIC method, it was confirmed that the crack initiated at the centre of discs in most BTD, so the tensile strength was calculated as Equation (4). However, the contact angle $\theta$ between the arc jaw and cycle disc was generally uncertain. As the suggestion of the ISRM, the contact angle was set to be approximately 10${}^{\circ }$ when the specimen loaded by two steel jaws was 1.5 times larger than the disc in diameter. In this study, the DIC method was applied to acquire the real contact angle $\theta$ by reading the displacement and strain field at the contacting region in the BTD as shown in Figure 12a. The real final contact angles $2\theta$ are described in

Table 5. Statistics of the maximum of load, loading rate, contact angle, and the tensile strength in the BTD.

Number	Loading Rate/kN/s	Maximum Load/kN	Contact Angle 2$\mathit{\theta }$/${}^{\circ }$	Tensile Strength/MPa
01	0.593	16.44	15.9	3.641
02	0.605	17.15	17.8	3.777
03	0.596	16.13	18.9	3.539
04	0.459	16.79	16.7	3.709
Average	0.563	16.63	17.3	3.667

, which were less than 10${}^{\circ }$, as recommended by ISRM. Hence, the real tensile strength was obtained according to Equation (4) using the real contact angle. According to Table 5, the average tensile strength of the BTD was 3.667 MPa, which was quite close to that of 3.405 MPa in the BTC. The average tensile strength of the BTD was only 7.69% larger than that of the BTC as shown in Figure 12b. This suggested that the tensile strength of brittle materials was slightly overestimated by Equation (4) based on the pure elastic theory and hyperbolic distributed loading conditions at the boundaries.

4.3. Numerical Analysis

Finite element simulations can be adopted to analyse the aforementioned tests, especially since the brittle nature of this material allows it to be considered as elastic up to the onset of a fracture. In this paper, finite element simulations were carried out on the BTD using ABAQUS software. ABAQUS [45] provides the capability of simulating the damage using one of the three crack models for reinforced concrete elements: (1) smeared-crack concrete model; (2) brittle-cracking concrete model; (3) concrete-damaged plasticity model. Among them, the brittle-cracking concrete model was selected for evaluating the tensile strength and failure process in Brazilian tests.

4.3.1. Material Model

The brittle cracking model in ABAQUS exhibits its great potential for modelling concrete in all types of structures: beams, trusses, shells, and solids. This model is designed for applications in which the behaviour is dominated by tensile cracking with the assumption that the compressive behaviour is always linearly elastic. The crack initiation criterion of this model states that the formation of cracks appears when the maximum principal tensile stress exceeds the tensile strength of brittle materials.

Table 6. Parameters of concrete in the brittle cracking model.

Materials	Density kg/m3	Elastic Modulus GPa	Poisson Ratio	Tensile Strength MPa	Fracture Energy N/m
Concrete	2400	30	0.2	3.2	120
Steel	7800	210	0.3

enumerates the parameters in the brittle cracking model [46]. Figure 13 depicts the stress–strain curve used in that model.

4.3.2. Geometry and Elements

Two plane-stress BTD models were carried out with different boundaries as shown in Figure 14. The geometric dimensions in simulation were identical as those in experiments. The parameters of the elastic model were listed in Table 6 with plates and arc jaws chosen as the material (steel). The mesh size of concrete disc was set to be 1 mm, which was small enough to meet the convergence of simulation.

4.3.3. Numerical Results and Analysis

Figure 15 presents the stress distribution of S11 (horizontal, $sigm{a}_{xx}$) and S22 (vertical, $sigm{a}_{yy}$) from the simulation in the Brazilian disc. The distribution of the horizontal tensile stress (Figure 15a,b) revealed that the maximum value was acquired at the centre of the discs and gradually decreased from the centre to two loading points. The maximum of teh vertical compressive stress was found at the loading point as shown in Figure 15c,d. With the same loading condition, the tensile stress in the BTD was larger than that in the BTC at the centre of the discs as shown in Figure 15a,b.

Figure 16 and Figure 17 show the horizontal $sigm{a}_{xx}$ and vertical $sigm{a}_{yy}$ normal stress distribution from the analytical, numerical solution, and experimental results in the BTC and BTD. Though both analytical and numerical solutions closely matched at the centre of the discs, there was an evident difference observed near the boundary, which was likely caused by the stress singularity occurring at the loading points. In this study, the experimental stress distribution is also described in Figure 16 and Figure 17 by calculating the strain from the DIC results as follows:

$$\left\{\begin{array}{c}{\sigma }_{xx}=E{\epsilon }_{xx}\hfill \\ {\sigma }_{yy}=E{\epsilon }_{yy}\hfill \end{array}\right.$$

(6)

In this equation, E represents the elastic modulus of concrete. Here, E was assumed to be the same for the tensile and the compression load, that is, E = 30 GPa.

As shown in Figure 16, the experimental normal compressive stress $sigm{a}_{yy}$ was quite close to the analytical and numerical solutions, while there was a larger tensile stress $sigm{a}_{xx}$ in the experiment than in the analytical and numerical solutions in Figure 17. The difference of $sigm{a}_{xx}$ could be expected because of the discrepancy in the E displayed in the tensile and compressive states. Yu [8] reported an asymmetric behaviour of concrete in the Young’s modulus during compressive and tensile stages. In this article, the Young’s modulus was about 10.2 GPa and 27.2 GPa during tensile and compressive stages, respectively, which was obviously different. Actually, a similar asymmetric behaviour of E during these two stages was also investigated and reported by Forquin and Erzar [47].

5. Conclusions

This study investigated the splitting tensile mechanical and fracture behaviours of concrete using the Brazilian test under quasi-static loads and further compared the tensile strength of concrete and the real-time evolution of crack patterns for Brazilian tests with concentrated and distributed boundary conditions. The conclusions can be summarised as follows:

(1)

The tensile strength and failure process was evaluated in Brazilian tests using a high-resolution camera and the DIC method.

(2)

In the BTC, cracks were confirmed to randomly initiate at the centre and boundary of the discs in most BTD with the validation of the DIC method.

(3)

The nominal tensile strength measured by the Brazilian test was related to the failure process of discs. The average nominal tensile strength of specimens cracking at the boundary was larger than that of specimens cracking at the centre in the BTC.

(4)

The tensile strength measured by the Brazilian test was associated with the boundary conditions. The average tensile strength of the BTD was close to that of the BTC. In addition, the equation recommended by the ISRM may slightly overestimate the tensile strength of brittle materials based on the pure elastic theory and hyperbolic-distributed loading conditions at the boundaries.