# Fracture Energy Analysis of Concrete considering the Boundary Effect of Single-Edge Notched Beams

Authors: Ping Xu; Jinyi Ma; Minxia Zhang; Yahong Ding; Lingqi Meng
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The method of determining concrete fracture energy recommended by RILEM has an obvious size effect, so determining fracture energy that is unaffected by size of the test specimen is difficult. In this study, 60 high-strength concrete single-edge notched beams (SENBs) of different sizes, crack length-to-depth ratios, and span-to-depth ratios were subjected to the three-point loading test as recommended by RILEM. Then, the influences of the boundary effect on the fracture energy were identified. Based on the SENB boundary effect model, a piecewise function of the interrelationships between the experimental test fracture energyGf, the local fracture energy gf, and the fracture energy unaffected by specimen size GF was established. The applicability of the boundary effect model was verified using the test results from this study and from the previously published research. The results show that the local fracture energy distribution in the boundary influence region was nonuniform. The smaller the local fracture energy was, the closer it was to the rear boundary of the specimen. The influence length al∗ of the boundary increased with the increasing specimen size. Based on the bilinear distribution model of the local fracture energy gf, the fracture energy unaffected by beam size GF can be obtained according to the fracture energy Gf measured for laboratory-scale small-sized SENB specimens. Furthermore, the model predictions are in good agreement with experimental observations.


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## Body

## 1. Introduction

Fracture energy is an important fracture property parameter of concrete and can directly reflect the crack resistance of this material [1–4]. Theoretically, the best method for determining the fracture energy of concrete is the direct tension test. However, due to the small deformation of concrete and the stiffness of a test specimen, the direct tension test is not as suitable as the standard method for testing the fracture energy of concrete. RILEM recommended using a simple three-point load on single-edge notched beams (SENBs) to determine the fracture energy of concrete [5]. However, it was found that the self-weight of the three-point load on an SENB has a great influence on the results, and the fracture energy determined from different sizes of specimens exhibits a significant size effect. Fracture energy that is not affected by the size of the test specimen cannot be obtained directly from the test [6, 7]. For this reason, the influencing mechanism of the specimen size on fracture energy during the test process is analyzed persistently, and determining the fracture energy of concrete unaffected by the size of test specimens has become a research focus [8, 9].Brameshuber and Hilsdorf [10] used finite element analysis to study the energy dissipation around the fracture process zone (FPZ) and found that, with the development of the FPZ, the nonlinear deformation outside the ligament increased and that about 10% of the total fracture energy was consumed due to nonlinear deformation. Wittmann et al. [11–13] observed that the crack opening displacement w0 of the critical crack tip with the change of the specimen size was the main factor that caused the size effect on the fracture energy of concrete. According to the test, the concrete fracture energy that is determined is not a material constant. Bažant et al. [14, 15] defined the fracture energy of concrete as the energy consumption of crack propagation in an infinite specimen; they found that the stress redistribution and energy release caused by the development of macrocracks or microcrack zones are the fundamental reason for the size effect on the fracture energy of concrete. Based on the fictitious crack model, Hu et al. [16–18] analyzed the influence of the FPZ on the fracture energy of concrete at the boundary of the fractured specimens. They found that the essence of the size effect on fracture energy is due to the influence of the boundary effect on the FPZ of concrete and proposed a quasi-brittle fracture model based on the boundary effect, revealing the influence of a specimen boundary on the fracture toughness.The cited research shows that the fracture energyGf determined using the RILEM-recommended three-point loading test of concrete SENB changes with the size of the specimen. To obtain a fracture energy that is not affected by specimen size GF, a sufficiently large specimen is required, which is difficult to achieve in an indoor laboratory. Therefore, revealing the effect of specimen size on the SENB fracture energy and analyzing the fracture energy unaffected by specimen size have important theoretical and practical value in evaluating the fracture performance of concrete components. In this study, the three-point loading of concrete SENB recommended by RILEM was used to determine the fracture energy of high-strength concrete SENBs of different sizes, span-to-depth ratios, and crack length-to-depth ratios. The specimens with a more wide range of crack length-to-depth ratios a (α = a/W = 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7) were designed in our tests, not just those geometrically similar specimens of different sizes used in the previous studies. This design was intended to more clearly reveal the effect of the specimen boundary on the fracture energy of concrete. Based on the SENB boundary effect model, the relationship between experimental test fracture energy Gf, local fracture energy gf, and fracture energy unaffected by specimen size GF was obtained. The applicability of the model was proved, and a method for analyzing the fracture energy unaffected by specimen size GF was established using laboratory-scale small-sized SENB specimens.


## 2. Experimental Investigation

### 2.1. Test Material and Concrete Mix Design

The test specimens were constructed using PO 42.5 Portland cement with a density of 3.11 g/cm3, first-grade fly ash, and silica fume with silicon content ≥95%. The fineness modulus of the medium sand comprising the fine aggregate was 2.7 with mud content <2% and an apparent density of 2.6 g/cm3. The gravel comprising the coarse aggregate was 5–10 mm in diameter with an apparent density of 2.70 g/cm3. A water-reducing agent containing a polycarboxylic superplasticizer was used with a reducing water rate >30%. The SENB specimens were cured by thermal treatment to accelerate the pozzolanic reaction of the matrix material, and the resulting hydration products were used to fill the micropores inside the specimens to increase the strength of concrete. The specimen mix is shown in Table 1. The basic physical and mechanical parameters of the configured high-strength concrete are shown in Table 2.Table 1
Concrete mix of test specimens (kg/m³).
Cement
Water
Fly ash
Silica fume
Medium sand
Crushed stone
Water-reducing admixture
895
250
268
90
626.5
268.5
15Table 2
Physical and mechanical parameters of concrete.
Slumps (mm)
Expansion (mm)
Compressive strength (28 d) (MPa)
Tensile strength (28 d) (MPa)
280
700
90
8.39


### 2.2. Specimen Preparation

To reveal the influence of SENB specimen size and boundary on the fracture energy of high-strength concrete during the fracturing process, 17 groups of 60 SENBs with different sizes, span-to-depth ratios, and crack length-to-depth ratios were designed. Typical test specimens are shown in Figure1. Nine groups (four specimens in each group, a total of 36) were small-sized SENB specimens with dimensions for length (L), depth (W), and width (B) of L × W × B = 515 × 100 × 100 mm, net span- (S-) to-depth ratio S/W=4, and various initial crack length- (a-) to-depth ratios α=a/W = 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. Likewise, the sizes of two groups of SENB specimens were 850 × 200 × 100 mm and 1250 × 300 × 100 mm, and both had a span-to-depth ratio S/W=4 and two crack length-to-depth ratios α=a/W = 0.2 and 0.5 (three specimens per group, 12 in total). The sizes of two other groups of SENB specimens were 675 × 50 × 100 mm and 1050 × 400 × 100 mm, and these had a span-to-depth ratio S/W=2.5 and two crack length-to-depth ratios α=a/W = 0.2 and 0.5 (three specimens per group, 12 in total). The quantity (W−a) is the fracture ligament length of SENB.Figure 1
SENBs with different sizes. (a) Small size withS/W=4. (b) Large size with S/W=2.5. (c) Geometrically similar specimens with S/W=4.
(a)
(b)
(c)


### 2.3. Fracture Energy Test of SENB

In this study, a three-point bending test was performed using a universal testing machine (Figure2) with a rated load of 100 kN and a sensitivity of 1.99010 mV/V. The load was applied at the rate of 0.02 mm/min and a loading head diameter of 30 mm and a length of 100 mm. The load and midspan deflection were recorded, and load-deflection curves were plotted graphically for each specimen during the test process. The test fracture energy Gf was obtained by using Equation (1) [5].Figure 2
Test equipment with a concrete beam in the test position.


## 2.1. Test Material and Concrete Mix Design

The test specimens were constructed using PO 42.5 Portland cement with a density of 3.11 g/cm3, first-grade fly ash, and silica fume with silicon content ≥95%. The fineness modulus of the medium sand comprising the fine aggregate was 2.7 with mud content <2% and an apparent density of 2.6 g/cm3. The gravel comprising the coarse aggregate was 5–10 mm in diameter with an apparent density of 2.70 g/cm3. A water-reducing agent containing a polycarboxylic superplasticizer was used with a reducing water rate >30%. The SENB specimens were cured by thermal treatment to accelerate the pozzolanic reaction of the matrix material, and the resulting hydration products were used to fill the micropores inside the specimens to increase the strength of concrete. The specimen mix is shown in Table 1. The basic physical and mechanical parameters of the configured high-strength concrete are shown in Table 2.Table 1
Concrete mix of test specimens (kg/m³).
Cement
Water
Fly ash
Silica fume
Medium sand
Crushed stone
Water-reducing admixture
895
250
268
90
626.5
268.5
15Table 2
Physical and mechanical parameters of concrete.
Slumps (mm)
Expansion (mm)
Compressive strength (28 d) (MPa)
Tensile strength (28 d) (MPa)
280
700
90
8.39


## 2.2. Specimen Preparation

To reveal the influence of SENB specimen size and boundary on the fracture energy of high-strength concrete during the fracturing process, 17 groups of 60 SENBs with different sizes, span-to-depth ratios, and crack length-to-depth ratios were designed. Typical test specimens are shown in Figure1. Nine groups (four specimens in each group, a total of 36) were small-sized SENB specimens with dimensions for length (L), depth (W), and width (B) of L × W × B = 515 × 100 × 100 mm, net span- (S-) to-depth ratio S/W=4, and various initial crack length- (a-) to-depth ratios α=a/W = 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. Likewise, the sizes of two groups of SENB specimens were 850 × 200 × 100 mm and 1250 × 300 × 100 mm, and both had a span-to-depth ratio S/W=4 and two crack length-to-depth ratios α=a/W = 0.2 and 0.5 (three specimens per group, 12 in total). The sizes of two other groups of SENB specimens were 675 × 50 × 100 mm and 1050 × 400 × 100 mm, and these had a span-to-depth ratio S/W=2.5 and two crack length-to-depth ratios α=a/W = 0.2 and 0.5 (three specimens per group, 12 in total). The quantity (W−a) is the fracture ligament length of SENB.Figure 1
SENBs with different sizes. (a) Small size withS/W=4. (b) Large size with S/W=2.5. (c) Geometrically similar specimens with S/W=4.
(a)
(b)
(c)


## 2.3. Fracture Energy Test of SENB

In this study, a three-point bending test was performed using a universal testing machine (Figure2) with a rated load of 100 kN and a sensitivity of 1.99010 mV/V. The load was applied at the rate of 0.02 mm/min and a loading head diameter of 30 mm and a length of 100 mm. The load and midspan deflection were recorded, and load-deflection curves were plotted graphically for each specimen during the test process. The test fracture energy Gf was obtained by using Equation (1) [5].Figure 2
Test equipment with a concrete beam in the test position.


## 3. Experimental Results

Considering the influence of the self-weight of the specimen on the results of the three-point bending test, Equation (1) was used to calculate the fracture energy Gf [5]. The test results for each specimen are shown in Table 3:(1)Gf=W0+m1S/L+2m2gδ0Alig,where W0 is the area enclosed by the load-deflection curve, m1 is the mass of the specimen, S is the span of the specimen between supports, L is the length of the specimen, m2 is the mass of the loading device (which does not touch the testing machine but always acts on the specimen), g is the acceleration of gravity, δ0 is the maximum deflection, and Alig is the area of the crack ligament.Table 3
Specimens with different sizes and the corresponding fracture energy.
SizeS×W×B (mm)
Specimen
a
/
W
(W−a) (mm)
G
f (N/m)
400×100×100
1-400-5-I
0.05
95
195.38
1-400-5-II
188.9
1-400-5-III
179.3
1-400-5-IV
190.25
2-400-10-I
0.1
90
162.3
2-400-10-II
173.4
2-400-10-III
180.54
2-400-10-IV
169.23
3-400-15-I
0.15
85
152.6
3-400-15-II
165.8
3-400-15-III
160.87
3-400-15-IV
169.23
4-400-20-I
0.2
80
150.75
4-400-20-II
155.39
4-400-20-III
162.8
4-400-20-IV
166.4
5-400-30-I
0.3
70
145.6
5-400-30-II
120
5-400-30-III
126.16
5-400-30-IV
135.7
6-400-40-I
0.4
60
125.68
6-400-40-II
123.34
6-400-40-III
110.25
6-400-40-IV
112.5
7-400-50-I
0.5
50
100.82
7-400-50-II
90.59
7-400-50-III
95.64
7-400-50-IV
88.74
8-400-60-I
0.6
40
85.64
8-400-60-II
70.28
8-400-60-III
75.96
8-400-60-IV
69.78
9-400-70-I
0.7
30
63.48
9-400-70-II
56.32
9-400-70-III
55.04
9-400-70-IV
47.86
800×200×100
10-800-40-I
0.2
160
357.05
10-800-40-II
298.59
10-800-40-III
323.64
11-800-100-I
0.5
100
200.12
11-800-100-II
180.32
11-800-100-III
210.53
1200×300×100
12-1200-60-I
0.2
240
413.69
12-1200-60-II
432.02
12-1200-60-III
450.34
13-1200-150-I
0.5
150
249.12
13-1200-150-II
278.69
13-1200-150-III
263.86
625×250×100
14-625-50-I
0.2
200
354.75
14-625-50-II
416.04
14-625-50-III
385.39
15-625-125-I
0.5
125
293.92
15-625-125-II
242.38
15-625-125-III
222.35
1000×400×100
16-1000-80-I
0.2
320
510.45
16-1000-80-II
480.72
16-1000-80-III
498.83
17-1000-200-I
0.5
200
312.72
17-1000-200-II
354.84
17-1000-200-III
344.81For Equation (1), W0 can be calculated using the load-deflection curve collected by the test system. Table 3 indicates that the fracture energy of the high-strength concrete SENB specimens with the same span-to-depth ratio decreased as the initial crack length of the specimen increased, and the distance W−a (fracture ligament length) from the initial crack length to the back boundary influenced the experimental test fracture energy. Fracture energy is the energy required for crack propagation per unit area. In practice, the fracture energy of concrete is regarded as constant, an assumption that is not consistent with the experimental test results. As shown in Table 3, the size of SENB specimens and the initial crack length both affect the fracture energy of the beams.Based on the fictitious crack model, Hu et al. [16–18] analyzed the FPZ of three-point loaded concrete SENB specimens and considered that the essence of the fracture energy affected by specimen size is due to the boundary zone in the FPZ of the specimens. The size of the initial FPZ from the specimen boundary is considered as W−a. To analyze the influence of the size of the SENB specimens and the initial crack length on the fracture energy Gf, the relationship between the tested fracture energy and W−a needs to be analyzed, as shown in Figure 3. Figure 3 shows that, at a span-to-depth ratio S/W=4, the test fracture energy Gf of SENB specimens (the average value of Gf in the same group of SENB specimens) changes with fracture ligament length W−a in beams of different depths. The fracture energy Gf of each specimen with the span-to-depth ratio S/W=4 decreased as the ligament length W−a decreased, and there was a clear linear correlation between the two. These results support the conclusion that the main reason for the variation of fracture energy Gf with W−a is that when the specimen is destroyed, the FPZ of the specimen boundary cannot be fully developed due to the presence of the specimen boundary influence zone. The closer the FPZ is to the back boundary of the specimen, the more limited the FPZ is and the smaller the fracture results are from the test; thus, the fracture energy Gf exhibits a correlation with specimen size. As the initial crack length a increases, the influence of the specimen boundary influence zone on the FPZ at the back end of the initial crack becomes greater, and the influence of the fracture energy on the specimen fracture energy Gf is more obvious.Figure 3
Fracture energyGf of SENBs with different fracture ligament lengths.Based on the experimental results, it can be seen that fracture ligament lengthW−a and specimen boundary are the main factors affecting changes in the fracture energy Gf. Therefore, to reveal the fracture energy of concrete unaffected by specimen size GF, it is necessary to effectively analyze the influence of the boundary effect of the SENB specimen during a test.


## 4. Analysis of Boundary Effect for SENB

### 4.1. Local Fracture Energygf Distribution

The fracture energy of concrete as defined by RILEM refers to a stable load-deflection curve obtained from the three-point bending test of SENB. The area enclosed by this curve and the horizontal axis represents the average distribution of the total energy absorbed by the concrete beam in the fracture ligament area during the development of the crack and is described by the following equation:(2)GF′=1B⋅W−a∫P⋅dδ,where B is the specimen thickness, a is the specimen initial crack length, W is the specimen depth, P is the total external load, and δ is the specimen deflection.The fracture energyGF′ in Equation (2) is the average value for the SENB fracture ligament area. Obviously, GF′ is affected by the size of the test specimen, and the GF′ obtained from Equation (2) is not the same as the fracture energy GF that is unaffected by specimen size. The results obtained in this study showed that the fracture energy is unevenly distributed in the fracture ligament area. Previously, to reveal the distribution of fracture energy in the fracture ligament area, Hu et al. [19] defined a type of fracture energy that is variable within the fracture area of the concrete SENB and called it the “local fracture energy” gf. Accordingly, gf is the fracture energy at each point along the crack propagation direction, as shown in Figure 4.Figure 4
Distribution characteristics ofgf and FPZ in the SENB fractured ligament area.As depicted in Figure4, gf is the local fracture energy corresponding to the formation of a new crack in the FPZ and refers to the energy absorbed when the surface opening of the fracture tip per unit area is displaced from 0 to wc, i.e.,(3)gf=∫0Wcσ⋅dw.Obviously, when the specimen size is small, the fracture ligament area will be located entirely in the SENB boundary influence zone. In this situation, the local fracture energy will decrease as the length of the fracture ligament area decreases, and the fracture energy is affected significantly by specimen size. If the SENB specimen size or the length of the fractured ligament is large, there will be an area in the specimen that is not affected by the boundary in the fracture ligament area and wheregf is constant, which is the fracture energy that is unaffected by specimen size GF, i.e., GF=gf. However, near the back boundary of the specimen, the FPZ cannot develop sufficiently, and gf gradually decreases as the crack expands. If the size of the test piece or the length of the fracture ligament area is sufficiently large, the boundary influence zone shown in Figure 4 is relatively negligible, and the average fracture energy GF′ in a test can be regarded as the fracture energy unaffected by specimen size GF. Therefore, based on the analysis of the local fracture energy gf and its distribution, the relationship between GF′ (as described in RILEM documentation) and the local fracture energy affected by the boundary gf can be established using the following equation:(4)GF′=1B⋅W−a∫P⋅dδ=1B⋅W−a∫0W−agf⋅B⋅dx,Gf=GF=gf=constantlargeenoughSENB.The above analysis shows that the change of the local fracture energygf in the boundary region and its effect on the average fracture energy GF′ are the fundamental reason that fracture energy is affected by the specimen size in the RILEM test. The SENB test specimens used in laboratory experiments usually cannot be sufficiently large to avoid this effect; thus, the local fracture energy gf is influenced by the boundary effect so that the fracture energy determined by the test is not the fracture energy that is independent of specimen size. However, using Equation (4), if the distribution of gf in the fracture ligament zone of an SENB is known, the fracture energy unaffected by specimen size GF can be obtained based on the measured fracture energy.


### 4.2. The Bilinear Model

Due to the boundary effect of SENB specimens, the local fracture energygf is not uniformly distributed within the fracture ligament zone W−a. To address the variation of gf, Karihaloo et al. [20] proposed a trilinear gf distribution model as shown in Figure 5. On the one hand, the trilinear gf distribution model predicts a fast-rising linear increase in gf at the front boundary; here, the front boundary of the SENB specimen has a small influence on the local fracture energy gf, and gf is mainly affected by the SENB back boundary. On the other hand, it is difficult to determine the location of the breakpoints for the trilinear model during the analysis, which is not conducive to practical application. To overcome this shortcoming, Hu et al. [19, 21] proposed a bilinear local fracture energy gf distribution model (also shown in Figure 5), and its bilinear distribution function can be regarded as follows:(5)gfx=GF=constant,al∗≤x≤W−a,g0+GF−g0xal∗,0≤x<al∗,where x represents the distance of a position in the ligament to the specimen back boundary, al∗ is the length of the boundary influence and varies with the specimen size, and g0 is the local fracture energy of the back boundary edge x=0. The analysis by Hu et al. [19] shows that g0 is small relative to GF and can generally be taken as zero. As shown in Figure 3, the experimentally determined test fracture energy varied linearly with changes of the fracture ligament length W−a; furthermore, because the regression line passing through the origin described the test results better than other regressions, g0=0. As shown in Figure 5, gf equals GF in the region that is not affected by the back boundary; furthermore, gf is a constant in this region, and the fracture energy of concrete is unaffected by specimen size.Figure 5
Distribution models describing local fracture energygf.Based on the analysis of the local fracture energygf, all of the experimental test fracture energy Gf contributes to the development of the FPZ, i.e.,(6)Gf=1B⋅W−a∫0W−aB⋅gfxdx.Incorporating Equation (5) into Equation (6), the relationship between the experimental test fracture energy Gf and the fracture energy unaffected by specimen size GF can be obtained as follows:(7)Gf=GF1−al∗2W−a,W−a>al∗,GFW−a2al∗,W−a≤al∗.According to the bilineargf distribution model, Equation (7) provides a method to analyze the fracture energy unaffected by specimen size GF using the experimental test fracture energy Gf. When the tip of the initial crack of the specimen is located in the zone unaffected by the boundary, the first relationship W−a>al∗ in Equation (7) can be used in theoretical calculations. When the tip of the initial crack of the specimen is located in the boundary zone, the second relationship W−a≤al∗ in Equation (7) can be used in theoretical calculations. So, the fracture energy unaffected by specimen size GF can be obtained according to the experimental test fracture energy gf of laboratory-scale, small-sized specimens.


## 4.1. Local Fracture Energygf Distribution

The fracture energy of concrete as defined by RILEM refers to a stable load-deflection curve obtained from the three-point bending test of SENB. The area enclosed by this curve and the horizontal axis represents the average distribution of the total energy absorbed by the concrete beam in the fracture ligament area during the development of the crack and is described by the following equation:(2)GF′=1B⋅W−a∫P⋅dδ,where B is the specimen thickness, a is the specimen initial crack length, W is the specimen depth, P is the total external load, and δ is the specimen deflection.The fracture energyGF′ in Equation (2) is the average value for the SENB fracture ligament area. Obviously, GF′ is affected by the size of the test specimen, and the GF′ obtained from Equation (2) is not the same as the fracture energy GF that is unaffected by specimen size. The results obtained in this study showed that the fracture energy is unevenly distributed in the fracture ligament area. Previously, to reveal the distribution of fracture energy in the fracture ligament area, Hu et al. [19] defined a type of fracture energy that is variable within the fracture area of the concrete SENB and called it the “local fracture energy” gf. Accordingly, gf is the fracture energy at each point along the crack propagation direction, as shown in Figure 4.Figure 4
Distribution characteristics ofgf and FPZ in the SENB fractured ligament area.As depicted in Figure4, gf is the local fracture energy corresponding to the formation of a new crack in the FPZ and refers to the energy absorbed when the surface opening of the fracture tip per unit area is displaced from 0 to wc, i.e.,(3)gf=∫0Wcσ⋅dw.Obviously, when the specimen size is small, the fracture ligament area will be located entirely in the SENB boundary influence zone. In this situation, the local fracture energy will decrease as the length of the fracture ligament area decreases, and the fracture energy is affected significantly by specimen size. If the SENB specimen size or the length of the fractured ligament is large, there will be an area in the specimen that is not affected by the boundary in the fracture ligament area and wheregf is constant, which is the fracture energy that is unaffected by specimen size GF, i.e., GF=gf. However, near the back boundary of the specimen, the FPZ cannot develop sufficiently, and gf gradually decreases as the crack expands. If the size of the test piece or the length of the fracture ligament area is sufficiently large, the boundary influence zone shown in Figure 4 is relatively negligible, and the average fracture energy GF′ in a test can be regarded as the fracture energy unaffected by specimen size GF. Therefore, based on the analysis of the local fracture energy gf and its distribution, the relationship between GF′ (as described in RILEM documentation) and the local fracture energy affected by the boundary gf can be established using the following equation:(4)GF′=1B⋅W−a∫P⋅dδ=1B⋅W−a∫0W−agf⋅B⋅dx,Gf=GF=gf=constantlargeenoughSENB.The above analysis shows that the change of the local fracture energygf in the boundary region and its effect on the average fracture energy GF′ are the fundamental reason that fracture energy is affected by the specimen size in the RILEM test. The SENB test specimens used in laboratory experiments usually cannot be sufficiently large to avoid this effect; thus, the local fracture energy gf is influenced by the boundary effect so that the fracture energy determined by the test is not the fracture energy that is independent of specimen size. However, using Equation (4), if the distribution of gf in the fracture ligament zone of an SENB is known, the fracture energy unaffected by specimen size GF can be obtained based on the measured fracture energy.


## 4.2. The Bilinear Model

Due to the boundary effect of SENB specimens, the local fracture energygf is not uniformly distributed within the fracture ligament zone W−a. To address the variation of gf, Karihaloo et al. [20] proposed a trilinear gf distribution model as shown in Figure 5. On the one hand, the trilinear gf distribution model predicts a fast-rising linear increase in gf at the front boundary; here, the front boundary of the SENB specimen has a small influence on the local fracture energy gf, and gf is mainly affected by the SENB back boundary. On the other hand, it is difficult to determine the location of the breakpoints for the trilinear model during the analysis, which is not conducive to practical application. To overcome this shortcoming, Hu et al. [19, 21] proposed a bilinear local fracture energy gf distribution model (also shown in Figure 5), and its bilinear distribution function can be regarded as follows:(5)gfx=GF=constant,al∗≤x≤W−a,g0+GF−g0xal∗,0≤x<al∗,where x represents the distance of a position in the ligament to the specimen back boundary, al∗ is the length of the boundary influence and varies with the specimen size, and g0 is the local fracture energy of the back boundary edge x=0. The analysis by Hu et al. [19] shows that g0 is small relative to GF and can generally be taken as zero. As shown in Figure 3, the experimentally determined test fracture energy varied linearly with changes of the fracture ligament length W−a; furthermore, because the regression line passing through the origin described the test results better than other regressions, g0=0. As shown in Figure 5, gf equals GF in the region that is not affected by the back boundary; furthermore, gf is a constant in this region, and the fracture energy of concrete is unaffected by specimen size.Figure 5
Distribution models describing local fracture energygf.Based on the analysis of the local fracture energygf, all of the experimental test fracture energy Gf contributes to the development of the FPZ, i.e.,(6)Gf=1B⋅W−a∫0W−aB⋅gfxdx.Incorporating Equation (5) into Equation (6), the relationship between the experimental test fracture energy Gf and the fracture energy unaffected by specimen size GF can be obtained as follows:(7)Gf=GF1−al∗2W−a,W−a>al∗,GFW−a2al∗,W−a≤al∗.According to the bilineargf distribution model, Equation (7) provides a method to analyze the fracture energy unaffected by specimen size GF using the experimental test fracture energy Gf. When the tip of the initial crack of the specimen is located in the zone unaffected by the boundary, the first relationship W−a>al∗ in Equation (7) can be used in theoretical calculations. When the tip of the initial crack of the specimen is located in the boundary zone, the second relationship W−a≤al∗ in Equation (7) can be used in theoretical calculations. So, the fracture energy unaffected by specimen size GF can be obtained according to the experimental test fracture energy gf of laboratory-scale, small-sized specimens.


## 5. Comparison between Experimental Results and Boundary Effect Model Predictions

Substituting the experimental test fracture energyGf of small-sized specimens numbered 1 to 9 (Table 3) into Equation (7), a binary equation system with fracture energy GF and boundary influence length al∗ can be obtained. Because the number of equations is greater than the number of unknowns, in order to minimize the sum of the squares of the error between the approximate solution of each equation and its mean value, a least squares method was used to obtain the optimal solution of the equations. Then, the optimal solutions for GF and al∗ were substituted into Equation (7) to yield the relationship between Gf and the crack length-to-depth ratio a/W. As shown in Figure 6, the theoretical fracture energy values obtained using Equation (7) agreed well with the experimental observations from testing 36 SENBs (each 100 mm × 100 mm × 515 mm) in nine sets of specimens. Because the sizes of the set of specimens were small, the length of the fracture ligament range W−a was within the range of influence of the back boundary. Thus, the larger the initial crack length, the closer the initial crack tip to the back boundary and the smaller the local fracture energy gf; consequently, the fracture energy Gf determined from the test was smaller.Figure 6
Theoretical values and test values of the fracture energy with differenta/W (W = 100 mm).Substituting the test fracture energyGf for each of the test specimens numbered 10–17 (Table 3) into Equation (7), the fracture energy unaffected by specimen size GF and the boundary influence length al∗ for multiple groups of specimens were calculated. The results of these calculations for the concrete specimens with various span-to-depth ratios are presented in Table 4. These results show that, with the change of SENB specimen size, the theoretical value GF obtained using Equation (7) was in the range 708.18–724.15 N/m; thus, the maximum change difference was only 2.2%. Furthermore, the fracture energy GF was independent of the size of the specimen and can be regarded as a material constant. The boundary influence length al∗ increased as the depth of the specimen increased when the specimen span-to-depth ratios were S/W=4 and S/W=2.5. That is to say, for the same span-to-depth ratios, the greater the depth of the specimens, the greater the influence range of the specimen’s boundary.Table 4
Fracture energyGF and the influence length al∗.
W (mm)
G
F (N/m)
a
l
∗ (mm)
200
708.18
180.94
250
724.15
187.12
300
717.6
196.6
400
716.7
215.2By substituting the high-strength concrete fracture energyGF and the boundary influence length al∗ (from Table 4) into Equation (7), the experimental test fracture energy Gf as a function of the crack length-to-depth ratio a/W can be obtained (Figure 7). As shown in Figure 7, the theoretical values from Equation (7) for test specimens of different sizes and span-to-depth ratios exhibited a good consistency with the experimental observations in both the boundary-affected zone and the uninfluenced zone. This shows that the bilinear boundary effect model can properly analyze the impact of an SENB specimen boundary on the fracture energy of high-strength concrete beams with different span-to-depth ratios and can properly determine the fracture energy GF of high-strength concrete that is unaffected by specimen size.Figure 7
Comparisons of theoretical with measured fracture energy for single-edge notched concrete beams having different depths (W) and span-to-depth (S/W) ratios. (a) Large size with S/W=4andW=200mm. (b) Large size with S/W=4andW=300mm. (c) Large size with S/W=2.5andW=250mm. (d) Large size with S/W=2.5andW=400mm.
(a)
(b)
(c)
(d)The foregoing comparative analysis of the fracture energy determined from SENB tests of high-strength concrete and the theoretical values of the bilinear model considering the boundary influence shows that the bilinear model predicts experimental observations well. To further examine and prove the adaptability of the bilinear model, it was used to analyze the fracture energy of ordinary concrete SENB with different strengths designed by Karihaloo et al. [22] and Muralidhara et al. [23].Karihaloo et al. [22] designed the three-point loading test of three sets of common concrete SENB with geometrical similarities but different depths (50, 100, and 200 mm) and a 28-day compressive strength of 57.1 MPa. Muralidhara et al. [23] designed the three-point loading test of two groups of common concrete SENB specimens with geometrical similarities and depths of 95 mm and 190 mm and a 28-day compressive strength of 52.2 MPa. The experimental test fracture energy Gf corresponding to the size of each specimen and the fracture energy unaffected by specimen size GF (calculated using Equation (7)) are shown in Table 5.Table 5
Specimens of different sizes and the corresponding calculated fracture energy [22, 23].
Size [22, 23] L×W×B (mm)
a/W
(W−a) (mm)
G
f (N/m)
a
l
∗ (mm)
G
F (N/m)
250×50×50
0.1
45
117.25
34.5
192.68
0.2
40
85.25
0.3
35
57.75
500×100×50
0.1
90
135.25
68.3
194.28
0.2
80
107.50
0.3
70
95.50
1000×200×50
0.1
180
153.00
95.4
193.28
0.2
160
126.25
0.3
140
109.25
375×95×47.5
0.25
71.25
151.9
30
194.3
190.9
0.35
61.75
157.5
154.5
750×190×95
0.25
142.5
165
55
197.4
147.2
167.4
159.7
0.35
123.5
153.7
129.5
179.2Data in Table5 show that the boundary influence length al∗ of the common concrete SENB specimens increases the depth of the specimen; that is, the greater the depth of the specimen, the greater the influence range of the boundary. As shown in Table 5, Equation (7) predicted the fracture energy GF for the Karihaloo test data to be between 192.68 and 194.28 N/m, a maximum difference of only 0.8%. Likewise, the predicted fracture energy GF for the Muralidhara test data is between 194.3 and 197.4 N/m, a maximum difference in GF of only 1.5%. Thus, the bilinear model of the boundary influence on concrete fracture energy GF for Karihaloo’s data and Muralidhara’s data indicates that the size of the specimen is basically irrelevant, and GF can be regarded as a material constant.Equation (7) also can be used with the fracture energy GF and the boundary influence length al∗ of the common concrete specimens described in Table 5 to determine the variation of the experimental test fracture energy Gf as a function of the crack length-to-depth ratio a/W, as shown in Figures 8 and 9. Because the fracture ligament length W−a of each specimen described in Table 5 is greater than the boundary influence length al∗, the tip of the initial crack of each specimen is located in zones unaffected by the boundary, and the first relationship W−a>al∗ in Equation (7) can be used in theoretical calculations. However, the theoretical value of the fracture energy Gf shown in Figure 8(a) is much different from the experimental observation. The main reason for this result is that the SENB specimen of this group is too small (the depth of the specimen is only 50 mm). When a/W is 0.2 and 0.3, the fracture ligament length W−a is only 40 mm and 35 mm, respectively, and the concrete aggregate particle size will have a significant influence on the test results. Notably, Figures 8 (except Figure 8(a)) and 9 show that the calculated values of Gf for specimens with different strengths and different sizes are in good agreement with the theoretical values. Thus, the analysis model considering the influence of the boundary effect (i.e., Equation (7)) can properly determine the fracture energy of concrete beams having different strengths, dimensions, and span-to-depth ratios. Therefore, through the use of Equation (7), the fracture energy that is not affected by specimen size GF can be calculated by using the fracture energy Gf of the small-sized specimen in a laboratory test, which provides a reference for the fracture design of engineering structures.Figure 8
Comparisons of the theoretical with measured fracture energy in the study of Karihaloo et al. [22]. (a) Small size with S/W=5andW=50mm. (b) Small size with S/W=5andW=100mm. (c) Large size with S/W=5andW=200mm.
(a)
(b)
(c)Figure 9
Comparisons of the theoretical with measured fracture energy in the study of Muralidhara et al. [23]. (a) Small size with S/W=4andW=95mm. (b) Large size with S/W=4andW=190mm.
(a)
(b)


## 6. Conclusions

A total of 60 high-strength concrete SENBs with different dimensions, crack length-to-depth ratios, and span-to-depth ratios were tested in this study. Using the experimental results of Karihaloo et al. [22] and Muralidhara et al. [23], the influence of the boundary on the fracture energy of the SENB specimen was revealed. Based on the three-point bending test results of the small-sized SENB, the fracture energy unaffected by the size of the concrete specimen GF was analyzed. A method for determining the fracture energy unaffected by specimen size GF was established using laboratory-scale small-sized SENB specimens. The major conclusions are summarized as follows:(1)
The fracture energyGf measured by the SENB three-point loading test decreases as the ligament length W−a decreases, and the two measures are linearly correlated. Because of the specimen boundary influence zone, the FPZ of the specimen boundary cannot be fully developed. The FPZ is more limited; the closer it is to the back boundary of the specimen, the smaller the fracture energy determined by the three-point test. The experimentally determined test fracture energy Gf shows a correlation with specimen size.(2)
In the RILEM-recommended three-point test, variation of the local fracture energygf in the boundary region is the fundamental cause of the specimen-size effect on fracture energy. Based on the bilinear distribution model of the local fracture energy gf, the relationship between the experimental test fracture energy Gf, the local fracture energy gf, and the fracture energy unaffected by specimen size GF can be established through testing, and the fracture energy unaffected by specimen size GF can be determined based on the experimental test fracture energy Gf.(3)
Based on the experiments carried out in this study and using the results of three-point loading tests of concrete SENB conducted by Karihaloo et al. [22] and Muralidhara et al. [23], the bilinear gf distribution model considering SENB boundary effects predicts theoretical values that are in good agreement with experimental observations, confirming the validity of this model in practical applications.



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