An Experimental Study on Fracture Energy of Plain Concrete
 Jaeha Lee^{1}Email author and
 Maria M. Lopez^{2}
https://doi.org/10.1007/s4006901400681
© The Author(s) 2014
Received: 22 May 2013
Accepted: 6 January 2014
Published: 20 May 2014
Abstract
In this study, the concrete fracture energy was obtained using the three point notched beam test method developed by Hillerborg et al. (Cem Concr Res 6(6):773–782, 1976). A total of 12 notched concrete beams were tested under two different loading conditions: constant stroke control and constant crack mouth opening displacement (CMOD) control. Despite individual fracture energies obtained from the two different loading conditions showing some variation, the average fracture energy from both loading conditions was very similar. Furthermore, the results obtained support the idea that a far tail constant “A” could change the true fracture energy by up to 11 %, if it is calculated using CMOD instead of LVDT. The far tail constant “A” is determined using a least squares fit onto a straight line according to Elices et al. (Mater Struct 25(148):212–218, 1992) and RILEM report (2007). It was also observed that the selection of the end point can produce variations of the true fracture energy. The end point indicates the point in the experiment at which to stop. An end point of 2 mm has been recommended, however, in this study other end points were also considered. The final form of the bilinear softening curve was determined based on Elices and Guinea’s methods (1992, 1994) and RILEM report (2007). This paper proposes a bilinear stress–crack opening displacement curve according to test results as well as the CEBFIP model code.
Keywords
1 Introduction
The tensile strength (f_{ t }) can be obtained from a splitting tensile test (ASTM C496 2005; ACI 446 2009). The size effect fracture energy (G_{ f }) and the true fracture energy (G_{ F }) can be found from a three point bending test of a notched concrete beam (Maturana et al. 1990; Guinea et al. 1994; Planas et al. 1999; Kitsutaka et al. 1998; Elices et al. 2002; RILEM 2007; ACI 446 2009). The direct method of determining the tensile softening curve of concrete is a stable tensile test, however this procedure has drawbacks such as the crack location not being known (Reinhardt et al. 1986; Bažant and Planas 1998). Accordingly, many researchers prefer indirect procedures such as the threepoint bending test. The threepoint bending test for the tensile properties of concrete is based upon the cohesive models of Hillerborg et al. (1976) which were further developed by Planas et al. (1999) and Guinea et al. (1994). Coronado and Lopez (2005, 2008) proposed experimental procedures for predicting the fracture behavior of externally bonded FRP strengthened concrete structures using three point bending tests and splitting tensile tests according to Guinea et al. (1994), Planas et al. (1999) and Elices et al. (2002). It was found that the bilinear softening curves approximated from the two tests successfully predicted the experimentally observed strain, strain distribution, failure loads and failure mode. Recently the ACI 446 committees collated research results and produced a draft ASTM test standard (ACI 446 2009) for the fracture toughness i.e. the fracture energy of concrete. The draft standard mentions that the precision and bias of the test methods are not yet known and that a consensus should be developed in order to publish an official ASTM standard for determining the tensile toughness of concrete (Gerstle 2010). Bažant and Yu (2011) recently pointed a nonuniqueness of cohesive softening law especially for small size concrete specimen (D = 25 mm). For larger specimens (D = 215 mm), differences of load–CMOD graphs were negligible no matter what cohesive softening curves are used. Therefore, fracture test of specimens of one size is insufficient especially for small sized concrete structures. However, size effect testing could be avoided if following values are known a priori. It was recommended that G_{ F }/G_{ f } and σ_{ 1 } (stress at kink point in the bilinear cohesive softening curve)/f_{ t } are 2.5 and 0.25, respectively.
2 Objectives
From the discussion above, it can be concluded that the standard method which is being developed for determining the fracture energy of concrete is not yet verified. Therefore, finding new test methods in order to develop a consensus on the standard test method are needed. This study aims to fulfill these needs. In this study, based on the RILEM (2007) and ACI 446 (2009) test methods, notched beams were tested using a three point bending frame. The sensitivity to the concrete fracture energy and the feasibility of the recommended end points (CMOD at 2 mm) were investigated using data obtained from both the LVDT and crack mouth opening displacement (CMOD). A bilinear stress–crack opening curve is proposed based on the test results obtained. Some important findings from the tests performed are summarized in the conclusion section.
3 Experiment Programs
A total of 12 notched beams and 13 concrete cylinders were tested at two different times. These specimens (12 notched beams and 13 concrete cylinders) came from different concrete mixture trucks, using the same concrete mixture design, being poured the same day and at the same construction site. It should be noted that the concrete properties will be different depending on the variations of each of the concrete trucks. The quantity of entrained air was 4.5 % of the volume of the concrete. A water to cement ratio of 0.5 was used. The maximum aggregate size was required to be equal or less than 25 mm. According to RILEM (1990) recommendations, aggregate sizes in the range 1 mm to 25 mm are allowable for the beam size selected in this study. Likewise, RILEM report (2007) recommend of using a ligament of 4 times the aggregate size. At a NSF sponsored workshop in 2001 (Florida, USA), key researchers in the field of fracture toughness testing of concrete gathered and decided testing standards. A 150 × 150 × 525 mm notched beam for a three point bending test and a 150 × 300 mm piece for a Brazilian splitting cylinder test were defined as the standard cases (ACI and ASTM, Gerstle 2010). The maximum allowable aggregate size for these tests was 25 mm.
4 Apparatus for the Bending Test
5 Splitting Tensile Test Setup
To obtain softening curves for the concrete’s tensional behavior, the maximum tensile strength of the concrete should be obtained at the same time and from the same batch. Instead of performing a direct tension test on plain concrete which is of needless difficulty, splitting tensile tests (ASTM C496) were conducted using 15 cm by 30 cm cylindrical concrete specimens. Some researchers have found that the test results could vary depending on the strip width/diameter of the cylinder (b/D), the loading rate, geometry and the size effect (Rocco et al. 2001; Coronado and Lopez 2008) and it is therefore necessary to carefully consider these factors. Rocco et al. (2001) concluded that the load bearing strip as recommended by the ASTM standard seem to be too wide, however it was found that if the width of the load bearing strip approaches zero then size effects vanish. Accordingly, Rocco et al. (2001) reported that those variations due to the strip width and the size effect could be minimized when the strip width is less than 4 % of either the diameter or the width of the concrete specimens. A similar study was also performed by Coronado and Lopez (2008), who found that the primary crack would be noticeable only with a strip width in the range of 4–8 % of the diameter of the cylinder. The primary crack was not arrested during the tests, indicating that the recorded maximum load could be regarded as a true maximum load. To minimize size effects, a diameter of 5 % of the concrete cylinder was selected as the strip width for both the top and bottom loading points. Based on recommendations from ASTM C496, a loading rate of 50–100 kN/min was used during the tests.
6 Three Point Bending Test Setup
For the stroke control loading procedure, it was programmed that a load of 5 kN was incrementally applied onto the notched beam using a rate of 2.5 kN/min to remove geometrical mismatches and gaps. Secondly, a constant stroke control with a rate of 0.018 mm/min was used until the end of the test. For the CMOD control loading procedure, a loading rate of 2 kN/min was used to remove geometrical mismatches and gaps until the load reached 2 kN. Then a constant CMOD loading rate (0.01 mm/min) was used until the end of the test. Test results based on both stroke control and CMOD control will be compared and discussed in detail in a later section.
7 Results of the Three Point Bending Tests
Figure 5b explains the CMOD control test. A loading rate of 2 kN/min was used to remove any geometrical mismatches and gaps until the load reached 2 kN, and then a constant CMOD loading rate (0.01 mm/min) was maintained until the end of the test. It can be seen that the rates of CMOD are constant until the end of the test, however the slopes for the LVDT and the stroke changed at the onset of the cracking. It should be noted that the test will never reach the loading level of zero as complete failure of the beam is approached asymptotically (Petersson 1981). This implies that the fracture test for a concrete beam is stopped before total energy dissipation. The draft ASTM standard recommends the use of 2 mm as an end point for the test (ACI 446 2009) and the fracture energy is calculated at this point.
Analysis of obtained test results with far tail constant A.
LVDT and CMOD data  LVDT data  CMOD data  

Specimen  E (GPa)  G_{ FM } (N/m)  A (N mm^{2})  G_{ F } (N/m)  G_{ f } (N/m)  A (N mm^{2})  G_{ F } (N/m)  G_{ f } (N/m) 
Stroke  
T51  38.87  122.29  353  154.84  25.83  730  173  26.05 
T52  35.34  133.35  419  171.25  32.3  787  184.14  32.15 
T201  37.41  127.92  315  157  71.23  626  168.35  71.08 
T202  42.67  175.46  530  223.07  153.64  999  240.09  153.17 
T301  34.55  185.99  839  261.82  73.7  1448  279.58  72.05 
T302  35.7  132.29  384  169.03  75.58  811  184.68  75.36 
Avg.  37.42  146.22  473  189.50  72.05  900  204.97  71.64 
CMOD  
T53  30.6  128.17  289  156.33  57.22  648  169.99  57.14 
T203  36.42  155.78  509  202.83  71.94  1074  221.61  71.71 
T204  29.14  143.19  648  202.23  52.34  1200  220.65  51.65 
T281  31.32  194.87  497  239.44  123.89  864  250.64  122.34 
T303  38.11  103.8  198  123.17  61  446  132.56  60.92 
Avg.  33.12  145.16  428  184.80  73.28  836  199.09  72.75 
8 Obtained Fracture Energy of Concrete
9 Sensitivity Study on the Far Tail Constant (A)
Originally, Petersson (1981) and Elices et al. (1992) used LVDT data to calculate the far tail constant (A), however the ACI 446 (2009) test method uses CMOD data to calculate the far tail constant (A). These studies led the authors to check the differences of the fracture energies (G_{ F } and G_{ f }), when the far tail constant A is calculated by LVDT or CMOD.
10 Sensitivity Study on Different Loading Conditions
11 Sensitivity Study on the End Points for the Three Point Bending Tests
Analysis of obtained test results with far tail constant A of CMOD.
Specimen  End point (mm)  G_{ FM } (N/m)  A (N mm^{2})  G_{ F } (N/m)  G_{ f } (N/m) 

T204  2  143.19  648  202.23  52.34 
T204  3  173.52  903  227.74  56.84 
T204  4  187.38  951  230.05  57.18 
T204  5  200.49  1092  239.7  58.28 
However, the final true fracture energy (G_{ F }) should be consistent, no matter what end point is chosen, provided that the test follows the same asymptotic curves. If the final true fracture energies depend on the end points and show some variation, then Peterson’s asymptotic assumption (Petersson 1981) to predict the tail fracture energy (G_{ tail }) is not fit for the specimens in this study, given their larger aggregate size (25 mm). The results obtained show that selections with a larger number of end points tend to yield larger true fracture energies (G_{ F }). It is also interesting to see that end points between 3 and 4 mm show relatively similar values to those at 1 and 5 mm. This indicates that there should be an appropriate end point for each different aggregate size. Therefore, if the size of the aggregate increases, then the end point should be increased in order to get the precise fracture energy. However, it was observed that the size effect fracture energy (G_{ f }) was not sensitive to the selection of the end point since the tail part of the load–CMOD curves are more related to the true fracture energy (G_{ F }).
12 Proposed Bilinear Stress–Crack Opening Displacement Curves
13 Conclusions

The average true fracture energies are 189 and 185 N/m from the stroke control and the CMOD control, respectively, after considering the possible sources of energy dissipation. The final forms of the bilinear stress–crack opening curves of concrete from both loading rates were also very similar.

It was found that the fracture energy could be sensitive to the far tail constant (A) value and to the selection of the end points used to calculate the true fracture energy. When CMOD data was used to calculate A, a maximum increase of 11 % of the true fracture energy (G_{ F }) was found. However, the size effect fracture energy (G_{ f }) did not change with the value of A.

It was also observed that an end point of 5 mm yields an 18 % larger true fracture energy (G_{ F }) than that of the 2 mm end point. Therefore, appropriate selection of the end point of the test should be considered along with the maximum aggregate size in order to obtain the true fracture energies. However, the size effect fracture energy (G_{ f }) was not influenced by the selection of the end point.

It is concluded that the loading rate is not a significant factor to control the fracture energy of the concrete as well as the final forms of the bilinear softening curves.

The overall shape of the bilinear stress crack opening curve estimated by the CEBFIP model is similar to the experimentally obtained bilinear curves. However, it tends to exhibit smaller tail fracture energy (G_{ tail }) and a larger size effect fracture energy (G_{ f }) when compared to the experimental ones.

In this study, a new bilinear curve is proposed based upon both the CEBFIP recommended bilinear curves and experimentally obtained bilinear curves. The proposed bilinear curve in this study fits the experimental bilinear curves well.

The accuracy of the size effect fracture energy (G_{ f }) determined using one size of notched beam and one size of cylinder has recently been brought into question. As a further study, a comparison of the size effect fracture energy (G_{ f }) as determined using multiple sizes of notched beams along with the results obtained from this study is recommended.

The biases of the various concrete toughness tests developed is still unknown. Sufficient data should be gathered and sufficient research conclusions should be collected in order to define a reliable test standard. It is hoped that results obtained from this study might be helpful for building a consensus on the concrete fracture toughness test method.
Declarations
Acknowledgment
This study is based upon work supported by the National Science Foundation under a CAREER Grant No. 0330592.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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