Experimental Investigation on the Effect of Specimen Size in Determining Fracture Parameters of Concrete

This paper presents the experimental results in investigating the effect of specimen size (ratio of beam width to aggregate size) on the value of stress intensity factor (KIC) and fracture energy (GF) using three-point bend (TPB). A test method recommended by RILEM was chosen to measure the KIC and the GF as fracture parameters. Three different specimen sizes of concrete beam with water/binder ratio of 0.2 and 0.30 were engaged in the experiments. Both qualitative and quantitative analyses based on the normalized stress against deflection curve, and the KIC and the GF were employed. Statistical analysis was carried out based on coefficient of variation of the measured value of fracture parameters in order to investigate the variability of corresponding results. It was found that specimen size have a relatively insensitive influence on the value of KIC, however, have a significant effect on the value of GF.


Introduction
Concrete is a common composite material that has been widely used in the construction industry to build structures such as high rise buildings, bridges, and towers.The mechanical and fracture behavior of structures are dependent on the type of material used.Fracture characteristic of concrete is determined by stress intensity factor, KIC [1,2] and total dissipated fracture energy, GF [3,4].As the KIC and the GF represent the fracture characteristic of a material, these are then considered to be fracture parameters.These parameters have been widely used to determine the fracture behavior of materials [1][2][3][4][5][6][7][8].Volume of concrete in the RC structures is mostly occupied by aggregates, up to 70% of total volume of concrete [9].As a consequence, the properties of aggregate induce significantly in the fracture process of the structure which can be examined by its fracture parameters, i.e. stress intensity factor, KIC [10] and total fracture energy, GF [11].Properties of aggregate such as size, shape, texture, and amount of aggregate have been widely investigated in terms of its effect on the fracture energy of hardened concrete [12][13][14][15][16][17].
Aggregate size as one of the properties of aggregate is then considered to predict the GF value of concrete as proposed by CEB-FIP Model Code 1990 [18], Bazant's formula [19], and CEB-FIP Model Code 2010 [20].It shows that aggregate size is an important variable influencing the fracture characteristic.Thus, the size of aggregate is engaged as a parameter for proposing an adequate size of specimen to determine the GF value of concrete.The fracture characteristic of material is dependent on the fracture mechanisms such as initiation of crack, propagation of crack growth, aggregate interlocks, and arrest of crack.These mechanisms will proceed appropriately when having an adequate space in the concrete.A beam having width of 100 mm and concrete mix using maximum size of 16 mm is recommended to be used in testing according to RILEM TC50-FCM recommendation [21].In the construction industry, concrete is not only manufactured by a definite size of aggregate, but it is dependent on availability of aggregate.Bazant and Oh [22] based on their experimental results, suggested to use a minimum beam width of three times maximum aggregate size (da) used in testing, as the width of fracture process zone of the concrete is three times maximum size of aggregate.As the specimen size has been proposed to be characterized by the aggregate size, therefore, the aim of this paper is to investigate the effect of specimen sizes in the fracture parameters of concrete.

Materials
The coarse aggregate used in this study was Thames Valley river gravel with the maximum and minimum particle size of 16 mm and 5 mm respectively.The sand fraction comprised a weathered river gravel of size ranging from 4 mm down to 0.30 mm.The Portland cement binder employed was a CEM Type I material, with a specific surface area of 338 m 2 /kg, incorporating Pulverized Fuel Ash (PFA) complying with BS EN450-1 [23].A slurry-based silica fume complying with BS EN 12363-1 [24] was employed in some of the mixes for producing high strength concrete.A poly-carboxylate polymer based superplasticizer was employed to allow adequate workability and ensure full compaction of the concrete by keeping the slump value for all the mixes as 120  20 mm.The proportions of each material used for the concrete mixes produced are shown in Table 1.

Specimen Preparation
The compressive strength of the hardened concrete was determined by crushing at least nine cubes (100 x 100 x 100 mm) following BS EN 12390-3 (2001) [25].The fracture parameters were calculated using beam specimens tested as shown in Table 2 and Figure 1.Three-point bend (TPB) test method proposed by RILEM TC 50-FCM Recommendation [21] is used.All specimens were de-molded approximately 24 hours after casting and water cured at 22  2 C for 30 days prior to testing.
Figure 1 shows the geometry of the beam specimen and the associated test arrangement.The width (t), the depth (D) and the total length (L) of the beam are presented in Table 2 with a support span (S).A notch depth (ao) was employed in all of the specimens and was created using a water-cooled diamond rotary cutter with a blade width of 2.5 mm.

Three-point Bend (TPB) Testing Set-up
The TPB test was carried out using a servo-hydraulic closed-loop testing machine with a maximum capacity of 600 kN in the material and concrete laboratory, University of Surrey, UK (Figure 2).A 10 kN capacity load cell was used to measure the applied load.The loading rate of all the specimens tested was controlled by applying a vertical displacement of 0.01 mm/s [26].The vertical deflection of the beam was measured at the loading point using a calibrated LVDT having a capacity and linearity of 7.5 mm and ± 0.0007 mm, respectively.The crack mouth opening displacement was measured using a clip gauge, Figure 1, with total range of 2 mm and accuracy of 0.001 mm.

Results
The compressive strength (fc), and the measured fracture characteristics of concrete, i.e. total fracture energy (GF) and critical stress intensity factor (KIC) are shown in Table 3.The fracture characteristics of concrete are discussed in detail in the following subsection.The value of GF of each specimen was calculated from experimental test based on the RILEM TC50-FCM recommendation [21], the effect of self-weight of beam was considered.The value of GF was calculated as follows where GF is the fracture energy, Wo is the area under the load-deflection curve, m is the total mass of specimen between supports, g is gravity, δo is the end deflection at P=0 when the specimen is totally fail, and D, ao and t are the height of the sample, depth of the notch, and width of the sample, respectively.

Stress Intensity Factor, KIC
KIC as a parameter of fracture toughness of material (indicating the initiation of crack) is a function of applied load, boundary conditions, crack length, and structural geometry [1].The value of KIC of each sample was carried out by adopting the RILEM TC89-FMT recommendation [27] on the basis of three-bend test method as follows in which where Pc is the critical maximum load, W is the selfweight of the beam, S is span of specimen, ac is critical effective elastic crack length, D is depth of specimen, and t is width of specimen.

Discussion
Figures 3 to 5 show the response of beam tested on the basis of the normalized stress-deflection curve of concrete for various sizes of specimens.The normalized curve is chosen to show the trend of softening curve [1] of each specimen in order to investigate qualitatively the effect of the ratio of beam width to maximum aggregate (t/da) on the fracture parameters (KIC and GF).The specimen size of 40 x 100 x 500 mm (A1-1) in which the ratio of beam width to maximum aggregate (t/da) was 2.5, Table 2, (close to the t/da ratio suggested by Bazant and Oh [22]) was used to investigate the behavior of beam specimens.Although it is a qualitative analysis on the stress/peak stress against deflection curve, it apparently shows that the curve drops sharply after peak stress and the tail of curve vanishes before deflection of 0.4 mm (Figure 3(a)).Pre-existing flaws such as capillary voids exist in the hardened concrete [28] and these defects initiate cracks [29] when stress intensity subjected by applied stress exceeds the critical stress intensity factors of material.A crack, then, propagates to search weakest link in the concrete.As aggregate is a crack growth arrester in concrete [30], the fracture process of concrete is then dependent on the strength of interfacial zone matrix and the aggregate.When the strength of the aggregate is higher than that of the interfacial zone matrix, the initial crack propagates in the vicinity of the aggregate particles.In this mechanism, the maximum size of aggregate, which controls the t/da ratio, then, dictates cracking path and affects the behavior of the beam specimen which is characterized by normalized stress-deflection curve.Since the t/da ratio is not properly provided by the specimens, the applied stress cannot be appropriately transferred to the particles in the concrete and vice versa.As a consequence, a steep decline curve of normalized stress against deflection is produced as shown on specimen size of 40 x 100 x 500 mm in Figure 3(a).However, specimen sizes of 53 x 133 x 600 mm (A1-2, t/da ratio of 3.3) and 80 x 200 x 900 (A1-3, t/da ratio of 5), demonstrate different softening curves which have longer tails and drop gradually.The change in the shape of the softening curve may influence the measured value of fracture parameters, i.e.KIC and GF.As these parameters are representation of fracture behavior of concrete before and after peak stress, the t/da ratio of the specimen should be considered in the testing (appropriate beam specimens) to measure specific fracture parameters of hardened concrete.
Observing the coefficient of variation (cv) of the measured value of the fracture parameters was intended to evaluate the effect of sample size on the variability of both KIC and GF values.It was computed by dividing the standard deviation, stdev, over the mean value, μ, of samples (stdev/μ).The cv of the measured KIC for sample sizes of 40 x 100 x 500 (A1-1), 53 x 133 x 600 (A1-2), and 80 x 200 x 900 (A1-3) are 8.5%, 9.3%, and 7.9% respectively.The current experimental results show that the variability of the measured value of KIC is relatively same as increasing t/da ratio of the specimens.This indicates that specimen size used in the test gives insignificant effect on the measured value of KIC.However, the cv of the measured value of GF of the concrete shows a different trend compared to the measured value of KIC, the cv of the measured value of GF for specimen sizes of 40 x 100 x 500 (A1-1), 53 x 133 x 600 (A1-2), and 80 x 200 x 900 (A1-3) are 15.7%, 14.6% and 8.4% on the basis of load -deflection curve, and 14.7%, 6.5%, and 7.9% on the basis of load -CMOD curve, respectively.The cv of specimens tested tends to decrease as the t/da ratio of the specimen increases.
As fracture characteristic of concrete after peak stress is dependent on the propagation of crack path, the t/da ratio of the specimen, which affects the distribution of the applied stress surrounding the aggregate particles, will play an important role in determining of the fracture energy of the concrete.
Whilst there has been an indication that specimen size affects the measured value of GF for the specimens made of mix A1 (using maximum aggregate size of 16 mm), the mix A2 which manufactured using maximum aggregate of 10 mm was engaged in order to compare with the current experimental results discussed in the previous paragraph.Figure 4 shows the typical normalized stress against deflection of beam specimens made of mix A2.For specimens of 40 x 100 x 500 mm (A2-1, t/da ratio of 4) show to have tendency of producing longer tail curves (Figure 4a), and also variation of the curves is less than that of specimens manufactured with t/da ratio of 2.5 (Figure 3a).Table 3 shows that variation of the measured value of KIC for specimen sizes of 40 x 100 x 500 (A2-1), 53 x 133 x 600 (A2-2) and 80 x 200 x 900 mm (A2-3) are 7.7 %, 8.6 % and 6.8 % respectively.These variations are thought to be relatively same in which those are mostly less than that of specimen manufactured with t/da ratio of 2.5 (as discussed in the previous paragraph).On the other hand, the measured value of GF of the concrete for specimens of 40 x 100 x 500 (A2-1), 53 x 133 x 600 (A2-2) and 80 x 200 x 900 mm (A2-3) provide variation of 12.9%, 9.5% and 7.4% respectively.These results demonstrate an indication of the influence of t/da ratio on calculating the GF of the concrete is significant; the variation of the value of GF reduces as t/da ratio of specimen increases.As concrete is a composite material in which the behavior of material will be dependent on the materials used, both low strength and high strength hardened concrete should be examined to have appropriate results.Mixes A1 and A2 which were represented of concrete with high w/b ratio (or low strength hardened concrete) and then, mix A3 was employed to analyze the effect of specimen size (t/da ratio) on the measured value of both KIC and GF for concrete with low w/b ratio (or high strength hardened concrete).The normalized stress-deflection curves of concrete for specimen sizes of 40 x 100 x 500 mm (A3-1, t/da ratio of 2.5), 80 x 200 x 900 mm 1 2 (A3-2, t/da ratio of 5), and 100 x 100 x 500 mm (A3-3, t/da ratio of 6.25) are shown in Figure 5.In high strength hardened concrete manufactured with maximum aggregate size of 16 mm shows that the variation of curves decreases with increasing t/da ratio of specimen.Variation of the measured value of KIC, Table 3, shows a similar result for all specimens and nevertheless variation of the measured value of GF decreases as t/da ratio of specimen increases.Taken together the results of both low and high strength concrete that t/da ratio provided by specimen was found to influence on the measured GF of concrete and was essentially independent to measure KIC.

Conclusion
Based on the investigation on the measured stress intensity factor and fracture energy of concrete through three different specimen sizes by the means of the coefficient of variation, some conclusions can be drawn as follows: 1.The specimen sizes tested in this investigation in deal with the t/da ratio of specimen does not influence significantly on variation of the measured KIC.This effect has been taken into account by the geometry factor of specimens tested which is proposed by RILEM TC89-FMT recommendation.
(b) (c) 2. Measuring the specific GF of concrete using Three Point Bend test method should consider t/da ratio of specimens as cv of the measured GF reduces when increasing t/da ratio of the specimen.3. Based on the experimental investigation results, specimen with t/da ratio of 4 is a minimum size recommended to use in determining the specific GF of concrete using Three Point Bend test method.

Figure 1 .
Figure 1.Geometry of Specimen and Test Set Up

Figure 5 .
Figure 5.Typical Normalized Stress-deflection of Specimens Made of Mix A3 for Sizes of (a) 40 x 100 x 500 mm, (b) 80 x 200 x 900 mm, and (c) 100 x 100 x 500 mm

Table 2 .
Dimension of beam specimens *Total water/binder ratio

Table 3 .
Mechanical and Fracture Parameters