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Determination of size-independent specific fracture energy of concrete mixes by two methods
Cement and Concrete Research 50 (2013) 19–25
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Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp
Determination of size-independent specific fracture energy of concrete mixes by two methods A. Ramachandra Murthy a, B.L. Karihaloo b,⁎, Nagesh R. Iyer a, B.K. Raghu Prasad c a b c
CSIR Structural Engineering Research Centre, Chennai 600113, India School of Engineering, Cardiff University, Cardiff CF24 3AA, UK Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 28 November 2012 Accepted 21 March 2013 Keywords: Toughness (C) Fiber reinforcement (E) High-performance concrete (E) Boundary effect (C)
a b s t r a c t The RILEM work-of-fracture method for measuring the specific fracture energy of concrete from notched three-point bend specimens is still the most common method used throughout the world, despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed in the literature to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a size-independent value (GF). It has also been proved recently, on the basis of a limited set of results on a single concrete mix with a compressive strength of 37 MPa, that when the size-dependent Gf measured by the RILEM method is corrected following either of these two methods, the resulting specific fracture energy GF is very nearly the same and independent of the size of the specimen. In this paper, we will provide further evidence in support of this important conclusion using extensive independent test results of three different concrete mixes ranging in compressive strength from 57 to 122 MPa. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The specific fracture energy of concrete is one of the most important properties required in the analysis of the mechanical behavior of cracked concrete structures. The work-of-fracture method recommended by RILEM [1] for measuring the specific fracture energy of concrete from notched three-point bend specimens of different sizes and notch to depth ratios is still the most common method used throughout the world [2,3], despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen [4–9]. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed by Elices and co-workers [10–12] and by Hu and Wittmann [13] to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a size-independent value (GF). Elices and co-workers [10–12] identified the sources of experimental error in the RILEM method and proposed a methodology for eliminating the major source of error, namely by including the work-of-fracture that is not measured in the RILEM method due to practical difficulties in capturing the tail part of the load–deflection plot. The second method proposed by Hu and Wittmann [13] recognized that the local specific energy varied during the propagation of a crack, the variation becoming more pronounced as the crack approached the stress-free back face boundary of the specimen. ⁎ Corresponding author. E-mail address: [email protected] (B.L. Karihaloo). 0008-8846/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cemconres.2013.03.015
Karihaloo et al. [14] and Abdalla and Karihaloo [15] extended the free boundary effect concept of Hu and Wittmann [13] and showed that the same size-independent specific fracture energy can also be obtained by testing only two specimens of the same size but with notches which are well separated, for example the notch to depth ratios in three-point bend specimens should be 0.05 and 0.5 or 0.1 and 0.6. Their method greatly reduces the number of test specimens and simplifies the determination of GF. In a recent paper, the first of its kind, Cifuentes et al. [16] showed that if the size-dependent Gf measured by the RILEM method is corrected following the methods of Elices and co-workers [10–12] and Hu and Wittmann [13] (or its simplification proposed by Karihaloo et al. [14], if the notch to depth ratios are well separated), then the resulting specific fracture energy GF is very nearly the same and independent of the size of the specimen. They reached this important conclusion on the basis of a limited set of results on a single concrete mix with a compressive strength of 37 MPa. In this paper, we will provide further evidence in support of this important conclusion using extensive independent test results of three different concrete mixes ranging in compressive strength from 57 to 122 MPa. 1.1. Boundary effect method (BEM) Hu and Wittmann [13] observed that the effect of the stress-free back boundary of the specimen is felt in the fracture process zone (FPZ) ahead of a real growing crack. The local fracture energy varies with the width of
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the fracture process zone. As the crack approaches the back stress-free face of the specimen the fracture process zone becomes more and more confined and hence the local fracture energy decreases [13]. Initially, when the crack grows from a pre-existing notch, the rate of decrease is negligible, but it accelerates as the crack approaches the end of the un-cracked ligament. This change in the local fracture energy (gf) is approximated by a bilinear function, as shown in Fig. 1. The transition from horizontal line to the sharply inclined line occurs at the transition ligament length, which depends on the material properties and specimen size and shape [17–19]. In the boundary effect model of Hu and Wittmann [13], the measured RILEM fracture energy, Gf, may be regarded as the average of the local fracture energy function (dotted line in Fig. 1) over the initial un-cracked ligament area. The relationship between all the involved variables is given by D−a
∫ g f ðxÞdx Gf ða; DÞ ¼
0
D−a
¼
8 > > > < GF 1− > > > :
GF
al =D ; 2ð1−a=DÞ
2ð1−a=DÞ ; 2al =D
1 a=D > al =D ð1Þ
1 a=D ≤ al =D:
In Eq. (1), D is the total depth of the specimen, a is the initial notch depth and al is the transition ligament length. This bi-linear variation in the local fracture energy is supported by independent acoustic emission measurements during the crack growth [20]. To obtain the values of GF and al of a concrete mix, the size-dependent specific fracture energy Gf of specimens of different sizes and a range of the notch to depth ratios is first determined by the RILEM work-offracture method. Then Eq. (1) is applied to each specimen depth and notch to depth ratio. This gives an over-determined system of equations which is solved by a least squares method to obtain the best estimates of GF and al. Hu and Duan [21] showed that although the measured values of Gf depend on D and a/D, the above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth. The simplified boundary effect method proposed by Karihaloo et al. [14] greatly reduces the number of test specimens and eliminates the least squares solution of the over-determined system of equations. It requires the testing of only two specimens of the same size but with notches which are well separated, for example the notch to depth ratios in three-point bend specimens should be 0.05 and 0.5 or 0.1 and 0.6. 1.2. Method proposed by Elices et al. (P–δ tail) Elices et al. [10], Guinea et al. [11], and Planas et al. [12] identified several sources of energy dissipation that influence the measurement
Fig. 1. Local fracture energy model of Hu and Wittmann [13].
of Gf by the RILEM work-of-fracture method. Most of these sources are due to experimental errors which affect the measured value of the work-of-fracture during the test. They classified these sources into three main groups: (i) the testing equipment and experimental setup, (ii) the energy dissipation in the specimen bulk, and (iii) the non-measured energy corresponding to the un-recorded tail of the load–deflection (P–δ) curve near the end of the test. The first source of error can be avoided with a proper calibration of the equipment and design of supports and loading system. The second source is due to crushing at supports and under the loading point, and due to high tensile stresses in the specimen bulk. The crushing dissipation can be corrected by adjusting the initial stiffness of the P–δ curve and measuring the mid-span deflection on the lower half of the beam depth. The dissipation in the specimen bulk, Wdb, due to high tensile stresses cannot be avoided; however, it was estimated that this causes an error of less than 2% [10]. The last source of error is introduced by the curtailment of the tail of the P–δ near the end of the test due to practical difficulties of unloading the cracked specimen fully in a stable manner and it has the most significant effect on the size dependency of the measured fracture energy. To estimate this non-measured energy when the test is stopped (Wnm = Wnm1 + Wnm2 in Fig. 2) at a very low load, it is necessary to model the beam behavior when the crack is close to the free surface [10]. Once the non-measured work of fracture has been estimated, the size-independent specific fracture energy of concrete can be calculated as [10]: δu
∫ Pdδ þ W nm GF ¼
0
bðD−aÞ
ð2Þ
where b(D − a) is the area of the ligament (Alig) of the test specimen that was un-cracked at the start of the test, and b is the width of the specimen. 2. Experimental investigation Prismatic notched specimens were subjected to three-point bending in accordance with the RILEM procedure [1]. Table 1 gives the geometrical dimensions of all test specimens [22]. The materials and mix proportions used in the normal strength (NSC), high strength concrete (HSC), and the ultra high strength concrete (UHSC) are given in Table 2. The specimen preparation was strictly controlled to minimize the scatter in the test results. The NSC specimens were demolded after 1 day and cured in a water tank at ambient temperature for 28 days. The HSC and UHSC specimens were also demolded after 1 day and immersed in water at ambient temperature for 2 days. They were then placed in an autoclave at 90 °C for 2 days and in an oven at 200 °C for 1 day. Thereafter they were air cooled for 6 h and placed in water at ambient
Fig. 2. P–δ curve in a three-point bend test and the measured (Wf) and non-measured (Wnm1 + Wnm2) work of fracture [16].
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25 Table 1 Geometrical properties of the notched specimens.
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Table 3 Mechanical properties of NSC, HSC and UHSC.
Mix
Beam dimensions (mm) (length × width × depth)
Notch to depth ratio
Mix
Cylinder compressive strength (MPa)
Split tensile strength (MPa)
Modulus of elasticity (GPa)
NSC
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
NSC HSC UHSC
57.1 87.7 122.5
4.0 15.4 20.7
35.8 37.9 43.0
500 × 50 × 100
1000 × 50 × 200
HSC
250 × 50 × 50
500 × 50 × 100
UHSC
250 × 50 × 50
400 × 50 × 80
650 × 50 × 130
3. Results 3.1. Adjustment of the P–δ tail Tables 4–6 show the NSC, HSC and UHSC specimen results. The size-independent specific fracture energy (G⁎F) has been obtained using Eq. (2). The tables also give the vertical displacement at the end of the test δu, the size-dependent specific fracture energy determined by RILEM work of fracture method Gf (i.e. Eq. (2) without the correction term for P–δ tail), the load P′ corresponding to δu, and the non-measured work-of-fracture Wnm. δ is the displacement corresponding to the fully unloaded broken specimen. In the present calculations the tail region of the P–δ curve was assumed to vary linearly from to P′ to zero load and δ was computed using the known slope of the P–δ at δu from the recorded readings. From Tables 4–6, it can be clearly seen that despite the large variation in the measured Gf with the specimen size and notch depth, the G⁎F values are almost the same for a particular mix i.e. NSC, HSC or UHSC irrespective of specimen size and notch depth. We have used here G⁎F only with a view to distinguish this sizeindependent specific fracture energy from that obtained by the boundary effect procedure, denoted by GF. We shall however see that the two values are practically the same.
temperature for a further 1 day before testing. Compression and split tensile tests were carried out on cylindrical specimens of 150 × 300 mm (diameter × height) in the case of NSC and on smaller cylinders 75 × 150 mm in the case of HSC and UHSC. Table 3 gives the mechanical properties of the three mixes. Notches (width approximately 3 mm) of various depths given in Table 1 were cut in beam specimens using a diamond saw. Four identical specimens for each beam size and each notch depth were cast. All tests were performed in a closed-loop servo-hydraulic testing machine, controlled by the crack mouth opening displacement (CMOD) measured with a clip gauge. A linearly varying displacement transducer (LVDT) was used to measure the mid-span vertical displacement, δ. The load–CMOD and load–displacement curves of all specimens were recorded. The ratio of the span between the supports to the depth of the specimen was maintained at 4 for all specimens. Fig. 3 shows typical load–deflection plots for NSC, HSC and UHSC mix.
3.2. Boundary effect method Tables 7–9 show the mean values of the specific fracture energy obtained from the three-point bend tests of NSC, HSC and UHSC specimens according to the RILEM procedure (the individual specimen results are given in Tables 4–6). An inspection of the entries in column 2 of Tables 7–9 highlights the dependency of the RILEM specific fracture energy on the notch depth or the size of the un-cracked ligament. The specific fracture energy increases with an increase in the beam depth for the same notch to depth ratio and it decreases with an increase in the notch to depth ratio for the same beam depth. Tables 7–9 also give the size-independent specific fracture energy (GF) and transition ligament length al obtained by solving the over-determined system of equations obtained by substituting Gf(a/D;D) from Tables 7–9 for each
Table 2 Mix proportions by mass (except for steel fiber which is by volume) of NSC, HSC and UHSC. Mix
Cement
Fine aggregate
Coarse aggregate
Silica fume
Quartz sand
Quartz powder
Steel fiber by vol. (length = 13 mm dia. = 0.18 mm)
w/c
SP, %
NSC HSC UHSC
1 1 1
1.25 – –
2.48 – –
– 0.25 0.25
– 1.5 1.1
– – 0.4
– 2% 2%
0.45 0.33 0.23
– 2.5 3.5
The maximum sizes of the aggregates used for the preparation of the above mixes are as follows: NSC Fine aggregate = 600 μm Coarse aggregate = 12 mm HSC Quartz sand = 600 μm UHSC Quartz powder = 75 μm Quartz sand = 600 μm.
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A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
practice. It might appear at first sight that these two methods use very different experimental procedures, but in reality they are closely interrelated. Both procedures apply some corrections to the final part of the P–δ diagram in the work-of-fracture test [23]. The local fracture energy model considers the influence when the crack approaches the back-face free boundary surface of the specimen towards the end of the test [24], whereas the method of Elices et al. [10] consists in determining the non-measured work-of-fracture by adjusting the tail of the P–δ curve that also corresponds to the final part of the test.
(a) NSC – 500x50x100mm 4.50 4.00
0.1d 0.2d 0.3d
Load, kN
3.50 3.00 2.50 2.00 1.50
4. Conclusions
1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5
0.6
Displacement, mm
(b) HSC – 250x50x50mm 4.50
0.1d 0.2d 0.3d 0.4d
4.00
Load, kN
3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
3
4
5
6
Displacement, mm
(c) UHSC – 250x50x50mm
Fig. 3. Typical load–deflection plots for NSC, HSC and UHSC.
D into Eq. (1) of boundary effect method. It is important to note that the solution of this over-determined system by the method of least squares may require the explicit imposition of the inequality constraint on al/D that appears in Eq. (1) in order to obtain the best estimates of GF and al. It should also be noted that the simplified boundary effect method proposed by Karihaloo et al. [14] cannot be used on the above test specimens because the notch to depth ratios are closely spaced and not well separated, as required by the simplified BEM. From Tables 7–9, it is clear that GF values are almost the same for a particular mix i.e. NSC, HSC or UHSC irrespective of specimen size and notch depth. Table 10 compares the mean size-independent specific fracture energy obtained by using two main methods. It is clear that both methods give nearly identical mean size-independent specific fracture energies of NSC, HSC and UHSC, so that either of them can be used in
From the above analysis of the extensive experimental data on three grades of concrete ranging in compressive strength from 57 to 122 MPa the following conclusions can be drawn. • In common with all previous investigations it has been found that the specific fracture energy Gf measured using the RILEM work-of-fracture procedure is highly dependent on the size of the specimen and the notch to depth ratio. For the NSC used in the present investigation it varied between 57.8 N/m for a specimen with D = 50 mm and a/D = 0.3 to 153.0 N/m for a specimen with D = 200 mm and a/D = 0.1 (Table 7). For the HSC (Table 8) it varied between 2923.0 N/m (D = 50 mm, a/D = 0.4) and 4396.3 N/m (D = 100 mm, a/D = 0.1) and for the UHSC (Table 9) between 4406.3 N/m (D = 50 mm, a/D = 0.4) and 11,945.3 N/m (D = 130 mm, a/D = 0.1). • The correction of the size-dependent Gf by the procedure proposed by Elices and his co-workers [10–12] or by the boundary effect procedure of Hu and Wittmann [13] result in nearly the same size-independent G⁎F or GF irrespective of the size of the specimen and the notch to depth ratio. For the NSC, G⁎F is 184.5 N/m and GF is 190.3 N/m (Table 10). For HSE the corresponding values are 6194.7 N/m and 6393.1 N/m and for UHSC the values are 13,760.7 N/m and 14,184.8 N/m. The maximum difference between G⁎F and GF in these three mixes is 3.1% (Table 10). Thus either of the two procedures may be used in practice. • In fact, in practice the simplified boundary effect procedure of Karihaloo et al. [14] may be followed. This procedure requires very few test specimens of the same size, say eight, four of which have a very small starter crack (notch to depth ratio 0.1) and the remaining four a deep starter crack (notch to depth ratio 0.6). The size-dependent Gf measured by the RILEM work-of-fracture method on these specimens of the same depth and two widely separated notch to depth ratios is substituted in Eq. (2) and the resulting two equations solved in the two unknowns GF and al. This eliminates the need for solving an over-determined system of equations by the least squares method. It is clear that the notch to depth ratios (0.1 to 0.4) used in the specimens of the present study (Table 1) did not meet the requirement of wide separation. Karihaloo et al. [14] have also shown that the size-independent specific fracture energy GF of a concrete mix is also independent of the shape of the test specimen (notched three-point bend or wedge splitting) and not only its size and notch to depth ratio. Finally, it should be added that for the analysis of a cracked concrete structure by the non-linear fictitious (or cohesive) crack model, it is not only necessary to know the (size-independent) specific fracture energy GF of the concrete mix, but also its cohesion–separation (tension softening) diagram. This diagram is commonly approximated by a bilinear function whose various parameters (i.e. the tensile strength, the critical crack opening, and the co-ordinates of the transition point) are inferred in an inverse manner. Details of this inverse parameter identification procedure can be found in [25,26]. Acknowledgments The first author's visit to Cardiff University is funded by Raman Research Fellowship awarded to him by CSIR, India. He acknowledges
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Table 4 Size-independent specific fracture energy of NSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig N/m
G⁎F, N/m
250 × 50 × 50
0.1
115 123 115 116 91 87 74 89 47 55 68 61 136 144 130 131 105 93 115 117 96 89 100 97 165 146 148 153 110 135 140 120 115 111 104 107
0.09 0.09 0.09 0.09 0.17 0.17 0.19 0.18 0.26 0.25 0.23 0.24 0.37 0.33 0.35 0.36 0.41 0.44 0.38 0.37 0.58 0.56 0.55 0.56 0.22 0.24 0.22 0.21 0.35 0.32 0.30 0.34 0.42 0.44 0.42 0.44
1.51 1.47 1.48 1.47 0.75 0.77 0.79 0.76 0.51 0.51 0.49 0.50 0.54 0.50 0.53 0.52 0.52 0.54 0.51 0.51 0.40 0.43 0.41 0.42 1.10 1.22 1.22 1.20 1.10 0.96 0.96 1.02 1.00 0.95 1.00 0.97
0.12 0.12 0.12 0.12 0.33 0.33 0.36 0.34 0.63 0.63 0.58 0.60 0.51 0.49 0.49 0.54 0.76 0.78 0.75 0.76 0.96 0.96 0.94 0.94 0.30 0.32 0.32 0.31 0.64 0.58 0.54 0.62 0.65 0.64 0.62 0.65
70.5 66.3 69.1 66.3 93.8 96.3 108.6 98.8 127.1 128.2 113.4 120.0 52.6 45.6 49.8 52.8 76.1 82.4 72.0 72.0 88.0 93.4 87.3 90.0 31.8 37.9 36.6 34.3 68.1 54.0 50.4 61.2 76.7 73.3 74.3 75.5
185.5 189.3 184.1 182.3 184.8 183.3 182.6 187.8 174.1 183.2 181.4 181.0 188.6 189.6 179.8 183.8 181.1 175.4 187.0 189.0 184.0 182.4 187.3 187.0 196.8 183.9 184.6 187.3 178.1 189.0 190.4 181.2 191.7 184.3 178.3 182.5
0.2
0.3
500 × 50 × 100
0.1
0.2
0.3
1000 × 50 × 200
0.1
0.2
0.3
Table 5 Size-independent specific fracture energy of HSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig, N/m
G⁎F, N/m
250 × 50 × 50
0.1
4157 4056 4102 4123 3464 3763 3880 3612 3685 3301 3410 3523 2894 2892 2988 2918 4811 4142 4200 4432 4516 4266 3829 4010 4623 3580 3865 4012 2898 3971 3407 3312
3.41 3.45 3.10 3.21 4.21 4.22 4.17 4.31 4.51 4.62 4.61 4.56 4.88 4.86 4.76 4.82 3.32 3.43 3.33 3.23 4.68 4.53 4.70 4.56 5.11 5.34 5.37 5.29 6.32 6.12 6.22 6.38
1.20 1.22 1.21 1.18 0.96 0.98 0.99 1.06 0.85 0.92 0.91 0.93 0.81 0.84 0.83 0.82 2.10 2.21 2.25 2.22 1.50 1.51 1.52 1.53 1.16 1.30 1.30 1.28 1.21 1.10 1.19 1.19
4.60 4.85 4.76 4.65 5.76 5.39 5.30 5.83 6.23 6.43 6.34 6.28 7.54 7.34 7.12 7.21 4.44 4.54 4.42 4.32 5.68 5.76 5.88 5.97 6.12 6.43 6.49 6.35 7.65 7.34 7.48 7.57
2136.3 2252.1 2120.2 2067.9 2393.8 2352.7 2344.9 2688.2 2608.8 2905.1 2848.3 2863.2 3361.4 3427.6 3296.9 3289.0 1809.7 1960.7 1941.3 1862.6 1946.2 1947.0 2010.4 2013.7 1854.9 2184.4 2199.8 2120.7 2818.9 2473.1 2727.2 2766.6
6293.3 6308.1 6222.2 6190.9 5857.8 6115.7 6224.9 6300.2 6293.8 6206.1 6258.3 6386.3 6255.4 6319.6 6284.9 6207.0 6620.7 6102.8 6141.3 6294.6 6462.2 6213.0 5839.4 6023.7 6477.9 5764.4 6064.8 6132.7 5716.9 6444.1 6134.2 6078.6
0.2
0.3
0.4
500 × 50 × 100
0.1
0.2
0.3
0.4
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Table 6 Size-independent specific fracture energy of UHSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig, N/m
G⁎F, N/m
250 × 50 × 50
0.1
10,505 10,349 10,376 10,391 8155 8308 7900 8231 6844 6925 6694 6732 11,435 11,557 11,354 11,465 8889 8701 8613 8732 7145 7172 6887 6943 12,052 11,944 11,829 11,956 8076 8893 8155 8103 6965 6889 7085 6987
3.45 3.35 3.21 3.25 5.34 5.43 5.54 5.51 6.43 6.32 6.44 6.39 3.12 3.09 3.10 2.98 5.65 5.80 5.76 5.70 6.76 6.57 6.65 6.84 3.43 3.13 3.11 3.10 5.79 5.44 5.68 5.81 6.98 6.79 6.54 6.62
2.02 1.98 1.97 1.98 1.88 1.86 1.88 1.87 1.75 1.75 1.76 1.76 2.51 2.50 2.48 2.49 2.41 2.46 2.43 2.45 2.45 2.39 2.46 2.47 3.12 3.10 3.09 3.23 4.23 4.15 4.21 4.35 4.10 4.05 4.01 4.08
4.38 4.54 4.41 4.32 6.45 6.34 6.73 6.32 7.65 7.41 7.35 7.43 4.12 3.98 4.00 3.85 6.88 7.23 7.29 8.35 8.35 8.54 8.52 8.55 4.64 4.42 4.39 4.22 7.43 7.23 7.40 7.43 8.77 8.68 8.32 8.42
3516.9 3467.2 3336.7 3334.2 5531.9 5486.6 5762.2 5546.7 7043.0 6859.0 6937.4 6929.2 2521.7 2449.5 2404.5 2359.2 4721.9 5003.4 4959.9 6421.7 6421.7 6465.2 6666.2 6789.4 2155.4 2022.5 1977.8 2019.5 5381.2 5055.4 5290.7 5539.5 7095.3 6883.5 6550.0 6744.1
14,021.9 13,816.2 13,712.7 13,725.2 13,686.8 13,794.7 13,662.2 13,777.7 13,887.0 13,784.5 13,631.4 13,661.2 13,956.7 14,006.5 13,758.5 13,824.2 13,610.8 13,704.4 13,572.9 13,566.7 13,566.7 13,637.2 13,553.2 13,732.4 14,207.4 13,946.5 13,806.8 13,975.5 13,457.2 13,948.4 13,445.7 13,642.6 14,060.3 13,772.5 13,634.9 13,731.1
0.2
0.3
400 × 50 × 80
0.1
0.2
0.3
650 × 50 × 130
0.1
0.2
0.3
Table 7 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of NSC. Beam dimensions, mm
Notch to depth ratio
Mean fracture energy, Gf, as per RILEM, N/m
Size independent fracture energy, GF, N/m
Transition ligament length, al, mm
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
117.3 85.3 57.8 135.3 107.5 95.5 153.0 126.3 109.3
188.2
34.5
500 × 50 × 100
1000 × 50 × 200
± ± ± ± ± ± ± ± ±
3.29% 9.03% 15.44% 4.73% 10.23% 4.87% 5.57% 10.91% 4.38%
Table 9 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of UHSC. Mean fracture Notch to depth energy, Gf, N/m ratio
Beam dimensions, mm 250 × 50 × 50
190.3
68.3
192.3
95.4
0.1 0.2 0.3 0.4 400 × 50 × 80 0.1 0.2 0.3 0.4 650 × 50 × 130 0.1 0.2 0.3 0.4
10,405.3 8148.5 6798.8 4406.3 11,452.8 8733.8 7036.8 4976.8 11,945.3 8306.8 6981.5 5961.5
± ± ± ± ± ± ± ± ± ± ± ±
Size independent Transition ligament fracture energy, length, al, mm GF, N/m
0.66% 14,103.1 2.17% 1.55% 2.67% 0.73% 14,212.8 1.32% 2.03% 1.98% 0.76% 14,238.6 4.72% 1.16% 1.78%
24.3
41.2
54.2
Table 8 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of HSC. Beam dimensions, mm
Notch to depth ratio
Mean fracture energy, Gf, N/m
Size independent fracture energy, GF, N/m
Transition ligament length, al, mm
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
4109.5 3679.8 3479.8 2923.0 4396.3 4155.3 4020.0 3397.0
6385.2
27.3
500 × 50 × 100
± ± ± ± ± ± ± ±
1.03% 4.91% 4.71% 1.54% 6.91% 7.22% 10.95% 13.01%
6401.0
Table 10 Comparison of size-independent specific fracture energy (N/m). Mix
RILEM Gf corrected for P–δ tail segment, G⁎F
Boundary Effect Method (BEM), GF
% diff compared to BEM
NSC HSC UHSC
184.5 ± 2.49% 6194.7 ± 3.25% 13,760.7 ± 1.26%
190.3 ± 1.08% 6393.1 ± 0.17% 14,184.8 ± 0.51%
3.0 3.1 3.0
51.2
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
with thanks the valuable technical suggestions and support provided by his colleagues, Dr. G.S. Palani, Mr S. Maheshwaran, Ms. Smitha Gopinath, V. Ramesh Kumar, and B. Bhuvaneswari, and by the staff of Advanced Materials Laboratory, CSIR-SERC during the course of the experimental investigation. His contribution to this paper is being published with the permission of the Director, CSIR-SERC, Chennai, India. References [1] RILEM 50-FMC, Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams, Mater. Struct. 18 (1985) 287–290. [2] Z. Pan, Fracture properties of geopolymer paste and concrete, Mag. Concr. Res. 63 (2011) 763–771. [3] Y. Sahin, F. Koksal, The influences of matrix and steel fibre tensile strengths on the fracture energy of high strength concrete, Constr. Build. Mater. 25 (2011) 1801–1806. [4] Z.P. Bazant, M.T. Kazemi, Size dependence of concrete fracture energy determined by RILEM work-of-fracture method, Int. J. Fract. 51 (1991) 121–138. [5] Z.P. Bazant, Analysis of work-of-fracture method for measuring fracture energy of concrete, ASCE J. Mater. Civil Eng. 122 (1996) 138–144. [6] P. Nallathambi, B.L. Karihaloo, B.S. Heaton, Various size effects in fracture of concrete, Cem. Concr. Res. 15 (1985) 117–126. [7] A. Carpinteri, B. Chiaia, Size effects on concrete fracture energy: dimensional transition from order to disorder, Mater. Struct. 29 (1996) 259–266. [8] X. Hu, F. Wittmann, Fracture energy and fracture process zone, Mater. Struct. 25 (1992) 319–326. [9] S. Mindess, The effect of specimen size on the fracture energy of concrete, Cem. Concr. Res. 14 (1984) 431–436. [10] M. Elices, G.V. Guinea, J. Planas, Measurement of the fracture energy using three-point bend tests: part 3—influence of cutting the P–δ tail, Mater. Struct. 25 (1992) 137–163. [11] G.V. Guinea, J. Planas, M. Elices, Measurement of the fracture energy using three-point bend tests: part 1—influence of experimental procedures, Mater. Struct. 25 (1992) 212–218. [12] J. Planas, M. Elices, G.V. Guinea, Measurement of the fracture energy using three-point bend tests: part 2—influence of bulk energy dissipation, Mater. Struct. 25 (1992) 305–312.
25
[13] X. Hu, F. Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65 (2000) 209–221. [14] B.L. Karihaloo, H.M. Abdalla, T. Imjai, A simple method for determining the true specific fracture energy of concrete, Mag. Concr. Res. 55 (2003) 471–481. [15] H.M. Abdalla, B.L. Karihaloo, Determination of size-independent specific fracture energy of concrete from three-point bend and wedge splitting tests, Mag. Concr. Res. 55 (2003) 133–141. [16] H. Cifuentes, M. Alcalde, F. Medina, Measuring the size independent fracture energy of concrete, Strain 49 (2013) 54–59. [17] X. Hu, Influence of fracture process zone height on fracture energy of concrete, Cem. Concr. Res. 34 (2004) 1321–1330. [18] H. Yuan, J. Chen, F.H. Wittmann, On size dependence of ductile material failure as predicted by a non-local damage model, in: H. Yuan, F.H. Wittmann (Eds.), Proc. Int. Workshop Non-local Modeling of Failure of Materials, Aedificatio Publishers, Freiburg, 2007, pp. 27–44. [19] K. Duan, X.Z. Hu, F.H. Wittmann, Size effect on specific fracture energy of concrete, Eng. Fract. Mech. 74 (2007) 87–96. [20] S. Muralidhara, B.K. Raghu Prasad, H. Eskandari, B.L. Karihaloo, Fracture process zone size and true fracture energy of plain concrete from acoustic emission catalogue, Constr. Build. Mater. 24 (2010) 479–486. [21] X. Hu, K. Duan, Size effect: influence of proximity of fracture process zone to specimen boundary, Eng. Fract. Mech. 74 (2007) 1093–1100. [22] A. Ramachandra Murthy, Fatigue and fracture behaviour of ultra high strength concrete beams. PhD thesis, Indian Institute of Science, Bangalore, India, 2011. [23] V. Vydra, K. Trtık, F. Vodak, Size independent fracture energy of concrete, Constr. Build. Mater. 26 (2012) 357–361. [24] Z. Zhao, S. Kwon, S.P. Shah, Effect of specimen size on fracture energy and softening curve of concrete: part I. Experiments and fracture energy, Cem. Concr. Res. 38 (2008) 1049–1060. [25] H.M. Abdalla, B.L. Karihaloo, A method for constructing the bilinear tension softening diagram of concrete corresponding to its true fracture energy, Mag. Concr. Res. 56 (2004) 597–604. [26] S. Kwon, Z. Zhao, S.P. Shah, Effect of specimen size on fracture energy and softening curve of concrete: part II. Inverse analysis and softening curve, Cem. Concr. Res. 38 (2008) 1061–1069.
Contents lists available at SciVerse ScienceDirect
Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp
Determination of size-independent specific fracture energy of concrete mixes by two methods A. Ramachandra Murthy a, B.L. Karihaloo b,⁎, Nagesh R. Iyer a, B.K. Raghu Prasad c a b c
CSIR Structural Engineering Research Centre, Chennai 600113, India School of Engineering, Cardiff University, Cardiff CF24 3AA, UK Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Article history: Received 28 November 2012 Accepted 21 March 2013 Keywords: Toughness (C) Fiber reinforcement (E) High-performance concrete (E) Boundary effect (C)
a b s t r a c t The RILEM work-of-fracture method for measuring the specific fracture energy of concrete from notched three-point bend specimens is still the most common method used throughout the world, despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed in the literature to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a size-independent value (GF). It has also been proved recently, on the basis of a limited set of results on a single concrete mix with a compressive strength of 37 MPa, that when the size-dependent Gf measured by the RILEM method is corrected following either of these two methods, the resulting specific fracture energy GF is very nearly the same and independent of the size of the specimen. In this paper, we will provide further evidence in support of this important conclusion using extensive independent test results of three different concrete mixes ranging in compressive strength from 57 to 122 MPa. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The specific fracture energy of concrete is one of the most important properties required in the analysis of the mechanical behavior of cracked concrete structures. The work-of-fracture method recommended by RILEM [1] for measuring the specific fracture energy of concrete from notched three-point bend specimens of different sizes and notch to depth ratios is still the most common method used throughout the world [2,3], despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen [4–9]. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed by Elices and co-workers [10–12] and by Hu and Wittmann [13] to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a size-independent value (GF). Elices and co-workers [10–12] identified the sources of experimental error in the RILEM method and proposed a methodology for eliminating the major source of error, namely by including the work-of-fracture that is not measured in the RILEM method due to practical difficulties in capturing the tail part of the load–deflection plot. The second method proposed by Hu and Wittmann [13] recognized that the local specific energy varied during the propagation of a crack, the variation becoming more pronounced as the crack approached the stress-free back face boundary of the specimen. ⁎ Corresponding author. E-mail address: [email protected] (B.L. Karihaloo). 0008-8846/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cemconres.2013.03.015
Karihaloo et al. [14] and Abdalla and Karihaloo [15] extended the free boundary effect concept of Hu and Wittmann [13] and showed that the same size-independent specific fracture energy can also be obtained by testing only two specimens of the same size but with notches which are well separated, for example the notch to depth ratios in three-point bend specimens should be 0.05 and 0.5 or 0.1 and 0.6. Their method greatly reduces the number of test specimens and simplifies the determination of GF. In a recent paper, the first of its kind, Cifuentes et al. [16] showed that if the size-dependent Gf measured by the RILEM method is corrected following the methods of Elices and co-workers [10–12] and Hu and Wittmann [13] (or its simplification proposed by Karihaloo et al. [14], if the notch to depth ratios are well separated), then the resulting specific fracture energy GF is very nearly the same and independent of the size of the specimen. They reached this important conclusion on the basis of a limited set of results on a single concrete mix with a compressive strength of 37 MPa. In this paper, we will provide further evidence in support of this important conclusion using extensive independent test results of three different concrete mixes ranging in compressive strength from 57 to 122 MPa. 1.1. Boundary effect method (BEM) Hu and Wittmann [13] observed that the effect of the stress-free back boundary of the specimen is felt in the fracture process zone (FPZ) ahead of a real growing crack. The local fracture energy varies with the width of
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A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
the fracture process zone. As the crack approaches the back stress-free face of the specimen the fracture process zone becomes more and more confined and hence the local fracture energy decreases [13]. Initially, when the crack grows from a pre-existing notch, the rate of decrease is negligible, but it accelerates as the crack approaches the end of the un-cracked ligament. This change in the local fracture energy (gf) is approximated by a bilinear function, as shown in Fig. 1. The transition from horizontal line to the sharply inclined line occurs at the transition ligament length, which depends on the material properties and specimen size and shape [17–19]. In the boundary effect model of Hu and Wittmann [13], the measured RILEM fracture energy, Gf, may be regarded as the average of the local fracture energy function (dotted line in Fig. 1) over the initial un-cracked ligament area. The relationship between all the involved variables is given by D−a
∫ g f ðxÞdx Gf ða; DÞ ¼
0
D−a
¼
8 > > > < GF 1− > > > :
GF
al =D ; 2ð1−a=DÞ
2ð1−a=DÞ ; 2al =D
1 a=D > al =D ð1Þ
1 a=D ≤ al =D:
In Eq. (1), D is the total depth of the specimen, a is the initial notch depth and al is the transition ligament length. This bi-linear variation in the local fracture energy is supported by independent acoustic emission measurements during the crack growth [20]. To obtain the values of GF and al of a concrete mix, the size-dependent specific fracture energy Gf of specimens of different sizes and a range of the notch to depth ratios is first determined by the RILEM work-offracture method. Then Eq. (1) is applied to each specimen depth and notch to depth ratio. This gives an over-determined system of equations which is solved by a least squares method to obtain the best estimates of GF and al. Hu and Duan [21] showed that although the measured values of Gf depend on D and a/D, the above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth. The simplified boundary effect method proposed by Karihaloo et al. [14] greatly reduces the number of test specimens and eliminates the least squares solution of the over-determined system of equations. It requires the testing of only two specimens of the same size but with notches which are well separated, for example the notch to depth ratios in three-point bend specimens should be 0.05 and 0.5 or 0.1 and 0.6. 1.2. Method proposed by Elices et al. (P–δ tail) Elices et al. [10], Guinea et al. [11], and Planas et al. [12] identified several sources of energy dissipation that influence the measurement
Fig. 1. Local fracture energy model of Hu and Wittmann [13].
of Gf by the RILEM work-of-fracture method. Most of these sources are due to experimental errors which affect the measured value of the work-of-fracture during the test. They classified these sources into three main groups: (i) the testing equipment and experimental setup, (ii) the energy dissipation in the specimen bulk, and (iii) the non-measured energy corresponding to the un-recorded tail of the load–deflection (P–δ) curve near the end of the test. The first source of error can be avoided with a proper calibration of the equipment and design of supports and loading system. The second source is due to crushing at supports and under the loading point, and due to high tensile stresses in the specimen bulk. The crushing dissipation can be corrected by adjusting the initial stiffness of the P–δ curve and measuring the mid-span deflection on the lower half of the beam depth. The dissipation in the specimen bulk, Wdb, due to high tensile stresses cannot be avoided; however, it was estimated that this causes an error of less than 2% [10]. The last source of error is introduced by the curtailment of the tail of the P–δ near the end of the test due to practical difficulties of unloading the cracked specimen fully in a stable manner and it has the most significant effect on the size dependency of the measured fracture energy. To estimate this non-measured energy when the test is stopped (Wnm = Wnm1 + Wnm2 in Fig. 2) at a very low load, it is necessary to model the beam behavior when the crack is close to the free surface [10]. Once the non-measured work of fracture has been estimated, the size-independent specific fracture energy of concrete can be calculated as [10]: δu
∫ Pdδ þ W nm GF ¼
0
bðD−aÞ
ð2Þ
where b(D − a) is the area of the ligament (Alig) of the test specimen that was un-cracked at the start of the test, and b is the width of the specimen. 2. Experimental investigation Prismatic notched specimens were subjected to three-point bending in accordance with the RILEM procedure [1]. Table 1 gives the geometrical dimensions of all test specimens [22]. The materials and mix proportions used in the normal strength (NSC), high strength concrete (HSC), and the ultra high strength concrete (UHSC) are given in Table 2. The specimen preparation was strictly controlled to minimize the scatter in the test results. The NSC specimens were demolded after 1 day and cured in a water tank at ambient temperature for 28 days. The HSC and UHSC specimens were also demolded after 1 day and immersed in water at ambient temperature for 2 days. They were then placed in an autoclave at 90 °C for 2 days and in an oven at 200 °C for 1 day. Thereafter they were air cooled for 6 h and placed in water at ambient
Fig. 2. P–δ curve in a three-point bend test and the measured (Wf) and non-measured (Wnm1 + Wnm2) work of fracture [16].
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25 Table 1 Geometrical properties of the notched specimens.
21
Table 3 Mechanical properties of NSC, HSC and UHSC.
Mix
Beam dimensions (mm) (length × width × depth)
Notch to depth ratio
Mix
Cylinder compressive strength (MPa)
Split tensile strength (MPa)
Modulus of elasticity (GPa)
NSC
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
NSC HSC UHSC
57.1 87.7 122.5
4.0 15.4 20.7
35.8 37.9 43.0
500 × 50 × 100
1000 × 50 × 200
HSC
250 × 50 × 50
500 × 50 × 100
UHSC
250 × 50 × 50
400 × 50 × 80
650 × 50 × 130
3. Results 3.1. Adjustment of the P–δ tail Tables 4–6 show the NSC, HSC and UHSC specimen results. The size-independent specific fracture energy (G⁎F) has been obtained using Eq. (2). The tables also give the vertical displacement at the end of the test δu, the size-dependent specific fracture energy determined by RILEM work of fracture method Gf (i.e. Eq. (2) without the correction term for P–δ tail), the load P′ corresponding to δu, and the non-measured work-of-fracture Wnm. δ is the displacement corresponding to the fully unloaded broken specimen. In the present calculations the tail region of the P–δ curve was assumed to vary linearly from to P′ to zero load and δ was computed using the known slope of the P–δ at δu from the recorded readings. From Tables 4–6, it can be clearly seen that despite the large variation in the measured Gf with the specimen size and notch depth, the G⁎F values are almost the same for a particular mix i.e. NSC, HSC or UHSC irrespective of specimen size and notch depth. We have used here G⁎F only with a view to distinguish this sizeindependent specific fracture energy from that obtained by the boundary effect procedure, denoted by GF. We shall however see that the two values are practically the same.
temperature for a further 1 day before testing. Compression and split tensile tests were carried out on cylindrical specimens of 150 × 300 mm (diameter × height) in the case of NSC and on smaller cylinders 75 × 150 mm in the case of HSC and UHSC. Table 3 gives the mechanical properties of the three mixes. Notches (width approximately 3 mm) of various depths given in Table 1 were cut in beam specimens using a diamond saw. Four identical specimens for each beam size and each notch depth were cast. All tests were performed in a closed-loop servo-hydraulic testing machine, controlled by the crack mouth opening displacement (CMOD) measured with a clip gauge. A linearly varying displacement transducer (LVDT) was used to measure the mid-span vertical displacement, δ. The load–CMOD and load–displacement curves of all specimens were recorded. The ratio of the span between the supports to the depth of the specimen was maintained at 4 for all specimens. Fig. 3 shows typical load–deflection plots for NSC, HSC and UHSC mix.
3.2. Boundary effect method Tables 7–9 show the mean values of the specific fracture energy obtained from the three-point bend tests of NSC, HSC and UHSC specimens according to the RILEM procedure (the individual specimen results are given in Tables 4–6). An inspection of the entries in column 2 of Tables 7–9 highlights the dependency of the RILEM specific fracture energy on the notch depth or the size of the un-cracked ligament. The specific fracture energy increases with an increase in the beam depth for the same notch to depth ratio and it decreases with an increase in the notch to depth ratio for the same beam depth. Tables 7–9 also give the size-independent specific fracture energy (GF) and transition ligament length al obtained by solving the over-determined system of equations obtained by substituting Gf(a/D;D) from Tables 7–9 for each
Table 2 Mix proportions by mass (except for steel fiber which is by volume) of NSC, HSC and UHSC. Mix
Cement
Fine aggregate
Coarse aggregate
Silica fume
Quartz sand
Quartz powder
Steel fiber by vol. (length = 13 mm dia. = 0.18 mm)
w/c
SP, %
NSC HSC UHSC
1 1 1
1.25 – –
2.48 – –
– 0.25 0.25
– 1.5 1.1
– – 0.4
– 2% 2%
0.45 0.33 0.23
– 2.5 3.5
The maximum sizes of the aggregates used for the preparation of the above mixes are as follows: NSC Fine aggregate = 600 μm Coarse aggregate = 12 mm HSC Quartz sand = 600 μm UHSC Quartz powder = 75 μm Quartz sand = 600 μm.
22
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
practice. It might appear at first sight that these two methods use very different experimental procedures, but in reality they are closely interrelated. Both procedures apply some corrections to the final part of the P–δ diagram in the work-of-fracture test [23]. The local fracture energy model considers the influence when the crack approaches the back-face free boundary surface of the specimen towards the end of the test [24], whereas the method of Elices et al. [10] consists in determining the non-measured work-of-fracture by adjusting the tail of the P–δ curve that also corresponds to the final part of the test.
(a) NSC – 500x50x100mm 4.50 4.00
0.1d 0.2d 0.3d
Load, kN
3.50 3.00 2.50 2.00 1.50
4. Conclusions
1.00 0.50 0.00 0
0.1
0.2
0.3
0.4
0.5
0.6
Displacement, mm
(b) HSC – 250x50x50mm 4.50
0.1d 0.2d 0.3d 0.4d
4.00
Load, kN
3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
3
4
5
6
Displacement, mm
(c) UHSC – 250x50x50mm
Fig. 3. Typical load–deflection plots for NSC, HSC and UHSC.
D into Eq. (1) of boundary effect method. It is important to note that the solution of this over-determined system by the method of least squares may require the explicit imposition of the inequality constraint on al/D that appears in Eq. (1) in order to obtain the best estimates of GF and al. It should also be noted that the simplified boundary effect method proposed by Karihaloo et al. [14] cannot be used on the above test specimens because the notch to depth ratios are closely spaced and not well separated, as required by the simplified BEM. From Tables 7–9, it is clear that GF values are almost the same for a particular mix i.e. NSC, HSC or UHSC irrespective of specimen size and notch depth. Table 10 compares the mean size-independent specific fracture energy obtained by using two main methods. It is clear that both methods give nearly identical mean size-independent specific fracture energies of NSC, HSC and UHSC, so that either of them can be used in
From the above analysis of the extensive experimental data on three grades of concrete ranging in compressive strength from 57 to 122 MPa the following conclusions can be drawn. • In common with all previous investigations it has been found that the specific fracture energy Gf measured using the RILEM work-of-fracture procedure is highly dependent on the size of the specimen and the notch to depth ratio. For the NSC used in the present investigation it varied between 57.8 N/m for a specimen with D = 50 mm and a/D = 0.3 to 153.0 N/m for a specimen with D = 200 mm and a/D = 0.1 (Table 7). For the HSC (Table 8) it varied between 2923.0 N/m (D = 50 mm, a/D = 0.4) and 4396.3 N/m (D = 100 mm, a/D = 0.1) and for the UHSC (Table 9) between 4406.3 N/m (D = 50 mm, a/D = 0.4) and 11,945.3 N/m (D = 130 mm, a/D = 0.1). • The correction of the size-dependent Gf by the procedure proposed by Elices and his co-workers [10–12] or by the boundary effect procedure of Hu and Wittmann [13] result in nearly the same size-independent G⁎F or GF irrespective of the size of the specimen and the notch to depth ratio. For the NSC, G⁎F is 184.5 N/m and GF is 190.3 N/m (Table 10). For HSE the corresponding values are 6194.7 N/m and 6393.1 N/m and for UHSC the values are 13,760.7 N/m and 14,184.8 N/m. The maximum difference between G⁎F and GF in these three mixes is 3.1% (Table 10). Thus either of the two procedures may be used in practice. • In fact, in practice the simplified boundary effect procedure of Karihaloo et al. [14] may be followed. This procedure requires very few test specimens of the same size, say eight, four of which have a very small starter crack (notch to depth ratio 0.1) and the remaining four a deep starter crack (notch to depth ratio 0.6). The size-dependent Gf measured by the RILEM work-of-fracture method on these specimens of the same depth and two widely separated notch to depth ratios is substituted in Eq. (2) and the resulting two equations solved in the two unknowns GF and al. This eliminates the need for solving an over-determined system of equations by the least squares method. It is clear that the notch to depth ratios (0.1 to 0.4) used in the specimens of the present study (Table 1) did not meet the requirement of wide separation. Karihaloo et al. [14] have also shown that the size-independent specific fracture energy GF of a concrete mix is also independent of the shape of the test specimen (notched three-point bend or wedge splitting) and not only its size and notch to depth ratio. Finally, it should be added that for the analysis of a cracked concrete structure by the non-linear fictitious (or cohesive) crack model, it is not only necessary to know the (size-independent) specific fracture energy GF of the concrete mix, but also its cohesion–separation (tension softening) diagram. This diagram is commonly approximated by a bilinear function whose various parameters (i.e. the tensile strength, the critical crack opening, and the co-ordinates of the transition point) are inferred in an inverse manner. Details of this inverse parameter identification procedure can be found in [25,26]. Acknowledgments The first author's visit to Cardiff University is funded by Raman Research Fellowship awarded to him by CSIR, India. He acknowledges
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
23
Table 4 Size-independent specific fracture energy of NSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig N/m
G⁎F, N/m
250 × 50 × 50
0.1
115 123 115 116 91 87 74 89 47 55 68 61 136 144 130 131 105 93 115 117 96 89 100 97 165 146 148 153 110 135 140 120 115 111 104 107
0.09 0.09 0.09 0.09 0.17 0.17 0.19 0.18 0.26 0.25 0.23 0.24 0.37 0.33 0.35 0.36 0.41 0.44 0.38 0.37 0.58 0.56 0.55 0.56 0.22 0.24 0.22 0.21 0.35 0.32 0.30 0.34 0.42 0.44 0.42 0.44
1.51 1.47 1.48 1.47 0.75 0.77 0.79 0.76 0.51 0.51 0.49 0.50 0.54 0.50 0.53 0.52 0.52 0.54 0.51 0.51 0.40 0.43 0.41 0.42 1.10 1.22 1.22 1.20 1.10 0.96 0.96 1.02 1.00 0.95 1.00 0.97
0.12 0.12 0.12 0.12 0.33 0.33 0.36 0.34 0.63 0.63 0.58 0.60 0.51 0.49 0.49 0.54 0.76 0.78 0.75 0.76 0.96 0.96 0.94 0.94 0.30 0.32 0.32 0.31 0.64 0.58 0.54 0.62 0.65 0.64 0.62 0.65
70.5 66.3 69.1 66.3 93.8 96.3 108.6 98.8 127.1 128.2 113.4 120.0 52.6 45.6 49.8 52.8 76.1 82.4 72.0 72.0 88.0 93.4 87.3 90.0 31.8 37.9 36.6 34.3 68.1 54.0 50.4 61.2 76.7 73.3 74.3 75.5
185.5 189.3 184.1 182.3 184.8 183.3 182.6 187.8 174.1 183.2 181.4 181.0 188.6 189.6 179.8 183.8 181.1 175.4 187.0 189.0 184.0 182.4 187.3 187.0 196.8 183.9 184.6 187.3 178.1 189.0 190.4 181.2 191.7 184.3 178.3 182.5
0.2
0.3
500 × 50 × 100
0.1
0.2
0.3
1000 × 50 × 200
0.1
0.2
0.3
Table 5 Size-independent specific fracture energy of HSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig, N/m
G⁎F, N/m
250 × 50 × 50
0.1
4157 4056 4102 4123 3464 3763 3880 3612 3685 3301 3410 3523 2894 2892 2988 2918 4811 4142 4200 4432 4516 4266 3829 4010 4623 3580 3865 4012 2898 3971 3407 3312
3.41 3.45 3.10 3.21 4.21 4.22 4.17 4.31 4.51 4.62 4.61 4.56 4.88 4.86 4.76 4.82 3.32 3.43 3.33 3.23 4.68 4.53 4.70 4.56 5.11 5.34 5.37 5.29 6.32 6.12 6.22 6.38
1.20 1.22 1.21 1.18 0.96 0.98 0.99 1.06 0.85 0.92 0.91 0.93 0.81 0.84 0.83 0.82 2.10 2.21 2.25 2.22 1.50 1.51 1.52 1.53 1.16 1.30 1.30 1.28 1.21 1.10 1.19 1.19
4.60 4.85 4.76 4.65 5.76 5.39 5.30 5.83 6.23 6.43 6.34 6.28 7.54 7.34 7.12 7.21 4.44 4.54 4.42 4.32 5.68 5.76 5.88 5.97 6.12 6.43 6.49 6.35 7.65 7.34 7.48 7.57
2136.3 2252.1 2120.2 2067.9 2393.8 2352.7 2344.9 2688.2 2608.8 2905.1 2848.3 2863.2 3361.4 3427.6 3296.9 3289.0 1809.7 1960.7 1941.3 1862.6 1946.2 1947.0 2010.4 2013.7 1854.9 2184.4 2199.8 2120.7 2818.9 2473.1 2727.2 2766.6
6293.3 6308.1 6222.2 6190.9 5857.8 6115.7 6224.9 6300.2 6293.8 6206.1 6258.3 6386.3 6255.4 6319.6 6284.9 6207.0 6620.7 6102.8 6141.3 6294.6 6462.2 6213.0 5839.4 6023.7 6477.9 5764.4 6064.8 6132.7 5716.9 6444.1 6134.2 6078.6
0.2
0.3
0.4
500 × 50 × 100
0.1
0.2
0.3
0.4
24
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
Table 6 Size-independent specific fracture energy of UHSC considering P–δ tail segment. Beam dimensions, mm
Notch to depth ratio
Gf as per RILEM, N/m
δu, mm
Load (P′) corresponding to δu, kN
Computed δ, mm
Wnm/Alig, N/m
G⁎F, N/m
250 × 50 × 50
0.1
10,505 10,349 10,376 10,391 8155 8308 7900 8231 6844 6925 6694 6732 11,435 11,557 11,354 11,465 8889 8701 8613 8732 7145 7172 6887 6943 12,052 11,944 11,829 11,956 8076 8893 8155 8103 6965 6889 7085 6987
3.45 3.35 3.21 3.25 5.34 5.43 5.54 5.51 6.43 6.32 6.44 6.39 3.12 3.09 3.10 2.98 5.65 5.80 5.76 5.70 6.76 6.57 6.65 6.84 3.43 3.13 3.11 3.10 5.79 5.44 5.68 5.81 6.98 6.79 6.54 6.62
2.02 1.98 1.97 1.98 1.88 1.86 1.88 1.87 1.75 1.75 1.76 1.76 2.51 2.50 2.48 2.49 2.41 2.46 2.43 2.45 2.45 2.39 2.46 2.47 3.12 3.10 3.09 3.23 4.23 4.15 4.21 4.35 4.10 4.05 4.01 4.08
4.38 4.54 4.41 4.32 6.45 6.34 6.73 6.32 7.65 7.41 7.35 7.43 4.12 3.98 4.00 3.85 6.88 7.23 7.29 8.35 8.35 8.54 8.52 8.55 4.64 4.42 4.39 4.22 7.43 7.23 7.40 7.43 8.77 8.68 8.32 8.42
3516.9 3467.2 3336.7 3334.2 5531.9 5486.6 5762.2 5546.7 7043.0 6859.0 6937.4 6929.2 2521.7 2449.5 2404.5 2359.2 4721.9 5003.4 4959.9 6421.7 6421.7 6465.2 6666.2 6789.4 2155.4 2022.5 1977.8 2019.5 5381.2 5055.4 5290.7 5539.5 7095.3 6883.5 6550.0 6744.1
14,021.9 13,816.2 13,712.7 13,725.2 13,686.8 13,794.7 13,662.2 13,777.7 13,887.0 13,784.5 13,631.4 13,661.2 13,956.7 14,006.5 13,758.5 13,824.2 13,610.8 13,704.4 13,572.9 13,566.7 13,566.7 13,637.2 13,553.2 13,732.4 14,207.4 13,946.5 13,806.8 13,975.5 13,457.2 13,948.4 13,445.7 13,642.6 14,060.3 13,772.5 13,634.9 13,731.1
0.2
0.3
400 × 50 × 80
0.1
0.2
0.3
650 × 50 × 130
0.1
0.2
0.3
Table 7 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of NSC. Beam dimensions, mm
Notch to depth ratio
Mean fracture energy, Gf, as per RILEM, N/m
Size independent fracture energy, GF, N/m
Transition ligament length, al, mm
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
117.3 85.3 57.8 135.3 107.5 95.5 153.0 126.3 109.3
188.2
34.5
500 × 50 × 100
1000 × 50 × 200
± ± ± ± ± ± ± ± ±
3.29% 9.03% 15.44% 4.73% 10.23% 4.87% 5.57% 10.91% 4.38%
Table 9 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of UHSC. Mean fracture Notch to depth energy, Gf, N/m ratio
Beam dimensions, mm 250 × 50 × 50
190.3
68.3
192.3
95.4
0.1 0.2 0.3 0.4 400 × 50 × 80 0.1 0.2 0.3 0.4 650 × 50 × 130 0.1 0.2 0.3 0.4
10,405.3 8148.5 6798.8 4406.3 11,452.8 8733.8 7036.8 4976.8 11,945.3 8306.8 6981.5 5961.5
± ± ± ± ± ± ± ± ± ± ± ±
Size independent Transition ligament fracture energy, length, al, mm GF, N/m
0.66% 14,103.1 2.17% 1.55% 2.67% 0.73% 14,212.8 1.32% 2.03% 1.98% 0.76% 14,238.6 4.72% 1.16% 1.78%
24.3
41.2
54.2
Table 8 Mean fracture energy and coefficient of variation, and size independent specific fracture energy of HSC. Beam dimensions, mm
Notch to depth ratio
Mean fracture energy, Gf, N/m
Size independent fracture energy, GF, N/m
Transition ligament length, al, mm
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
4109.5 3679.8 3479.8 2923.0 4396.3 4155.3 4020.0 3397.0
6385.2
27.3
500 × 50 × 100
± ± ± ± ± ± ± ±
1.03% 4.91% 4.71% 1.54% 6.91% 7.22% 10.95% 13.01%
6401.0
Table 10 Comparison of size-independent specific fracture energy (N/m). Mix
RILEM Gf corrected for P–δ tail segment, G⁎F
Boundary Effect Method (BEM), GF
% diff compared to BEM
NSC HSC UHSC
184.5 ± 2.49% 6194.7 ± 3.25% 13,760.7 ± 1.26%
190.3 ± 1.08% 6393.1 ± 0.17% 14,184.8 ± 0.51%
3.0 3.1 3.0
51.2
A. Ramachandra Murthy et al. / Cement and Concrete Research 50 (2013) 19–25
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