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Determination of size-independent specific fracture energy of concrete mixes by the tri-linear model
Cement and Concrete Research 49 (2013) 82–88
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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 the tri-linear model B.L. Karihaloo a,⁎, A. Ramachandra Murthy b, Nagesh R. Iyer b a b
School of Engineering, Cardiff University, Cardiff CF24 3AA, UK CSIR Structural Engineering Research Centre, Chennai 600113, India
a r t i c l e
i n f o
Article history: Received 10 January 2013 Accepted 13 March 2013 Keywords: Fiber reinforcement (E) High-performance concrete (E) Toughness (C)
a b s t r a c t The methods proposed by Elices and co-workers [1–3] and by Hu and Wittmann [4] are commonly used to determine the size-independent specific fracture energy (GF) of concrete by correcting the size-dependent specific fracture energy (Gf) measured by the RILEM work-of-fracture method. In the boundary effect model of Hu and Wittmann [4], the change in the local fracture energy (gf) is approximated by a bilinear function, whereas the method of Elices et al. [1] consists in determining the non-measured work-of-fracture by adjusting the tail of the P-δ curve that corresponds to the final part of the test. Acoustic emission (AE) experiments on notched specimens (Muralidhara et al. [5,6]) have revealed that under loading the AE events follow approximately a tri-linear distribution; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen, reminiscent of the development of R-curve in a finite size specimen. This paper exploits this observation and proposes a tri-linear model for the determination of the size-independent specific fracture energy for three different concrete mixes ranging in compressive strength from 57 to 122 MPa. Remarkably, it is found that the resulting size-independent specific fracture energy GF determined by this tri-linear model and by the bi-linear model of Hu and Wittmann [4] is very nearly the same and independent of the size of the specimen. © 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 [7] 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 [8,9], despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen [10–15]. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed by Elices and co-workers [1–3] and by Hu and Wittmann [4] to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a sizeindependent value (GF). Elices and co-workers [1–3] 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
⁎ 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.2013.03.010
plot. The second method proposed by Hu and Wittmann [4] 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. Karihaloo et al. [16] and Abdalla and Karihaloo [17] extended the free boundary effect concept of Hu and Wittmann [4] 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. The present authors have recently carried out extensive studies to determine the size-independent specific fracture energy by using the boundary effect method and the RILEM work of fracture method with P-δ tail correction for three different concrete mixes ranging in compressive strength from 57 to 122 MPa [18]. The acoustic emission (AE) technique has been applied to the study of the fracture of concrete, with a focus on the properties of crack extension during the fracture process [19–22]. In recent AE experiments on notched concrete specimens, Muralidhara et al. [5,6] observed that under loading the AE events follow approximately a tri-linear distribution, reminiscent of the development of R-curve in a finite size specimen; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen. This paper exploits this observation and improves on the tri-linear model proposed by Muralidhara [6] for the determination
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of the size-independent specific fracture energy for three different concrete mixes ranging in compressive strength from 57 to 122 MPa. Remarkably, it is found that the resulting size-independent specific fracture energy GF determined by this tri-linear model and by the bi-linear model of Hu and Wittmann is very nearly the same and independent of the size of the specimen. 1.1. Boundary effect method (BEM) Hu and Wittmann [14] 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 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 [4]. 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 [23]. In the boundary effect model of Hu and Wittmann [4], 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 W−a
∫ gf ðxÞdx Gf ða; W Þ ¼
¼
0
8 W−a " > > > > GF 1− < > > > > :
al =W
#
; 1 a=W > a1 =W 2ð1−a=W Þ 2ð1−a=W Þ GF ; 1 a=W≤a1 =W 2al =W
ð1Þ
in which Gf is the specific fracture energy or size dependent fracture energy (RILEM), GF is the true or size-independent fracture energy,
Fig. 2. Histogram of acoustic emission events along the ligament length ahead of the notch tip [6].
W is the overall depth of the beam, a is the initial notch depth and a⁎l is the transition ligament length. To obtain the values of GF and a⁎l of a concrete mix, the sizedependent specific fracture energy Gf of the specimens of different sizes and a range of the notch to depth ratios is first determined by the RILEM work-of-fracture 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 a⁎l . Hu and Duan [24] showed that although the measured values of Gf depend on W and a/W, the above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth. 1.2. Tri-linear model It is known that the fracture process zone (FPZ) ahead of a crack tip in concrete is due to the formation of micro-cracks, some of which later link up to form a macro-crack. The formation and growth of micro-cracks are associated with the release of elastic strain waves as energy waves called the acoustic emission (AE) waves. Studies by Colombo et al. [25] have shown that micro-cracks emit waves with smaller amplitudes, whereas the waves from macro-cracks have larger amplitudes. Pioneering work by Landis [26], established the relationship between the AE events and the evolution of fracture process in concrete. The AE monitoring system can record the energy released during the fracture process in concrete and in case no material attenuation is considered, the recorded AE energy is almost equal to the fracture energy of concrete. It was noted that there
W
a
Fig. 1. Variation of local fracture energy gf and GF over the ligament length [17].
b* l
c* l
a* l
Fig. 3. Tri-linear model showing the trend of gf variation over the un-notched ligament length [6].
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Table 1 Geometrical properties of the notched specimens.
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
The tri-linear model is essentially an extension of the bi-linear model of Hu and Wittmann [4] but it allows for the rise in the local fracture energy at the beginning of the crack growth from the starter notch. It contains three variables, namely the size-dependent fracture energy (Gf), a⁎l and b⁎l . Muralidhara et al. [6] determined size-independent fracture energy by substituting the values of a⁎l and b⁎l estimated from the AE data in Eq. (2). Although, their tri-linear model represents the real physical behavior of crack growth in a notched specimen, it requires AE experimental data in order to find size-independent specific fracture energy GF, thus limiting its practical usefulness. In this paper, we propose an improved tri-linear model in the spirit of the bi-linear model to compute the size-independent fracture energy (GF), and the ligament lengths a⁎l and b⁎l in order to overcome this serious limitation. To obtain the values of GF, a⁎l and b⁎l of a concrete mix, the sizedependent 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-of-fracture method. Then Eq. (2) 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, a⁎l and b⁎l .
is a correlation between AE energy and fracture energy [27,28]. It was observed in the AE experiments conducted recently [5,6,28,29] on notched three-point bend specimens that the histogram of AE events along the ligament at the notch tip is reminiscent of the R-curve behavior (Fig. 2) of a finite size specimen; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen. Muralidhara et al. [6] approximated the AE histogram by a tri-linear variation of local fracture energy, as shown in Fig. 3. The local fracture energy gf reduces from a constant GF whilst approaching both the back boundary and the notch tip. From AE experiments it has also been found that the transition ligament length b⁎l is smaller than a⁎l [6]. Muralidhara et al. [6] proposed a tri-linear model based on AE data along the ligament length ahead of the notch tip (Fig. 3) to determine the size-independent fracture energy GF. The relationship between local fracture energy and the size-independent fracture energy GF for the tri-linear model is given in Eq. (2).
2. Experimental investigation Prismatic notched specimens were subjected to three-point bending in accordance with the RILEM procedure [7]. Table 1 gives the geometrical dimensions of all test specimens [30]. 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 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 gage. 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
bl =W for ð1 a=WÞ≥b1 =W 2ð1−a=W Þ a =W þ bl =W for ð1 a=WÞ≥a1 =W Gf ¼ GF 1− l 2ð1 a=WÞ Gf ¼ GF
ð2Þ
in which b⁎l and a⁎l are the transition ligament lengths near the notch tip and near the back boundary respectively and a is the starter notch depth.
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
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Table 4 Mean fracture energy and size independent specific fracture energy of NSC. Beam dimensions, mm
Notch to depth ratio Mean fracture energy, Gf, N/m Size independent fracture energy, GF, N/m Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
500 × 50 × 100
1000 × 50 × 200
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 117.25 85.25 57.75 135.25 107.50 95.50 153.00 126.25 109.25
192.68
31.63
1.02
194.28
63.5
2.11
193.28
94.2
recorded. The ratio of the span between the supports to the depth of the specimen was maintained at 4 for all specimens. 3. Results Tables 4–6 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. An inspection of the entries in column 3 of Tables 4–6 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 4–6 also give the size-independent specific fracture energy (GF) and transition ligament lengths a⁎l and b⁎l obtained by solving the over-determined system of equations resulting from the substitution of Gf(a/W;W) from Tables 4–6 for each W into Eq. (2) of the tri-linear model. 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 constraints on a⁎l /W
24.3
12.35 7.35 2.35 24.39 14.39 4.39 61.5 41.5 21.5
34.5
68.3
95.4
and b⁎l /W that appear in Eq. (2) in order to obtain the best estimates of GF and a⁎l and b⁎l . It should also be noted that the simplified boundary effect method proposed by Karihaloo et al. [16] 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 4–6, 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. Tables 4–6 also contain the parameters such as a⁎l , b⁎l and c⁎l of the tri-linear model (Fig. 3). It can be observed that for a particular mix and a beam size, the values of a⁎l , b⁎l are constant whereas the value of c⁎l decreases with an increase in the notch depth, i.e. a decrease in the un-notched ligament. In other words, the window where the local fracture energy remains constant narrows as the un-notched ligament size decreases. For example, for the case of NSC, W = 50 mm, the values of a⁎l , b⁎l are 31.63 mm and 1.02 mm and c⁎l values decreasing from 12.35 mm (a/W = 0.1) to 2.35 mm (a/W = 0.3). It can also be observed from Tables 4–6 that for a particular mix, the values of a⁎l , b⁎l increase with increasing beam size. For example, for the case of NSC, the values of a⁎l , b⁎l increase from 31.63 mm and 1.02 mm (W = 50 mm) to 94.2 mm and
Table 5 Mean fracture energy 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 Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
500 × 50 × 100
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 4109.50 3679.75 3479.75 2923.00 4396.25 4155.25 4020.00 3397.00
6432.43
24.12
3.67
6426.93
49.03
5.81
17.21 12.21 7.21 2.21 35.16 25.16 15.16 5.16
27.3
51.2
Table 6 Mean fracture energy size independent specific fracture energy of UHSC. Beam dimensions, mm
Notch to depth ratio Mean fracture energy, Gf, N/m Size independent fracture energy, GF, N/m Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
400 × 50 × 80
650 × 50 × 130
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 10,405.25 8148.50 6798.75 4406.25 11,452.75 8733.75 7036.75 4976.75 11,945.25 8306.75 6981.5 5961.5
14,221.3
22.69
3.86
14,278.60
37.69
5.02
14,263.9
52.7
11.23
18.45 13.45 8.45 3.45 29.29 21.29 13.29 5.29 53.07 40.07 27.03 14.07
24.30
41.20
54.20
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24.3 mm (W = 200 mm). Fig. 4 shows typical variations of the local fracture energy with the size of the specimen and notch depth for several concrete mixes. It can be seen that (i) window of constant fracture energy is very narrow in small size specimens especially with deeper starter notches (notch to depth ratio 0.3 for the case of NSC and 0.4 for the case of HSC or UHSC); and (ii) the influence of the stress free boundary is profound in specimens of all sizes and all notches;
it dominates the influence of the rising part of the local energy variation. Table 7 compares the mean size-independent specific fracture energy obtained by using the bi-linear boundary effect model of Hu and Wittmann [14] with that obtained by the proposed tri-linear model. The size-independent specific fracture energy (GF) according to the bi-linear model is obtained by solving the over-determined
Fig. 4. Schematic representation of local fracture energy variation for different specimen sizes, a/W ratios and concrete mixes (all dimensions in mm).
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Table 7 Comparison of size-independent specific fracture energy (N/m). Mix
NSC HSC UHSC
Size independent specific fracture energy GF, N/m Tri-linear
Bi-linear
193.4 ± 0.42% 6429.7 ± 0.06% 14,254.6 ± 0.21%
190.3 ± 1.08% 6393.1 ± 0.17% 14,184.8 ± 0.51%
system by the method of least squares may require the explicit imposition of the inequality constraint on a⁎l /W that appears in Eq. (1) in order to obtain the best estimates of GF and a⁎l . From Table 7, it can be observed that the mean size-independent specific fracture energies of NSC, HSC and UHSC obtained using the bi-linear and tri-linear models are nearly the same. This is not surprising in view of the above observation, namely that the effect of the stress free boundary dominates the variation in the local fracture energy over the un-notched ligament. This is further confirmed by the fact that the transition ligament size a⁎l is nearly the same in both the bi-linear and tri-linear models (refer Tables 4–6). Thus whilst the tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, the bi-linear model of Hu and Wittmann [14] or its simplified version proposed by Karihaloo et al. [16] is quite adequate in practice to determine the size-independent fracture energy of concrete mixes, especially if small notched specimens are used in which b⁎l is very small, i.e. ascending branch of the local fracture energy variation over the un-notched ligament is nearly vertical (see Fig. 4a,b). 4. Conclusions 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 workof-fracture procedure is highly dependent on the size of the specimen and the notch to depth ratio. • An improved tri-linear model has been proposed to determine the size-independent specific fracture energy from the size-dependent RILEM fracture energy. The tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, as evidenced by acoustic emission data [5,6]. • The size-independent specific fracture energy (GF) obtained by the improved tri-linear model and the bi-linear model of Hu and Wittmann [4] resulted in nearly the same value irrespective of the size of the specimen and the notch to depth ratio. This is due to the fact that the effect of the stress free boundary dominates the variation in the local fracture energy over the un-notched ligament. As a result, the transition ligament size a⁎l is nearly the same in both the bi-linear and tri-linear models (Tables 4–6). • Whilst the tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, the bi-linear model of Hu and Wittmann [14] or its simplified version proposed by Karihaloo et al. [16] is quite adequate in practice to determine the size-independent fracture energy of concrete mixes, especially if small notched specimens are used in which b⁎l is very small, i.e. ascending branch of the local fracture energy variation over the un-notched ligament is nearly vertical (see Fig. 4a,b). Fig. 4 (continued).
system of equations resulting from the substitution of Gf(a/W;W) from Tables 4–6 for each W into Eq. (1) of boundary effect method. It is important to note that the solution of this over-determined
Acknowledgments The second author's visit to Cardiff University is funded by Raman Research Fellowship awarded to him by CSIR, India. He acknowledges with thanks the valuable technical suggestions and support
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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] 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. [2] 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. [3] 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. [4] X. Hu, F. Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65 (2000) 209–221. [5] 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. [6] S. Muralidhara, B.K. Raghu Prasad, B.L. Karihaloo, R.K. Singh, Size independent fracture energy in plain concrete beams using tri-linear model, Constr. Build. Mater. 25 (2011) 3051–3058. [7] RILEM TCM-85, 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. [8] Z. Pan, Fracture properties of geopolymer paste and concrete, Mag. Concr. Res. 63 (2011) 763–771. [9] 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. [10] 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. [11] Z.P. Bazant, Analysis of work-of-fracture method for measuring fracture energy of concrete, ASCE J. Mater. Civ. Eng. 122 (1996) 138–144. [12] P. Nallathambi, B.L. Karihaloo, B.S. Heaton, Various size effects in fracture of concrete, Cem. Concr. Res. 15 (1985) 117–126. [13] A. Carpinteri, B. Chiaia, Size effects on concrete fracture energy: dimensional transition from order to disorder, Mater. Struct. 29 (1996) 259–266.
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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 the tri-linear model B.L. Karihaloo a,⁎, A. Ramachandra Murthy b, Nagesh R. Iyer b a b
School of Engineering, Cardiff University, Cardiff CF24 3AA, UK CSIR Structural Engineering Research Centre, Chennai 600113, India
a r t i c l e
i n f o
Article history: Received 10 January 2013 Accepted 13 March 2013 Keywords: Fiber reinforcement (E) High-performance concrete (E) Toughness (C)
a b s t r a c t The methods proposed by Elices and co-workers [1–3] and by Hu and Wittmann [4] are commonly used to determine the size-independent specific fracture energy (GF) of concrete by correcting the size-dependent specific fracture energy (Gf) measured by the RILEM work-of-fracture method. In the boundary effect model of Hu and Wittmann [4], the change in the local fracture energy (gf) is approximated by a bilinear function, whereas the method of Elices et al. [1] consists in determining the non-measured work-of-fracture by adjusting the tail of the P-δ curve that corresponds to the final part of the test. Acoustic emission (AE) experiments on notched specimens (Muralidhara et al. [5,6]) have revealed that under loading the AE events follow approximately a tri-linear distribution; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen, reminiscent of the development of R-curve in a finite size specimen. This paper exploits this observation and proposes a tri-linear model for the determination of the size-independent specific fracture energy for three different concrete mixes ranging in compressive strength from 57 to 122 MPa. Remarkably, it is found that the resulting size-independent specific fracture energy GF determined by this tri-linear model and by the bi-linear model of Hu and Wittmann [4] is very nearly the same and independent of the size of the specimen. © 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 [7] 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 [8,9], despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen [10–15]. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed by Elices and co-workers [1–3] and by Hu and Wittmann [4] to correct the measured size-dependent specific fracture energy (Gf) in order to obtain a sizeindependent value (GF). Elices and co-workers [1–3] 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
⁎ 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.2013.03.010
plot. The second method proposed by Hu and Wittmann [4] 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. Karihaloo et al. [16] and Abdalla and Karihaloo [17] extended the free boundary effect concept of Hu and Wittmann [4] 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. The present authors have recently carried out extensive studies to determine the size-independent specific fracture energy by using the boundary effect method and the RILEM work of fracture method with P-δ tail correction for three different concrete mixes ranging in compressive strength from 57 to 122 MPa [18]. The acoustic emission (AE) technique has been applied to the study of the fracture of concrete, with a focus on the properties of crack extension during the fracture process [19–22]. In recent AE experiments on notched concrete specimens, Muralidhara et al. [5,6] observed that under loading the AE events follow approximately a tri-linear distribution, reminiscent of the development of R-curve in a finite size specimen; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen. This paper exploits this observation and improves on the tri-linear model proposed by Muralidhara [6] for the determination
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of the size-independent specific fracture energy for three different concrete mixes ranging in compressive strength from 57 to 122 MPa. Remarkably, it is found that the resulting size-independent specific fracture energy GF determined by this tri-linear model and by the bi-linear model of Hu and Wittmann is very nearly the same and independent of the size of the specimen. 1.1. Boundary effect method (BEM) Hu and Wittmann [14] 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 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 [4]. 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 [23]. In the boundary effect model of Hu and Wittmann [4], 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 W−a
∫ gf ðxÞdx Gf ða; W Þ ¼
¼
0
8 W−a " > > > > GF 1− < > > > > :
al =W
#
; 1 a=W > a1 =W 2ð1−a=W Þ 2ð1−a=W Þ GF ; 1 a=W≤a1 =W 2al =W
ð1Þ
in which Gf is the specific fracture energy or size dependent fracture energy (RILEM), GF is the true or size-independent fracture energy,
Fig. 2. Histogram of acoustic emission events along the ligament length ahead of the notch tip [6].
W is the overall depth of the beam, a is the initial notch depth and a⁎l is the transition ligament length. To obtain the values of GF and a⁎l of a concrete mix, the sizedependent specific fracture energy Gf of the specimens of different sizes and a range of the notch to depth ratios is first determined by the RILEM work-of-fracture 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 a⁎l . Hu and Duan [24] showed that although the measured values of Gf depend on W and a/W, the above procedure indeed leads to a GF value that is essentially independent of the specimen size and relative notch depth. 1.2. Tri-linear model It is known that the fracture process zone (FPZ) ahead of a crack tip in concrete is due to the formation of micro-cracks, some of which later link up to form a macro-crack. The formation and growth of micro-cracks are associated with the release of elastic strain waves as energy waves called the acoustic emission (AE) waves. Studies by Colombo et al. [25] have shown that micro-cracks emit waves with smaller amplitudes, whereas the waves from macro-cracks have larger amplitudes. Pioneering work by Landis [26], established the relationship between the AE events and the evolution of fracture process in concrete. The AE monitoring system can record the energy released during the fracture process in concrete and in case no material attenuation is considered, the recorded AE energy is almost equal to the fracture energy of concrete. It was noted that there
W
a
Fig. 1. Variation of local fracture energy gf and GF over the ligament length [17].
b* l
c* l
a* l
Fig. 3. Tri-linear model showing the trend of gf variation over the un-notched ligament length [6].
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Table 1 Geometrical properties of the notched specimens.
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
The tri-linear model is essentially an extension of the bi-linear model of Hu and Wittmann [4] but it allows for the rise in the local fracture energy at the beginning of the crack growth from the starter notch. It contains three variables, namely the size-dependent fracture energy (Gf), a⁎l and b⁎l . Muralidhara et al. [6] determined size-independent fracture energy by substituting the values of a⁎l and b⁎l estimated from the AE data in Eq. (2). Although, their tri-linear model represents the real physical behavior of crack growth in a notched specimen, it requires AE experimental data in order to find size-independent specific fracture energy GF, thus limiting its practical usefulness. In this paper, we propose an improved tri-linear model in the spirit of the bi-linear model to compute the size-independent fracture energy (GF), and the ligament lengths a⁎l and b⁎l in order to overcome this serious limitation. To obtain the values of GF, a⁎l and b⁎l of a concrete mix, the sizedependent 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-of-fracture method. Then Eq. (2) 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, a⁎l and b⁎l .
is a correlation between AE energy and fracture energy [27,28]. It was observed in the AE experiments conducted recently [5,6,28,29] on notched three-point bend specimens that the histogram of AE events along the ligament at the notch tip is reminiscent of the R-curve behavior (Fig. 2) of a finite size specimen; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen. Muralidhara et al. [6] approximated the AE histogram by a tri-linear variation of local fracture energy, as shown in Fig. 3. The local fracture energy gf reduces from a constant GF whilst approaching both the back boundary and the notch tip. From AE experiments it has also been found that the transition ligament length b⁎l is smaller than a⁎l [6]. Muralidhara et al. [6] proposed a tri-linear model based on AE data along the ligament length ahead of the notch tip (Fig. 3) to determine the size-independent fracture energy GF. The relationship between local fracture energy and the size-independent fracture energy GF for the tri-linear model is given in Eq. (2).
2. Experimental investigation Prismatic notched specimens were subjected to three-point bending in accordance with the RILEM procedure [7]. Table 1 gives the geometrical dimensions of all test specimens [30]. 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 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 gage. 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
bl =W for ð1 a=WÞ≥b1 =W 2ð1−a=W Þ a =W þ bl =W for ð1 a=WÞ≥a1 =W Gf ¼ GF 1− l 2ð1 a=WÞ Gf ¼ GF
ð2Þ
in which b⁎l and a⁎l are the transition ligament lengths near the notch tip and near the back boundary respectively and a is the starter notch depth.
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
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Table 4 Mean fracture energy and size independent specific fracture energy of NSC. Beam dimensions, mm
Notch to depth ratio Mean fracture energy, Gf, N/m Size independent fracture energy, GF, N/m Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
500 × 50 × 100
1000 × 50 × 200
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 117.25 85.25 57.75 135.25 107.50 95.50 153.00 126.25 109.25
192.68
31.63
1.02
194.28
63.5
2.11
193.28
94.2
recorded. The ratio of the span between the supports to the depth of the specimen was maintained at 4 for all specimens. 3. Results Tables 4–6 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. An inspection of the entries in column 3 of Tables 4–6 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 4–6 also give the size-independent specific fracture energy (GF) and transition ligament lengths a⁎l and b⁎l obtained by solving the over-determined system of equations resulting from the substitution of Gf(a/W;W) from Tables 4–6 for each W into Eq. (2) of the tri-linear model. 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 constraints on a⁎l /W
24.3
12.35 7.35 2.35 24.39 14.39 4.39 61.5 41.5 21.5
34.5
68.3
95.4
and b⁎l /W that appear in Eq. (2) in order to obtain the best estimates of GF and a⁎l and b⁎l . It should also be noted that the simplified boundary effect method proposed by Karihaloo et al. [16] 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 4–6, 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. Tables 4–6 also contain the parameters such as a⁎l , b⁎l and c⁎l of the tri-linear model (Fig. 3). It can be observed that for a particular mix and a beam size, the values of a⁎l , b⁎l are constant whereas the value of c⁎l decreases with an increase in the notch depth, i.e. a decrease in the un-notched ligament. In other words, the window where the local fracture energy remains constant narrows as the un-notched ligament size decreases. For example, for the case of NSC, W = 50 mm, the values of a⁎l , b⁎l are 31.63 mm and 1.02 mm and c⁎l values decreasing from 12.35 mm (a/W = 0.1) to 2.35 mm (a/W = 0.3). It can also be observed from Tables 4–6 that for a particular mix, the values of a⁎l , b⁎l increase with increasing beam size. For example, for the case of NSC, the values of a⁎l , b⁎l increase from 31.63 mm and 1.02 mm (W = 50 mm) to 94.2 mm and
Table 5 Mean fracture energy 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 Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
500 × 50 × 100
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 4109.50 3679.75 3479.75 2923.00 4396.25 4155.25 4020.00 3397.00
6432.43
24.12
3.67
6426.93
49.03
5.81
17.21 12.21 7.21 2.21 35.16 25.16 15.16 5.16
27.3
51.2
Table 6 Mean fracture energy size independent specific fracture energy of UHSC. Beam dimensions, mm
Notch to depth ratio Mean fracture energy, Gf, N/m Size independent fracture energy, GF, N/m Model parameters
250 × 50 × 50
0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
400 × 50 × 80
650 × 50 × 130
a⁎l of bi-linear model, mm
a⁎l , mm b⁎l , mm c⁎l , mm 10,405.25 8148.50 6798.75 4406.25 11,452.75 8733.75 7036.75 4976.75 11,945.25 8306.75 6981.5 5961.5
14,221.3
22.69
3.86
14,278.60
37.69
5.02
14,263.9
52.7
11.23
18.45 13.45 8.45 3.45 29.29 21.29 13.29 5.29 53.07 40.07 27.03 14.07
24.30
41.20
54.20
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24.3 mm (W = 200 mm). Fig. 4 shows typical variations of the local fracture energy with the size of the specimen and notch depth for several concrete mixes. It can be seen that (i) window of constant fracture energy is very narrow in small size specimens especially with deeper starter notches (notch to depth ratio 0.3 for the case of NSC and 0.4 for the case of HSC or UHSC); and (ii) the influence of the stress free boundary is profound in specimens of all sizes and all notches;
it dominates the influence of the rising part of the local energy variation. Table 7 compares the mean size-independent specific fracture energy obtained by using the bi-linear boundary effect model of Hu and Wittmann [14] with that obtained by the proposed tri-linear model. The size-independent specific fracture energy (GF) according to the bi-linear model is obtained by solving the over-determined
Fig. 4. Schematic representation of local fracture energy variation for different specimen sizes, a/W ratios and concrete mixes (all dimensions in mm).
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Table 7 Comparison of size-independent specific fracture energy (N/m). Mix
NSC HSC UHSC
Size independent specific fracture energy GF, N/m Tri-linear
Bi-linear
193.4 ± 0.42% 6429.7 ± 0.06% 14,254.6 ± 0.21%
190.3 ± 1.08% 6393.1 ± 0.17% 14,184.8 ± 0.51%
system by the method of least squares may require the explicit imposition of the inequality constraint on a⁎l /W that appears in Eq. (1) in order to obtain the best estimates of GF and a⁎l . From Table 7, it can be observed that the mean size-independent specific fracture energies of NSC, HSC and UHSC obtained using the bi-linear and tri-linear models are nearly the same. This is not surprising in view of the above observation, namely that the effect of the stress free boundary dominates the variation in the local fracture energy over the un-notched ligament. This is further confirmed by the fact that the transition ligament size a⁎l is nearly the same in both the bi-linear and tri-linear models (refer Tables 4–6). Thus whilst the tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, the bi-linear model of Hu and Wittmann [14] or its simplified version proposed by Karihaloo et al. [16] is quite adequate in practice to determine the size-independent fracture energy of concrete mixes, especially if small notched specimens are used in which b⁎l is very small, i.e. ascending branch of the local fracture energy variation over the un-notched ligament is nearly vertical (see Fig. 4a,b). 4. Conclusions 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 workof-fracture procedure is highly dependent on the size of the specimen and the notch to depth ratio. • An improved tri-linear model has been proposed to determine the size-independent specific fracture energy from the size-dependent RILEM fracture energy. The tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, as evidenced by acoustic emission data [5,6]. • The size-independent specific fracture energy (GF) obtained by the improved tri-linear model and the bi-linear model of Hu and Wittmann [4] resulted in nearly the same value irrespective of the size of the specimen and the notch to depth ratio. This is due to the fact that the effect of the stress free boundary dominates the variation in the local fracture energy over the un-notched ligament. As a result, the transition ligament size a⁎l is nearly the same in both the bi-linear and tri-linear models (Tables 4–6). • Whilst the tri-linear model is closer to how the local fracture energy varies during crack growth in a notched concrete specimen, the bi-linear model of Hu and Wittmann [14] or its simplified version proposed by Karihaloo et al. [16] is quite adequate in practice to determine the size-independent fracture energy of concrete mixes, especially if small notched specimens are used in which b⁎l is very small, i.e. ascending branch of the local fracture energy variation over the un-notched ligament is nearly vertical (see Fig. 4a,b). Fig. 4 (continued).
system of equations resulting from the substitution of Gf(a/W;W) from Tables 4–6 for each W into Eq. (1) of boundary effect method. It is important to note that the solution of this over-determined
Acknowledgments The second author's visit to Cardiff University is funded by Raman Research Fellowship awarded to him by CSIR, India. He acknowledges with thanks the valuable technical suggestions and support
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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] 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. [2] 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. [3] 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. [4] X. Hu, F. Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65 (2000) 209–221. [5] 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. [6] S. Muralidhara, B.K. Raghu Prasad, B.L. Karihaloo, R.K. Singh, Size independent fracture energy in plain concrete beams using tri-linear model, Constr. Build. Mater. 25 (2011) 3051–3058. [7] RILEM TCM-85, 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. [8] Z. Pan, Fracture properties of geopolymer paste and concrete, Mag. Concr. Res. 63 (2011) 763–771. [9] 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. [10] 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. [11] Z.P. Bazant, Analysis of work-of-fracture method for measuring fracture energy of concrete, ASCE J. Mater. Civ. Eng. 122 (1996) 138–144. [12] P. Nallathambi, B.L. Karihaloo, B.S. Heaton, Various size effects in fracture of concrete, Cem. Concr. Res. 15 (1985) 117–126. [13] A. Carpinteri, B. Chiaia, Size effects on concrete fracture energy: dimensional transition from order to disorder, Mater. Struct. 29 (1996) 259–266.
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