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Size-independent fracture energy in plain concrete beams using tri-linear model
Construction and Building Materials 25 (2011) 3051–3058
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Size-independent fracture energy in plain concrete beams using tri-linear model S. Muralidhara a,b,⇑, B.K. Raghu Prasad b, B.L. Karihaloo c, R.K. Singh d a
BMS College of Engineering, Bangalore, India Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India c School of Engineering, Cardiff University, Queen’s Buildings, P.O. Box 925, Cardiff CF24 OYF, UK d Reactor Safety Division, BARC, Mumbai, India b
a r t i c l e
i n f o
Article history: Received 10 April 2010 Received in revised form 9 December 2010 Accepted 11 January 2011 Available online 18 February 2011 Keywords: Local fracture energy Tri-linear model Fictitious boundary Size-independent fracture energy Concrete
a b s t r a c t The boundary effect or the size effect on the fracture properties of concrete has been studied assuming a bi-linear model for the distribution of local fracture energy concept. Boundary effect is observed not only near the back boundary but also near the notch tip, where a fictitious boundary seems to exist, separating the linear and non-linear fracture zones. In this paper a tri-linear function is assumed for the local fracture energy distribution along the ligament and expressions relating RILEM fracture energy and the size-independent fracture energy are developed. Transition ligament length measurements based on the acoustic emission (AE) histogram of events are used to obtain size-independent fracture energy. Length of the fracture process zone is identified in the AE histogram and compared with the value obtained from softening beam model. There seems to be a good agreement between the results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Concrete, which is a quasi-brittle material, exhibits a post-peak softening behavior which lies between a brittle and a ductile material behavior. Unlike in metals, the plastic zone in concrete is a scatter of micro-cracks around the crack tip, called the process zone. In concrete, a solitary crack initiation and propagation is seldom seen and the energy is consumed in the formation of micro-cracks in the process zone. The parameter to characterize the fracture is the energy consumed per unit area of crack, also called the fracture energy. RILEM has quantified this parameter as specific fracture energy based on work of fracture. However this is beset with size effects. The size effects are being explained through different models. Bazant’s size effect law, based on the dimensional analysis of the fracture energy released during crack propagation in geometrically similar specimens, has been at the forefront to explain the size effect [1,2]. Size effect on the specific fracture energy has also been linked to the influence of ligament length on fracture process zone (FPZ). Small FPZ associated with smaller ligament length leads to lower specific fracture energy. Local fracture energy model based on the size/width of FPZ was developed by Hu and Duan [3], Duan et al. [4]. In this model, fracture dissipation has been assumed to be ⇑ Corresponding author. Present address: Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India. Tel.: +91 080 2293 2435; fax: +91 080 2360 0404. E-mail addresses: [email protected], [email protected] (S. Muralidhara). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.01.003
different at different locations along the crack path. This nonconstant local fracture energy dissipation has been shown to be the cause for the size effect in the fracture energy calculated according to RILEM FMC-50 [5]. 1.1. Review of boundary effect model As is well known, specific fracture energy is the energy required to create a crack of unit area. This parameter is useful to characterize the fracture process in concrete in conjunction with the tensile strength ft. RILEM has recommended an expression for the specific fracture energy based on the work of fracture. One important assumption here is that all the energy expended goes into crack initiation and propagation. The specific fracture energy is found to vary with specimen dimensions. If we use gf to indicate the specific fracture energy relevant to a small surface area along the crack path, the Gf definition given by RILEM has practically assumed that the local fracture energy gf is constant along the crack path. The localized specific fracture energy gf is generated at the crack tip region during crack growth after crack width w increases from 0 to wc. This process is repeated for a constant local fracture energy distribution along the crack path. This constant local fracture energy is construed as size independent and many researchers have tried to obtain this size-independent specific fracture energy using various models. The size effect model based on Bazant’s size effect law has been adopted extensively and the fracture parameters are obtained from this model. The effect of specimen size on fracture energy was studied using three-point bend test on beams by
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Mindess [6], who concluded that the self-weight of the beam has little effect on the specific fracture energy Gf even in large beams. The ligament area used in the calculation of RILEM Gf, is only the projected area of the fracture surface which underestimates the actual fracture area. Nallathambi and Karihaloo [7] have tried with modified models of FCM of Hillerborg and Blunt crack band model of Bazant, incorporating slow crack growth and tortuosity, but found that the specific fracture energy is still size dependent. Guinea et al. [8] and Elices et al. [9] calculated the energy expended for fracturing by excluding energy losses through friction near supports, crushing near supports and bulk energy dissipation in the compression zone from the measured work of fracture and obtained size-independent fracture energy. Bazˇant and Li [10] have proposed a new size effect model wherein only one size of the specimen instead of the conventional three sizes, is sufficient to determine fracture parameters. The size dependence of the RILEM fracture energy was explained by Guo and Gilbert [11] through the concept of partial fracture energy (Gf). The partial fracture energy depends on the width (Wp) of the fracture process zone (FPZ); the larger the width, dG the higher is the Gf and dWfp decreases with Wp. If the specimen is large enough to allow the formation of full FPZ, then the fracture energy is not size dependent. Research studies to correlate elastic and plastic fracture conditions in metals to those in concrete specimens have been pursued by Hu and Wittmann [12] to arrive at an asymptotic function in fracture energy and critical stress intensity factor. They have concluded that reference crack size or the characteristic length lch has implication in size effect and therefore in fracture toughness and fracture energy. A deterministic size effect formula was proposed by Bazˇant [2] which was based on the energy release due to boundary layer cracking. For large specimens, however, the formula was found to be inadequate. Attempts have been made to ascertain the effect of aggregate content on the fracture energy and fracture process zone size. Critical volume fraction of aggregates to obtain maximum fracture energy seems to be non-existent, which seems to be rather surprising. Darwin et al. [13] have tried to relate the fracture energy with the age of concrete and have concluded that fracture energy is almost constant as a function of water-cement ratio, age and compressive strength. Trunk and Wittmann [14] have presented a model to explain the nature of size and geometry dependence of specific fracture energy Gf. Further studies on the size dependence of fracture energy by Duan et al. [15] based on the asymptotic approach have contributed in identifying the influence of the ratio of process zone size over its distance from the free surface. Ever since the specific fracture energy was identified as a fracture parameter for the nonlinear fracture theory applied to concrete, earnest attempts are being made to isolate the true fracture energy parameter which is independent of size and shape of the test specimen. Karihaloo et al. [16] have been actively pursuing studies to determine sizeindependent specific fracture energy of concrete, adopting concepts of local fracture energy and boundary effect induced by free surface, introduced by Duan et al. [17], using beam and wedge splitting specimens. The averaged fracture energy Gf as recommended by RILEM is given by the following equation, based on the concept of work of fracture.
Gf ¼
1 BðW aÞ
Z
Pdd
GF ¼
Z
wc
rdw
ð2Þ
0
In which Wc is the critical crack opening and r is the cohesive stress. The FPZ width reduces while approaching the back boundary, which has lead to characterize local fracture energy dissipation pattern as a bi-linear function to make it simple to analyze. The bi-linear function consists of a horizontal straight line of the intrinsic fracture energy GF and a declining straight line that reduces to zero at the back boundary. Within the zone in which FPZ has constant width, the fracture energy is size independent while nearer to the outer boundary, the FPZ width decreases and hence the fracture energy also decreases. However, it is not still clear from all the previous investigations in literature, how one can conclude that constancy of fracture energy over the un-cracked ligament implies constancy over different sizes to call it size independent. The RILEM fracture energy considers both inner and outer zones for the calculation to get averaged fracture energy over the ligament length and hence is size dependent. The relationship between FPZ characteristics and the RILEM fracture energy is developed based on the variation of FPZ width along the ligament length. Process zone around a propagating crack is considered to have two regions, an inner softening region (Wsf) and an outer region (Wf) with micro-cracks as shown in Fig. 1. Wsf region comprises main crack with branches of larger cracks, which add to concrete softening. In the outer region the cracks are not inter-connected and do not account for softening. It is very obvious to expect higher fracture energy dissipation in the inner region than in the outer region. The widths Wsf and Wf vary depending on the stress field at the crack tip which apparently influence critical crack width Wc. As the crack approaches the back boundary of the specimen, the size of the ligament area diminishes, apparently reducing Wsf, Wf, Wc and finally the fracture energy. The local fracture energy gf is associated only to r–w relationship at a particular location. If r– w is unique then GF = gf = Constant. Another observation here is that the above fact in terms of Wsf and Wf is not experimentally validated. The local fracture energy gf is connected to Wsf, Wf, and Wc as given below
W sf aW f ðxÞ wc aW sf ðxÞ
ð3Þ
g f awc ðxÞ Fracture energy Gf and gf are related by the following expression:
ð1Þ
In which B is the thickness of the specimen, P is the applied load and d is the load point displacement. The fracture energy GF is related to cohesive stress–crack opening (r–w) curve by
Fig. 1. FPZ and bridging stresses. FPZ with inner softening zone and outer micro fracture zone [18].
S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058
g f ðaÞ ¼ Gf ðaÞ ðW aÞ
dGf ðaÞ da
ð4Þ
In which gf(a) is the local fracture energy at crack tip, and Gf(a) is the averaged fracture energy from a specimen with initial crack length of a. The boundary effect model of Duan et al. [18] has been able to account for the size effect of fracture energy through the back boundary effect assuming a bi-linear variation of local fracture energy gf over the un-notched ligament length (Fig. 2). According to the boundary effect concept, the dissipation of fracture energy decreases sharply when a crack is approaching to a free boundary but remains constant at locations away from a free boundary. Using a bi-linear energy distribution function to approximate the fracture energy distribution, it has been shown that the common size effect is, in fact, due to the influence of a free specimen boundary. The intersection of these two straight lines is defined as the transition ligament al , measured along the un-notched ligament. The relationship between local fracture energy gf and the size-independent fracture energy GF from the boundary effect model is developed considering the relationships given in Eqs. (5) and (6). However, the expressions relating specific fracture energy Gf (RILEM) and GF are more convenient for calculations.
Gf ðxÞ ¼ GF
for x < ðW a al Þ
ðW a xÞ g f ðxÞ ¼ GF for x P ðW a al Þ al al for ðW aÞ > al Gf ¼ GF 1 2ðW aÞ W a forðW aÞ 6 al Gf ¼ GF 2al
ð5Þ
ð6Þ
In which Gf is the specific fracture energy or size dependent fracture energy (RILEM), GF is the true or size-independent fracture energy, W is the overall depth of the beam, a is the initial notch depth and al is the transition ligament length. Hu and Duan [19] while comparing boundary effect model with SEL of Bazant [20], have emphasized the role of FPZ in different sizes of specimens. For a constant notch/depth ratio, in small or large specimens, FPZsmall < FPZLarge but
FPZsmall Dsmall
>
FPZLarge . DLarge
Hence they
claimed that the condition of geometric similarity is inapplicable to FPZ. Hence the size effect is the size of FPZ or in other words due to distance of FPZ from structure boundaries. Here again it seems that the fact is a foregone conclusion. Yu et al. [21] have compared boundary effect model with SEL, considering notched and un-notched beams and influence on fracture energy Gf, critical stress intensity factor KIC, tensile strength ft for different geometries. They have claimed that the boundary effect model has a few deficiencies and cannot be applied to all sizes. In fact the basic hypothesis that size effect is due to back boundary is questioned. It has been indicated that with in a certain parametric range of 0.15–0.6 in notch/depth and 0.1–0.9 in brittleness
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number, the values of KIC and Gf from boundary effect model and SEL match well. It has also been indicated that Gf obtained from boundary effect model is the initial fracture energy i.e. area under the initial tangent in the softening portion of r–x curve. In fact the value Gf from boundary effect model seems to be similar to the one from SEL, which has been claimed to be size independent. Abdalla and Karihaloo [22] through their studies could observe that the transition ligament length first increased with an increase in the specimen size but at a reducing rate and then stabilized. They also pointed out that testing large specimens is unnecessary and true fracture energy could be determined if the ligament length is greater than transition ligament length. Adopting regression analysis, based on the bi-linear boundary effect model, on a few set of data of beams with different depths and notch to depth ratio, and found the size-independent fracture energy and transition ligament lengths for different depths of beams. Finally they concluded that fracture energy is almost constant but the transition ligament length varies with the depth of the beam. Size effects on specific fracture energy have been studied and reported. The bi-linear variation of local fracture energy has been extended to specimens having larger thickness and there has been negligible effect on specimens with thickness larger than four times the maximum aggregate size, as reported by Duan et al. [18]. Shilang et al. [23], obtained two fracture energy quantities corresponding to stable and unstable crack growth by conducting wedge splitting tests and observed that the fracture energy during the stable crack growth remained constant and was size independent. Yanhua et al. [24] introduced two concepts viz. the back boundary affected length and overall boundary affected length, to explain non-uniform local fracture energy along the ligament length through tests on WS specimens. It was observed that the back boundary affected length decreases linearly with the crack evolution, whereas the overall boundary affected length increased at early stages but decreased later. 1.2. Acoustic emission technique In concrete, the fracture process zone ahead of a crack tip is the consequence of the formation of micro-cracks, a few of which later link up to form a macro-crack. The formation and growth of cracks is 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 works by Landis [26] have shed light on the relationship between the AE events and the evolution of fracture process in concrete. Formation of FPZ in mortar by AE techniques was studied by Keru Wu et al. [27]. Muralidhara et al. [28] have estimated the size of the fracture process zone based on the distribution of AE events and their energies and also the size-independent fracture energy using the bi-linear model with inputs from the AE event histograms. 1.3. Review of softening beam model
Fig. 2. Variation of local fracture energy gf and Gf over the ligament length [18].
Accurately modeling the fracture process and the analyzing the behavior of concrete structures has been the core of the research in concrete fracture. One such model is the one-dimensional model also called softening beam model proposed by Ananthan et al. [29]. Through this model fracture behavior of notched and unnotched plain concrete slender beams subjected to three-point or four-point bending is analyzed. Structural size effect in changing the fracture type from brittle to plastic collapse is explained through the stress distribution across the un-cracked ligament obtained by varying the strain softening modulus. It is shown that the length of the fracture process zone depends on the value of the
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strain softening modulus. In addition non-linear fracture parameters such as CTOD, CMOD and fracture energy are computed for a wide variety of beam specimens. The complete stress–crack opening relation is necessary for any model to be able to characterize fracture process. The fracture behavior is characterized by the slope of the postpeak softening portion in the stress–strain curve in tension. In softening beam model the post-peak behavior is idealized to be linear, as could be seen in Fig. 3, and the slope is termed as softening modulus ET. The ratio ET/E is E⁄ which is used to characterize the nature of the material failure. For example, for E⁄ = 1, the fracture is brittle and for E⁄ = 0, it is ductile. The geometrical parameters like depth, span/depth ratio and notch/depth ratio have a bearing on the nature of specimen fracture. With the backing of experimental studies, it has been shown that E⁄ varies between 0 and 1 for different geometrical dimensions of concrete specimens. Further, cohesive stress variation along the process zone length is assumed linear and non-linear variation of stress in the compression zone, as given in Fig. 4. At the onset of fracture instability, the cohesive stress at the beginning of the process zone reduces to zero while at the end of the process zone attains a maximum value equal to the tensile strength ft of concrete as shown in Fig. 5. Making use of the stress and cohesive stress variations as given in Figs. 4 and
Fig. 5. Stress diagram at the onset of fracture instability [29].
5, an expression relating the maximum process zone length LPmax and E⁄ is obtained [29] as
2 E LP max ¼ 1 E þ 1 ðW aÞ
ð7Þ
In which (W a) is the ligament length of the specimen. The results from the model have correlated reasonably well with values from the literature because of its simplicity. 1.4. Research significance and objectives
Fig. 3. Stress–strain diagram in concrete under tension [29].
Fracture energy has been the most suitable fracture parameter to characterize the fracture in concrete. The energy criterion based analysis suits well to concrete due to the non-linearity of mechanics associated with concrete. This is mainly due to the formation of fracture process zone, which is a scatter of micro-cracks around the notch/crack tip. However this fracture energy parameter is beset with size effect, which has made it a size dependent. Different models viz. Hillerborg’s cohesive crack model [30], Bazˇant’s size effect model Bazˇant [2], Karihaloo’s size effect model [31], Boundary effect model based on non-uniform energy distribution along the ligament length due to Duan and Wittmann [17], Carpinteri’s fractal concept for size effect [32] have been developed to determine size-independent fracture energy. The objective of the present study is to obtain size-independent fracture energy and the length of the fracture process zone. In this paper, the bi-linear boundary effect model is improved further to a tri-linear model based on the observations made in the acoustic emission data in front of the notch, on a plain concrete beam tested under three-point bend condition to estimate size-independent fracture energy. The trilinear model appears to showcase a more accurate energy distribution along the ligament length and facilitates a better estimate of size-independent fracture energy. From this model, a part of the ligament length has been interpreted as the fracture process zone length and compared with the one obtained using softening beam model. The study reaffirms the use of the AE techniques to obtain fracture parameters with out resorting to any regression analysis. 2. Experimental setup 2.1. Test setup
Fig. 4. Stress block for 1 > E⁄ > 0 [29].
Plain concrete single edged notched beam specimens of compressive strength 45 MPa were cast. The proportions of the concrete ingredients used (based on
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S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058 mix design using Indian standard code IS-10262 [33] (1982)) and the geometrical dimensions of the beams are as given in Tables 1 and 2. The beams were tested under three-point bending condition. In Table 2 the specimens are identified with symbols for e.g. D1P20UC01. D1 is the type of beam, P stands for pour mix, 20 for aggregate size as 20 mm, U for un-reinforced, C stands for notch/depth ratio of 0.33 and 01 is the sample number. T stands for trial mix and B for notch/depth ratio of 0.25. Trial mix and pour mix are of same compressive strength. Notch cutting in beams forms an important and delicate part of the specimen preparation. The notches were cut using concrete tile cutting machine with a blade of thickness 3 mm and diameter of 150 mm. The notch thus had a width of 3 mm at its tip, which is less than 10 mm as suggested by RILEM FMC-50. A 500kN capacity servo controlled DARTEC machine under crack mouth opening displacement (CMOD) control was employed. The central deflection of the beam was recorded by an LVDT which could measure up to 0.1 lm. The clip gauge for the measurement of CMOD had a resolution of 0.1lm. The test was performed keeping the CMOD rate at 0.0005 mm/s. The acquisition of loading and displacement parameters along with the acoustic emission data were simultaneous. 2.2. AE Monitoring Equipment The AE equipment used was from Physical Acoustic Corporation, Princeton, New Jersey, USA. Four sensors with a resonant frequency of 60 kHz were used for the AE acquisition and were arranged on one face of the specimen as shown in Fig. 6. The locations of events have the origin of reference at the bottom left corner of the specimen. The surface of specimen where the positions of the sensors were marked was initially cleaned with acetone and then the AE sensors were fixed to the specimen surface using special vacuum grease/gel. The sensors were initially tested for their sensitivity by pencil lead breaking. Further, automatic sensor testing (AST) available in the AE software was employed to check the fixity of the sensors to the concrete surface and to ensure that there is no gap between the sensor and the surface.
Fig. 7. Histogram of events along the ligament length at the notch from a D2 type beam.
gram (Fig. 7) along the ligament length. The number of times an AE event occurs (number of AE events) increases with increase in distance from the notch tip and stabilize over a certain distance. Later it reduces towards the back boundary. A similar declining trend could be seen towards the notch tip where of course there is no physical boundary. This could be explained by the tri-linear variation of local fracture energy as shown in Fig. 8. The local fracture energy gf reduces from a constant GF while approaching both the back boundary and the notch tip. Because the events die down near the notch tip one can imagine that a boundary like the back boundary exists. There and here after it could be called as fictitious boundary. The fictitious boundary separating linear and non-linear zones seems to be another source of the boundary effect. Transition ligament length bl is found to be smaller than al . The relationship between local fracture energy and the size-independent fracture energy GF developed from the tri-linear model is given in Eqs. (8) and (9).
2.3. Tri-linear model for local fracture energy distribution The fracture process zone size could be estimated using the event locations from AE catalogue. An interesting observation could be made from AE event histoTable 1 Quantity of materials per m3 of concrete. Property
Mix-with 20 mm and down size coarse aggregates
Mix-with 12.5 mm and down size coarse aggregates
Cement (kg) (C) Coarse aggregate (kg) 20 mm (CA) 12.5 mm Fine aggregate (kg) (FA) Water (kg) Super plasticizer (% weight of cement content) Water/cement ratio Mix proportion (C:FA:CA) 28 days compressive strength (N/mm2)
400 492 492 902 152 1.4
435 – 949 870 165 1.4
0.38 1:2.26:2.46 52.23
0.38 1:2:2.18 54.3
Table 2 Geometric properties of specimens. Type
Length (mm)
Depth (mm)
Width (mm)
Span (mm)
D1 D2
375 750
95 190
47.5 95
282 564
Y
Fig. 8. Tri-linear model showing the trend of gf variation over un-notched ligament length.
25 3
190 m m
60
100
S e n so rs 1
60
O rig in
375 m m
190 m m
2
4
N o tch 564 m m E L E V A T IO N
Fig. 6. Profile of the beam showing position of sensors.
X 95 m m S ID E V IE W
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g f ðxÞ ¼ GF
S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058
x bl
g f ðxÞ ¼ GF
for x 6 bl
for
bl
6 x < ðW a
ðW a xÞ g f ðxÞ ¼ GF al
al Þ
for x P ðW a al Þ
ð8Þ
In which bl and al are the transition ligament lengths near the notch tip and near the back boundary respectively of the beam specimen and a is the notch depth. The above relationship could be expressed in terms of specific fracture energy, size-independent fracture energy and transition ligament lengths bl and al as
bl for ðW aÞ 6 bl 2ðW aÞ 1 al bl for ðW aÞ P al Gf ¼ GF 2ðW aÞ Gf ¼ GF
ð9Þ
3. Results and discussion The pattern of AE event histogram was alike in all the specimens tested. Hence results from a sample beam D2T20UB02, of dimensions 750 mm 190 mm 95 mm with a notch to depth ratio of 0.25 D (47.5 mm) are analyzed and presented here. It is known that the number of AE events directly depends on the stress level in the material. With an increase in stress level, the number of events also increases and in the post-peak softening regime, the event number decreases as the stress reduces. During this period, the crack initiation and propagation takes place from the notch tip towards the specimen back boundary. From Fig. 7, it is observed that, the number of events reduces as the crack propagates towards the back boundary. The histogram clearly follows the energy trend shown by the boundary effect model. The length of the horizontal projection of the post-peak sloping portion of the histogram is the transition ligament length. It is also stated by Duan et al. [18] that gf and FPZ length/width rapidly decrease when the crack is approaching back boundary of the specimen. The histogram explicitly shows that FPZ width/length rather than FPZ length/width decreases rapidly when crack approaches back boundary. A critical observation of the AE event histogram indicates that the variation of local fracture energy across the ligament length in TPB specimens is more of tri-linear pattern than bi-linear. It also appears that a fictitious boundary similar to the back boundary is present near the notch tip. The number AE activities decline near the fictitious boundary giving the impression of a dual boundary effect. The Local fracture energy distribution is steeper near the notch tip than at the back boundary, which is perceptibly due to stress singularity at the crack tip. It also confirms the fictitious crack model where the stress at the crack tip decreases to zero at the time of initial collapse. It is also observed that the back boundary transition ligament length through AE histogram reasonably satisfied the boundary effect model.
Arguments over the validity of SEL and boundary effect model are becoming louder over the past couple of years. Recent discussions by Yu et al. [21] about the comparative studies between SEL and boundary effect model have disagreed some of the facts put forth in boundary effect model. In spite of wide disagreements, there have been commonalities in the values of some fracture parameters calculated from SEL and boundary effect model with in a certain range of geometric proportions. The value of KIC and Gf from both SEL and boundary effect model are found to be reasonably similar for notch/depth in the range of 0.15–0.6 and brittleness number 0.1–0.9. In the discussion, the fracture energy calculated based on boundary effect model is claimed to be equal to the area under the initial tangent of softening portion of the r–x curve. In fact the fracture energy calculated using SEL has been reported as size independent and compared with the area under initial tangent of post-peak softening portion of r–x curve. The above observation strongly confirms the fact that the fracture energy calculated using boundary effect model is size independent and of course with in the parametric range mentioned above. In the light of the above discussion on boundary effect model, the geometries of the concrete beam specimens used were checked for notch/depth ratio and brittleness number. The notch/depth ratio of the beam specimens tested for D1 type and D2 type was 0.25 and 0.33, and brittleness numbers [34] were 0.27–0.54. All the values calculated are well with in the range of validity. The validated geometric conditions reaffirm that the calculated fracture energy using boundary effect model is size independent. The RILEM fracture energy Gf obtained from TPB tests on D1 and D2 type beams with different notch lengths are as given in Table 3. The true specific fracture energy has been calculated using the Eq. (8), because the ligament length (W a) is greater than al . The RILEM fracture energy obtained from the tests, transition ligament lengths al and bl measured from the AE event histogram are used in Eq. (8) and the true fracture energy or size-independent fracture energy GF is calculated. Measured transition ligament lengths al and bl from AE histograms, and calculated size-independent fracture energy GF from both bi-linear model [28] and tri-linear models are as given in Table 3. Co-efficient of variation of specific fracture energy and the true fracture energy are obtained from statistical analysis of the results as given in Table 5. Bar-chart showing nearly constant fracture energy against RILEM fracture energy from both bi-linear model and tri-linear model are given in Fig. 10. The fracture energy variation along the ligament holds the trace about the length of fracture process zone. It is obvious to find the density of AE events to be more along the crack path than away from it. This reaffirms the fact that there are two portions in FPZ, one inner zone with micro- and macro-cracks and the other with micro-cracks called the outer zone. It is very difficult to demarcate these zones distinctly and hence the FPZ length. In the tri-linear model the length of FPZ (LP) is assumed as the distance between
Table 3 RILEM Gf and GF from bi-linear and tri-linear models of different specimens. Sl. no.
Specimen ID
Notch depth a (mm)
Beam depth W (mm)
Transition ligament length al (mm)
Transition ligament length bl (mm)
RILEM Gf (N/m)
Bi-linear GF (N/m)
Tri-linear GF (N/m)
1 2 3 4 5 6 7 8 9 10 11
D1T20UB01 D1P20UB01 D1P20UC01 D1T12.5UC01 D2P20UC02 D2T20UC01 D2T12.5UB03 D2T20UB02 D2T12.5UB02 D2T12.5UC03 D2T20UB03
23.8 24 31 29 66 67 49 49 50 68 49
95 95 95 95 190 190 190 190 190 190 190
37.5 10 30 27.5 55 75 50 60 50 24.5 55
6.2 10 9 6 22 21 31 28 25 10 25
151.9 190.9 157.5 154.5 153.7 129.5 165 147.2 167.4 179.2 159.7
206.2 205.4 205.7 195.2 197.5 186.3 200.6 187.0 203.8 199.2 198.4
219.2 222.2 226.5 207.0 222.9 212.4 231.5 214.0 228.6 208.7 222.9
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S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058 Table 4 Process zone length from tri-linear model and softening beam model. Sl. no.
Specimen ID
Effective ligament length LP (mm)
Average LP (mm)
Softening beam model Lpmax
Average Lpmax (mm)
W (mm)
LP/W
Average LP/W
1 2 3 4 5 6 7 8 9 10 11
D1T20UB01 D1P20UB01 D1P20UC01 D1T12.5UC01 D2P20UC02 D2T20UC01 D2T12.5UB03 D2T20UB02 D2T12.5UB02 D2T12.5UC03 D2T20UB03
42.1 57.0 38.0 43.7 71.3 57.7 87.0 80.3 86.7 97.7 87.7
45.2
51.0 56.9 45.0 51.0 83.9 57.8 93.6 75.2 86.0 84.0 83.3
51.0
95 95 95 95 190 190 190 190 190 190 190
0.44127 0.59926 0.39863 0.45847 0.37405 0.30179 0.45647 0.42121 0.45474 0.52382 0.46
0.47441
81.2
80.5
0.42744
Table 5 Results from statistical analysis of RILEM Gf and GF from bi-linear and tri-linear models. Fracture energy
Mean value
Standard deviation
Co-efficient of variation (COV)
Gf (N/m) GF (bi-linear model) (N/m) GF (tri-linear model) (N/m)
159.7 198.7 219.6
16.2 6.9 8.1
0.102 0.035 0.037
the centroids of triangles near the notch tip and the back boundary in the histogram of AE events (Fig. 9). It may seem proper to assume the length of the central portion cl , over which the distribution of fracture energy is almost uniform, as equivalent to FPZ length but recognizing the transition zones on either side it is suitable to assume LP as the FPZ length. Softening beam model is used to determine FPZ length taking into account the geometries of the beams tested. The values of (lFPZ)max calculated using Eq. (7) for the beams with maximum loads from the experiments and the geometries as the input to the model are given in Table 4. The values of FPZ lengths for the beams from tri-linear model and softening beam model seem to compare well as seen in Fig. 11. The FPZ lengths from smaller (D1 type) are less than those from larger beams (D2 type). However the ratio FPZ length/ligament length is larger for smaller beams and vice versa, which is obvious and is reaffirmed through AE results. Yu et al. [21] in their paper have argued that when the depth of the beam and the ligament length are much larger than the FPZ length, there will not be any interaction between FPZ and back
Fig. 10. Bar chart showing nearly constant fracture energy against RILEM fracture energy from bi-linear and tri-linear models.
Fig. 11. Bar chart showing process zone lengths from softening beam model.
Fig. 9. Distance between centroids of triangles is the effective fracture process zone length LP.
boundary and hence size effect should not exist but the reality is contradictory. Thus interaction between FPZ and the boundary does not have any influence on size effect. The AE histogram of events, in the tri-linear model, exhibits declining patterns one near the notch tip and the other at the back boundary of the specimen. It may become essential to verify the effect of fictitious boundary near the notch tip on the size effect when the FPZ size is very small in comparison to depth and the ligament length of the specimen.
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4. Conclusions 1. The variation of local fracture energy across the ligament length in TPB specimens appears to be tri-linear instead of bi-linear. 2. The transition ligament length depends on specimen geometry and notch to depth ratio. 3. A dual boundary effect, due to specimen back boundary and fictitious boundary near the notch tip, could be a factor influencing the size effect on the fracture energy. Of course the effect of fictitious boundary needs to be verified. 4. Local fracture energy distribution is steeper near the notch tip than at the back boundary, which is obviously due to stress singularity at the crack tip. It also confirms the fictitious crack model where the stress at the crack tip decreases to zero at the time of initial collapse. 5. The observed back boundary transition ligament length through AE histogram reasonably satisfied the boundary effect model. 6. The values of FPZ lengths measured from AE histogram from trilinear model agree well with the values calculated from softening beam model. In D1 and D2 type beams the average value of FPZ length from tri-linear model is 45.5 mm and 81.2 mm respectively and from softening beam model is 51 mm and 80.5 mm respectively. 7. From tri-linear model, the conditions FPZsmall < FPZLarge and FPZLarge FPZsmall > DLarge is reaffirmed. D small
Acknowledgement Authors gratefully acknowledge M/s BARC, Mumbai, India for the financial support rendered towards the experimentation. References [1] Bazˇant ZP. Size effect aspects of measurement of fracture characteristics of quasibrittle material. Adv Cem Based Mater 1996;4:128–37. [2] Bazˇant ZP. Size effect. Int J Solids Struct 2000;37:69–80. [3] Hu X, Duan K. Influence of fracture process zone height on fracture energy of concrete. Cem Concr Res 2004;34:1321–30. [4] Duan K, Hu X, Wittmann FH. Size effect on specific fracture energy of concrete. Eng Fract Mech 2007;74:87–96. [5] RILEM FMC-50. Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Mater Struct 1985;18(4):287–90. [6] Mindess S. The effect of specimen size on the fracture energy of concrete. Cem Concr Res 1984;14:431–6. [7] Nallathambi P, Karihaloo BL. Prediction of load–deflection behavior of plain concrete from fracture energy. Cem Concr Res 1986;16:373–82. [8] Guinea GV, Planas J, Elices M. Measurement of the fracture energy using threepoint bend tests: Part 1 – influence of experimental procedures. Mater Struct 1992;25:212–8.
[9] Elices M, Guinea GV, Planas J. Measurement of the fracture energy using threepoint bend tests: Part 3 – influence of cutting the P–d tail. Mater Struct 1992;25:327–34. [10] Bazˇant ZP, Li Z. Zero-brittleness size–effect method for one-size fracture test of concrete. J Eng Mech ASCE 1996;122:458–68. [11] Guo XH, Gilbert RI. The effect of specimen size on the fracture energy and softening function of concrete. Mater Struct 2000;33:309–16. [12] Hu X, Wittmann FH. Size effect on toughness induced by crack close to free surface. Eng Fract Mech 2000;65:209–21. [13] Darwin David, Barham Shawn, Kozul Rozalija, Luan Shuguang. Fracture energy of high-strength concrete. ACI Mater J 2001:410–7. [14] Trunk B, Wittmann FH. Influence of size on fracture energy of concrete. Mater Struct 2001;34:260–5. [15] Duan K, Hu X, Wittmann FH. Explanation of size effect in concrete fracture using non-uniform energy distribution. Mater Struct 2002;35:326–31. [16] Karihaloo BL, Abdalla HM, Imjai T. A simple method for determining the true specific fracture energy of concrete. Mag Concr Res 2003;55:471–81. [17] Duan K, Hu X, Wittmann FH. Boundary effect on concrete fracture and nonconstant fracture energy distribution. Eng Fract Mech 2003;70:2257–68. [18] Duan K, Hu X, Wittmann FH. Thickness effect on fracture energy of cementitious materials. Cem Concr Res 2003;33:499–507. [19] Hu X, Duan K. Mechanism behind the size effect phenomenon. J Eng Mech ASCE 2010;136(1):60–8. [20] Bazant ZP. Size effect on blunt fracture: concrete, rock, metal. J Eng Mech 1984;110(4):518–35. [21] Yu Qiang, Le Jia-Liang, Christian GH, Bazant ZP. Problems with Hu-Duan boundary effect model and its comparison to size–shape effect law for quasibrittle fracture. J Mater Civil Eng ASCE 2010;136(1):40–50. [22] Abdalla HM, Karihaloo BL. Determination of size-independent specific fracture energy of concrete from three-point bend and wedge splitting tests. Mag Concr Res 2003;55:133–41. [23] Shilang Xu, Yanhua Zhao, Zhimin Wu. Study on the average fracture energy for crack propagation in concrete. J Mater Civil Eng ASCE 2006;18:817–24. [24] Yanhua Zhao, Shilang Xu, Zhimin Wu. Variation of fracture energy dissipation along evolving fracture process zones in concrete. J Mater Civil Eng ASCE 2007;19:625–33. [25] Colombo S, Main IG, Forde MC. Assessing damage of reinforced concrete beam using ‘‘b-value’’ analysis of acoustic emission signals. J Mater Civil Eng ASCE 2003:280–6. [26] Landis EN. Micro–macro fracture relationships and acoustic emissions in concrete. Constr Build Mater 1999;13:65–72. [27] Wu Keru, Chen Bing, Yao Wu. Study on the AE characteristics of fracture process of mortar, concrete and steel–fiber-reinforced concrete beams. Cem Concr Res 2000;30:1495–500. [28] Muralidhara S, Raghu Prasad BK, Hamid Eskandari, Karihaloo BL. Fracture process zone size and true fracture energy of concrete using acoustic emission. Constr Build Mater 2010;24(4):479–86. [29] Ananthan H, Raghu Prasad BK, Sundara Raja Iyengar KT. Influence of strain softening on the fracture of plain concrete beams. Int J Fract 1990;45:195–219. [30] Hillerborg A. Results of three comparative test series for determining the fracture energy GF of concrete. Mater Struct 1985;18(107):407–12. [31] Karihaloo BL, Xiao QZ. Size effect in the strength of concrete structures. Sãdhanã, vol. 27, no. 4; 2002. p. 449–59. [32] Carpinteri A. Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int J Solids Struct 1989;25(4):407–29. [33] IS-10262. Recommended guidelines for concrete mix design. Bureau of Indian standards, New Delhi; 1982. [34] Bazant Z, Kazemi MT. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int J Fract 1990;44(2):111–31.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Size-independent fracture energy in plain concrete beams using tri-linear model S. Muralidhara a,b,⇑, B.K. Raghu Prasad b, B.L. Karihaloo c, R.K. Singh d a
BMS College of Engineering, Bangalore, India Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India c School of Engineering, Cardiff University, Queen’s Buildings, P.O. Box 925, Cardiff CF24 OYF, UK d Reactor Safety Division, BARC, Mumbai, India b
a r t i c l e
i n f o
Article history: Received 10 April 2010 Received in revised form 9 December 2010 Accepted 11 January 2011 Available online 18 February 2011 Keywords: Local fracture energy Tri-linear model Fictitious boundary Size-independent fracture energy Concrete
a b s t r a c t The boundary effect or the size effect on the fracture properties of concrete has been studied assuming a bi-linear model for the distribution of local fracture energy concept. Boundary effect is observed not only near the back boundary but also near the notch tip, where a fictitious boundary seems to exist, separating the linear and non-linear fracture zones. In this paper a tri-linear function is assumed for the local fracture energy distribution along the ligament and expressions relating RILEM fracture energy and the size-independent fracture energy are developed. Transition ligament length measurements based on the acoustic emission (AE) histogram of events are used to obtain size-independent fracture energy. Length of the fracture process zone is identified in the AE histogram and compared with the value obtained from softening beam model. There seems to be a good agreement between the results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Concrete, which is a quasi-brittle material, exhibits a post-peak softening behavior which lies between a brittle and a ductile material behavior. Unlike in metals, the plastic zone in concrete is a scatter of micro-cracks around the crack tip, called the process zone. In concrete, a solitary crack initiation and propagation is seldom seen and the energy is consumed in the formation of micro-cracks in the process zone. The parameter to characterize the fracture is the energy consumed per unit area of crack, also called the fracture energy. RILEM has quantified this parameter as specific fracture energy based on work of fracture. However this is beset with size effects. The size effects are being explained through different models. Bazant’s size effect law, based on the dimensional analysis of the fracture energy released during crack propagation in geometrically similar specimens, has been at the forefront to explain the size effect [1,2]. Size effect on the specific fracture energy has also been linked to the influence of ligament length on fracture process zone (FPZ). Small FPZ associated with smaller ligament length leads to lower specific fracture energy. Local fracture energy model based on the size/width of FPZ was developed by Hu and Duan [3], Duan et al. [4]. In this model, fracture dissipation has been assumed to be ⇑ Corresponding author. Present address: Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India. Tel.: +91 080 2293 2435; fax: +91 080 2360 0404. E-mail addresses: [email protected], [email protected] (S. Muralidhara). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.01.003
different at different locations along the crack path. This nonconstant local fracture energy dissipation has been shown to be the cause for the size effect in the fracture energy calculated according to RILEM FMC-50 [5]. 1.1. Review of boundary effect model As is well known, specific fracture energy is the energy required to create a crack of unit area. This parameter is useful to characterize the fracture process in concrete in conjunction with the tensile strength ft. RILEM has recommended an expression for the specific fracture energy based on the work of fracture. One important assumption here is that all the energy expended goes into crack initiation and propagation. The specific fracture energy is found to vary with specimen dimensions. If we use gf to indicate the specific fracture energy relevant to a small surface area along the crack path, the Gf definition given by RILEM has practically assumed that the local fracture energy gf is constant along the crack path. The localized specific fracture energy gf is generated at the crack tip region during crack growth after crack width w increases from 0 to wc. This process is repeated for a constant local fracture energy distribution along the crack path. This constant local fracture energy is construed as size independent and many researchers have tried to obtain this size-independent specific fracture energy using various models. The size effect model based on Bazant’s size effect law has been adopted extensively and the fracture parameters are obtained from this model. The effect of specimen size on fracture energy was studied using three-point bend test on beams by
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Mindess [6], who concluded that the self-weight of the beam has little effect on the specific fracture energy Gf even in large beams. The ligament area used in the calculation of RILEM Gf, is only the projected area of the fracture surface which underestimates the actual fracture area. Nallathambi and Karihaloo [7] have tried with modified models of FCM of Hillerborg and Blunt crack band model of Bazant, incorporating slow crack growth and tortuosity, but found that the specific fracture energy is still size dependent. Guinea et al. [8] and Elices et al. [9] calculated the energy expended for fracturing by excluding energy losses through friction near supports, crushing near supports and bulk energy dissipation in the compression zone from the measured work of fracture and obtained size-independent fracture energy. Bazˇant and Li [10] have proposed a new size effect model wherein only one size of the specimen instead of the conventional three sizes, is sufficient to determine fracture parameters. The size dependence of the RILEM fracture energy was explained by Guo and Gilbert [11] through the concept of partial fracture energy (Gf). The partial fracture energy depends on the width (Wp) of the fracture process zone (FPZ); the larger the width, dG the higher is the Gf and dWfp decreases with Wp. If the specimen is large enough to allow the formation of full FPZ, then the fracture energy is not size dependent. Research studies to correlate elastic and plastic fracture conditions in metals to those in concrete specimens have been pursued by Hu and Wittmann [12] to arrive at an asymptotic function in fracture energy and critical stress intensity factor. They have concluded that reference crack size or the characteristic length lch has implication in size effect and therefore in fracture toughness and fracture energy. A deterministic size effect formula was proposed by Bazˇant [2] which was based on the energy release due to boundary layer cracking. For large specimens, however, the formula was found to be inadequate. Attempts have been made to ascertain the effect of aggregate content on the fracture energy and fracture process zone size. Critical volume fraction of aggregates to obtain maximum fracture energy seems to be non-existent, which seems to be rather surprising. Darwin et al. [13] have tried to relate the fracture energy with the age of concrete and have concluded that fracture energy is almost constant as a function of water-cement ratio, age and compressive strength. Trunk and Wittmann [14] have presented a model to explain the nature of size and geometry dependence of specific fracture energy Gf. Further studies on the size dependence of fracture energy by Duan et al. [15] based on the asymptotic approach have contributed in identifying the influence of the ratio of process zone size over its distance from the free surface. Ever since the specific fracture energy was identified as a fracture parameter for the nonlinear fracture theory applied to concrete, earnest attempts are being made to isolate the true fracture energy parameter which is independent of size and shape of the test specimen. Karihaloo et al. [16] have been actively pursuing studies to determine sizeindependent specific fracture energy of concrete, adopting concepts of local fracture energy and boundary effect induced by free surface, introduced by Duan et al. [17], using beam and wedge splitting specimens. The averaged fracture energy Gf as recommended by RILEM is given by the following equation, based on the concept of work of fracture.
Gf ¼
1 BðW aÞ
Z
Pdd
GF ¼
Z
wc
rdw
ð2Þ
0
In which Wc is the critical crack opening and r is the cohesive stress. The FPZ width reduces while approaching the back boundary, which has lead to characterize local fracture energy dissipation pattern as a bi-linear function to make it simple to analyze. The bi-linear function consists of a horizontal straight line of the intrinsic fracture energy GF and a declining straight line that reduces to zero at the back boundary. Within the zone in which FPZ has constant width, the fracture energy is size independent while nearer to the outer boundary, the FPZ width decreases and hence the fracture energy also decreases. However, it is not still clear from all the previous investigations in literature, how one can conclude that constancy of fracture energy over the un-cracked ligament implies constancy over different sizes to call it size independent. The RILEM fracture energy considers both inner and outer zones for the calculation to get averaged fracture energy over the ligament length and hence is size dependent. The relationship between FPZ characteristics and the RILEM fracture energy is developed based on the variation of FPZ width along the ligament length. Process zone around a propagating crack is considered to have two regions, an inner softening region (Wsf) and an outer region (Wf) with micro-cracks as shown in Fig. 1. Wsf region comprises main crack with branches of larger cracks, which add to concrete softening. In the outer region the cracks are not inter-connected and do not account for softening. It is very obvious to expect higher fracture energy dissipation in the inner region than in the outer region. The widths Wsf and Wf vary depending on the stress field at the crack tip which apparently influence critical crack width Wc. As the crack approaches the back boundary of the specimen, the size of the ligament area diminishes, apparently reducing Wsf, Wf, Wc and finally the fracture energy. The local fracture energy gf is associated only to r–w relationship at a particular location. If r– w is unique then GF = gf = Constant. Another observation here is that the above fact in terms of Wsf and Wf is not experimentally validated. The local fracture energy gf is connected to Wsf, Wf, and Wc as given below
W sf aW f ðxÞ wc aW sf ðxÞ
ð3Þ
g f awc ðxÞ Fracture energy Gf and gf are related by the following expression:
ð1Þ
In which B is the thickness of the specimen, P is the applied load and d is the load point displacement. The fracture energy GF is related to cohesive stress–crack opening (r–w) curve by
Fig. 1. FPZ and bridging stresses. FPZ with inner softening zone and outer micro fracture zone [18].
S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058
g f ðaÞ ¼ Gf ðaÞ ðW aÞ
dGf ðaÞ da
ð4Þ
In which gf(a) is the local fracture energy at crack tip, and Gf(a) is the averaged fracture energy from a specimen with initial crack length of a. The boundary effect model of Duan et al. [18] has been able to account for the size effect of fracture energy through the back boundary effect assuming a bi-linear variation of local fracture energy gf over the un-notched ligament length (Fig. 2). According to the boundary effect concept, the dissipation of fracture energy decreases sharply when a crack is approaching to a free boundary but remains constant at locations away from a free boundary. Using a bi-linear energy distribution function to approximate the fracture energy distribution, it has been shown that the common size effect is, in fact, due to the influence of a free specimen boundary. The intersection of these two straight lines is defined as the transition ligament al , measured along the un-notched ligament. The relationship between local fracture energy gf and the size-independent fracture energy GF from the boundary effect model is developed considering the relationships given in Eqs. (5) and (6). However, the expressions relating specific fracture energy Gf (RILEM) and GF are more convenient for calculations.
Gf ðxÞ ¼ GF
for x < ðW a al Þ
ðW a xÞ g f ðxÞ ¼ GF for x P ðW a al Þ al al for ðW aÞ > al Gf ¼ GF 1 2ðW aÞ W a forðW aÞ 6 al Gf ¼ GF 2al
ð5Þ
ð6Þ
In which Gf is the specific fracture energy or size dependent fracture energy (RILEM), GF is the true or size-independent fracture energy, W is the overall depth of the beam, a is the initial notch depth and al is the transition ligament length. Hu and Duan [19] while comparing boundary effect model with SEL of Bazant [20], have emphasized the role of FPZ in different sizes of specimens. For a constant notch/depth ratio, in small or large specimens, FPZsmall < FPZLarge but
FPZsmall Dsmall
>
FPZLarge . DLarge
Hence they
claimed that the condition of geometric similarity is inapplicable to FPZ. Hence the size effect is the size of FPZ or in other words due to distance of FPZ from structure boundaries. Here again it seems that the fact is a foregone conclusion. Yu et al. [21] have compared boundary effect model with SEL, considering notched and un-notched beams and influence on fracture energy Gf, critical stress intensity factor KIC, tensile strength ft for different geometries. They have claimed that the boundary effect model has a few deficiencies and cannot be applied to all sizes. In fact the basic hypothesis that size effect is due to back boundary is questioned. It has been indicated that with in a certain parametric range of 0.15–0.6 in notch/depth and 0.1–0.9 in brittleness
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number, the values of KIC and Gf from boundary effect model and SEL match well. It has also been indicated that Gf obtained from boundary effect model is the initial fracture energy i.e. area under the initial tangent in the softening portion of r–x curve. In fact the value Gf from boundary effect model seems to be similar to the one from SEL, which has been claimed to be size independent. Abdalla and Karihaloo [22] through their studies could observe that the transition ligament length first increased with an increase in the specimen size but at a reducing rate and then stabilized. They also pointed out that testing large specimens is unnecessary and true fracture energy could be determined if the ligament length is greater than transition ligament length. Adopting regression analysis, based on the bi-linear boundary effect model, on a few set of data of beams with different depths and notch to depth ratio, and found the size-independent fracture energy and transition ligament lengths for different depths of beams. Finally they concluded that fracture energy is almost constant but the transition ligament length varies with the depth of the beam. Size effects on specific fracture energy have been studied and reported. The bi-linear variation of local fracture energy has been extended to specimens having larger thickness and there has been negligible effect on specimens with thickness larger than four times the maximum aggregate size, as reported by Duan et al. [18]. Shilang et al. [23], obtained two fracture energy quantities corresponding to stable and unstable crack growth by conducting wedge splitting tests and observed that the fracture energy during the stable crack growth remained constant and was size independent. Yanhua et al. [24] introduced two concepts viz. the back boundary affected length and overall boundary affected length, to explain non-uniform local fracture energy along the ligament length through tests on WS specimens. It was observed that the back boundary affected length decreases linearly with the crack evolution, whereas the overall boundary affected length increased at early stages but decreased later. 1.2. Acoustic emission technique In concrete, the fracture process zone ahead of a crack tip is the consequence of the formation of micro-cracks, a few of which later link up to form a macro-crack. The formation and growth of cracks is 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 works by Landis [26] have shed light on the relationship between the AE events and the evolution of fracture process in concrete. Formation of FPZ in mortar by AE techniques was studied by Keru Wu et al. [27]. Muralidhara et al. [28] have estimated the size of the fracture process zone based on the distribution of AE events and their energies and also the size-independent fracture energy using the bi-linear model with inputs from the AE event histograms. 1.3. Review of softening beam model
Fig. 2. Variation of local fracture energy gf and Gf over the ligament length [18].
Accurately modeling the fracture process and the analyzing the behavior of concrete structures has been the core of the research in concrete fracture. One such model is the one-dimensional model also called softening beam model proposed by Ananthan et al. [29]. Through this model fracture behavior of notched and unnotched plain concrete slender beams subjected to three-point or four-point bending is analyzed. Structural size effect in changing the fracture type from brittle to plastic collapse is explained through the stress distribution across the un-cracked ligament obtained by varying the strain softening modulus. It is shown that the length of the fracture process zone depends on the value of the
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strain softening modulus. In addition non-linear fracture parameters such as CTOD, CMOD and fracture energy are computed for a wide variety of beam specimens. The complete stress–crack opening relation is necessary for any model to be able to characterize fracture process. The fracture behavior is characterized by the slope of the postpeak softening portion in the stress–strain curve in tension. In softening beam model the post-peak behavior is idealized to be linear, as could be seen in Fig. 3, and the slope is termed as softening modulus ET. The ratio ET/E is E⁄ which is used to characterize the nature of the material failure. For example, for E⁄ = 1, the fracture is brittle and for E⁄ = 0, it is ductile. The geometrical parameters like depth, span/depth ratio and notch/depth ratio have a bearing on the nature of specimen fracture. With the backing of experimental studies, it has been shown that E⁄ varies between 0 and 1 for different geometrical dimensions of concrete specimens. Further, cohesive stress variation along the process zone length is assumed linear and non-linear variation of stress in the compression zone, as given in Fig. 4. At the onset of fracture instability, the cohesive stress at the beginning of the process zone reduces to zero while at the end of the process zone attains a maximum value equal to the tensile strength ft of concrete as shown in Fig. 5. Making use of the stress and cohesive stress variations as given in Figs. 4 and
Fig. 5. Stress diagram at the onset of fracture instability [29].
5, an expression relating the maximum process zone length LPmax and E⁄ is obtained [29] as
2 E LP max ¼ 1 E þ 1 ðW aÞ
ð7Þ
In which (W a) is the ligament length of the specimen. The results from the model have correlated reasonably well with values from the literature because of its simplicity. 1.4. Research significance and objectives
Fig. 3. Stress–strain diagram in concrete under tension [29].
Fracture energy has been the most suitable fracture parameter to characterize the fracture in concrete. The energy criterion based analysis suits well to concrete due to the non-linearity of mechanics associated with concrete. This is mainly due to the formation of fracture process zone, which is a scatter of micro-cracks around the notch/crack tip. However this fracture energy parameter is beset with size effect, which has made it a size dependent. Different models viz. Hillerborg’s cohesive crack model [30], Bazˇant’s size effect model Bazˇant [2], Karihaloo’s size effect model [31], Boundary effect model based on non-uniform energy distribution along the ligament length due to Duan and Wittmann [17], Carpinteri’s fractal concept for size effect [32] have been developed to determine size-independent fracture energy. The objective of the present study is to obtain size-independent fracture energy and the length of the fracture process zone. In this paper, the bi-linear boundary effect model is improved further to a tri-linear model based on the observations made in the acoustic emission data in front of the notch, on a plain concrete beam tested under three-point bend condition to estimate size-independent fracture energy. The trilinear model appears to showcase a more accurate energy distribution along the ligament length and facilitates a better estimate of size-independent fracture energy. From this model, a part of the ligament length has been interpreted as the fracture process zone length and compared with the one obtained using softening beam model. The study reaffirms the use of the AE techniques to obtain fracture parameters with out resorting to any regression analysis. 2. Experimental setup 2.1. Test setup
Fig. 4. Stress block for 1 > E⁄ > 0 [29].
Plain concrete single edged notched beam specimens of compressive strength 45 MPa were cast. The proportions of the concrete ingredients used (based on
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S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058 mix design using Indian standard code IS-10262 [33] (1982)) and the geometrical dimensions of the beams are as given in Tables 1 and 2. The beams were tested under three-point bending condition. In Table 2 the specimens are identified with symbols for e.g. D1P20UC01. D1 is the type of beam, P stands for pour mix, 20 for aggregate size as 20 mm, U for un-reinforced, C stands for notch/depth ratio of 0.33 and 01 is the sample number. T stands for trial mix and B for notch/depth ratio of 0.25. Trial mix and pour mix are of same compressive strength. Notch cutting in beams forms an important and delicate part of the specimen preparation. The notches were cut using concrete tile cutting machine with a blade of thickness 3 mm and diameter of 150 mm. The notch thus had a width of 3 mm at its tip, which is less than 10 mm as suggested by RILEM FMC-50. A 500kN capacity servo controlled DARTEC machine under crack mouth opening displacement (CMOD) control was employed. The central deflection of the beam was recorded by an LVDT which could measure up to 0.1 lm. The clip gauge for the measurement of CMOD had a resolution of 0.1lm. The test was performed keeping the CMOD rate at 0.0005 mm/s. The acquisition of loading and displacement parameters along with the acoustic emission data were simultaneous. 2.2. AE Monitoring Equipment The AE equipment used was from Physical Acoustic Corporation, Princeton, New Jersey, USA. Four sensors with a resonant frequency of 60 kHz were used for the AE acquisition and were arranged on one face of the specimen as shown in Fig. 6. The locations of events have the origin of reference at the bottom left corner of the specimen. The surface of specimen where the positions of the sensors were marked was initially cleaned with acetone and then the AE sensors were fixed to the specimen surface using special vacuum grease/gel. The sensors were initially tested for their sensitivity by pencil lead breaking. Further, automatic sensor testing (AST) available in the AE software was employed to check the fixity of the sensors to the concrete surface and to ensure that there is no gap between the sensor and the surface.
Fig. 7. Histogram of events along the ligament length at the notch from a D2 type beam.
gram (Fig. 7) along the ligament length. The number of times an AE event occurs (number of AE events) increases with increase in distance from the notch tip and stabilize over a certain distance. Later it reduces towards the back boundary. A similar declining trend could be seen towards the notch tip where of course there is no physical boundary. This could be explained by the tri-linear variation of local fracture energy as shown in Fig. 8. The local fracture energy gf reduces from a constant GF while approaching both the back boundary and the notch tip. Because the events die down near the notch tip one can imagine that a boundary like the back boundary exists. There and here after it could be called as fictitious boundary. The fictitious boundary separating linear and non-linear zones seems to be another source of the boundary effect. Transition ligament length bl is found to be smaller than al . The relationship between local fracture energy and the size-independent fracture energy GF developed from the tri-linear model is given in Eqs. (8) and (9).
2.3. Tri-linear model for local fracture energy distribution The fracture process zone size could be estimated using the event locations from AE catalogue. An interesting observation could be made from AE event histoTable 1 Quantity of materials per m3 of concrete. Property
Mix-with 20 mm and down size coarse aggregates
Mix-with 12.5 mm and down size coarse aggregates
Cement (kg) (C) Coarse aggregate (kg) 20 mm (CA) 12.5 mm Fine aggregate (kg) (FA) Water (kg) Super plasticizer (% weight of cement content) Water/cement ratio Mix proportion (C:FA:CA) 28 days compressive strength (N/mm2)
400 492 492 902 152 1.4
435 – 949 870 165 1.4
0.38 1:2.26:2.46 52.23
0.38 1:2:2.18 54.3
Table 2 Geometric properties of specimens. Type
Length (mm)
Depth (mm)
Width (mm)
Span (mm)
D1 D2
375 750
95 190
47.5 95
282 564
Y
Fig. 8. Tri-linear model showing the trend of gf variation over un-notched ligament length.
25 3
190 m m
60
100
S e n so rs 1
60
O rig in
375 m m
190 m m
2
4
N o tch 564 m m E L E V A T IO N
Fig. 6. Profile of the beam showing position of sensors.
X 95 m m S ID E V IE W
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g f ðxÞ ¼ GF
S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058
x bl
g f ðxÞ ¼ GF
for x 6 bl
for
bl
6 x < ðW a
ðW a xÞ g f ðxÞ ¼ GF al
al Þ
for x P ðW a al Þ
ð8Þ
In which bl and al are the transition ligament lengths near the notch tip and near the back boundary respectively of the beam specimen and a is the notch depth. The above relationship could be expressed in terms of specific fracture energy, size-independent fracture energy and transition ligament lengths bl and al as
bl for ðW aÞ 6 bl 2ðW aÞ 1 al bl for ðW aÞ P al Gf ¼ GF 2ðW aÞ Gf ¼ GF
ð9Þ
3. Results and discussion The pattern of AE event histogram was alike in all the specimens tested. Hence results from a sample beam D2T20UB02, of dimensions 750 mm 190 mm 95 mm with a notch to depth ratio of 0.25 D (47.5 mm) are analyzed and presented here. It is known that the number of AE events directly depends on the stress level in the material. With an increase in stress level, the number of events also increases and in the post-peak softening regime, the event number decreases as the stress reduces. During this period, the crack initiation and propagation takes place from the notch tip towards the specimen back boundary. From Fig. 7, it is observed that, the number of events reduces as the crack propagates towards the back boundary. The histogram clearly follows the energy trend shown by the boundary effect model. The length of the horizontal projection of the post-peak sloping portion of the histogram is the transition ligament length. It is also stated by Duan et al. [18] that gf and FPZ length/width rapidly decrease when the crack is approaching back boundary of the specimen. The histogram explicitly shows that FPZ width/length rather than FPZ length/width decreases rapidly when crack approaches back boundary. A critical observation of the AE event histogram indicates that the variation of local fracture energy across the ligament length in TPB specimens is more of tri-linear pattern than bi-linear. It also appears that a fictitious boundary similar to the back boundary is present near the notch tip. The number AE activities decline near the fictitious boundary giving the impression of a dual boundary effect. The Local fracture energy distribution is steeper near the notch tip than at the back boundary, which is perceptibly due to stress singularity at the crack tip. It also confirms the fictitious crack model where the stress at the crack tip decreases to zero at the time of initial collapse. It is also observed that the back boundary transition ligament length through AE histogram reasonably satisfied the boundary effect model.
Arguments over the validity of SEL and boundary effect model are becoming louder over the past couple of years. Recent discussions by Yu et al. [21] about the comparative studies between SEL and boundary effect model have disagreed some of the facts put forth in boundary effect model. In spite of wide disagreements, there have been commonalities in the values of some fracture parameters calculated from SEL and boundary effect model with in a certain range of geometric proportions. The value of KIC and Gf from both SEL and boundary effect model are found to be reasonably similar for notch/depth in the range of 0.15–0.6 and brittleness number 0.1–0.9. In the discussion, the fracture energy calculated based on boundary effect model is claimed to be equal to the area under the initial tangent of softening portion of the r–x curve. In fact the fracture energy calculated using SEL has been reported as size independent and compared with the area under initial tangent of post-peak softening portion of r–x curve. The above observation strongly confirms the fact that the fracture energy calculated using boundary effect model is size independent and of course with in the parametric range mentioned above. In the light of the above discussion on boundary effect model, the geometries of the concrete beam specimens used were checked for notch/depth ratio and brittleness number. The notch/depth ratio of the beam specimens tested for D1 type and D2 type was 0.25 and 0.33, and brittleness numbers [34] were 0.27–0.54. All the values calculated are well with in the range of validity. The validated geometric conditions reaffirm that the calculated fracture energy using boundary effect model is size independent. The RILEM fracture energy Gf obtained from TPB tests on D1 and D2 type beams with different notch lengths are as given in Table 3. The true specific fracture energy has been calculated using the Eq. (8), because the ligament length (W a) is greater than al . The RILEM fracture energy obtained from the tests, transition ligament lengths al and bl measured from the AE event histogram are used in Eq. (8) and the true fracture energy or size-independent fracture energy GF is calculated. Measured transition ligament lengths al and bl from AE histograms, and calculated size-independent fracture energy GF from both bi-linear model [28] and tri-linear models are as given in Table 3. Co-efficient of variation of specific fracture energy and the true fracture energy are obtained from statistical analysis of the results as given in Table 5. Bar-chart showing nearly constant fracture energy against RILEM fracture energy from both bi-linear model and tri-linear model are given in Fig. 10. The fracture energy variation along the ligament holds the trace about the length of fracture process zone. It is obvious to find the density of AE events to be more along the crack path than away from it. This reaffirms the fact that there are two portions in FPZ, one inner zone with micro- and macro-cracks and the other with micro-cracks called the outer zone. It is very difficult to demarcate these zones distinctly and hence the FPZ length. In the tri-linear model the length of FPZ (LP) is assumed as the distance between
Table 3 RILEM Gf and GF from bi-linear and tri-linear models of different specimens. Sl. no.
Specimen ID
Notch depth a (mm)
Beam depth W (mm)
Transition ligament length al (mm)
Transition ligament length bl (mm)
RILEM Gf (N/m)
Bi-linear GF (N/m)
Tri-linear GF (N/m)
1 2 3 4 5 6 7 8 9 10 11
D1T20UB01 D1P20UB01 D1P20UC01 D1T12.5UC01 D2P20UC02 D2T20UC01 D2T12.5UB03 D2T20UB02 D2T12.5UB02 D2T12.5UC03 D2T20UB03
23.8 24 31 29 66 67 49 49 50 68 49
95 95 95 95 190 190 190 190 190 190 190
37.5 10 30 27.5 55 75 50 60 50 24.5 55
6.2 10 9 6 22 21 31 28 25 10 25
151.9 190.9 157.5 154.5 153.7 129.5 165 147.2 167.4 179.2 159.7
206.2 205.4 205.7 195.2 197.5 186.3 200.6 187.0 203.8 199.2 198.4
219.2 222.2 226.5 207.0 222.9 212.4 231.5 214.0 228.6 208.7 222.9
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S. Muralidhara et al. / Construction and Building Materials 25 (2011) 3051–3058 Table 4 Process zone length from tri-linear model and softening beam model. Sl. no.
Specimen ID
Effective ligament length LP (mm)
Average LP (mm)
Softening beam model Lpmax
Average Lpmax (mm)
W (mm)
LP/W
Average LP/W
1 2 3 4 5 6 7 8 9 10 11
D1T20UB01 D1P20UB01 D1P20UC01 D1T12.5UC01 D2P20UC02 D2T20UC01 D2T12.5UB03 D2T20UB02 D2T12.5UB02 D2T12.5UC03 D2T20UB03
42.1 57.0 38.0 43.7 71.3 57.7 87.0 80.3 86.7 97.7 87.7
45.2
51.0 56.9 45.0 51.0 83.9 57.8 93.6 75.2 86.0 84.0 83.3
51.0
95 95 95 95 190 190 190 190 190 190 190
0.44127 0.59926 0.39863 0.45847 0.37405 0.30179 0.45647 0.42121 0.45474 0.52382 0.46
0.47441
81.2
80.5
0.42744
Table 5 Results from statistical analysis of RILEM Gf and GF from bi-linear and tri-linear models. Fracture energy
Mean value
Standard deviation
Co-efficient of variation (COV)
Gf (N/m) GF (bi-linear model) (N/m) GF (tri-linear model) (N/m)
159.7 198.7 219.6
16.2 6.9 8.1
0.102 0.035 0.037
the centroids of triangles near the notch tip and the back boundary in the histogram of AE events (Fig. 9). It may seem proper to assume the length of the central portion cl , over which the distribution of fracture energy is almost uniform, as equivalent to FPZ length but recognizing the transition zones on either side it is suitable to assume LP as the FPZ length. Softening beam model is used to determine FPZ length taking into account the geometries of the beams tested. The values of (lFPZ)max calculated using Eq. (7) for the beams with maximum loads from the experiments and the geometries as the input to the model are given in Table 4. The values of FPZ lengths for the beams from tri-linear model and softening beam model seem to compare well as seen in Fig. 11. The FPZ lengths from smaller (D1 type) are less than those from larger beams (D2 type). However the ratio FPZ length/ligament length is larger for smaller beams and vice versa, which is obvious and is reaffirmed through AE results. Yu et al. [21] in their paper have argued that when the depth of the beam and the ligament length are much larger than the FPZ length, there will not be any interaction between FPZ and back
Fig. 10. Bar chart showing nearly constant fracture energy against RILEM fracture energy from bi-linear and tri-linear models.
Fig. 11. Bar chart showing process zone lengths from softening beam model.
Fig. 9. Distance between centroids of triangles is the effective fracture process zone length LP.
boundary and hence size effect should not exist but the reality is contradictory. Thus interaction between FPZ and the boundary does not have any influence on size effect. The AE histogram of events, in the tri-linear model, exhibits declining patterns one near the notch tip and the other at the back boundary of the specimen. It may become essential to verify the effect of fictitious boundary near the notch tip on the size effect when the FPZ size is very small in comparison to depth and the ligament length of the specimen.
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4. Conclusions 1. The variation of local fracture energy across the ligament length in TPB specimens appears to be tri-linear instead of bi-linear. 2. The transition ligament length depends on specimen geometry and notch to depth ratio. 3. A dual boundary effect, due to specimen back boundary and fictitious boundary near the notch tip, could be a factor influencing the size effect on the fracture energy. Of course the effect of fictitious boundary needs to be verified. 4. Local fracture energy distribution is steeper near the notch tip than at the back boundary, which is obviously due to stress singularity at the crack tip. It also confirms the fictitious crack model where the stress at the crack tip decreases to zero at the time of initial collapse. 5. The observed back boundary transition ligament length through AE histogram reasonably satisfied the boundary effect model. 6. The values of FPZ lengths measured from AE histogram from trilinear model agree well with the values calculated from softening beam model. In D1 and D2 type beams the average value of FPZ length from tri-linear model is 45.5 mm and 81.2 mm respectively and from softening beam model is 51 mm and 80.5 mm respectively. 7. From tri-linear model, the conditions FPZsmall < FPZLarge and FPZLarge FPZsmall > DLarge is reaffirmed. D small
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