An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams

Engineering Fracture Mechanics 73 (2006) 2166–2191 www.elsevier.com/locate/engfracmech

An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams Zhimin Wu

a,*

, Shutong Yang a, Xiaozhi Hu b, Jianjun Zheng

c

a

b

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, PR China Department of Mechanical and Materials Engineering, University of Western Australia, Nedlands, Perth, WA 6907, Australia c School of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou, 310014, PR China Received 14 October 2005; received in revised form 15 March 2006; accepted 3 April 2006 Available online 24 May 2006

Abstract An analytical model to predict the effective fracture toughness K sIC of concrete was proposed based on the fictitious crack model. Firstly, the equilibrium equations of forces in the section were formed in combination with the plane section assumption. Then a Lagrange function was presented through the equilibrium equations and the relationship formula between the effective crack length and crack tip opening displacement. Taking into account Lagrange Multiplier Method, the maximum load Pmax was obtained, as well as the critical effective crack length ac. Furthermore, K sIC was gained in an analytical manner. Subsequently, some material and structural parameters from other literatures were adopted into the proposed model for the calculation. Compared with the experimental results, most of the calculated values show a good agreement for Pmax and ac. In order to study the influence of the softening curve in the fictitious crack on the calculated fracture parameters, three series of constants determining the shape of the softening curve were chosen in the calculation. The results show that the calculated fracture parameters are not sensitive to the shape of the softening curve. Therefore, only if the elastic modulus Ec and flexural tensile strength fr were measured, Pmax, ac and K sIC can be predicted accurately using the proposed model. Finally, the variations of the calculated fracture parameters with the specimen size and a0/h (i.e., the ratio of the initial crack length to the depth of the specimen) were studied. It was found that both K sIC and the pre-critical crack propagation length Dac increase with the specimen size. However, the two parameters increase to the maximums and then decrease gradually with a0/h. Moreover, the theories of free surface effect were utilized to explain the observed size effects. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Effective fracture toughness; Lagrange Multiplier Method; Flexural tensile strength; Size effect; Free surface

1. Introduction Fracture mechanics has been developed and applied for many decades. Its scope of application has been extended into numerous fields of materials, such as metal, ceramic, concrete, etc. Since Kaplan firstly *

Corresponding author. Tel.: +86 411 84709842. E-mail addresses: [email protected] (Z. Wu), [email protected] (S. Yang).

0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.04.001

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Nomenclature a effective crack length a0 initial crack length a0/h ratio of initial crack length to depth of beam ac critical effective crack length aec measured ac in the experiment b width of beam c1 one of material constants c2 one of material constants CTODc critical crack tip opening displacement Ec elastic modulus of concrete fc0 cylindrical compressive strength of concrete ft uniaxial tensile strength of concrete fr flexural tensile strength of concrete beam with a0/h = 0 H0 thickness of clip gauge holder h depth of beam hc distance from the fictitious crack tip to the position of neutral axis K ini initial fracture toughness IC K sIC effective fracture toughness K un unstable fracture toughness IC L span of beam M applied bending moment in the section of midspan P applied load Pini initial cracking load Pmax maximum applied load P emax measured maximum applied load W self-weight of beam w crack opening displacement w0 maximum crack opening width when the cohesive stress becomes zero wCMOD crack mouth opening displacement wt crack tip opening displacement Dac pre-critical crack propagation length rc compressive stress of concrete at the top extremity of specimen rw cohesive stress in the fictitious crack

introduced fracture mechanics into concrete beam to measure fracture toughness in 1961 [1], more and more investigations have been performed for fracture mechanics of concrete. Moreover, the fundamental research and application in concrete structures have attracted increasing attention since early 1980s [2]. According to the quasi-brittle manner of concrete, various fracture models have been introduced to study the crack propagation in concrete structures, such as fictitious crack model (FCM) [3], crack band model (CBM) [4], two parameter fracture model (TPFM) [5], size effect model (SEM) [6], effective crack model (ECM) [7–10], double-K fracture model [11–15] and cohesive-force-based KR curve [16]. Based on the above mentioned different fracture models, the corresponding methods for calculating fracture parameters are also different from each other. Bazˇant and Oh simulated the fracture as a blunt smeared crack band justified by the random nature of the microstructure [4]. Jenq and Shah [17] defined an effective crack length as the sum of the initial crack length plus an effective crack extension at the peak load. Subsequently, a two parameter fracture model [5] was proposed to characterize the fracture process of concrete known as the critical stress intensity factor and critical crack tip opening displacement CTODc. Effective crack model [9,10] utilized highly accurate regression equations based on a very large data for determining

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the effective crack length at the peak load. Besides, Elices and Planas presented the equivalent elastic crack concept [18]. As the development of concrete performance, high strength concrete has been applied widely in Civil Engineering. Therefore, fracture analysis of high strength concrete structures has attracted increasing attention of researchers. Navalurkar and Hsu developed a nonlinear model for the fracture analysis of high strength concrete members through the experimental data and finite element analysis [19]. Results show that the shape of the softening curve plays a crucial role on the overall post peak load–deflection and load–CMOD (crack mouth opening displacement) responses. Recently, Xu and Reinhardt have presented the double-K criterion for crack propagation in quasi-brittle fracture [11–15]. In the proposed model, the initial fracture toughness K ini IC and the unstable fracture toughness K un were selected as the controlling parameters to describe the crack initiation, stable propagation and IC unstable propagation [12]. As an example of application for double-K criterion, it could be applied into three-point bending notched beams [13], CT-specimens and wedge splitting specimens [14]. Additionally, a simplified method was proposed to determine the double-K fracture parameters of three-point bending test [15]. Moreover, the crack extension resistance and fracture properties were studied in detail in the form of KR curve for quasi-brittle materials like concrete with a softening traction-separation law by investigating the complete fracture process [16]. Then this proposed cohesive stress based KR curves and the double-K fracture criterion were applied to negative geometry specimens [20]. Besides, Zhao and Xu proposed a double-G criterion characterized by two fracture parameters in terms of energy release rate [21], which can be thought as a supplement to the double-K fracture criterion in describing the crack propagation in concrete materials. Furthermore, the size effect is also a key problem that deserves to be extensively studied. The research work has been carried out for more than 20 years [22–34]. Bazˇant pointed out that the main physical mechanism that causes the size effect is the crack front blunting of any type based on his pioneer work-crack band model [4,22]. Subsequently, he made the parameters of the size effect law be identified by linear regression [23]. In addition, based on the size effect law, the specific energy was defined [24], which is a unique material property, independent of specimen size, shape and type of loading. Recently, Bazˇant and Yu presented six problems on size effect [25]. They pointed out that the introduction of size effect into the specifications in concrete design codes is of the greatest practical importance. Issa et al. performed a lot of investigations on size effects in concrete fracture [26,27]. Through the analysis of test results [27], they pointed out that the rate of increase in fracture toughness with respect to an increase of the dimensionless crack length (the crack length normalized by the specimen width) increases as both the specimen size and the maximum aggregate size increase. Karihaloo et al. [29] pointed out that the deterministic strength size effect becomes stronger as the size of the cracked structure increases but weakens as the size of the crack reduces relative to the size of the structure. Recently, size effect owing to free surface effect of the specimen has been extensively investigated by Hu et al. [30–34]. As the crack approaches the back end of a specimen, the fracture process zone becomes more and more confined and hence the local fracture energy decreases [30]. If a bridging zone or damage zone is not small, there will be free edge effects in toughness [31]. Using the asymptotic analysis, it showed that the size effect on fracture toughness and energy of a heterogeneous material such as concrete will be inevitable if the crack length or remained ligament length is too close to the proposed reference crack length a*. Subsequently, the size effect on fracture properties was further developed to consider the influence of both the front and back free surfaces of small sized specimens [34], i.e., the fracture process zone (FPZ) can be so close to the front or back surface that it is limited and cannot fully develop. In this paper, taking into account the fictitious crack model and the plane section assumption, an analytical model was presented to predict the effective fracture toughness K sIC for standard three-point bending notched beams using Lagrange Multiplier Method. Subsequently, the fundamental material and structural parameters obtained from other literatures were inserted into the proposed model to calculate the fracture parameters. Moreover, the computed values were compared with the experimental ones for the critical effective crack length ac and maximum applied load Pmax. Besides, the sensitivity of the calculated fracture parameters to the shape of the softening curve in the fictitious crack was analyzed. Finally, the size effect for the computed fracture parameters by the proposed model was studied. And the theories of free surface effect [30–34] were utilized to explain the observed results.

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2. Analytical derivation 2.1. Fundamental assumptions In the whole analytical derivation, the fundamental assumptions are expressed as follows: (a) The uncracked part of the cracked section in the beam remains plane during the loading process, i.e., the strain in the uncracked part distributes linearly along the height of the beam. (b) The crack surface (i.e., stress-free crack surface plus the equivalent elastic fictitious crack surface) remains plane, i.e., the crack opening displacement at different position of the crack surface varies linearly along the height of the beam. (c) The compressive stress and the tensile stress ahead of the fictitious crack tip distribute linearly along the height of the beam. (d) The tensile elastic modulus of concrete is equal to the compressive one. 2.2. Analytical derivation A standard three-point bending notched beam is shown in Fig. 1, with width b, depth h, span L, initial crack length a0 in the midspan and L/h = 4. When the applied load attains the initial cracking load Pini, the performed crack starts to crack initially [11–15]. Then the crack develops steadily as the increase of the applied load until the latter attains the maximum applied load Pmax. During the crack stable propagation, the stress and strain distributions in the cracked section of the beam are shown in Figs. 2 and 3, respectively. The origin and x-axis are also formed in the figures. According to the third assumption, the distributions of the compressive stress and tensile stress ahead of the fictitious crack tip are linear along the height of the beam. Therefore, the relationships between the compressive stress and strain and between the tensile stress and strain are also linear by the use of the first assumption. Moreover, the tensile stress at the fictitious crack tip is thought as the flexural tensile strength fr of the beams with a0 = 0 instead of the uniaxial tensile strength ft. The reason will be discussed in the next section. The expression proposed by Reinhardt et al. [35] is presented as following for the relationship between the cohesive stress rw in the fictitious crack and the fictitious crack opening displacement w  rw ¼ f t

  c31 3 wc2 w ð1 þ c31 Þ ec2 1þ 3w e 0  w w0 w0

ð1Þ

where c1 and c2 are the material constants, w0 is the maximum crack opening width when the stress becomes zero. In this section, ft in Eq. (1) should be replaced by fr and then Eq. (1) is expressed in the form of Eq. (2) as following:   rw ¼ fr ð1 þ aw3 Þ ebw  cw

ð2Þ

P

h

h a0

L

a0

b

Fig. 1. The standard three-point bending notched beam and details of the cracked section.

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σc

M

0

hc

h

fr

σw a a0

x

Fig. 2. The stress distribution in the cracked section where hc is the distance from the fictitious crack tip to the position of neutral axis.

σc Ec h – a – hc

0

hc

fr Ec

a

x

Fig. 3. The strain distribution in the cracked section where hc is the distance from the fictitious crack tip to the position of neutral axis.

where a¼

c31 ; w30

b¼

c2 ; w0

c¼

ð1 þ c31 Þ ec2 w0

According to the second assumption, the fictitious crack opening displacement w along the height of the beam can be expressed in the following equation: w¼

x  hc wt a  a0

ðhc 6 x 6 hc þ a  a0 Þ

ð3Þ

where wt is the crack tip opening displacement. Substituting Eq. (3) into Eq. (2), the formula for the cohesive stress rw along the height of the beam is obtained as following: (" rw ¼ fr

1þ

aw3t



x  hc a  a0

3 # e

bwt



xhc aa0



  cwt

x  hc a  a0

) ðhc 6 x 6 hc þ a  a0 Þ

ð4Þ

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Considering the strain distribution in Fig. 3, the following equation is gained: rc Ec fr Ec

¼

h  a  hc fr ðh  a  hc Þ ) rc ¼ hc hc

ð5Þ

The equilibrium equation for the stresses in the cracked section in Fig. 2 is expressed as 1 1 rc bðh  a  hc Þ ¼ fr bhc þ 2 2

Z

hc þaa0

rw b dx

ð6Þ

hc

Upon substitution of Eqs. (4) and (5) into Eq. (6), we obtain      2 6a 2 3 3 2 2 3 bwt hc  ðh  aÞ2 ¼ 0 2ðh  aÞ þ ða  a0 Þ 1 þ 3  cwt  4 ab wt þ 3ab wt þ 6abwt þ 6a þ b e bwt b b wt ð7Þ

Moreover, the applied bending moment M in Fig. 2 can be expressed as following: 1 1 M ¼ rc bðh  a  hc Þ2 þ fr bh2c þ 3 3

Z

hc þaa0

rw bx dx

ð8Þ

hc

Substituting Eqs. (4) and (5) into Eq. (8) and denoting F = M/frb, we have     ðh  aÞ3 a  a0 a  a0 1 1 1 2  ðh  aÞ þ ðh  aÞhc þ hc þ F ¼  cwt ða  a0 Þ hc þ a  a0 2 3 3 3hc bwt bwt   2 6ahc ða  a0 Þ 24aða  a0 Þ a  a0 a  a0 bwt þ  hc þ a  a0 þ þ e bwt bwt b5 w2t b4 wt 

 ahc ða  a0 Þ 3 3 b wt þ 3b2 w2t þ 6bwt þ 6 ebwt 4 b wt



 aða  a0 Þ2 4 4 b wt þ 4b3 w3t þ 12b2 w2t þ 24bwt þ 24 ebwt 5 2 b wt

ð9Þ

When M attains the maximum, F also attains the maximum. Besides, the applied load P = 4M/L  W/2 (W is the self-weight of the beam), so P also attains the maximum Pmax. The next goal is to seek the maximum of F. Firstly, the relationship between wt and a should be obtained. The empirical formula proposed by Tada et al. [36] is presented for the crack mouth opening displacement wCMOD and a as following: wCMOD

"  2  3  2 # 6PLa a þ H0 a þ H0 a þ H0 h þ H0 ¼ 2 0:76  2:28 þ 3:87  2:04 þ 0:66 h þ H0 h þ H0 h þ H0 ha h bEc

ð10Þ

where H0 is thickness of clip gauge holder. According to the second assumption, wCMOD can be expressed as a function of wt shown in the following equation: wt ¼

a  a0 wCMOD a þ H0

ð11Þ

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Substituting P = 4M/L  W/2, Eqs. (9) and (11) into Eq. (10), we obtain  3 2 8T ðh  aÞ ða  a0 Þ 24T ða  a0 Þ a  a0 2  24T ðh  aÞ ða  a0 Þ þ 24hc T ðh  aÞða  a0 Þ þ hc þ hc bwt bwt   2 3 1 1 1 144T ahc ða  a0 Þ 576T aða  a0 Þ þ  24T cwt ða  a0 Þ2 hc þ a  a0 þ 2 3 3 b5 w2t b4 wt    24T ða  a0 Þ2 a  a0 bwt 24T ahc ða  a0 Þ2 3 3  hc þ a  a0 þ  b wt þ 3b2 w2t þ 6bwt þ 6 ebwt e 4 bwt bwt b wt   3  bw 3WLT ða  a0 Þ 24T aða  a0 Þ 4 4 H0 3 3 2 2 2 t  m0 h 1 þ  b wt þ 4b wt þ 12b wt þ 24bwt þ 24 e  wt ¼ 0 fr b a b5 w2t ð12Þ where T = 0.76  2.28(a + H0)/(h + H0) + 3.87(a + H0)2/(h + H0)2  2.04(a + H0)3/(h + H0)3 + 0.66(h + H0)2/ (h  a)2 and m0 = Ec/fr. Taking into account Lagrange Multiplier Method, a Lagrange function U(a, hc, wt, k1, k2) is established in combination with Eqs. (7), (9) and (12) as following:   3 ðh  aÞ a  a0 a  a0 2 Uða; hc ; wt ; k1 ; k2 Þ ¼  ðh  aÞ þ ðh  aÞhc þ hc þ 3hc bwt bwt   2 1 1 1 6ahc ða  a0 Þ 24aða  a0 Þ a  a0  cwt ða  a0 Þ hc þ a  a0 þ þ  2 3 3 bwt b5 w2t b4 wt  

 a  a0 bwt ahc ða  a0 Þ 3 3  hc þ a  a0 þ  b wt þ 3b2 w2t þ 6bwt þ 6 ebwt e 4 bwt b wt 2

 aða  a0 Þ  b4 w4t þ 4b3 w3t þ 12b2 w2t þ 24bwt þ 24 ebwt 5 2 bw t    2 6a þ k1 2ðh  aÞ þ ða  a0 Þ 1þ 3 bwt b 

 2 3 3 2  cwt  4 ab wt þ 3ab2 w2t þ 6abwt þ 6a þ b3 ebwt hc  ðh  aÞ b wt " 3 8T ðh  aÞ ða  a0 Þ 2  24T ðh  aÞ ða  a0 Þ þ 24hc T ðh  aÞða  a0 Þ þ k2 hc    2 24T ða  a0 Þ a  a0 1 1 2 1 þ hc þ a  a0 hc þ  24T cwt ða  a0 Þ 2 3 3 bwt bwt  2 3 2 144T ahc ða  a0 Þ 576T aða  a0 Þ 24T ða  a0 Þ a  a0 bwt þ þ  hc þ a  a0 þ e bwt bwt b5 w2t b4 wt  24T ahc ða  a0 Þ2 3 3  b wt þ 3b2 w2t þ 6bwt þ 6 ebwt 4 b wt  24T aða  a0 Þ3 4 4  b wt þ 4b3 w3t þ 12b2 w2t þ 24bwt þ 24 ebwt 5 2 b wt   # 3WLT ða  a0 Þ H0 2  m0 h 1 þ  ð13Þ wt fr b a where k1 and k2 are the unknown parameters to be solved. Using oU/oi = 0 (i = a, hc, wt, k1, k2), five equations can be obtained. Subsequently, the five equations are solved using mathematical software Matlab and the values of five unknown parameters a, hc, wt, k1 and k2 are obtained. Because of the complexity and high order of the equations, there must be more than one series of solutions for the five unknown parameters. In fact, after the performed crack propagates initially, it turns into

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crack steadily propagation. During this process, the applied load P increases with the crack extension until P attains Pmax. Then the crack will propagate unsteadily. Moreover, according to the above analysis, P and F attain the maximums simultaneously. Therefore, if there is one series of solutions satisfying a > a0 and h  a > 2hc, it will make F attain the maximum. In other words, this series of solutions will make the obtained P attain Pmax. Correspondingly, a and wt can be denoted as the critical effective crack length ac and critical crack tip opening displacement CTODc, respectively. For the calculation of effective fracture toughness K sIC , the following formula of stress intensity factor for three-point notched beam [36] is utilized: K ¼ 1:5

PL pffiffiffi aF ðnÞ bh2

ð14Þ

where a n¼ ; h

F ðnÞ ¼

 1:99  nð1  nÞ 2:15  3:93n þ 2:7n2 ð1 þ 2nÞð1  nÞ

3=2

If Pmax and ac are inserted into Eq. (14) and the self-weight W of the beam is considered, the effective fracture toughness K sIC can be expressed as following: K sIC ¼ 1:5

ðP max þ 0:5W ÞL pffiffiffiffiffi ac F ðnc Þ bh2

ð15Þ

where nc ¼

ac ; h

F ðnc Þ ¼

 1:99  nc ð1  nc Þ 2:15  3:93nc þ 2:7n2c ð1 þ 2nc Þð1  nc Þ

3=2

3. Analysis of calculated results 3.1. Comparison between the calculated values and experimental data In this section, the fundamental material and structural parameters of three-point bending notched beams are obtained from the literatures [37], i.e., for Series B, b = 76 mm, h = 203 mm, L = 762 mm, H0 = 3.2 mm and Ec = 38.4 GPa; for Series C, b = 76 mm, h = 305 mm, L = 1143 mm, H0 = 3.2 mm and Ec = 39.3 GPa. The ratios of the initial crack length to the depth of the beam a0/h are shown in Table 1 for Series B and Series C. Besides, the constants c1, c2 and w0 to determine the shape of the cohesive stress distribution curve vary with the type and grade of concrete. Without the experimental values of these parameters, c1 = 3, c2 = 7 and w0 = 160 lm are adopted in the present calculation. Also are there no measured values for the flexural tensile strength fr and uniaxial tensile strength ft. However, the cylindrical compressive strength fc0 is presented 1=2 1=2 and fr, ft can be expressed as fr ¼ 0:62ðfc0 Þ , ft ¼ 0:4983ðfc0 Þ , respectively. Therefore, for Series B, fr = 4.52 MPa, ft = 3.63 MPa; for Series C, fr = 4.57 MPa, ft = 3.68 MPa. Herein, 2450 kg/m3 is adopted as the density of concrete. If the tensile stress at the fictitious crack tip is equal to ft rather than fr, all the material and structural parameters mentioned above are inserted into the proposed model and the comparisons between the calculated values and experimental ones for ac/h and Pmax are shown in Figs. 4–7, respectively. Herein, aec and P emax refer to the measured ac and Pmax in the experiment, respectively. The solid line in the figure represents the function aec ¼ ac or P emax ¼ P max . Moreover, they will have the same meanings as mentioned above when they appear again in the following figures. It can be seen that the scattered points are all distributed on the left of the solid line in Fig. 6. Some points are even far away from the line. In Fig. 7, only two scattered points are located under the solid line and others

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Table 1 a0/h for Series B and C Numbers of specimen

a0/h

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18 B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40 B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16

0.383 0.442 0.319 0.558 0.648 0.636 0.706 0.654 0.775 0.775 0.815 0.376 0.309 0.362 0.478 0.495 0.459 0.626 0.631 0.665 0.617 0.774 0.756 0.73 0.748 0.738 0.45 0.673 0.369 0.575 0.748 0.431 0.455 0.631 0.588 0.49 0.713 0.703 0.863 0.594 0.55 0.403 0.437 0.44 0.478 0.546 0.515 0.597 0.631 0.638 0.648 0.745 0.75 0.772 0.82 0.453 0.507

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Table 1 (continued) Numbers of specimen

a0/h

C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

0.556 0.549 0.504 0.465 0.381 0.379 0.426 0.818 0.529 0.616 0.639 0.612 0.611 0.774 0.352 0.551 0.712

1.0 0.9

ace /h

0.8 0.7 0.6 0.5 0.4 0.4

0.5

0.6

0.7

0.8

0.9

1.0

ac /h Fig. 4. Comparison for ac/h of Series B when the tensile stress at crack tip is equal to ft.

1.0 0.9

ace /h

0.8 0.7 0.6 0.5 0.4 0.4

0.5

0.6

0.7

0.8

0.9

1.0

ac /h Fig. 5. Comparison for ac/h of Series C when the tensile stress at crack tip is equal to ft.

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3

e

Pmax /kN

4

2 1 0 0

1

2

3

4

5

6

Pmax /kN Fig. 6. Comparison for Pmax of Series B when the tensile stress at crack tip is equal to ft.

8 7 6

4

e

Pmax /kN

5

3 2 1 0 0

1

2

3

4

5

6

7

8

Pmax /kN Fig. 7. Comparison for Pmax of Series C when the tensile stress at crack tip is equal to ft.

all appear on the left of the line. This phenomenon shows that nearly all of the measured values are larger than the calculated ones. So it can be concluded that the proposed model underestimates the value of the maximum applied load if the tensile stress at the fictitious crack tip is thought to be the uniaxial tensile strength ft. Therefore, the value of the tensile stress at the crack tip should be larger than ft. If the tensile stress is thought to be flexural tensile strength fr, the results of comparisons are shown in Figs. 8–11, respectively. For ac/h, the calculated values show a very good agreement with the measured ones. The scatter points are located on the left and right of the solid line symmetrically in Figs. 8 and 9, respectively. They almost approach the line. The accuracy of the maximum applied load Pmax is inferior to the one of ac/h. However, most values of Pmax compare very well with the measured ones especially for Series C. It also can be seen from Figs. 10 and 11 that more than 80% of the points scatter symmetrically on the left and right of the solid line. Moreover, the relative errors are larger mainly for the beams with lower depth or higher a0/h. It is because the applied load will be so low when h decreases or a0/h becomes higher that the value of Pmax is difficult to control and the measured values might be discrete seriously. Therefore, it is very difficult to obtain a good agreement

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191 1.0 0.9

ace /h

0.8 0.7 0.6 0.5 0.4 0.4

0.5

0.6

0.7

0.8

0.9

1.0

ac/h Fig. 8. Comparison for ac/h of Series B when the tensile stress at crack tip is equal to fr.

1.0 0.9

ace /h

0.8 0.7 0.6 0.5 0.4 0.4

0.5

0.6

0.7

0.8

0.9

1.0

ac /h Fig. 9. Comparison for ac/h of Series C when the tensile stress at crack tip is equal to fr.

8 7 6

4

e

Pmax /kN

5

3 2 1 0 0

1

2

3

4

5

6

7

8

Pmax /kN Fig. 10. Comparison for Pmax of Series B when the tensile stress at crack tip is equal to fr.

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e

Pmax /kN

6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

8

9

10

Pmax /kN

Fig. 11. Comparison for Pmax of Series C when the tensile stress at crack tip is equal to fr.

between the computed values and measured ones for these specimens. It also explains why the results in Series C have better accuracy than in Series B. There are also other factors leading to the larger errors for some specimens such as the flexural tensile strength fr. Because there are no measured values for fr, the formula fr ¼ 0:62ðfc0 Þ1=2 is adopted in the calculation. However, it origins from the code, so there is probably a large discrepancy between the calculated 1=2 fr and the actual fr for some specimens. In other words, the computed fr using fr ¼ 0:62ðfc0 Þ cannot reflect the actual value of fr accurately in some cases. Therefore, the mentioned discrepancy would lead to larger errors directly. According to the relationship between the scattered points and the solid line in Figs. 8–11, the total tendency of the points is aec ¼ ac or P emax ¼ P max except for few cases. It shows that there is a significant improvement for the proposed model if ft is replaced by fr. Moreover, other fracture parameters can be obtained from the proposed model, i.e., CTODc and K sIC . The calculated results are shown in Table 2. 3.2. The study on the sensitivity of the calculated fracture parameters to the shape of the softening curve In the determination of the relationship between the cohesive stress and crack width in the fictitious crack, the parameters c1 = 3, c2 = 7, w0 = 160 lm are adopted determining the shape of the softening curve. However, the parameters vary with the grade and type of concrete. In other words, for concrete with different mixtures, the shape of the softening curve is changed. In order to study the sensitivity of the calculated fracture parameters to the shape of the softening curve, different series of c1, c2 and w0 are inserted into the calculation. Then the results using each series of c1, c2 and w0 are compared with each other. Herein, besides c1 = 3, c2 = 7, w0 = 160 lm mentioned above, two other series of c1, c2 and w0 are adopted, i.e., c1 = 2, c2 = 6.3, w0 = 140 lm and c1 = 4, c2 = 8.2, w0 = 200 lm. The comparisons for the fracture parameters using these three series of c1, c2 and w0 are shown in Tables A.1–A.4, respectively. When the distribution of cohesive stress in the fictitious crack varies, the calculated fracture parameters are also changed correspondingly. However, it can be seen from Tables A.1–A.4 that the difference is marginal between the calculated values using three series of c1, c2 and w0, respectively. There is little effect of c1, c2 and w0 on the calculated results. In other words, the obtained fracture parameters using the proposed model are not sensitive to the shape of the softening curve. Above all, the characteristics of the proposed model can be summarized as follows: (a) The calculated fracture parameters are not sensitive to the shape of the softening curve. Therefore, the fracture parameters can be predicted even without the exact curve of the cohesive stress distribution.

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191 Table 2 Results for CTODc and K sIC Numbers of specimen

CTODc (lm)

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18 B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40 B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16

13.1 12.8 13.4 12.2 11.6 11.7 11.2 11.6 10.4 10.4 9.8 13.1 13.4 13.2 12.7 12.6 12.7 11.8 11.8 11.5 11.9 10.4 10.6 10.9 10.7 10.8 12.8 11.4 13.2 12.1 10.7 12.9 12.8 11.8 12 12.6 11.1 11.2 8.7 12 12.3 13.7 13.6 13.5 13.4 13 13.2 12.7 12.5 12.4 12.3 11.4 11.3 11 10.1 13.5 13.2

2179

pffiffiffiffi K sIC ðMPa mÞ 1.588 1.575 1.595 1.538 1.502 1.507 1.481 1.5 1.472 1.472 1.49 1.589 1.595 1.591 1.565 1.559 1.571 1.511 1.508 1.496 1.514 1.472 1.472 1.475 1.473 1.474 1.573 1.493 1.589 1.531 1.473 1.578 1.571 1.508 1.526 1.561 1.479 1.482 1.6 1.524 1.54 1.773 1.762 1.762 1.747 1.718 1.731 1.693 1.676 1.673 1.669 1.643 1.644 1.652 1.72 1.756 1.735 (continued on next page)

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Table 2 (continued) Numbers of specimen

CTODc (lm)

pffiffiffiffi K sIC ðMPa mÞ

C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

13 13 13.2 13.4 13.8 13.8 13.6 10.2 13.1 12.6 12.4 12.6 12.6 11 13.9 13 11.8

1.712 1.716 1.737 1.752 1.778 1.779 1.766 1.713 1.725 1.683 1.673 1.686 1.686 1.652 1.784 1.715 1.646

(b) The model needs only a few of fundamental parameters for the calculation. Besides the geometrical parameters which can be measured conveniently, only two material parameters need measuring, i.e., elastic modulus Ec and flexural tensile strength fr. The two parameters can also be obtained using ordinary experimental method without any difficulty. Therefore, if the fundamental geometrical sizes, elastic modulus and flexural tensile strength of specimen are obtained, the maximum applied load can be gained accurately using the proposed model without any experiment, as well as ac, CTODc and K sIC .

4. Study on size effect Size effect of fracture parameters is a very important problem in fracture mechanics of concrete. In this section, investigations are carried out on the calculated fracture parameters, i.e., the effective fracture toughness K sIC and pre-critical crack propagation length Dac, using the proposed model. In order to study the size effect of these parameters, the parameters varying with the depth of the specimen but fixed a0/h or varying with a0/h but fixed depth are calculated. When a0/h is fixed as 0.2, 0.3, 0.4 and 0.5, the depth of the specimen varies from 200 mm to 2000 mm. And when the depth of specimen is fixed as 200 mm, 400 mm, 600 mm, 800 mm, 1000 mm, 1200 mm, 1400 mm, 1600 mm, 1800 mm, 2000 mm, a0/h varies from 0.1 to 0.5. Herein, if the specimen size is large enough, for example h P 1000 mm, the self-weight is so heavy that the specimen would be cracked rapidly under its own weight without any force on it especially for the specimens with high a0/h. Therefore, the maximum of a0/h is limited to 0.5. Other parameters are adopted as b = 200 mm, L = 4h, Ec = 30 GPa, H0 = 2 mm, c1 = 3, c2 = 7, w0 = 160 lm. The density of concrete is adopted as 2450 kg/m3. Taking into account the flexural tensile strength fr varying with the depth of the specimen, the relationship expression between the two variables   49:25 fr ¼ ft 1 þ h

ð16Þ

is utilized in the calculation, which was presented by Bazˇant and Li [38] according to Hillerborg et al.’s experimental data [3]. In Eq. (16), ft is adopted as 2.41 MPa. Then the curves of the fracture parameters K sIC and Dac varying with the depth of the specimen and a0/h are shown in Figs. 12–15, respectively.

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191 2.6 2.4

a0/h=0.2

2.2

a0/h=0.3

KICs /MPa·m1/2

2.0

a0/h=0.4

1.8

a0/h=0.5

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

400

800

1200

1600

2000

2400

2800

h/mm

Δac /mm

Fig. 12. The response of K sIC to h.

180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10

a0/h=0.2 a0/h=0.3 a0/h=0.4 a0/h=0.5

0

400

800

1200

1600

2000

2400

2800

h/mm Fig. 13. The response of Dac to h.

2.4 h=200mm h=400mm h=600mm h=800mm h=1000mm h=1200mm h=1400mm h=1600mm h=1800mm h=2000mm

2.2 2.0

s

K IC /MPa·m

1/2

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.1

0.2

0.3

0.4

0.5

0.6

a0 /h Fig. 14. The response of K sIC to a0/h.

0.7

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Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191 180 h=200mm h=400mm h =600mm h=800mm h=1000mm h=1200mm h=1400mm h=1600mm h=1800mm h=2000mm

160

Δac /mm

140 120 100 80 60 40 20 0.1

0.2

0.3

0.4

0.5

0.6

0.7

a0/h Fig. 15. The response of Dac to a0/h.

2.4

a0/h=0.2

2.2

a0/h=0.3

2.0

a0/h=0.4

s

KIC /MPa·m

1/2

1.8

a0/h=0.5

1.6

a0/h=0.6

1.4

a0/h=0.7

1.2

a0/h=0.8

1.0 0.8 0.6 0.4 0.2 0.0 0

400

800

1200

1600

2000

2400

2800

h/mm Fig. 16. The response of K sIC to h (neglecting self-weight).

a0/h=0.2

45

a0/h=0.3 a0/h=0.4

40

a0/h=0.5 35

Δac /mm

2182

a0/h=0.6 a0/h=0.7

30

a0/h=0.8

25 20 15 400

800

1200

1600

2000

2400

2800

h/mm Fig. 17. The response of Dac to h (neglecting self-weight).

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191

h=200mm h=400mm h=600mm h=800mm h=1000mm h=1200mm h=1400mm h=1600mm h=1800mm h =2000mm

2.0

KICs /MPa·m1/2

2183

1.5

1.0

0.5

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

a0/h

Δac /mm

Fig. 18. The response of K sIC to a0/h (neglecting self-weight).

46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12

h=200mm h=400mm h =600mm h=800mm h=1000mm h=1200mm h=1400mm h=1600mm h=1800mm h=2000mm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

a0/h Fig. 19. The response of Dac to a0/h (neglecting self-weight).

If the self-weight of the specimen is neglected, a0/h is allowed to vary from 0.1 to 0.8. Keeping other parameters constant, the curves of the fracture parameters mentioned above varying with h and a0/h are shown in Figs. 16–19, respectively. Both K sIC and Dac have similar tendencies of variation. As the increase of h, both K sIC and Dac increase. They also tend to be higher when a0/h increases. However, if the self-weight is neglected, the shapes of the curves are changed. Both K sIC and Dac increase with a0/h until they attain the maximums, respectively. Then as the increase of a0/h, the two fracture parameters begin to decrease gradually. Moreover, it is also found that the variation of K sIC or Dac with a0/h is more significant as the increase of specimen size. When h 6 500 mm, the variation is marginal. Herein, the theories of free surface effect [30–34] are introduced to explain the observed size effects. The front free surface still refers to the bottom surface of the specimen and the back free surface represents the top surface. As the increase of h, the initial crack tip or fracture process zone (FPZ) is gradually far

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away from the front free surface for fixed a0/h. Therefore, the influence of front free surface is weaker and FPZ can fully develop. It leads to the increase of Dac and K sIC . In addition, if the specimen size keeps constant, the distance from the initial crack tip to the front free surface becomes larger with a0/h and the influence of front free surface is weaker. Then FPZ can develop so fully that the values of Dac and K sIC become larger. However, as the increase of a0/h, the influence of back free surface becomes more significant and the developing of FPZ is limited. Therefore, it can be seen from Figs. 18 and 19 that both Dac and K sIC increase to the maximums and then begin to decrease with a0/h. Moreover, the variations of both Dac and K sIC are not significant in the vicinity of the maximums. It means that FPZ is far away from both front and back free surfaces in this region so that the effect of the two free surfaces is marginal. It should be noted that the influence of back free surface would not appear if a0/h is limited to 0.5. This is why the descending parts of curves are not found in Figs. 14 and 15. Besides, if the specimen size is not large enough, FPZ is always affected by the influence of either front or back free surface. Therefore, it cannot fully develop all the time. The values of Dac and K sIC are lower but the variations of the two parameters are marginal.

5. Conclusions In combination with Lagrange Multiplier Method, an analytical model was proposed to calculate the effective fracture toughness K sIC based on the fictitious crack model and plane section assumption. Upon the comparison with the experimental values, the calculated results show a good agreement for the maximum load Pmax and critical effective crack length ac. Moreover, the calculated fracture parameters were not sensitive to the shape of the softening curve. Therefore, it was concluded that Pmax, ac and K sIC can be obtained accurately using this model only if the flexural tensile strength fr and elastic modulus Ec were gained. Subsequently, the size effects of K sIC and pre-critical crack propagation length Dac were analyzed and explained according to the theories of free surface effect [30–34]. The conclusions can be summarized as follows: (a) If a0/h is kept constant, as the increase of h, FPZ is gradually far away from the front free surface so that it can fully develop due to the weaker effect of front free surface. Therefore, both K sIC and Dac increase with h. (b) If h is fixed, as the increase of a0/h, FPZ can fully develop due to the weaker effect of front free surface so that both K sIC and Dac become larger. However, if a0/h continues increasing, the influence of back free surface would appear gradually. Therefore, the developing of FPZ is limited and it cannot fully develop so that the values of K sIC and Dac decrease gradually. (c) For small sized specimens with fixed h, as the increase of a0/h, FPZ is not far away from the front free surface or back free surface so that it cannot fully develop all the time. Therefore, the variations of K sIC and Dac are marginal.

Acknowledgement The authors gratefully acknowledge that the National Nature Science Foundation of China (Grant No. 50578025) has supported this work.

Appendix A. Tables for studying the sensitivity of the calculated fracture parameters to the softening curve See Tables A.1–A.4

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191

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Table A.1 Comparison for ac/h using three series of c1, c2 and w0 Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18 B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40 B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16

0.492 0.547 0.431 0.653 0.733 0.723 0.784 0.739 0.843 0.843 0.877 0.485 0.422 0.472 0.58 0.595 0.562 0.714 0.718 0.748 0.706 0.843 0.827 0.805 0.82 0.812 0.554 0.755 0.478 0.668 0.82 0.536 0.559 0.718 0.68 0.591 0.79 0.782 0.917 0.685 0.646 0.489 0.522 0.525 0.561 0.626 0.596 0.673 0.705 0.711 0.721 0.809 0.814 0.834 0.877 0.537 0.589

0.489 0.544 0.428 0.651 0.732 0.721 0.783 0.737 0.842 0.842 0.876 0.482 0.419 0.469 0.578 0.593 0.56 0.712 0.717 0.747 0.704 0.842 0.826 0.804 0.819 0.811 0.552 0.754 0.476 0.666 0.819 0.534 0.556 0.717 0.678 0.589 0.789 0.78 0.917 0.684 0.644 0.487 0.52 0.523 0.559 0.624 0.594 0.672 0.703 0.71 0.719 0.808 0.813 0.833 0.877 0.535 0.587

0.497 0.551 0.437 0.657 0.736 0.726 0.787 0.742 0.845 0.845 0.879 0.49 0.427 0.477 0.584 0.6 0.567 0.717 0.722 0.751 0.709 0.844 0.829 0.807 0.823 0.814 0.559 0.758 0.484 0.672 0.823 0.541 0.563 0.722 0.684 0.595 0.793 0.784 0.918 0.689 0.65 0.494 0.526 0.529 0.565 0.629 0.6 0.677 0.708 0.715 0.724 0.812 0.816 0.836 0.879 0.542 0.593 (continued on next page)

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Table A.1 (continued) Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

0.635 0.628 0.586 0.549 0.468 0.466 0.511 0.876 0.61 0.691 0.712 0.687 0.686 0.839 0.44 0.63 0.779

0.633 0.626 0.584 0.547 0.466 0.464 0.509 0.875 0.608 0.689 0.711 0.686 0.685 0.835 0.437 0.628 0.778

0.639 0.632 0.59 0.553 0.473 0.471 0.516 0.877 0.613 0.694 0.715 0.691 0.69 0.838 0.445 0.634 0.782

Table A.2 Comparison for Pmax using three series of c1, c2 and w0 unit: kN Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18 B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40

5.44 4.48 6.58 2.85 1.82 1.94 1.26 1.75 0.72 0.72 0.46 5.56 6.77 5.8 3.94 3.69 4.22 2.05 1.99 1.64 2.15 0.72 0.85 1.06 0.91 0.99 4.36 1.57 5.68 2.64 0.91 4.66 4.28 1.99 2.48 3.77

5.42 4.47 6.56 2.84 1.81 1.93 1.25 1.75 0.71 0.71 0.46 5.54 6.74 5.78 3.93 3.68 4.21 2.04 1.99 1.64 2.14 0.72 0.85 1.05 0.91 0.99 4.34 1.56 5.66 2.63 0.91 4.64 4.27 1.99 2.47 3.75

5.48 4.52 6.63 2.87 1.83 1.96 1.27 1.77 0.72 0.72 0.46 5.61 6.81 5.85 3.97 3.73 4.26 2.07 2.01 1.66 2.17 0.73 0.86 1.07 0.92 1 4.4 1.6 5.73 2.66 0.92 4.69 4.32 2.01 2.5 3.8

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191

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Table A.2 (continued) Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

1.2 1.29 0.21 2.41 2.95 7.39 6.58 6.51 5.66 4.28 4.89 3.36 2.8 2.7 2.54 1.26 1.2 0.97 0.53 6.21 5.05 4.09 4.23 5.12 5.95 7.93 7.98 6.84 0.54 4.61 3.05 2.68 3.11 3.13 0.95 8.68 4.19 1.65

1.19 1.28 0.21 2.4 2.94 7.36 6.56 6.49 5.65 4.27 4.87 3.35 2.79 2.68 2.53 1.25 1.2 0.96 0.52 6.19 5.04 4.08 4.21 5.1 5.93 7.9 7.95 6.81 0.54 4.6 3.03 2.67 3.1 3.11 0.94 8.65 4.17 1.65

1.21 1.3 0.21 2.43 2.98 7.44 6.63 6.56 5.71 4.32 4.93 3.39 2.83 2.72 2.57 1.27 1.22 0.98 0.53 6.26 5.09 4.13 4.26 5.16 5.99 7.98 8.03 6.89 0.55 4.65 3.07 2.7 3.14 3.16 0.96 8.73 4.22 1.67

Table A.3 Comparison for CTODc using three series of c1, c2 and w0 unit: lm Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18

13.1 12.8 13.4 12.2 11.6 11.7 11.2 11.6 10.4 10.4 9.8 13.1 13.4 13.2 12.7

12.6 12.4 12.9 11.8 11.2 11.3 10.8 11.2 10.1 10.1 9.5 12.6 12.9 12.7 12.2

14.1 13.8 14.4 13.1 12.4 12.5 11.9 12.4 11.1 11.1 10.4 14.1 14.5 14.2 13.6 (continued on next page)

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Table A.3 (continued) Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40 B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

12.6 12.7 11.8 11.8 11.5 11.9 10.4 10.6 10.9 10.7 10.8 12.8 11.4 13.2 12.1 10.7 12.9 12.8 11.8 12 12.6 11.1 11.2 8.7 12 12.3 13.7 13.6 13.5 13.4 13 13.2 12.7 12.5 12.4 12.3 11.4 11.3 11 10.1 13.5 13.2 13 13 13.2 13.4 13.8 13.8 13.6 10.2 13.1 12.6 12.4 12.6 12.6 11 13.9 13 11.8

12.1 12.3 11.4 11.4 11.1 11.5 10.1 10.3 10.6 10.4 10.5 12.3 11.1 12.7 11.7 10.4 12.4 12.3 11.4 11.6 12.1 10.7 10.8 8.4 11.6 11.8 13.2 13 13 12.8 12.5 12.7 12.2 12 12 11.9 11 11 10.7 9.8 13 12.7 12.5 12.5 12.7 12.9 13.3 13.3 13.1 9.9 12.6 12.1 12 12.1 12.2 10.6 13.4 12.5 11.4

13.5 13.7 12.6 12.6 12.3 12.7 11.1 11.3 11.7 11.4 11.6 13.7 12.2 14.1 13 11.4 13.8 13.7 12.6 12.9 13.5 11.8 11.9 9.2 12.9 13.1 14.8 14.6 14.6 14.4 14 14.2 13.6 13.4 13.3 13.2 12.2 12.1 11.8 10.8 14.5 14.2 13.9 14 14.2 14.4 14.9 14.9 14.6 10.8 14.1 13.5 13.3 13.5 13.5 11.7 15 13.9 12.6

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191

2189

Table A.4 pffiffiffiffi Comparison for K sIC using three series of c1, c2 and w0 unit: MPa m Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

B1 B3 B4 B5 B7 B8 B9 B10 B11 B12 B13 B15 B16 B17 B18 B19 B20 B21 B22 B24 B25 B26 B27 B28 B29 B30 B31 B32 B33 B34 B35 B36 B37 B38 B39 B40 B41 B42 B43 B44 B45 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16

1.588 1.575 1.595 1.538 1.502 1.507 1.481 1.5 1.472 1.472 1.49 1.589 1.595 1.591 1.565 1.559 1.571 1.511 1.508 1.496 1.514 1.472 1.472 1.475 1.473 1.474 1.573 1.493 1.589 1.531 1.473 1.578 1.571 1.508 1.526 1.561 1.479 1.482 1.6 1.524 1.54 1.773 1.762 1.762 1.747 1.718 1.731 1.693 1.676 1.673 1.669 1.643 1.644 1.652 1.72 1.756 1.735

1.569 1.556 1.576 1.518 1.483 1.488 1.462 1.481 1.452 1.452 1.47 1.57 1.577 1.572 1.546 1.541 1.551 1.492 1.489 1.477 1.495 1.452 1.452 1.455 1.453 1.454 1.554 1.473 1.571 1.512 1.453 1.559 1.553 1.489 1.507 1.543 1.46 1.463 1.58 1.504 1.522 1.753 1.743 1.742 1.729 1.698 1.712 1.674 1.657 1.653 1.649 1.623 1.623 1.63 1.697 1.737 1.716

1.625 1.613 1.632 1.576 1.542 1.546 1.521 1.539 1.513 1.513 1.533 1.626 1.632 1.628 1.603 1.598 1.608 1.55 1.548 1.535 1.553 1.513 1.513 1.515 1.513 1.513 1.611 1.532 1.628 1.57 1.513 1.616 1.61 1.548 1.565 1.6 1.52 1.523 1.645 1.562 1.58 1.81 1.8 1.799 1.785 1.756 1.77 1.733 1.716 1.714 1.709 1.686 1.687 1.695 1.765 1.795 1.774 (continued on next page)

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Table A.4 (continued) Numbers of specimen

c1 = 3, c2 = 7, w0 = 160 lm

c1 = 2, c2 = 6.3, w0 = 140 lm

c1 = 4, c2 = 8.2, w0 = 200 lm

C17 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34

1.712 1.716 1.737 1.752 1.778 1.779 1.766 1.713 1.725 1.683 1.673 1.686 1.686 1.652 1.784 1.715 1.646

1.693 1.697 1.717 1.733 1.759 1.759 1.747 1.691 1.706 1.664 1.653 1.666 1.666 1.632 1.765 1.696 1.625

1.752 1.755 1.775 1.791 1.815 1.816 1.804 1.761 1.764 1.723 1.713 1.726 1.726 1.697 1.821 1.755 1.688

References [1] Kaplan MF. Crack propagation and the fracture of concrete. J Am Concrete Inst 1961;58(5):591–610. [2] Wittmann FH, editor. Fracture mechanics of concrete. Amsterdam: Elsevier; 1983. [3] Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 1976;6:773–82. [4] Bazˇant ZP, Oh BH. Crack band theory for fracture of concrete. Mater Struct 1983;16(93):155–77. [5] Jenq YS, Shah SP. Two parameter fracture model for concrete. J Engng Mech 1985;111(10):1227–41. [6] Bazˇant ZP, Kazemi MT. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int J Fracture 1990;44:111–31. [7] Swartz SE, Go CG. Validity of compliance calibration to cracked concrete beams in bending. Exp Mech 1984;24(2):129–34. [8] Swartz SE, Refai TME. Influence of size on opening mode fracture parameters for precracked concrete beams in bending. In: Shah SP, Swartz SE, editors. Proceedings of SEM-RILEM international conference on fracture of concrete and rock, Houston, Texas; 1987. p. 242–54. [9] Karihaloo BL, Nallathambi P. An improved effective crack model for the determination of fracture toughness of concrete. Cement Concrete Res 1989;19:603–10. [10] Karihaloo BL, Nallathambi P. Effective crack model for the determination of fracture toughness ðK sIC Þ of concrete. Engng Fract Mech 1990;35(4/5):637–45. [11] Xu SL, Reinhardt HW. Determination of the double-K fracture parameters in standard three-point bending notched beams. In: Mihashi H, Rokugo K, editors. Fracture mechanics of concrete structures, proceedings FRAMCOS-3. Germany: Aedificatio Publishers; 1998. p. 431–40. [12] Xu SL, Reinhardt HW. Determination of double-K criterion for crack propagation in quasi-brittle materials, part I: experimental investigation of crack propagation. Int J Fracture 1999;98:111–49. [13] Xu SL, Reinhardt HW. Determination of double-K criterion for crack propagation in quasi-brittle materials, part II: analytical evaluating and practical measuring methods for three-point bending notched beams. Int J Fracture 1999;98:151–77. [14] Xu SL, Reinhardt HW. Determination of double-K criterion for crack propagation in quasi-brittle materials, part III: compact tension specimens and wedge splitting specimens. Int J Fracture 1999;98:179–93. [15] Xu SL, Reinhardt HW. A simplified method for determining double-K fracture parameters for three-point bending tests. Int J Fracture 2000;104:181–208. [16] Xu SL, Reinhardt HW. Crack extension resistance and fracture properties of quasi-brittle the softening materials like concrete based on the complete process of fracture. Int J Fracture 1998;92:71–99. [17] Jenq YS, Shah SP. A fracture toughness criterion for concrete. Engng Fract Mech 1985;21(5):1055–69. [18] Elices M, Planas J. The equivalent elastic crack: 1. Load-Y equivalences. Int J Fracture 1993;61:159–72. [19] Navalurkar RK, Hsu C-TT. Fracture analysis of high strength concrete members. J Mater Civil Engng 2001;13(3):185–93. [20] Xu SL, Reinhardt HW. Double-K parameters and the cohesive-stress-based KR curve for the negative geometry. In: Li et al., editors. Proceedings of the fifth international conference on fracture mechanics of concrete and concrete structures, USA; 2004. p. 423–30. [21] Zhao YH, Xu SL. Determination of double-G energy fracture criterion for concrete materials. In: Li et al., editors. Proceedings of the fifth international conference on fracture mechanics of concrete and concrete structures, USA; 2004. p. 431–8. [22] Bazˇant ZP. Size effect in blunt fracture: concrete, rock, metal. J Engng Mech 1984;110(4):518–35.

Z. Wu et al. / Engineering Fracture Mechanics 73 (2006) 2166–2191

2191

[23] Bazˇant ZP, Kim J-K, Pfeiffer PA. Nonlinear fracture properties from size effect tests. J Struct Engng 1986;112(2):289–307. [24] Bazˇant ZP, Pfeiffer PA. Determination of fracture energy from size effect and brittleness number. ACI Mater J 1987;84(6):463–80. [25] Bazˇant ZP, Yu Q. Size effect in fracture of concrete specimens and structures: new problems and progress. In: Li et al., editors. Proceedings of the fifth international conference on fracture mechanics of concrete and concrete structures, USA; 2004. p. 153–62. [26] Issa Mohsen A, Issa Mahmoud A, Islam Mohammad S, Chudnovsky A. Size effects in concrete fracture, part I: experimental setup and observations. Int J Fracture 2000;102:1–24. [27] Issa Mohsen A, Issa Mahmoud A, Islam Mohammad S, Chudnovsky A. Size effect in concrete fracture, part II: analysis of test results. Int J Fracture 2000;102:25–42. [28] Zaitsev YV, Kovler KL. Notch sensitivity of concrete and size effect, part I: effect of specimen size and crack length by 3-point bending. Cement Concrete Res 1985;15:979–87. [29] Karihaloo BL, Xiao QZ, Abdalla HM. Strength size effect in quasi-brittle structures. In: Li et al., editors. Proceedings of the fifth international conference on fracture mechanics of concrete and concrete structures, USA; 2004. p. 163–71. [30] Hu XZ, Wittmann FH. Fracture energy and fracture process zone. Mater Struct 1992;25:319–26. [31] Hu XZ. Toughness measurements from crack close to free edge. Int J Fracture 1997;86:L63–8. [32] Hu XZ, Wittmann FH. Size effect on toughness induced by crack close to free surface. Engng Fract Mech 2000;65(2):209–21. [33] Hu XZ. An asymptotic approach to size effect on fracture toughness and fracture energy of composites. Engng Fract Mech 2002;69(5):555–64. [34] Duan K, Hu XZ, Wittmann FH. Size effect on fracture resistance and fracture energy of concrete. Mater Struct 2003;36(256):74–80. [35] Reinhardt HW, Cornelissen HAW, Hordijk DA. Tensile tests and failure analysis of concrete. J Struct Engng 1986;112(11):2462–77. [36] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. St. Louis, MO: Paris Productions Incorporated; 1985. [37] Refai TME, Swartz SE. Fracture behavior of concrete beams in three-point bending considering the influence of size effects. Report No. 190, Engineering Experiment Station, Kansas State University. [38] Bazˇant ZP, Li ZZ. Modulus of rupture: size effect due to fracture initiation in boundary layer. J Struct Engng 1995;121(4):739–46.