Fracture of 0.1 and 2 m long mortar beams under three-point-bending

Accepted Manuscript Fracture of 0.1 and 2m long mortar beams under three-pointbending

Junfeng Guan, Xiaozhi Hu, Xiaohua Yao, Qiang Wang, Qingbin Li, Zhimin Wu PII: DOI: Reference:

S0264-1275(17)30751-7 doi: 10.1016/j.matdes.2017.08.005 JMADE 3264

To appear in:

Materials & Design

Received date: Revised date: Accepted date:

21 May 2017 19 July 2017 3 August 2017

Please cite this article as: Junfeng Guan, Xiaozhi Hu, Xiaohua Yao, Qiang Wang, Qingbin Li, Zhimin Wu , Fracture of 0.1 and 2m long mortar beams under three-point-bending, Materials & Design (2017), doi: 10.1016/j.matdes.2017.08.005

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ACCEPTED MANUSCRIPT Fracture of 0.1 and 2 m long mortar beams under three-point-bending

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Junfeng Guana,b , Xiaozhi Hu b*, Xiaohua Yao a, Qiang Wang a, Qingbin Li c, Zhimin Wu d a School of Civil Engineering and Communication, North China University of Water Resources and Electric Power, Zhengzhou 450045, P. R. China b School of Mechanical and Chemical Engineering, University of Western Australia, Perth WA 6009, Australia c State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, P. R. China d State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

Introduction

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Abstract: Two hugely different mortar beams with the volume ratio close to 1:1000 were tested under three-point-bending (3-p-b) to verify fracture predictions for the large and long (LL) structures from results of the small and short (SS) specimens. They differed by both size and geometry: SS specimens with span length S = 0.1m, width W = 40 mm and S/W = 2.5, and LL beams with S = 2m, W = 500 mm and S/W = 4. This study performed and analysed quasi-brittle fracture of 0.1m SS specimens with initial notch a0 from 1 to 25 mm for determination of tensile strength ft and fracture toughness KIC, from which fracture of 2m LL beams with a0 of 250 mm was accurately predicted. Experiment results from the two vastly different 3-p-b beams, highly heterogeneous SS and nearly homogeneous LL in comparison to aggregates, were analysed by the boundary effect model (BEM). Keywords: size effect; boundary effect model (BEM); aggregate size; tensile strength; fracture toughness

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Real mortar and concrete structures and notched samples tested in laboratories are rarely geometrically similar besides their obvious difference in size. Fracture predictions for large engineering structures based on experiments from small laboratory samples are possible only after a number of differences between the two are carefully considered and properly modelled. These include size, geometry, thickness, loading condition, size ratio by comparing structure and aggregates, and crack length and structure/specimen boundary conditions, as emphasized separately in a number of publications [e.g. 1–20]. The significance of those variables is already evident even we consider small and large three-point-bending (3-p-b) concrete specimens. If the small and large 3-p-b specimens have different span and width (S/W) ratios, or short and long beams, they are not geometrically similar. Two notches/cracks (a0) of a given length in the small and large specimens have different effects on fracture because the un-notched ligaments (W-a0) are different, which has been elucidated by the boundary effect model (BEM) [12, 13]. Similarly, if concrete has a maximum aggregate dmax, the small and large 3-p-b specimens are different by at least one size ratio, either in terms of a0/dmax or (Wa0)/dmax. It is possible that in some cases real mortar or concrete structures under safety consideration may contain short cracks comparable to the notches in small test samples, but in other cases they may even contain well-developed cracks much larger than the size of common laboratory specimens. To address the above issues, in this study, 3-p-b tests of very large and very small mortar _______________________________________________________________________________

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Corresponding author: Tel/Fax: 61-8-6488-2812/61-8-6488-1024.

Email: [email protected]

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specimens (their volume ratio is close to 1000:1) with the maximum sand diameter of 2.5 mm were designed and performed to simulate the task of fracture predictions for large structures using experimental results from small test samples. The small and short (SS) beams have span S = 0.1m, and width W = 40 mm. The large and long (LL) beams or “assumed structures” have S = 2m, and W = 500 mm. It should be emphasized that the logic behind selecting non-geometrically-similar SS and LL beams (S/W = 2.5 and 4, and thickness B = 40 and 150 mm) in this study is that real engineering structures and the common specimens tested in laboratories are seldom geometrically similar and they rarely have the same thickness. Fracture predictions for large structures using experimental results of small and non-geometrically-similar specimens are not only more desirable in engineering practice, but also more challenging to any design models. The non-size-effect tests of the SS beams with W = 40 mm involve different notches, from 1 to 25 mm, so that the influence of notch a0 and un-notched ligament (W-a0) on quasi-brittle fracture of the mortar samples can be methodically studied. The boundary effect model (BEM) [12, 13, 21] dealing with a0 and (W-a0) is capable of analysing quasi-brittle fracture of non-size-effect specimens (W = constant). The latest developments of BEM [22–25], with consideration the maximum aggregate size dmax and the fictitious crack formation Δafic at the peak load in modelling, show the effects of heterogeneous concrete material structures on quasi-brittle fracture can now be studied in terms of the size ratios, a0/dmax, (W-a0)/dmax, and Δafic/dmax. It should also be mentioned that BEM on quasi-brittle fracture of a large plate with a small edge crack [e.g. 12, 13, 25] is consistent with the well-received theory of critical distances (TCD) and finite fracture mechanics developed for elastic and plastic fracture of metals with blunt notches or shallow cracks [e.g. 26–28]. Although TCD and BEM were proposed using different assumptions, the key equations of both TCD and BEM are almost identical as they all contain an added crack component to the initial crack in the common form of stress intensity factors. In other words, the general non- Linear Elastic Fractures, either quasi-brittle fracture of concrete or elastic and plastic fracture of metals can be described by both TCD and BEM. Design of Experimental small/short (SS), and large/long (LL) mortar beams

2.1.

Non-geometrically-similar specimen designs and rationale

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In this study, we tried to make SS and LL beams, illustrated in Fig. 1, as different as possible in terms of size and test geometry although both are under 3-p-b loading condition. The volume ratio of SS: LL beams is close to 1000:1 (938:1). Furthermore, they are nongeometrically-similar since SS samples have S/W = 2.5 and B = 40 mm while LL “structures” have S/W = 4 and B = 150 mm. The non-size-effect tests of the SS specimens with constant width W were performed with different notches from around 1 to 25 mm. The notch of LL structures, a0 = 250 mm, is far greater than any dimension of SS specimens. The LL and SS beam design with a huge volume ratio ≈ 1000:1, rarely reported in literature, aims at addressing the following key questions: (1) Can quasi-brittle fracture of small specimens with highly heterogeneous material structures be used to determine the tensile strength ft and fracture toughness KIC of mortar or concrete-like materials? Here KIC is valid for large concrete structures with very long notch >> dmax. (2) Can non-size-effect test results of small specimens (W = constant) be used to predict failure of large structures of different geometries (close to 1000 times larger in volume)?

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ACCEPTED MANUSCRIPT (3) Does a single fracture relation exist for very small and very large notched beams, which are not geometrically similar? P

LL structure

relatively homogeneous

W/dmax = 200

W = 500 mm

S/W = 4

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S = 2000 mm

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B = 150 mm

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W/dmax = 16 W = 40 mm a0 = 1- 25 mm

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SS specimens

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L = 2200 mm

highly heterogeneous

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a0 = 250 mm

Notch

L = 160 mm S = 100 mm

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Fig.1. 2m LL beam and 0.1m SS specimens with the LL: SS volume ratio close to 1000:1. LL structure with S/W = 4 and SS specimens with S/W = 2.5 are not geometrically similar. Thickness B is also different, 150 mm for LL structure and 40 mm for SS specimen. LL is approximately homogeneous, but SS is highly heterogeneous.

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The above questions and associated beam designs in Fig. 1 are important for both design applications, and fracture testing and modelling as geometrically similar specimens (a0/W = constant, B = constant) have been routinely adopted to study size effect on quasi-brittle fracture of concrete even up to now [e.g. 2, 8, 18, 24, 25, 29, 30]. Since large and complicated structures and small and simple samples tested in laboratories are different in every way one can imagine, a thorough study on the above three questions is fully warranted. It should be mentioned that besides brittle homogeneous materials Linear elastic fracture mechanics (LEFM) principles also apply to large mortar or concrete structures if W/dmax is sufficiently large, e.g. 1000. Since KIC (the asymptotic limit for large mortar or concrete structures) does not apply to quasi-brittle fracture of normal concrete specimens tested in laboratory (e.g. W/dmax < 20), there are seemly two obvious options for modeling of the quasi-brittle fracture: (1) to establish or assume a new non-LEFM criterion for quasi-brittle fracture[e.g. 31, 32], and (2) to approximate the quasi-brittle fracture of small mortar or concrete specimens using the asymptotic limit KIC of large mortar or concrete structures and tensile strength [e.g. 18–25, 29, 30, 33]. Both Options (1) and (2) or approximations need to be verified by experiments. This study adopts Option (2), i.e. to use the two asymptotic limits (KIC of large mortar or concrete structures with long notches and ft for un-notched concrete specimens) to approximate the quasi-brittle fracture of small mortar or concrete specimens commonly tested in laboratories. It should be emphasized experimental results are required to verify both options.

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ACCEPTED MANUSCRIPT 2.2.

Specimen preparations and test setups

Dimension of 2m LL specimen/structure (W= 500 mm, W/dmax = 200) Length L (mm)

Length S (mm)

LL1 LL2

2200

2000

Width B (mm) 152 154

Depth W (mm) 503 510

Notch a0 (mm) 250 241

a0/W

W/dmax

a0/dmax

(W- a0)/dmax

0.50 0.47

201 204

100 96

101 108

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Large specimens

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Table 1.

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All the mortar beams were cast in the same mix proportion, and the water/cement/sand ratio by weight is 0.4/1.0/2.4. The sand used in the test is river sand with the maximum aggregate size of 2.5 mm and a fineness module of 1.2. The cement used in the experiment is No. 32.5 ordinary Portland cement (Ruifeng cement Co, LTD, China), in which the compressive strength is 32.5 MPa at 28 days. The density of mortar specimen is 2225 kg/m3. A 2.5 mm thick steel plate was fixed in the mould for LL beams before casting to make the notch. Notches of small specimens were cut after casting and curing; the notch width was approximately 2.5 mm. The LL and SS specimens are illustrated proportionally in Fig. 1. Because of the relatively high cost and difficulty in large beam preparation and test, only three LL beams were prepared in one go. Unfortunately, one was broken accidentally while being moved to the laboratory (which somewhat highlights the need for testing of small specimens). As a result, only two large specimens LL1, LL2 were tested, and their dimension details were listed in Table 1. Luckily both LL samples yielded almost identical test results.

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In this study, compressive strength, splitting strength and Young's modulus were carried out according to the Chinese Standard JGJ/T 70 2009[34]. LL beams were cast on December 8, 2015 and tested on June 10, 2016. The age of specimens is 186 day. The average compressive strength fcu from three 150 mm × 150 mm × 150 mm cubes is 28.507 MPa, and the deviation coefficient is 0.106. The average splitting strength fts from three 150 mm × 150 mm × 150 mm cubes cast on the same day is 2.433 MPa, and the deviation coefficient is 0.030. The average Young’s modulus E from three 70.7 mm × 70.7 mm × 210 mm cubes cast on the same day is 23.745 GPa, and the deviation coefficient is 0.029. Two batches of SS specimens were cast and tested on different dates. SS-I was cast on November 8, 2015 and tested on June 12, 2016, and SS-II was cast on November 11, 2015 and tested on June 16, 2016. SS-I has 17 specimens denoted from 1 to 17, and SS-II has 24 specimens denoted by 18 to 41. The specimen details and notch lengths are listed in Tables 2 and 3. It should be mentioned that the total combined volume of 41 SS-I and SS-II specimens is barely 4% of one LL beam. Table 2.

SS-I specimens details (W= 40 mm, W/dmax = 16) SS-I 1 2 3 4 5 6 7 8 9 10

Notch a0 (mm) 1.7 2.6 3.0 4.2 4.2 5.8 7.5 8.1 9.6 11.0

a0/W 0.04 0.07 0.08 0.11 0.11 0.15 0.19 0.20 0.24 0.28

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a0/dmax 0.68 1.04 1.20 1.68 1.68 2.32 3.00 3.24 3.84 4.40

(W- a0)/dmax 15.32 14.96 14.80 14.32 14.32 13.68 13.00 12.76 12.16 11.60

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0.31 0.34 0.39 0.53 0.59 0.61 0.64

4.92 5.44 6.24 8.40 9.48 9.80 10.24

11.08 10.56 9.76 7.60 6.52 6.20 5.76

SS-II specimen details (W = 40 mm, W/dmax = 16)

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(W- a0)/dmax 15.48 14.60 14.48 14.32 13.64 13.48 13.32 12.96 12.88 12.88 12.20 12.12 11.88 11.36 11.36 11.32 9.76 9.68 8.20 8.16 8.16 6.60 6.52 6.48

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a0/dmax 0.52 1.40 1.52 1.68 2.36 2.52 2.68 3.04 3.12 3.12 3.80 3.88 4.12 4.64 4.64 4.68 6.24 6.32 7.80 7.84 7.84 9.40 9.48 9.52

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a0/W 0.03 0.09 0.10 0.11 0.15 0.16 0.17 0.19 0.20 0.20 0.24 0.24 0.26 0.29 0.29 0.29 0.39 0.40 0.49 0.49 0.49 0.59 0.59 0.60

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Notch a0 (mm) 1.3 3.5 3.8 4.2 5.9 6.3 6.7 7.6 7.8 7.8 9.5 9.7 10.3 11.6 11.6 11.7 15.6 15.8 19.5 19.6 19.6 23.5 23.7 23.8

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SS-II 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

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For SS-I specimens, the average compressive strength fcu from five 150 mm × 150 mm × 150 mm cubes is 27.781 MPa, with a deviation coefficient of 0.130. The average splitting strength fts from five 150 mm × 150 mm × 150 mm cubes is 2.444 MPa, with a deviation coefficient of 0.110. The average Young's modulus E from five 70.7 mm × 70.7 mm × 210 mm prisms is 23.398 GPa, with a deviation coefficient of 0.103. The average compressive strength fcu,2 from four 70.7 mm × 70.7 mm × 70.7 mm cubes is 37.167 MPa, with a deviation coefficient of 0.067. The average splitting strength fts,2 from four 70.7 mm × 70.7 mm × 70.7 mm cubes is 4.018 MPa, with a deviation coefficient of 0.085. For the SS-II specimens, the average compressive strength fcu from five 150 mm × 150 mm × 150 mm cubes is 29.492 MPa, with a deviation coefficient of 0.141. The average splitting strength fts from five 150 mm × 150 mm × 150 mm cubes is 2.440 MPa, with a deviation coefficient of 0.129. The average Young's modulus E from five 70.7 mm × 70.7 mm × 210 mm prisms is 25.086 GPa, with a deviation coefficient of 0.184. The average compressive strength fcu,2 from four 70.7 mm × 70.7 mm × 70.7 mm cubes is 38.230 MPa, with a deviation coefficient of 0.053. The average splitting strength fts,2 from four 70.7 mm × 70.7 mm × 70.7 mm cubes is 4.138 MPa, with a deviation coefficient of 0.049.

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ACCEPTED MANUSCRIPT The test setups of SS and LL beams are shown in Fig. 2. The SS specimens were tested using a computer-controlled servo-hydraulic closed-loop testing machine with 600 N load capacity. The loading rate was 10 N/s. Two LL specimens were tested by vertical counterforce testing system. All load values are measured using the load cell set. The readings from the load cell were collected in a data acquisition system.

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W = 0.5 m, S = 2 m

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Fig. 2. (a) Experimental setup of LL beam (S = 2 m, B = 150 mm) (b) Test machine and SS specimen (S = 0.1 m, B = 40 mm). The LL: SS volume ratio is close to 1000:1. Determination of ft and KIC from SS Beams using BEM

3.1.

Fictitious crack formation ∆afic at the peak load Pmax and influence of dmax

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Quasi-brittle fracture implies there is micro-crack formation around aggregates at a notch tip at Pmax, and the fictitious crack formation ∆afic cannot be ignored in small concrete specimens. ∆afic at Pmax can also be understood indirectly from what happens after Pmax. Because of its relatively large size, the fully-developed fracture process zone FPZ in concrete after the peak load Pmax has been mapped in detail by acoustic emission and X-ray measurements. The 3D images show that FPZ is strongly influenced by the maximum aggregate size dmax, and micro-crack formations in front of the notch a0 are highly irregular and discontinuous [35–38]. The fictitious crack growth Δafic in concrete at the peak load Pmax is much shorter than the fully-developed FPZ. Recent measurements using a digital image technique [39] show Δafic ≈ dmax in 3-p-b concrete specimens with a0 = 12 and 48 mm and W = 40 and 80 mm (W/dmax = 5 and 10), and dmax = 8 mm. Acoustic emission measurements of 3-p-b granite specimens with the average grain size dav = 10 mm (dav < dmax) show that at the peak loads Δafic ≈ 0.85dav for W/dav = 5, Δafic ≈ 1.4dav for W/dav = 10, Δafic ≈ 2.7dav for W/dav = 20, and Δafic ≈ 3.9dav for W/dav = 40 [40]. We also found in our recent work [22–25] that Δafic ≈ 1-2dmax for concrete specimens with W/dmax ≤ 20, and in most cases the average Δafic ≈ 1.0dmax is sufficient. Δafic around 1-2dmax at the peak load Pmax in a 3-p-b mortar specimen with W/dmax ≤ 20 is illustrated in Fig. 3. The assumed straight-line Δafic in such a highly heterogeneous mortar

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dmax (W-a0) /dmax ≈ 3-15 W-a 0

W/dmax ≈ 5-20

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Fig. 3. Fictitious crack growth Δafic at Pmax is linked to dmax because of the coarse aggregate structure and limited specimen size W (e.g. W/dmax < 20). A discrete number β (= 0, 1, 2) provides a convenient description on discontinuous micro-crack formation around aggregates.

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To reflect the randomness of the coarse aggregate structure and its strong influence on irregular micro-crack formation, Δafic at Pmax needs to be linked to dmax and a discrete number β, i.e. (e.g. β = 0, 0.5, 1.0, 1.5, 2.0 …)

(1)

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Δafic = β·dmax

3.2.

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Even for a given mortar mix and dmax, seemingly identical mortar specimens can still have different aggregate distributions at the notch tips, so the discrete number β is necessary for various micro-crack formations in different specimens. β can also be justified from the viewpoint of aggregate distributions. For instance, even for a given dmax, a mortar mix can still be varied by percentage of smaller aggregates. In this study, the average discrete number β = 1 is adopted for simplicity for SS specimens with W/dmax = 16 while the β-range between 1 and 2 is also considered. Larger β values up to 10 are considered for LL beams to study the influence of different β values for SS and LL beams on quasi-brittle fracture predictions. Boundary effect model (BEM) with dmax and Δafic at Pmax

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The methodology for determination of tensile strength ft and fracture toughness KIC by using small 3-p-b specimens (W/dmax ≤ 20) has recently been reported in detail [22–25]; thus only the final equations and essential explanations are included in the present study. The SS and LL beams considered in this study have S/W = 2.5 and 4. They are not geometrically similar and have different geometry factors. The single BEM equation fully determined by dmax, ft and KIC and a0/W ratio is given below together with the geometry factor Y. Other details can be found in [22–25] and relevant background information on the boundary effect model (BEM) can be found in [12, 13, 21]. Useful relations for 3-p-b concrete specimens [12, 13, 22–25, 41, 42] are provided here.  a   n  f t 1  e  a  

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(2)

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 (1   ) 2  Y ( )  ae     a0 1.12   2 3 (1  2.5  4.49  3.98  1.33 4 ) Y (   (1   )3 / 2 Y (  

for 3-p-b specimen (3)

1.99   (1   )(2.15  3.93  2.7 2 )  (1  2 )(1   )3 / 2 K  a  0.25   IC   ft 

for S/W = 2.5

(4a)

for S/W = 4

(4b)

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The derivation of Eq. (2) for Δafic = 0 and relevant background information was provided in [12, 13, 21], where σn is the nominal strength at the notch tip shown in Fig. 3. The equivalent crack ae is determined by the initial notch a0, α-ratio (=a0/W) and Y(α). The equivalent crack ae instead of the initial notch a0 is used in Eq. (2) because the crack tip conditions are generally influenced by both the front and back specimen boundaries for small concrete specimens. Naturally, if α-ratio is close to zero, the initial notch a0 can be inserted into Eq. (2) to replace the equivalent crack ae. The geometry factor Y(α) for S/W = 2.5 and 4 can be found in [41, 42]. The characteristic crack a*∞ is a material constant and is fully determined by ft and KIC. From Eqs. (2) and (5), it can be shown that KIC = 2σn√(ae + a*∞), which is very similar to one of the key relations of TCD [e.g. 26–28]. Here the approximation 1.18√π ≈ 2 has been used. In both models, it has been realized that the added crack component a*∞ is a material constant. Possibly, TCD and BEM are the closest models available in the literature in both mathematical forms and physical explanations of their parameters. For Δafic ≠ 0 as shown in Fig. 3, the nominal stress σn is given below, which has been used to analyse quasi-brittle fracture of concrete and rock with dmax from 2 to 20 mm [22–25]. S Pmax B

σ n (Pmax ，  d max  

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(6)  2   33   2 4  6(  d max    2 2   1 2  2(  d max  2 3 1 S and B are known for SS and LL beams, and Pmax measurements can be easily obtained from 3-p-b tests of SS specimens. γ1 = W - a0; γ2 = γ1 - ( β·dmax ); γ3 = γ1 + ( β·dmax ). β can be determined if the discrete number β is given, e.g. assuming β = 1 or 2. If ft and KIC are known, quasi-brittle fracture of mortar or Pmax of 3-p-b beams can be predicted by Eq. (2) without the need of curve fitting. Direct measurements of ft and KIC for concrete are not easy, but they can be determined indirectly by analysing 3-p-b test results as shown in [22–25]. A linear relation is obtained by rearranging Eq. (2), i.e. 1 1 1 a 1 4a  2  2  e  2  2e  ( Pmax ，  d max ) f t f t a f t K IC 2 n

(7)

The nominal strength σn can be obtained for every Pmax measurement for a given discrete number β (e.g. = 1). The equivalent crack ae is determined by Eq. (3) with the geometry factor Y(α). Therefore, ft and KIC can be determined from the linear relation of Eq. (7) through detailed data analyses considering possible variations in the discrete number β (e.g. = 1-2). In

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Interpretations of SS and LL beam fracture from ASTM standard and BEM

4.1

ASTM standard for KIC and associated boundary conditions

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The ASTM standard [42] on measurements of fracture toughness KIC under the Linear Elastic Fracture Mechanics (LEFM) conditions given below has effectively specified the front and back specimen boundary zones (BZ) in terms of the initial crack/notch a0 and unnotched ligament (W - a0). B  2  K IC     10  a a0 (8)   2.5 f t   W  a0  Different to the plane stress and plane strain conditions for metal fracture, thickness B influences mortar or concrete fracture in a different way [43], which will be discussed later. The front and back BZ conditions relevant to the SS and LL beam fracture are illustrated in Fig. 4. KIC applies only if a mortar or concrete specimen/structure satisfies both the front and back BZ conditions specified by a0 and (W - a0). Although metals and concrete are different, Eq. (8) can still give an indication on BZ in mortar or concrete specimens. Assuming ft = 4 MPa and KIC = 1.0 MPa·m1/2 for the mortar specimens considered in this study, BZ estimated from the ASTM standard is around 156 mm, or W should be at least over 300 mm. SS specimens only have W = 40 mm, so direct measurement of KIC is not possible. LL beams have W = 500 mm and a0 = 250 mm, so there should be a better chance to have a good estimation of KIC even without detailed knowledge of Δafic at Pmax. However, quasi-brittle fracture of LL beams is certain if the notch is too short or too long, or within the front or back BZ. Fig. 4 schematically illustrates the fracture conditions of SS and LL beams using the concept of BZ [25] based on the ASTM standard and the rough estimations of ft and KIC. Boundary zone ( BZ )

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< 2 BZ




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( ft & KIC )

( ft & KIC )

( KIC )

( ft & KIC )

BZ


BZ

( ft & KIC ) α1 =

=

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α2 =

=

= 0.5

α3 =

=

= 0.75

Quasi-Brittle region

LEFM region

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ACCEPTED MANUSCRIPT Fig.4. Influence of front and back Boundary Zone (BZ) [25] on fracture of SS specimens (WS < 2BZ) and LL structures (WL > 2BZ). Quasi-brittle fracture is observed if the crack tip is within either the front or back BZ.

W1

Small specimens: W = constant & S/W = 2.5

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a2

W1

Tensile strength ft

W1

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KIC

10

σn/ft

100 W/a∞*

1000

10000

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0.1

Large structure: S/W = 4 and a0 = 250 mm

Fracture toughness

Quasi-brittle fracture

0.01

a4

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a3

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a1

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Eq. (2) is schematically illustrated in Fig. 5 together with SS and LL beams, and their fracture conditions as indicated by the ASTM standard. Different geometry factors have been considered by Eqs. (3) and (4) so that both SS and LL beams can be described by a single design curve. While quasi-brittle fracture of SS specimens with different notches would be mostly close to the strength criterion region, fracture of LL beams with the notch tip at the centre (a0 = 250 mm) would be fairly close to the fracture toughness criterion region.

10

100

1000

10000

ae/a∞*

PT E

D

Fig.5. Quasi-brittle fracture of 0.1m SS specimens (S/W = 2.5) is closer to the tensile strength region, and fracture of 2m LL structures (S/W = 4) is closer to the toughness region.

AC

CE

The condition ae/a*∞ ≥ 10 or a0 ≥ 2.5(KIC/ft)2, as specified by the ASTM standard Eq. (8), for a valid KIC measurement or application [22–25] is shown in Fig. 5. Similarly, Eq. (2) can also be used to specify the region where the tensile strength criterion applies, i.e. for ae/a*∞ ≤ 0.1, which is also marked in Fig. 5. The quasi-brittle fracture region is between 0.1 < ae/a*∞ < 10. The equivalent crack ae is used in Fig. 5 and Eq. (2) because the two ASTM requirements on the initial notch a0 and un-cracked ligament (W - a0) are combined into one measurement – the equivalent crack ae. In general, ae < a0 because of the combined boundary influence. A mortar or concrete structure can be approximately considered as “homogeneous” if both the initial notch a0 and un-cracked ligament (W - a0) are much larger than the maximum aggregate dmax. Considering both the ASTM standard or Eq, (8) and Eq, (2), it is clear the characteristic crack length a*∞ > dmax. Our recent studies [22–25] show the value of a*∞ /dmax should be at least around 2 or larger. Supposing 10·a*∞ = 30·dmax (a*∞ = 3·dmax), and if a0/dmax ≥ 50 and (W - a0)/dmax ≥ 50; KIC will be the dominant fracture criterion so that LEFM applies. If a0/dmax ≤ 0.05 and (W - a0)/dmax ≤ 0.05, ft will be the dominant failure criterion. 4.2

Estimation of KIC based on tests of 2m long LL beams

Supposing that a*∞ is around 3·dmax, the ASTM standard considering the front and back boundaries separately would show the initial notch a0 and un-notched ligament (W - a0) should be at least 300 mm. The 2m LL beams have a0 and (W - a0) around 250 mm. Although

10

ACCEPTED MANUSCRIPT

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250 mm may not satisfy the ASTM standard, it could be close enough to give a reasonable estimation of KIC. Based on the ASTM standard [42], the fracture toughness KIC, can be calculated using the peak load Pmax, i.e.  P S  K IC   max3 / 2  f ( ) (9a)  BW  where 1.99   (1   )(2.15  3.93  2.7 2 )  f (   3 1/ 2  (9b)  2(1  2 )(1   ) 3 / 2   The fracture toughness KIC approximately estimated by Eq. (9) and experimentally measured Pmax from the two 2m LL beams (W = 500 mm) were 0.85 and 0.88 MPa·m1/2, respectively. The true fracture toughness KIC for the LL beams of 150 mm in thickness should be slightly higher than that for the SS specimens of 40 mm in thickness [43]. Data analysis of 0.1m SS specimens and fracture prediction for 2m LL beams

5.1

Determination of mortar properties, ft and KIC, using 0.1m SS specimens

NU

SC

5.



1/2

ft = 4.81 MPa KIC= 0.81 MPam a = 7.1 mm

0.1

0.0 Curve fitting

AC

Curve fitting

SS-I test data

-0.1 0.5 0.2

1.0

SS-I test data

SS-II test data

1.5

2.0

-0.1 0.5

2.5



SS-II test data

1.5

2.0

2.5

ae 1/2

0.2

 = 1.5

1/2

ft = 3.95 MPa KIC= 0.95 MPam 

a = 14.6 mm

 = 2.0





1/n

0.1

1/n

0.1

0.0

0.0 Curve fitting SS-I test data

-0.1 0.5

1.0

ae

ft = 4.34 MPa KIC= 0.86 MPam

a = 9.8 mm

 = 1.0



= 0.0

CE

0.1

1/n

0.2





a = 4.0 mm

0.0

1/2

ft = 6.09 MPa KIC= 0.77 MPam

1/n

0.2

PT E

D

MA

The maximum fracture loads of 0.1m SS test specimens were analysed by Eqs. (1) to (7) to determine the mortar properties ft and KIC. The discrete number  in Eq. (6) needs to be preselected so that the nominal strength n can be evaluated. According to our recent studies [22–25],  is fairly close to one if W/dmax < 20. In this study, the 0.1m SS specimens have W = 40 mm and dmax about 2.5 mm so that W/dmax = 16 and  ≈ 1 should be a good approximation. However, to show the general trend and influence of the discrete number  on estimations of ft and KIC, results for  = 0, 1, 1.5 and 2 are shown in Fig. 6 for the combined SS-I and SSII data, listed in Tables 2 and 3.

1.0

1.5

Curve fitting SS-II test data 2.0

SS-I test data 2.5

-0.1 0.5

1.0

1.5

SS-II test data 2.0

2.5

ae ae Fig. 6. Determination of ft and KIC from combined SS-I and SS-II test samples,

11

ACCEPTED MANUSCRIPT for different Δafic (β = 0.0, 1.0, 1.5 and 2.0) ( W = 40 mm, W/dmax = 16 )

1/2

0.2

ft = 4.75 MPa KIC= 0.77 MPam 



 = 1.0

a = 6.5 mm

a = 5.6 mm 0.1



1/n



1/n

0.1

1.0

1.5

2.0

 = 1.0

Curve fitting

SS-I test data 2.5

-0.1 0.5

NU

Curve fitting -0.1 0.5

SC

0.0

0.0

1/2

ft = 5.01 MPa KIC= 0.75 MPam

RI

0.2

PT

It is clear from the Fig. 6 that ft is overestimated if the fictitious crack growth Δafic is not considered, or real quasi-brittle fracture behaviour is approximated by brittle fracture. The specific fracture energy of concrete specimens is lower if the thickness is reduced [43]. Similarly, the fracture toughness KIC of 0.1m SS specimens with thickness B = 40 mm should be lower than that of 2m LL beams with thickness B = 150 mm. Using the estimations in Section 4.2, KIC = 0.85 – 0.88 MPa·m1/2 for B = 150 mm, the estimation of KIC = 0.81 MPa·m1/2 with β = 1.0 for thickness B = 40 mm appears to be reasonable. This is consistent with our other recent studies [22–25].

1.0

1.5

SS-II test data 2.0

2.5

MA

ae ae Fig. 7. Determination of ft and KIC separately from SS-I and SS-II (for β = 1.0).

CE

PT E

D

For the purpose of comparison, SS-I and SS-II test samples have also been analysed separately and shown in Fig. 7 for β = 1.0. In comparison with ft = 4.81 MPa and KIC = 0.81 MPam1/2 determined from the combined SS-I and SS-I specimens with β = 1.0, the separate results in Fig. 7 from either SS-I or SS-II are varied by only around 7%. Therefore, the combined results in Fig. 6 are adopted in this study. To have a better understanding of the discrete number β, and its influence on estimations of ft and KIC, β = 0 – 3 with a fine increment of 0.1 are listed in Table 4 together with estimated ft and KIC values, and the characteristic crack a*∞ . As previously discussed, the ratio a*∞ /dmax should be at least around 2 or larger. If a*∞/dmax = 2.0 – 3.0dmax, it would indicate a*∞ = 5.0 – 7.5 mm, so that the results in Table 4 with β < 1.0 can be excluded.

AC

Table 4. Estimation of ft and KIC from SS samples (W = 40 mm, dmax = 2.5 mm, W/dmax = 16) β 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 1.3 1.4

ft (MPa) 6.09 5.93 5.78 5.64 5.51 5.38 5.26 5.14 5.03 4.91 4.81 4.71 4.61 4.52 4.43

KIC (MPa·m1/2) 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.79 0.80 0.80 0.81 0.82 0.83 0.84 0.85

12

a*∞ (mm) 4.0 4.2 4.5 4.7 5.0 5.2 5.6 5.9 6.3 6.7 7.1 7.5 8.1 8.6 9.3

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.4 2.6 2.8 3.0

4.34 4.26 4.17 4.10 4.02 3.95 3.88 3.81 3.68 3.56 3.45 3.34

0.86 0.88 0.89 0.91 0.93 0.95 0.99 1.01 1.08 1.20 1.35 1.58

9.8 10.6 11.5 12.4 13.4 14.6 16.2 17.6 21.7 28.1 38.2 55.9

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ACCEPTED MANUSCRIPT

1.2

[45] This study

fcu,1 (MPa)

Ottawa Sand 9.5 2.5

fcu,2 (MPa) 33.76 47.75 70.07

*fc (MPa)

fts,1 (MPa)

fts,2 (MPa)

28.637

Fracture toughness (MPa·m1/2) 0.773 – 1.137 0.992 – 1.444 1.053 – 1.855

46.6

0.452 – 0.487

56.1

0.731 – 0.800 0.81 – 0.86

D

[44]

Value of fracture toughness of mortar from different researchers

PT E

Reference

dmax (mm)

MA

Table 5.

NU

SC

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Considering KIC estimations from 2m LL beams based on Eq. (9), the results in Table 4 suggests that ft and KIC for β = 1.0 – 1.5 should be considered, which are in fact fairly close to the values commonly used for mortar materials (in this study, the average experimental splitting strength fts from ten 150 mm × 150 mm × 150 mm cubes is 2.442 MPa; the average experimental splitting strength fts,2 from eight 70.7 mm × 70.7 mm × 70.7 mm cubes is 4.078 MPa). In a way, 3-p-b tests of SS specimens with W = B = 40 mm can be considered to be similar to splitting of 40 × 40 mm cubes, which would have even higher fts.

37.696

2.442

4.078

Method Double K model ASTM BEM

*sample size is not given

Failure or design curves of the mortar constructed by the estimated ft and KIC

CE

5.2

AC

As discussed in the previous sections, ft and KIC can be fully determined from Pmax measurements after the condition β = 1 is affirmed, which is the case to be emphasized in this study. However, in order to give readers more general pictures, the failure curves or design diagrams for β = 1.0 and 2.0 are shown in Figs. 8 and 9. By using Eq. (2), we can define the region where ft is dominant (ae/a*∞ < 0.1), the quasibrittle fracture region (0.1 < ae/a*∞ < 10), and the LEFM region where 2m LL mortar beams are close to pure elastic and “homogeneous” (ae/a*∞ > 10). Although two measurements were obtained from LL beams, they are very consistent as shown in Figs. 8 and 9. This is because the initial notch a0 is around 250 mm, any uncertainty in notching and associated microcracks would be limited to around the maximum aggregate dmax = 2.5 mm, which is only 1% of a0.

13

ACCEPTED MANUSCRIPT

1.00

W = 1.3 m (S/W = 2.5) W = 1.1 m (S/W = 4)  = 0.2



a = 7.1 mm

0.50



ae / a  = 10

0.00 -0.25

SS-I SS-II LL with =  LL with =  LL with =  LL with =  LL with = 

-0.50 -0.5

0.0

0.5

1/2

KIC = 0.81 MPam

1.0

1.5 2.0 logae

(a)

Prediction curve

NU

0.40

D

0.30

1.3

PT E

0.25 1.2

3.0

3.5

LL with =  LL with =  LL with =  LL with =  LL with =  1/2

KIC = 0.81 MPam

MA

logn

0.45

0.35

2.5

SC

0.50

PT

0.25

RI

logn

0.75

 = 1.0

dmax = 2.5 mm

ft = 4.81 MPa  ae / a  = 0.1

1.4

1.5

1.6

1.7

logae

(b)

CE

Fig. 8. (a) Failure prediction of 2m LL “structures” using the mortar properties determined from 0.1m SS specimens, i.e. ft = 4.81 MPa and KIC = 0.81 MPa·m1/2 (assuming the discrete number β = 1.0); (b) enlarged details for the LL beam results. We have examined the situations that larger specimens have longer Δafic [25], i.e. although

AC

 = 1 is a good approximation for 0.1m SS specimens,  should be much bigger for 2m LL beams. Fig. 8 (b) shows that with increasing , the LL results from Eq. (6) are indeed getting closer to the prediction based on ft and KIC determined from 0.1m SS specimens. Since 2m LL beams have thickness B = 150 mm, KIC (B = 150 mm) should be higher than KIC (B = 40 mm) for 0.1m SS specimens according to the fracture energy measurements [43]. Fig. 8 (b) seems to suggest  = 4 – 8 is more appropriate. Based on the estimated  range, Δafic = 10 – 20 mm so that Δafic/a0 = 4 – 8 %, which suggests that 2m LL beams with W = 500 mm should be very close to the LEFM conditions. The LEFM conditions based on Eq. (2) are marked by ae/a*∞ = 10 in Fig. 8 (a), which shows W > 1 m. For comparison, Fig. 9 shows the estimations for  = 2 for 0.1m SS specimens, where KIC = 0.95 MPa·m1/2 is already higher than the estimations of 0.85 – 0.88 MPa·m1/2 from Eq. (9). It should not be the case as KIC of thicker LL beams should be higher than that of thinner SS specimens. The details in Fig. 9 (b) suggest  has to be limited to 6 for 2m LL beams. However, Fig. 9 (a) indicates W > 2 m for the LEFM conditions. Considering all the results in

14

ACCEPTED MANUSCRIPT Figs. 8 and 9 and estimations from Eq. (9), it is clear that  = 1.0 for 0.1m SS specimens is more reasonable, and  can go up to 8.0 for 2m LL beams.



ae / a  = 0.1 

0.75

a = 14.6 mm

0.50

W = 2.7 m (S/W = 2.5) W = 2.3 m (S/W = 4)  = 0.2 

0.25

ae / a  = 10

-0.25 -0.50 -0.5

0.0

0.5

1/2

KIC = 0.95 MPam

1.0

1.5 2.0 logae

(a)

NU MA

logn

0.45 0.40

PT E

D

0.35

0.25 1.2

2.5

LL LL LL LL LL

0.50

0.30

RI

0.00

PT

SS-I SS-II LL with =  LL with =  LL with =  LL with =  LL with = 

1.3

3.0

SC

logn

 = 2.0

dmax = 2.5 mm

ft = 3.95 MPa

1.00

3.5

with =  with =  with =  with =  with =  1/2

KIC = 0.95 MPam

Predction curve

1.4

1.5 logae

1.6

1.7

1.8

5.3

AC

CE

(b) Fig. 9. (a) Failure prediction of 2m LL “structures” using the mortar properties determined from 0.1m SS specimens, i.e. ft = 3.95 MPa and KIC = 0.95 MPa·m1/2 (assuming the discrete number β = 2.0); (b) enlarged details for the LL beam results. Determination of ft and KIC using both 0.1m SS and 2m LL beams

The linear relation shown in Figs. 6 and 7 indicates that two points, which are sufficiently apart, can determine ft and KIC reliably. In other words, one can test two groups of very small and very large specimens to extrapolate ft and KIC. Karihaloo et al. [46] have used a similar technique for evaluation of the size independent fracture energy, which also relies on two points that are sufficiently apart. The 0.1m SS and 2m LL beams considered in the present study are sufficiently apart except the concerns that they have different fracture toughness values, KIC (B = 150 mm) > KIC(B = 40 mm), and they have different β values. According to the specific fracture energy variations analysed by Duan and Hu [43], KIC (B = 150 mm) could be around 10% higher than KIC (B = 40 mm). If we ignore the limited difference, Eq. (7) can still be used to estimate an averaged KIC for both SS and LL beams.

15

ACCEPTED MANUSCRIPT Fig. 10 shows the linear relations of Eq. (7) for SS specimens with β = 1.0 and LL beams with β = 3, 6, 8 and 10. Following the suggested β range from Fig. 9, β = 6 and 8 is selected for LL beams, so that ft = 4.68 – 4.72 MPa and KIC = 0.92 – 0.88 MPa·m1/2 as the averaged properties for both SS and LL beams. This is reasonable in comparison with that ft = 4.81 MPa and KIC = 0.81 MPa·m1/2 for SS specimens with B = 40 mm. Fig. 10 also suggests some variations in the discrete number β for LL beams do not alter the estimation much, this is because the ratio (a0 + Δafic)/W does not vary much with β for LL beams. 0.6

1/2

ft = 4.61 MPa KIC= 0.99 MPam 

a = 11.5 mm

1/2

ft = 4.68 MPa KIC= 0.92 MPam 

 = 1.0 — 3.0

a = 9.5 mm 



1/n

0.3

1/n

0.3

0.0 W = 40 mm W = 500 mm

0

5

10

W = 40 mm W = 500 mm

Curve fitting

15

20

25

-0.3

SC

-0.3

30

1/2

ft = 4.72 MPa KIC= 0.88 MPam 

 = 1.0 — 8.0

10

15

20

25

30

0.6

1/2

ft = 4.77 MPa KIC= 0.84 MPam 

a = 7.7 mm

 = 1.0 — 10.0

0.3



1/n



1/n

0.3

5

Curve fitting

ae (b) β = 1.0 for SS and β = 6.0 for LL

MA

a = 8.6 mm

0

NU

ae (a) β = 1.0 for SS and β = 3.0 for LL 0.6

RI

0.0

 = 1.0 — 6.0

PT

0.6

W = 40 mm W = 500 mm

0

5

10

0.0 W = 40 mm W = 500 mm

Curve fitting

15

20

PT E

-0.3

D

0.0

25

30

ae (c) β = 1.0 for SS and β = 8.0 for LL

-0.3

0

5

10

Curve fitting

15

20

25

30

ae (d) β = 1.0 for SS and β = 10.0 for LL

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Fig. 10. Estimation of ft and KIC for β = 1.0 for SS specimens and different β values for LL beams

AC

Again for comparison, estimations with β = 2.0 for SS specimens and different β values for LL beams are shown in Fig. 11. Overall, ft appears to be lower and the acceptable KIC = 0.89 – 0.85 MPa·m1/2 in comparison with the estimations from Eq. (9) is for the β range of 10 – 12. However, Fig. 9 (b) suggests β cannot be less than 6. This awkward unexplainable situation is due to the wrong assumption of β = 2 for SS specimens.

16

ACCEPTED MANUSCRIPT 0.6

0.6

1/2

ft = 3.93 MPa KIC= 1.00 MPam 

1/2

ft = 3.96 MPa KIC= 0.94 MPam 

 = 2.0 — 6.0

a = 16.2 mm

a = 14.2 mm 



1/n

0.3

1/n

0.3

 = 2.0 — 8.0

0.0

0.0 W = 40 mm W = 500 mm

5

10

15

20

25

30

ae (a) β = 2.0 for SS and β = 6.0 for LL 0.6

0.6

1/2



a = 11.3 mm

15

20

25

30

0.0 W = 40 mm W = 500 mm

0

5

Curve fitting

10

15

20

25

-0.3

30

0

W = 40 mm W = 500 mm

5

10

Curve fitting

15

20

25

30

ae (d) β = 2.0 for SS and β = 12.0 for LL

MA

ae (c) β = 2.0 for SS and β = 10.0 for LL

NU

0.0

1/2

 = 2.0 — 12.0

SC





1/n

0.3

1/n

0.3

-0.3

10

ft = 4.02 MPa KIC= 0.85 MPam

 = 2.0 — 10.0

a = 12.6 mm

5

ae (b) β = 2.0 for SS and β = 8.0 for LL

ft = 3.99 MPa KIC= 0.89 MPam 

0

Curve fitting

PT

0

-0.3

RI

-0.3

W = 40 mm W = 500 mm

Curve fitting

Fig. 11. Estimation of ft and KIC for β = 2.0 for SS specimens and different β values for LL beams

PT E

D

For the purpose of comparison, the estimations using various β values for both SS and LL beams are listed in Tables 8 and 9 so that readers can check the corresponding variations in ft and KIC. Table 8. Estimation of ft and KIC for individual discrete number β (W = 40 and 500 mm )

AC

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β = 1.0 for W = 40 mm; β = 2.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 3.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 4.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 5.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 6.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 8.0 for W = 500 mm β = 1.0 for W = 40 mm; β = 10.0 for W = 500 mm β = 2.0 for W = 40 mm; β = 6.0 for W = 500 mm β = 2.0 for W = 40 mm; β = 8.0 for W = 500 mm β = 2.0 for W = 40 mm; β= 10.0 for W = 500 mm β = 2.0 for W = 40 mm; β = 12.0 for W = 500 mm

ft (MPa) 4.59 4.61 4.64 4.66 4.68 4.72 4.77 3.93 3.96 3.99 4.02

KIC (MPa·m1/2) 1.03 0.99 0.96 0.94 0.92 0.88 0.84 1.00 0.94 0.89 0.85

a*∞ (mm) 12.5 11.5 10.8 10.2 9.5 8.6 7.7 16.2 14.2 12.6 11.3

Table 9. Peak load Pmax of all specimens and σn with different β Specimens

Pmax (KN)

β = 0.0

β = 1.0

LL1

8.80

2.71

2.66

σn (MPa) β = 1.0 and 10.0 β = 2.0 for W = 40 and 500 mm 2.61 2.27

17

β = 2.0 and 8.0 for W = 40 and 500 mm 2.34

ACCEPTED MANUSCRIPT

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2.36 3.35 3.08 3.45 3.54 3.60 3.45 3.28 3.29 3.67 4.23 3.34 4.11 4.24 4.47 3.36 4.84 4.74 3.65 3.40 3.49 3.33 4.11 3.45 4.28 3.32 3.81 3.92 3.74 3.97 3.73 4.01 3.24 3.57 4.62 3.28 4.84 4.91 3.62 4.82 4.22 3.32

PT

RI

2.29 3.74 3.44 3.87 3.98 4.04 3.89 3.72 3.74 4.19 4.85 3.85 4.77 4.96 5.40 4.15 6.02 5.96 4.07 3.81 3.91 3.73 4.64 3.89 4.83 3.76 4.32 4.45 4.27 4.53 4.27 4.61 3.73 4.10 5.40 3.84 5.79 5.87 4.33 5.95 5.21 4.11

SC

NU

2.61 3.35 3.08 3.45 3.54 3.60 3.45 3.28 3.29 3.67 4.23 3.34 4.11 4.24 4.47 3.36 4.84 4.74 3.65 3.40 3.49 3.33 4.11 3.45 4.28 3.32 3.81 3.92 3.74 3.97 3.73 4.01 3.24 3.57 4.62 3.28 4.84 4.91 3.62 4.82 4.22 3.32

MA

2.66 3.74 3.44 3.87 3.98 4.04 3.89 3.72 3.74 4.19 4.85 3.85 4.77 4.96 5.40 4.15 6.02 5.96 4.07 3.81 3.91 3.73 4.64 3.89 4.83 3.76 4.32 4.45 4.27 4.53 4.27 4.61 3.73 4.10 5.40 3.84 5.79 5.87 4.33 5.95 5.21 4.11

D

2.71 4.23 3.90 4.39 4.53 4.61 4.46 4.29 4.33 4.88 5.69 4.54 5.67 5.98 6.82 5.42 7.97 8.04 4.60 4.33 4.45 4.26 5.32 4.47 5.56 4.34 4.99 5.14 4.96 5.28 4.98 5.42 4.39 4.83 6.51 4.64 7.20 7.31 5.39 7.75 6.81 5.38

PT E

10.07 1.66 1.46 1.60 1.55 1.58 1.39 1.21 1.17 1.20 1.28 0.93 1.05 0.95 0.66 0.38 0.51 0.44 1.84 1.54 1.56 1.45 1.65 1.35 1.64 1.21 1.38 1.42 1.23 1.29 1.17 1.17 0.94 1.03 1.03 0.72 0.81 0.81 0.60 0.56 0.48 0.38

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LL2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Using the averaged ft and KIC for both SS and LL beams, the failure or design diagrams for β = 1.0 and 5.0, β = 1.0 and 10.0, β = 2.0 and 6.0, and β = 2.0 and 12.0 are shown in Figs. 12 and 13. It is interesting to observe that while the data variations among the SS specimens are very small in Figs. 10 and 11, they appear to be more scattered in Figs. 12 and 13 in the form of design diagrams.

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1.00



ae / a  = 0.1

W = 1.9 m (S/W = 2.5) W = 1.6 m (S/W = 4)  = 0.2

0.50



ae / a  = 10 0.25 1/2

KIC = 0.94 MPam

SS-I SS-II LL

-0.25 -0.50 -0.5

0.0



a = 10.2 mm 0.5

1.0

1.5 2.0 logae

2.5

(a)

3.0

 = 1.0 - 10.0

dmax = 2.5 mm

ft = 4.77 MPa 

ae / a  = 0.1

W = 1.4 m (S/W = 2.5) W = 1.2 m (S/W = 4)  = 0.2

NU

0.75 0.50

3.5

SC

1.00

PT

0.00

RI

logn

0.75



ae / a  = 10

0.00

MA

0.25 SS-I SS-II LL

-0.25 -0.50 -0.5

0.5

a = 7.7 mm

1.0

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0.0

1/2

KIC = 0.84 MPam



D

logn

 = 1.0 - 5.0

dmax = 2.5 mm

ft = 4.66 MPa

1.5 2.0 logae

2.5

3.0

3.5

(b)

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Fig. 12. (a) Fracture transition using ft = 4.66 MPa and KIC = 0.94 MPa·m1/2 (β = 1.0 and 5.0 for W = 40 and 500 mm) (b) Fracture transition using ft = 4.77 MPa and KIC = 0.84 MPa·m1/2 (β = 1.0 and 10.0 for W = 40 and 500 mm) 1.00

ft = 3.93 MPa

dmax = 2.5 mm  = 2.0 - 6.0 

ae / a  = 0.1

logn

0.75 0.50

W = 3 m (S/W = 2.5) W = 2.5 m (S/W = 4)  = 0.2 

ae / a  = 10

0.25

1/2

KIC = 1.00 MPam

0.00 -0.25 -0.50 -0.5

SS-I SS-II LL

0.0



a = 16.2 mm 0.5

1.0

1.5 2.0 logae

19

2.5

3.0

3.5

ACCEPTED MANUSCRIPT (a) ft = 4.02 MPa  ae / a  = 0.1

1.00

dmax = 2.5 mm  = 2.0 - 12.0 W = 2.1 m (S/W = 2.5) W = 1.8 m (S/W = 4)  = 0.2

0.75

logn

0.50



ae / a  = 10

0.25

1/2

SS-I SS-II LL

-0.50 -0.5

0.0



a = 11.3 mm 0.5

1.0

1.5 2.0 logae

3.0

3.5

SC

(b)

2.5

RI

-0.25

PT

KIC = 0.85 MPam

0.00

Conclusions

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Fig. 13. (a) Fracture transition using ft = 3.93 MPa and KIC = 1.00 MPa·m1/2 (β = 2.0 and 6.0 for W = 40 and 500 mm) (b) Fracture transition using ft = 4.02 MPa and KIC = 0.85 MPa·m1/2 (β = 2.0 and 12.0 for W = 40 and 500 mm)

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Following are the three questions raised at the beginning of this study: (1) Can quasi-brittle fracture of small specimens with highly heterogeneous material structures be used to determine the tensile strength ft and fracture toughness KIC of mortar or concrete-like materials? (2) Can non-size-effect test results of small specimens (W = constant) be used to predict failure of large structures of different geometries (close to 1000 times larger in volume)? (3) Does a single fracture relation exist for very small and very large notched beams, which are not geometrically similar? In this study two different groups of 0.1m SS mortar specimens have been used to determine the tensile strength ft and fracture toughness KIC (valid for large mortar or concrete structures with very long notches). These SS specimens are designed for the boundary effect study as the size W = constant, and they only have a small W/dmax ratio of 16. Although the volume ratio of SS and LL beams is close to 1:1000, quasi-brittle fracture (close to LEFM) of the LL beams has been successfully predicted. The fracture predictions for the LL beams are performed using ft and KIC from the SS specimens despite of the facts that they not geometrically similar, and are different in thickness. The predictions are possible because Eq. (2) is able to describe the strength-controlled failure, toughness-controlled brittle fracture and quasi-brittle fracture in between. Therefore, the above three questions have been addressed in the present study. This study is different to normal size effect studies in which geometrically similar specimens (α = a0/W = constant, size W ≠ constant) are commonly used [e.g. 8, 18, 29, 30, 47]. In the most recent tests on geometrically similar specimens [18, 29, 30, 47], dmax = 10 mm, W = 40 – 500 mm, B = 40 mm, and S/W = 2.5. The volume ratio of smallest to largest specimens is around 1:160, in comparison with 1:1000 in the present study. The size ratio of dmax to size W of the largest specimen is 1:50, in comparison with 1:200 in the present study. This study and our recent publications [22–25] show that the boundary effect model (BEM)

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are more flexible in terms of specimen designs, and more importantly BEM is a predictive model, i.e. curve-fitting is not required if the fundamental material properties, ft and KIC, are known. The results in Figs. 10 and 11 warrant further study since it is enough to use two points, sufficiently away from each other, to determine a straight-line relation (which defines both the tensile strength ft and fracture toughness KIC). The thickness effect on ft and KIC should also be conducted as the linear function in Figs. 10 and 11 is for constant ft and KIC, or for one thickness. It should be pointed out that this preliminary study is based on a limited number of mortar test results and more mortar tests from SS and LL beams will be followed. Finally, it should be emphasized that although BEM and TCD [26–28] were developed by adopting different assumptions, they arrived at almost identical solutions. That is from Eqs. (2) and (5), it can be shown that KIC = 2σn√(ae + a*∞), which is almost identical to one of the key relations of TCD. It can be found in literature that both BEM and TCD have been compared and confirmed by various experimental results from concrete, ceramics and metals.

Acknowledgements

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JF Guan would like to thank the University of Western Australia (UWA) for a Visiting Professorship at UWA from 2015 to 2016. XZ Hu would like to thank Australian Research Council for two discovery Grants between 2000 and 2007 on quasi-brittle fracture of coarse micro-structured composites. The financial supports for this research provided by the National Natural Science Foundation of China (No.51209094; No. 51339003; No.51679092) and the Science and Technology Planning Project of Zhengzhou City of China (No.153PKJGG111) are gratefully acknowledged.

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References

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[1] A. Hillerborg, M. Modéer, P.E. Petersson, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cem. Concr. Res. 6 (6) (1976) 773-781. [2] Z.P. Bažant, Size effect in blunt fracture: concrete, rock, metal, J. Eng. Mech. ASCE 110 (4) (1984) 518-535. [3] A. Carpinteri, B. Chiaia, G. Ferro, Multifractal scaling law: an extensive application to nominal strength size effect of concrete structures, Atti del Dipartimento di Ingegneria Strutturale No. 50, vol. 2. Italy: Politecnico de Torino; 1995. [4] F. H. Wittmann, F. H. Roelfstra, H. Sadouki, Simulation and analysis of composite structures. Mater. Sci. Eng. 68(2) (1985) 239-248. [5] S. Mindess, Fracture toughness testing of cement and concrete. In: Fracture Mechanics of Concrete: Material characterization and testing, Springer Netherlands; 1984. p. 67-110. [6] X.Z. Hu, F.H. Wittmann, Fracture energy and fracture process zone. Mater. Struct. 25(6) (1992) 319-326. [7] S.E. Swartz, T.M.E. Refai, Influence of size effects on opening mode fracture parameters for precracked concrete beams in bending. In: Shah SP, Swartz SE,editors. Fracture of concrete and rock: SEM-RILEM international conference, Houston, Texas, USA, June 17-19 1987. p. 242-254. [8] B.L. Karihaloo, H.M. Abdalla, Q.Z.Xiao, Size effect in concrete beams, Eng. Fract. Mech. 70 (2003) 979-993. [9] B.L. Karihaloo, A. R. Murthy, N.R. Iyer, Determination of size-independent specific

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fracture energy of concrete mixes by the tri-linear model, Cem. Concr. Res. 49(50) (2013) 82-88. [10] M.R.A. van Vliet, J.G.M.van Mier, Experimental investigation of size effect in concrete and sandstone under uniaxial tension. Eng. Fract. Mech. 65(2) (2000)165-188. [11] A.F. Canteli, L. Castanon-Jano, H. Cifuentes, M. Muñiz, E.Castillo, Fitting the fracture curve of concrete as a density function pertaining to the generalized extreme value family, Mater. Des. (2017) doi: 10.1016/j.matdes.2017.05.030 [12] X.Z. Hu, F.H.Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65(2-3) (2000) 209-221. [13] X.Z. Hu, K.Duan, Size effect and quasi-brittle fracture: the role of FPZ, Int. J. Fract. 154 (1) (2008) 3-14. [14] Q.B. Li, J.F. Guan, Z.M. Wu, W. Dong, S.W. Zhou, Fracture behavior of site-casting dam concrete, ACI Mater. J. 111(1) (2015)11-20. [15] J.F. Guan, Q.B. Li, Z.M. Wu, S.B. Zhao, W. Dong, S.W. Zhou, Fracture parameters of site-casting and sieved concrete, Mag. Concr. Res. 68(1) (2016) 43-54. [16] J.F. Guan, Q.B. Li, Z.M. Wu, S.B. Zhao, W. Dong, S.W. Zhou, Minimum specimen size for fracture parameters of site-casting dam concrete, Constr. Build. Mater. 93 (2015) 973982. [17] Q.B. Li, J.F. Guan, Z.M. Wu, W. Dong, S.W Zhou, Equivalent maturity for ambient temperature effect on fracture parameters of site-casting dam concrete, Constr. Build. Mater. 120 (2016) 293-308. [18] Y. Çağlar, S. Şener, Size effect tests of different notch depth specimens with support rotation measurements, Eng. Fract. Mech. 157 (2016) 43-55. [19] X.F. Gao, G. Koval, C. Chazallon, A size and boundary effects model for quasi-brittle fracture, Mater.9(12) (2016) 1030; doi:10.3390/ma9121030 [20] X.F. Gao, G. Koval, C. Chazallon, Energetical formulation of size effect law for quasibrittle fracture, Eng. Fract. Mech.175 (2017) 279–292 [21] X.Z. Hu, K. Duan, Mechanism behind the size effect phenomenon, J. Eng. Mech. ASCE 136(1) (2010) 60-68. [22] Y.S. Wang, X.Z. Hu, L. Liang, W. C. Zhu, Determination of tensile strength and fracture toughness of concrete using notched 3-p-b specimens, Eng. Fract. Mech. 160 (2016) 6777. [23] Y.S. Wang, X.Z. Hu, Determination of tensile strength and fracture toughness of granite using notched three-point-bend samples, Rock. Mech. Rock. Eng. 50 (2017) 17-28. [24] J.F. Guan, X.Z. Hu, Q.B. Li, In-depth analysis of notched 3-p-b concrete fracture, Eng. Fract. Mech. 165 (2016) 57-71. [25] X.Z. Hu, J.F. Guan, Y.S. Wang, A. Keating, S.T. Yang, Comparison of boundary and size effect models based on new developments, Eng. Fract. Mech. 175 (2017) 146-167. http://dx.doi.org/10.1016/j.engfracmech.2017.02.005. [26] D. Taylor, Geometrical effects in fatigue: a unifying theoretical model, Int. J. Fatigue. 21 (1999) 413-420. [27] D. Taylor, P. Cornetti, N. Pugno, The fracture mechanics of finite crack extension, Eng. Fract. Mech.72 (2005) 1021-1038. [28] D. Taylor, The theory of critical distances, Eng. Fract. Mech.75 (2008) 1696-1705. [29] C.G. Hoover, Z.P. Bažant, J. Vorel, R. Wendner, M.H. Hubler, Comprehensive concrete fracture tests: description and results, Eng. Fract. Mech. 114 (2013) 92-103. [30] C.G. Hoover, Z.P. Bažant, Comparison of the Hu-Duan boundary effect model with the size-shape effect law for quasi-brittle fracture based on new comprehensive fracture tests, J. Eng. Mech. ASCE 140 (3) (2014) 480-486. [31] Y.S. Jenq, S.P. Shah, Two parameter fracture model for concrete, J. Eng. Mech. 111(10)

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(1985) 1227-1241. [32] B.L. Karihaloo Size effect in shallow and deep notched quasi-brittle structures, Int. J. Fract. 95 (1999) 379-390. [33] Z.P. Bažant, B.H. Oh, Crack band theory for fracture concrete, Mater. Struct. 16 (3) (1983) 155-177. [34] JGJ/T 70 2009. Standard for test method of basic properties of construction mortar. Ministry of Housing and Urban-Rural Construction, Beijing, 2009 [35] K. Otsuka, H. Date, Fracture process zone in concrete tension specimen, Eng. Fract. Mech. 65 (2000) 111-131. [36] S. Muralidhara, B. K. Raghu Prasad, H. Eskandari, B. L. Karihaloo, Fracture process zone size and true fracture energy of concrete using acoustic emission, Constr. Build. Mater. 24 (2010) 479-486. [37] K. Ohno, K. Uji, A. Ueno, M. Ohtsu. Fracture process zone in notched concrete beam under three-point-bending by acoustic emission. Constr. Build. Mater. 67 (2014) 139-145. [38] I. Kumpova, T. Fila, D. Vavik, Z. Kersner, X-ray dynamic observation of the evolution of the fracture process zone in a quasi-brittle specimen, J. Instrum. (2015) C08004. [39] Z.M. Wu, H. Rong, J.J. Zheng, F. Xu, W. Dong, An experimental investigation on the FPZ properties in concrete using digital image correlation technique, Eng. Fract. Mech. 78(17) (2011) 2978-2990. [40] A. Tarokh, R.Y. Makhnenko, A. Fakhimi, J. F. Labuz, Scaling of the fracture process zone in rock, Int. J. Fract. 204(2) (2017) 191-204. [41] RILEM Draft Recommendations, TC89-FMT Fracture Mechanics of Concrete Test Methods, Mater. Struct. 23(1991) 461-465. [42] ASTM E399-12e2. Standard test method for linear-elastic plane-strain fracture toughness testing of high strength metallic materials. American Society for Testing and Material, Philadelphia, 2013. [43] K. Duan, X.Z. Hu, F.H. Wittmann, Thickness effect on fracture energy of cementitious materials, Cem. Concr. Res. 33(2003) 499-507. [44] S.L. Xu, Y. Zhu, Experimental determination of fracture parameters for crack propagation in hardening cement paste and mortar, Int. J. Fract. 157 (2009) 33-43. [45] S. Mindess, J.S. Nadeau, Effect of notch width on KIC for mortar and concrete, Cem. Concr. Res. 6(4) (1976) 529-534. [46] B.L. Karihaloo, H.M. Abdalla, T. Imjai, A simple method for determining the true specific fracture energy of concrete, Mag. Concr. Res. 55(5) (2003) 471-481. [47] C.G. Hoover, Z.P. Bažant, Comprehensive concrete fracture tests: size effects of type 1 & 2, crack length effect and postpeak, Eng. Fract. Mech. 110 (2013) 281-289.

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Experiment on specimens small specimens with 40mm depth

P

P 0.35

Pmax

∆afic = β· dmax

B

0.2

Curve fitting

0.1

test data

-0.25



ae / a  = 10 SS-I SS-II LL with =  LL with =  LL with =  LL with =  LL with = 

-0.50 -0.5

0.0

0.5

Fracture toughne 1/2 0.81 MPam 

a = 7.1 mm 1.0

1.5 2.0 logae

P

SC

Large structure with 500mm depth and 2000mm long



1/n

1.5 ae

dmax = 2.5 mm  = 1.0

Prediction structure fracture

0.0

1.0

1.7

RI

S

-0.1 0.5

1.6

Determine design curve

a0

a0

2.0

2.5

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Graphical Abstract

AC

σn =

1.5

PT

σn

1.4

NU

W

0.25 0.00

1.3

logae

dmax

Using Boundary effect model

Prediction curve

0.25 1.2

Determine material constant

ae / a  = 0.1

0.75 Fracture toughness 1/2 0.81 MPam 0.50

0.40

0.30



1.00

logn

P

0.45

logn

P

Tensile strength 4.81 MPa

=  =  =  =  = 

0.50

24

2.5

3.0

3.5

ACCEPTED MANUSCRIPT Highlights 1. Tensile strength & fracture toughness of mortar were determined by small 0.1m-long specimens 2. Failure of 2m-long structures were successfully predicted by 0.1m specimens, although their volume ratio was close to 1:1000

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3. A single design curve was obtained for small and large mortar beams with and without notch

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