An arc bending notched specimen for determining the mechanical and fracture parameters of concrete based on the FET

Engineering Fracture Mechanics 220 (2019) 106639

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An arc bending notched specimen for determining the mechanical and fracture parameters of concrete based on the FET

T

⁎

Longbang Qinga, , Yuehua Chenga, Xiangqian Fanb, Ru Mua, Shaoqiang Dinga a b

School of Civil Engineering and Transportation, Hebei University of Technology, Tianjin 300401, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing Hydraulic Research Institute, Nanjing 210098, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Arc bending notched specimen Fracture extreme theory Tensile strength Effective fracture toughness Concrete

An arc bending notched specimen was used to determine the mechanical and fracture parameters of concrete in this study. Compared with the traditional arc bending notched specimen, the lateral thrust generated at the bottom supports in the specimen can counteract the influence of self-weight on the crack propagation. Therefore, relative stable crack propagation can be obtained. The expressions of the stress intensity factor K and crack mouth opening displacement CMOD of the specimen were derived via the superposition method. Based on the fracture extreme theory (FET), a theoretical method for determining the tensile strength ft and effective fracture toughness KeIC of concrete was established, and only the experimental peak load is required. The rationality and applicability of the proposed theoretical method were verified by comparing the calculated values of ft to experimental data. The sensitivity of KeIC calculated by the proposed method to the initial notch ratio a0/h was analysed.

1. Introduction A fracture process zone (FPZ) exists at the crack tip in quasi-brittle materials, such as concrete. The FPZ results in the size effect of fracture parameters if the FPZ is not small enough compared with specimen size. Hence, the linear elastic fracture mechanics cannot be directly applied to the quasi-brittle fracture analysis [1]. Since Kaplan [2] firstly employed fracture mechanics in concrete beam to test the fracture toughness of concrete in 1961, the fracture behavior of concrete has been extensively investigated. Three-point bending specimen [3,4], compact tensile specimen [5,6], wedge splitting specimen [7,8], and central notched splittension specimen [9,10] are commonly used to determine the fracture parameters of concrete in fracture tests. Among them, the three-point bending specimen is the most widely used. The experimental setup of the three-point bending specimen is relatively simple; it is composed of bottom supports and upper loading equipment. In large-sized three-point bending specimen, the self-weight may affect the stability of crack propagation. Therefore, in the case of large specimens, the compact tension and wedge splitting specimens are used to determine the fracture parameters of concrete. The three-point bending specimen and compact tension specimen have been listed as the standard specimens in ASTM E399-74 [11]. Because of its compact and lightweight, the central notched split-tension specimen was used to study the size effect of fracture parameters [10]. Additionally, there is an arc bending specimen, which is more commonly used in fracture tests for metal materials. Kapp [12,13] derived the expressions of the stress intensity factor K and crack mouth opening displacement CMOD for the arc bending specimen. For the traditional arc bending notched specimen, there is no lateral thrust at the bottom supports, which is very similar to the three-point bending specimen. For large specimen, the self-weight significantly affects the stability of the specimen. In particular, for a large specimen with a large initial notch ratio, the ⁎

Corresponding author. E-mail address: [email protected] (L. Qing).

https://doi.org/10.1016/j.engfracmech.2019.106639 Received 13 December 2018; Received in revised form 20 August 2019; Accepted 21 August 2019 Available online 22 August 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 220 (2019) 106639

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Nomenclature a a0 ac Δa B h D H L S r1 r2 CMOD CMODc CTOD CTODc K Γ Δ

W c1, c2 w0

effective crack length initial crack length critical effective crack length at peak load crack propagation length specimen thickness radial thickness of specimen vertical distance from supports to loading point specimen depth horizontal distance between the two supports for the arc bending notched specimen horizontal distance between the two supports for the traditional arc bending notched specimen inner radius outer radius crack mouth opening displacement critical crack mouth opening displacement crack tip opening displacement critical crack tip opening displacement stress intensity factor at the crack tip stress intensity coefficient crack mouth opening displacement coefficient

μ E ν fc fcu ft fr hc KIC e KIC s KIC P Pmax σc σw w

self-weight of the specimen material constants for nonlinear softening function critical value of crack opening displacement at which the σw = 0 friction coefficient elastic modulus of concrete Poisson's ratio cylinder compressive strength of concrete cube compressive strength of concrete tensile strength of concrete flexural tensile strength distance from the fictitious crack tip to the position of neutral axis fracture toughness effective fracture toughness critical stress intensity factor in two parameter fracture model external load peak load compressive stress of concrete at the bottom of extremity of specimen cohesive stress fictitious crack opening displacement

self-weight may cause material damage in the crack tip or lead to crack propagation. Several nonlinear fracture models have been proposed to consider the PFZ of concrete, such as the fictitious crack model [14], crack band model [15], two parameter fracture model [3], effective crack model [16], size effect model [17,18], double-K fracture model [19], and boundary effect model [20,21], etc. Corresponding theoretical methods for determining the fracture parameters s have been developed by using different types of specimens. Jenq and Shah [3] determined the critical stress intensity factor KIC and critical crack tip opening displacement CTODc of concrete using the P-CMOD curves obtained by three-point bending beam fracture tests and introduced them as two material fracture parameters. Karihaloo and Nallthambi [16] determined the critical effective crack e length ac and effective fracture toughness KIC of concrete using the load–deflection (P–δ) curves obtained via three-point bending beam fracture tests. Wu et al. [22] proposed an analytical model to predict the peak load Pmax of three-point bending notched beams e and determined KIC of concrete using the tensile strength. According to the boundary effect model, the tensile strength ft and fracture toughness KIC of concrete were determined by using Pmax of two three-point bending concrete beams [23] or wedge splitting specimens [24] with different initial notch lengths. Xu and Reinhardt [25,26] determined the double-K fracture parameters of concrete in the double-K model. Ince determined the fracture parameters of concrete by using the central notched split-tension specimen based on the two-parameter fracture model, size effect model [10], and double-K fracture model [27], respectively. Recently, Qing et al. [28,29] proposed the fracture extreme theory (FET) of concrete. According to a typical P-a (where P represents the external load, a represents the effective crack length) curve of quasi-brittle material fracture, the FET considers that the partial derivative of P with respect to a is equal to 0 when P reaches its maximum value Pmax. Based on the FET, the fracture parameters of concrete were determined using compact tension specimen and wedge splitting specimen [28,30], three-point bending e specimen [31,32], and central notched split-tension specimen [33]. Recently, the FET was used to determine ft and KIC of concrete using three-point bending beams [29] and wedge splitting specimens [34]. The FET does not need to measure the critical crack mouth opening displacement CMODc, and only the experimental Pmax is needed to determine the fracture parameters of concrete. e In the present study, an arc bending notched specimen was used to determine ft and KIC of concrete. Using the superposition

h

a0

x

r1

D

H

r2

B

L Fig. 1. Test setup for the arc bending notched specimen. 2

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method, the expressions of K and CMOD of the specimen were derived. Based on the FET, a theoretical method was proposed to e determine ft and KIC of concrete. Four groups of specimens with different initial notch ratios a0/h were tested. The calculated values of ft were compared with those by the experimental data to verify the rationality and applicability of the proposed theoretical method. 2. Determination of expressions for K and CMOD Fig. 1 shows the arc bending notched specimen in this study, where H represents the depth of the specimen; D represents the vertical distance from the supports to the loading point; h represents the radial thickness of the specimen; B represents the thickness of the specimen; a0 represents the initial crack length; r1 and r2 represent the inner and outer radii of the specimen, respectively; L represents the horizontal distance between the two supports. The test setup of the specimen is similar to that of the traditional arc bending notched specimen. During the loading process, lateral thrust is generated at the bottom supports. Compared with the traditional arc bending notched specimen, the specimen is more stable. The lateral thrust in the specimen can reduce the effect of the self-weight on the crack propagation. Compared with the traditional arc bending notched specimen, the specimen in this study lacks two parts surrounded by the dotted line in Fig. 2. By performing a stress analysis on these parts, the stress state at the crack tip of the specimen is equivalent to that of the traditional arc bending notched specimen as shown in Fig. 2, where W represents the self-weight of the specimen, μ represents the friction coefficient between the support and the contact surface, and S represents the horizontal distance between the two supports for the traditional arc bending notched specimen. In this study, the superposition method was employed to derive the expressions of K and CMOD for the specimen. According to the expressions of K for different types of specimens given by the existing literature, the stress state of traditional arc bending notched specimen in Fig. 2 can be expressed by a superposition of the three specimens shown in Fig. 3. Thus, expressions of K and CMOD can be expressed as follows:

K = K3a − K3b + K3c

(1)

CMOD = CMOD3a − CMOD3b + CMOD3c

(2)

where K3a, K3b, and K3c represent the values of K for specimens in Fig. 3(a), (b), and (c), respectively; CMOD3a, CMOD3b, and CMOD3c represent the values of CMOD for specimens in Fig. 3(a), (b), and (c), respectively. Kapp [13] proposed expressions of K3a and CMOD3a for the traditional specimen with S/h = 4 in Fig. 3(a). Gross and Srawley [35] derived the expressions of K3b, CMOD3b and K3c, CMOD3c for the traditional specimens in Fig. 3(b) and (c), respectively. Therefore, the suggested specimen size for the specimen in this study is r1:r2:H = 2:3:1.55. The forces N and N1 are both applied to the midpoint of the boundary. In this case, the ratio of S/h is equal to 4, L1 = 0.28 h, and L2 = 0.35 h. The expression of K3a for the specimen with S/h = 4 in Fig. 3(a) is expressed as follows [13]:

K3a =

P (S /h) a/h [1 + g (r1/r2) k (a/h)] f (a/h) B h (1 − a/ h)3/2

(3)

g (r1/ r2) = 0.832 − 1.376r1/ r2 + 0.544(r1/ r2

)2

(4)

k (a/ h) = 1.003 − 4.680a/ h + 9.953(a/ h)2 − 10.030(a/ h)3 + 3.754(a/ h) 4

(5)

17.157(a/ h)2

(6)

f (a/ h) = 2.978 − 8.240a/ h +

−

18.073(a/ h)3

+

7.174(a/ h) 4

When the specimen shown in Fig. 3(b) is subjected to a pair of horizontal forces at the bottom, the expression of K3b at the crack tip can also be derived by using the superposition method, as shown in Fig. 4. P a

F

F

x

r1

r2

N

N

F

L2

L1

F1

N

M N1

P

F1

M N1

F1 S

M N1

Fig. 2. Comparing the specimen with the traditional arc bending notched specimen. 3

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P

+

+ F1

F1

N1

N1

(a)

M

M

(c)

(b)

Fig. 3. Determination of the expression of K and CMOD for the specimen in Fig. 1 via the superposition method.

+ F1

M

F1

(a)

M

M

(b)

M

Fig. 4. Determination of the expression of K3b and CMOD3b for the specimen shown in Fig. 3(b) via the superposition method.

(7)

K3b = K 4a + K 4b

where K4a and K4b represent the values of K for specimens in Fig. 4(a) and (b), respectively. In this case, the expressions of K4a and K4b are obtained through the boundary collocation method [35]. The stress intensity coefficient Γ can be expressed as follows:

Γ= where σP =

K (σP + σM ) a (1 − a/ h) P B (h − a)

(8)

6M . P and M represent the horizontal force and bending B (h − a)2 (1 − μ)(P + W )(r 1 − x ) (1 − μ)(P + W )[r 1 + (h + a) / 2] and MII = MI − . 2(1 + μ) 2(1 + μ)

and σM =

moment, respectively, at the bottom of the

specimen. In Fig. 4, MI = According to the superposition method, K3b can be obtained as follows:

= K 4a + K 4b =

K3b =

(

ΓP (1 − a / h) 2(a / h + 2 + 3x / h)

+

(

ΓP σ P + ΓMσ M σP + σM

) (σ

P

+ σM ) a (1 − a/ h)

3ΓM (1 + a / h + 2x / h) 2(a / h + 2 + 3x / h)

) (σ

P

+ σM ) a (1 − a/ h)

(9)

where ΓP and ΓM are the stress intensity coefficients for specimens in Fig. 4(a) and (b), respectively. The values of the two coefficients for different a/h and r2/r1 are determined by the boundary collocation method [35]. x represents the vertical distance from the intersection of the crack and the inner diameter of the specimen to the load line at the bottom. The expression of K3c for the specimen shown in Fig. 3(c) was derived by Gross and Srawley [35].

K3c = ΓMσM a (1 − a/ h)

(10)

Finally, the expression of K at the crack tip for the specimen in Fig. 1 can be obtained as follows:

K = K3a − K3b + K3c = K3a − (K 4a + K 4b) + K3c

(11)

Similar to the determination of K, the expression of CMOD for the specimen in Fig. 1 can be derived using the superposition method.

CMOD = CMOD3a − CMOD3b + CMOD3c = CMOD3a − (CMOD4a + CMOD4b) + CMOD3c

(12)

where CMOD4a and K4b represent the CMOD of specimens in Fig. 4(a) and (b), respectively. The expression of CMOD3a for the traditional specimen with S/h = 4 shown in Fig. 3(a) can be obtained [13].

CMOD3a =

P (S / h)[1 + g (r1/ r2 ) k1 (a/ h)] f1 (a/ h) E′B (1 − a/ h)2

(13)

k1 (a/ h) = 1.007 − 3.249a/ h + 4.517(a/ h)2 − 2.275(a/ h)3

(14) 4

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f1 (a/ h) = 8.747 − 25.576a/ h + 66.722(a/ h)2 − 93.010(a/ h)3 + 63.579(a/ h) 4 − 16.509(a/h)5

(15)

Here, E' represents the elastic modulus E for plane stress or E/(1 − v ) for plane strain. ν represents the Poisson’s ratio. Similar to the method for K3b, the expression of CMOD3b can be obtained. 2

(16)

CMOD3b = CMOD4a + CMOD4b

The expressions of CMOD4a and CMOD4b were derived by Gross and Srawley [35]. The crack mouth opening displacement coefficient Δ can be expressed as follows:

Δ=

E′CMOD (σP + σM ) a

(17) 6M . B (h − a)2

P B (h − a)

where σP = and σM = P and M represent the horizontal force and bending moment at the bottom of the specimen, respectively. In this case, the expression of CMOD3b can be expressed as follows:

CMOD3b = CMOD4a + CMOD4b = =

(

ΔP (1 − a / h) 2(a / h + 2 + 3x / h)

+

(

ΔP σ P + ΔMσ M σP + σM

3ΔM (1 + a / h + 2x / h) 2(a / h + 2 + 3x / h)

) (σ ) (σ

P

+ σM ) a/E′

P

+ σM ) a/ E′

(18)

Here, ΔP and ΔM are the crack mouth opening displacement coefficients for the specimen shown in Fig. 4(a) and (b), respectively. The values of the two coefficients for different a/h and r2/r1 values are determined by the boundary collocation method [35]. CMOD3c can be expressed as follows [35]:

CMOD3c = ΔM σM a/E′

(19)

e based on the FET 3. Determination of ft and KIC

3.1. Fracture extreme theory In the concrete fracture test, with an increase in CMOD, P gradually increases and then decreases. When P reaches its maximum value Pmax, the crack tip opening displacement CTOD, CMOD, and crack propagation length Δa all simultaneously increase to their critical values. Assuming the partial derivative of P with respect to a is continuous, P and a at the peak load state can be expressed as follows [28–34]:

∂P ∂a

=0

(20)

a = ac

3.2. Theoretical method Based on the assumptions of plane section and crack surface remaining plane [22,29], the distributions of the stress and strain in the midspan for the arc bending notched specimen are shown in Fig. 5. σc represents the compressive stress at the top of the specimen and hc represents the distance from the fictitious crack tip to the position of the neutral axis. Reinhardt et al. [36] proposed a nonlinear softening curve describing the relationship between the cohesive stress σw and the fictitious crack opening displacement w,

ıc

ıc/E

M

h 0

fr

hc

h-a-hc 0

fr/E

ıw a0 a

hc

a

x (b) strain distribution

x (a) stress distribution

Fig. 5. Distributions of stress (a) and strain (b) in the midspan section. 5

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c2 (1 + c13 ) e−c2 ⎫ c3 w σw = fr ⎧ ⎡1 + 13 w 3⎤ e− w0 w − ⎢ ⎥ ⎨ ⎬ w0 w0 ⎦ ⎩⎣ ⎭

(21)

where c1 and c2 are the material parameters, and w0 represents the critical value of the crack opening displacement at which σw = 0. The tensile strength ft is replaced by the flexural tensile strength fr in Eq. (21) [22,29]. According to the assumption that the crack surface remains plane, the distribution of w along the height of the specimen can be obtained as follows:

w=

x − hc CTOD a − a0

(h c ⩽ x ⩽ h c + a − a 0)

(22)

The relationship between CTOD and CMOD can also be obtained.

CTOD =

a − a0 CMOD a

(23)

By substituting Eq. (22) into Eq. (21), σw can be expressed as follows: 3

c3 (1 + c13) exp(−c2 ) x − hc ⎞ ⎤ c x − hc ⎞ ⎞ x − hc ⎞ ⎫ ⎧ σw = fr ⎡ 1 + 13 (CTOD)3 ⎛ exp ⎜⎛− 2 CTOD ⎛ CTOD ⎛ ⎟ − ⎢ ⎥ ⎨ − − − a0 ⎠ ⎬ a a w a a w a w 0 0 0 0 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ ⎦ ⎩⎣ ⎭ ⎜

⎟

⎜

⎟

⎜

⎟

(h c ⩽ x ⩽ (24)

hc + a − −a0) In Fig. 5, σc can be obtained using the plane section assumption:

f (h − a − hc ) σc/ E h − a − hc = ⇒ σc = r fr / E hc hc

(25)

According to the stress distribution in Fig. 5, the stress equilibrium equation is expressed as follows:

1 1 σc B (h − a − h c ) − fr Bh c − 2 2

∫h

hc + a − a0

c

σw B dx =

(1 − μ)(P + W ) 2(1 + μ)

(26)

Moreover, the moment equilibrium equation is expressed as follows:

1 1 σc B (h − a − hc )2 + fr Bhc2 + 3 3

∫h

hc + a − a0

(1 − μ)(P + W ) L (D − h + hc + a) (P + W ) − MW − 2(1 + μ) 4

σw Bx dx =

c

(27)

where MW is the moment generated by the self-weight of the specimen. By substituting Eqs. (24) and (25) into Eq. (26), hc can be expressed as follows:

(h − a)2

hc =

{

(1 − μ)(P + W ) fr B (1 + μ) 2w04

−

c24 CTOD

2w 0 ⎛1 + + 2(h − a) + (a − a0 ) ⎡ c CTOD ⎢ 2 ⎝ ⎣

c13 c23

⎛ ⎝

w06

(CTOD)3 + 3

c13 c22

(CTOD)2 + 6

w05

c13 c 2 w04

6c13 c23

⎞− ⎠

(1 + c13) exp(−c 2 ) w0 c3

c23

0

w03

CTOD + 6 w13 +

CTOD

(

)

⎞ exp − c2 CTOD ⎤ ⎫ w0 ⎠ ⎦⎬ ⎭

(28)

By substituting Eqs. (24) and (25) into Eq. (27), the relationship between P and a can be expressed as follows: (P + W ) L 4fr B

= − + − −

−

(h − a)3 3h c

MW fr B

−

(1 − μ)(P + W )(D − h + hc + a) 2fr B (1 + μ) (a − a0 ) w 0 c 2 CTOD

− (h − a)2 + (h − a) hc +

(1 + c13) exp(−c 2 ) w0

CTOD (a − a 0 )

6c13 w 0 h c (a − a0 )

+

c24 CTOD c13 w 0 h c (a − a0 ) c24 CTOD c13 w02 (a − a0 )2 c25 (CTOD)2

⎛ ⎝

⎛ ⎝

(

24c13 w02 (a − a0 )2 c25 (CTOD)2

c23 (CTOD)3 w03

c24 (CTOD) 4 w04

+ +

1 h 2 c

−

+

−

(a − a0 ) w 0 c 2 CTOD

3c22 (CTOD)2 w02 4c23 (CTOD)3 w03

1 a 3

+ +

(h

c

1 a 3 0

(h

c

+

)

)

) exp (− + 6⎞ exp (− ) ⎠ + + 24⎞ exp (− ⎠

+ a − a0 +

6c 2 CTOD w0

12c22 (CTOD)2 w02

(a − a0 ) w 0 c 2 CTOD

(a − a0 ) w 0 c 2 CTOD

c 2 CTOD w0

)

c 2 CTOD w0

24c 2 CTOD w0

c 2 CTOD w0

)

(29)

Table 1 Mixture proportions of concrete. w/c

Water (kg/m3)

Cement (kg/m3)

Sand (kg/m3)

Coarse aggregate (kg/m3)

0.53

190

360

660

1190

6

Engineering Fracture Mechanics 220 (2019) 106639

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Fig. 6. Experimental setup for the arc bending notched specimen. Table 2 Calculated results for the arc bending notched specimens. Specimen

0.13-1 0.13-2 0.13-3 Mean Stdev 0.24-1 0.24-2 0.24-3 Mean Stdev 0.35-1 0.35-2 0.35-3 Mean Stdev 0.41-1 0.41-2 0.41-3 Mean Stdev

a0/h

0.13

0.24

0.35

0.41

Pmax (kN)

3.052 3.104 2.922 3.026 0.077 2.231 2.159 2.057 2.149 0.071 1.761 1.928 1.666 1.785 0.108 1.186 1.228 1.477 1.297 0.128

Calculated results ac (mm)

CTODc (μm)

e KIC (MPa·m1/2)

ft (MPa)

20.22 20.22 20.62 20.35 0.189 25.26 25.56 25.56 25.46 0.141 29.70 29.20 29.90 29.60 0.294 33.24 33.24 32.34 32.94 0.424

14.02 14.25 14.01 14.09 0.111 12.81 12.90 12.30 12.67 0.264 12.15 12.26 11.85 12.09 0.173 11.16 11.56 11.90 11.54 0.302

0.954 0.970 0.924 0.949 0.019 0.954 0.935 0.891 0.927 0.026 1.017 1.080 0.967 1.021 0.046 0.866 0.896 1.022 0.928 0.068

3.35 3.42 3.18 3.32 0.101 3.09 2.97 2.81 2.96 0.115 3.26 3.62 3.05 3.31 0.235 2.53 2.64 3.26 2.81 0.321

The expression of P can be obtained by substituting Eqs. (12), (23), and (28) into Eq. (29). Then, by substituting a = ac and P = Pmax into Eqs. (20) and (29), the values of fr and ac can be determined. Eq. (30) describes the relationship between fr and ft [37].

fr = 0.62(fc )1/2

(30a)

ft = 0.4983(fc )1/2

(30b)

where fc represents the cylinder compressive strength of concrete. e Finally, by substituting the calculated values of ac and Pmax into Eq. (11), the KIC of concrete can be calculated. 7

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5.0

1.8

calculated mean values test values

4.5

1.5

KeIC / (MPa·m1/2)

4.0

ft / MPa

calculated values mean values

3.5 3.0 2.5 2.0

1.2 0.9 0.6

1.5 1.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 a0 / h

0.3 0.10

0.15

0.20

(a) ft

0.25 0.30 a0 / h

0.35

0.40

0.45

(b) K ICe

e Fig. 7. Variations of ft (a) and KIC (b) with respect to a0/h.

5.0 4.5

ȝ = 0.3 ȝ = 0.4 ȝ = 0.5

1.5

KeIC / (MPa·m1/2)

4.0

ft / MPa

1.8

ȝ = 0.3 ȝ = 0.4 ȝ = 0.5 test values

3.5 3.0 2.5 2.0

1.2 0.9 0.6

1.5 1.0 0.10

0.15

0.20

0.25 0.30 a0 / h

0.35

0.40

0.3 0.10

0.45

0.15

(a) ft

0.20

0.25 0.30 a0 / h

0.35

0.40

0.45

(b) K ICe e Fig. 8. Effects of μ on ft (a) and KIC (b).

4. Experimental investigations The fracture tests were conducted to verify the rationality of the proposed theoretical method for calculating the mechanical and fracture parameters of concrete. The mixture proportions are presented in Table 1. The specimens were made of Grade R42.5 Portland cement, natural river sand with a density of 2500 kg/m3, coarse aggregate with a particle-size range of 5–10 mm and a density of 2750 kg/m3, and tap water. Four groups of specimens with different a0/h values were used to conduct the fracture test at the age of 28 d. To investigate the sensitivity of the fracture parameters to a0/h, the following a0/h were used: 0.15, 0.25, 0.35, and 0.4. Fig. 6 shows the setup for the concrete fracture tests. The dimensions of the specimens were as follows: r1 = 100 mm, r2 = 154 mm, h = 54 mm, and B = 62 mm. The specimens were tested on a 20-kN testing machine with displacement control at a rate of 0.02 mm/min. The external load of each specimen was recorded. The cube compressive strength fcu and elastic modulus E were 44.7 MPa and 29.8 GPa, respectively. 5. Results and discussion The peak load Pmax of the specimens with different a0/h were presented in Table 2. Owing to the experimental error, the actual values of a0/h of the tested specimens were 0.13, 0.14, 0.35, and 0.41, respectively. Using the proposed theoretical method, the critical effective crack length ac, critical crack tip opening displacement CTODc, tensile strength ft, and effective fracture toughness e KIC were determined and listed in Table 2. The parameters in Eq. (21) were c1 = 3, c2 = 7, and w0 = 160 μm [36]. The value of μ was approximately 0.4 between the steel 8

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and concrete [38]. The experimental value of ft was 2.96 MPa, as calculated using Eq. (30) according to the cylinder compressive strength fc = 0.79fcu = 35.31 MPa. As shown in Table 2, the values of ft calculated using the proposed theoretical method for the specimens with a0/h of 0.13, 0.24, 0.35, and 0.41 were 3.32 MPa, 2.96 MPa, 3.31 MPa, and 2.81 MPa, respectively. It can be seen from Fig. 7(a) that the mean values of ft calculated for each group specimens with different a0/h have slightly fluctuation. The mean value for all the specimens was 3.10 MPa, and the relative error was only 4.73% compared with the experimental value, confirming the rationality and applicability of the proposed method. e e In Fig. 7(b), with an increase in a0/h, the changes of KIC are not obvious. This indicates that KIC was insensitive to a0/h. Additionally, in Table 2, the changes of CTODc with an increase in a0/h are not obvious, confirming that CTODc is a material fracture parameter in the two-parameter fracture model [3]. Moreover, the effects of μ on the mechanical and fracture parameters were analysed. The calculation results obtained using the proposed theoretical method were shown in Fig. 8 for μ = 0.3, 0.4, and 0.5. As shown in Fig. 8, μ effected the calculated mechanical and fracture parameters; the calculated parameters increased with μ. When μ was 0.4 [38], the proposed theoretical method obtained relatively accurate mechanical and fracture parameters of the concrete. 6. Conclusions An arc bending notched specimen was used to determine the mechanical and fracture parameters of concrete based on the FET. The applicability of the proposed theoretical method was verified by comparing the calculated results with experimental data. The following conclusions are drawn. (1) The arc bending notched specimen in this study can exhibit relatively stable crack propagation. Thus, it can be used to determine the fracture parameters of concrete in the case of a large-sized specimen. (2) The expressions of K and CMOD for the arc bending notched specimen were determined using the superposition method. e (3) Based on the FET, a theoretical method for determining ft and KIC of concrete was proposed, and only the experimental Pmax is required in the calculation. The calculated values of ft were close to the experimental values, confirming the applicability of the e proposed method. Additionally, KIC was insensitive to a0/h. Acknowledgements This work has been supported by the National Key R&D Program of China (No. 2016YFC0401907), National Natural Science Foundation of China (Nos. 51779069, 51679150, 51578208) and Hebei Province Graduate Innovation Funding Project (No. CXZZSS2018027). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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