Mixed mode fracture characterization of GFRP-concrete bonded interface using four-point asymmetric end-notched flexure test

Accepted Manuscript Mixed Mode Fracture Characterization of GFRP-concrete Bonded Interface Using Four-point Asymmetric End-notched Flexure Test Qinghui Liu, Pizhong Qiao PII: DOI: Reference:

S0167-8442(17)30101-5 http://dx.doi.org/10.1016/j.tafmec.2017.06.009 TAFMEC 1891

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

28 February 2017 5 June 2017 12 June 2017

Please cite this article as: Q. Liu, P. Qiao, Mixed Mode Fracture Characterization of GFRP-concrete Bonded Interface Using Four-point Asymmetric End-notched Flexure Test, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.tafmec.2017.06.009

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Mixed Mode Fracture Characterization of GFRP-concrete Bonded Interface Using Four-point Asymmetric End-notched Flexure Test Qinghui Liu a, Pizhong Qiao a,b,∗ a

State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced

Ship and Deep-Sea Exploration, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China b

Department of Civil and Environmental Engineering, Washington State University, Sloan Hall 117, Pullman, WA 99164-2910, USA

Abstract: A combined analytical and experimental study using an asymmetric end-notched flexure specimen under four-point bend loading is conducted to characterize the fracture of glass fiber reinforced polymer (GFRP)-concrete bonded interface. Both the classic composite (rigid joint) and interface deformable (flexible joint) bi-layer beam models are used to calculate the compliance and energy release rate (ERR) of the proposed four-point asymmetric end-notched flexure (4-AENF) specimen. The validity and accuracy of the models are obtained by comparison with the numerical finite element results. The results show that the flexible joint model predicts more accurately the compliance and ERR compared with those of rigid joint model in 4-AENF specimen due to the attribute of crack tip deformation. Moreover, the calculated ERR by the flexible joint model can be reduced to that of the rigid joint one when the specimen is properly sized. Then, the designed 4-AENF specimens are utilized to characterize the fracture toughness of GFRP-concrete bonded ∗

Corresponding author. E-mail addresses: [email protected] and [email protected] (P. Qiao). 1

interface. To overcome the obstacle of low tensile strength and cracking behavior of concrete and prevent the premature fracture of concrete substrate before debonding of bonded interface takes place, a reduced section scheme is adopted and the steel bars are used to reinforce the concrete substrate beams. An aluminum beam with different thickness is bonded to the thin GFRP layer so as to change the stiffness of the composite GFRP/aluminum substrate, resulting in different fracture mode mixities. The fracture toughness values of GFRP-concrete bonded interface under three different mode ratios are obtained. The proposed 4-AENF specimen and data reduction procedures for interface fracture toughness evaluation can be used to effectively characterize mixed mode and mode-II dominated fracture of hybrid material bonded interface. Keywords: interface debonding; glass fiber reinforced polymer; concrete; compliance; end-notched flexure (ENF) specimen; energy release rate; fracture toughness. 1. Introduction Fiber reinforced polymer (FRP) composites have been increasingly used in civil engineering for strengthening, retrofitting and repairing of traditional concrete structures as well as new constructions (e.g., FRP-concrete systems) due to their high specific strength, high specific stiffness, high flexibility in design and superior environmental durability [1-4]. Although satisfactory performance (such as, stiffness and strength) have been achieved by use of FRP materials, there is a concern about reliable performance of bonded interface, which is susceptible to debonding [5,6]. To assess this type of failure modes, fracture mechanics-based approaches have been widely employed, in which the energy release rate

2

(ERR or G) or stress intensity factor (SIF or K) is predicted and compared with its critical value Gc or Kc [7,8]. However, for the bi-material interface of FRP and concrete, Gc is a function of fracture mode ratios. Thus, fracture tests have to be conducted to obtain fracture toughness values over a wide range of mode mixities. In the present work, determination of the mixed mode and mode-II dominated fracture toughness of FRP-concrete bonded interface is addressed. To characterize the fracture behavior of bi-material interface, beam-type fracture specimens are most widely used. The mixed mode bending (MMB) specimen is very popular for determination of mixed mode envelops, and it has been adopted as a standard by ASTM [9]. However, this standard is limited to FRP composites with high tensile strength. The single leg bending (SLB) [10], asymmetric double cantilever beam (ADCB) [11], and asymmetric end notched flexure (AENF) [12] specimens are all good choices for bi-material bonded interface fracture toughness determination over a different range of mode mixities. Even though ERR can be obtained from the above three typical fracture specimens under different mode mixities, only one crack initiation value can be obtained at a time per specimen because of rapid debonding growth. To overcome the above difficulty, crack steady types of specimens, such as four point end notched flexure (4-ENF) [13], tapered end notched flexural (TENF) [14,15], single contoured cantilever beam (SCCB) [7] and over-leg bending (OLB) [16,17] specimens, were proposed. It has to be mentioned that due to the fact that only bending moments exists within the zone in front of the interface crack tip for the 4-ENF specimen, it was usually deemed that the conventional composite bi-layer beam theory (rigid

3

joint model) would give the same ERRs as those of semi-rigid joint and flexible joint models [18,19]. For fracture characterization of FRP and concrete bonded interface, usually peel test [20-24], shear test [24-28], and combination of the two test methods [6,29,30] were adopted due to the low tensile strength of concrete. The peel test is usually used to measure interface fracture toughness under mode-I loading. Ye et al. [21] and Giurgiutiu et al. [23] tested the fracture toughness of FRP and concrete bonded interface using the peel test based on the linear elastic fracture mechanics. However, usually large-deformation existed in the un-bonded portion of FRP as shown by Kimpara et al. [22] and Lorenzis and Zavarise [31], and the linear elastic fracture mechanics is thus not suitable for evaluation of ERR of FRP-concrete bonded interface. To overcome the above obstacles, Kimpara et al. [22] obtained the ERR of FRP-concrete bonded interface using the area method and membrane peeling method. The shear tests are usually used to obtain the mode II fracture behavior of FRP-concrete interface [25,26]. However, different shear test set-ups can lead to significantly different test results. Moreover, majority of existing shear tests were concerned with predictions of ultimate load and effective bond length [27]. Considering that the fracture mode is usually mixed in practical application, the effect of combined pulling and peeling on FRP-concrete interface debonding was studied by Pan and Leung [30] using a novel experimental set-up, from which a new theoretical model for debonding analysis was proposed. Recently, the bond behavior of FRP and concrete under mixed-mode I/II loading was experimentally investigated by Ghorbani et al. [6], and the FRP was loaded at different

4

angles with respect to the concrete substrate in order to experience mixed-mode I/II loading. The results showed that the bond strength would decrease or increase over the control specimen when the interface experiences mode I component of loading in the form of normal out-of-plane tensile or compressive stresses, respectively. It can be seen from the existing studies that the beam-typed specimens have seldom be used to measure fracture toughness between FRP and concrete due to low tensile strength and cracking behavior of concrete materials. Moreover, the above mentioned 4-ENF specimen is usually used for characterization of symmetric sublayers with identical material [13,32-34]. The 4-AENF specimen was first proposed by Qiao et al. [8] to characterize mixed-mode fracture behavior of Carbon FRP-concrete bonded interface. The critical loads for crack initiation and crack arrest and corresponding critical energy release rates were obtained. However, the concrete beam substrate easily failed in tension before the interface propagation took place in their test. To overcome the difficulties of low tensile strength of concrete, a reduced section scheme, similar to the I-section proposed by Yoshihara [33] to prevent bending failure in mode II fracture toughness measurement of wood, is adapted in this study. In this study, the fracture behavior of GFRP-concrete bonded interface is analytically and experimentally studied using the four-point asymmetric end-notched flexure (4-AENF) specimen. First, both the rigid joint and flexible joint models are used to calculate the compliance and energy release rate (ERR) of the proposed 4-AENF specimen. Then, validity and accuracy of the models are obtained by comparing the results of the two joint models and finite element analysis (FEA). Finally, the designed 4-AENF specimens are utilized to

5

experimentally characterize the fracture toughness of GFRP-concrete bonded interface. To overcome the obstacle of low tensile strength and cracking behavior of concrete and prevent the premature fracture of concrete substrate before the bonded interface takes place, a reduced section scheme is adopted and the steel bars are used to reinforce the concrete substrate beams. Three groups of specimens with different thickness of aluminum plates are considered to achieve three different fracture mode I/II mixities, under which the crack initiation and arrest loads and corresponding fracture toughness values are obtained. 2. Analysis of 4-AENF specimen A four-point asymmetric end-notched flexural (4-AENF) specimen with an interface delamination length a and a clear span of L loaded by P/2 at the corresponding points with distance d from the left and right supports as shown in Fig. 1 is considered. Following the same procedure provided in the literature [34], the concept of crack-tip element proposed by Davidson et al. [35] is adopted, in which a cracked bi-layer beam lies along the interface of top beam 1 and bottom beam 2 with thickness h 1 and h2 and width b1 and b2, respectively. Each substrate layer may be composed of two or more layers with different materials. The configuration in Fig. 2 illustrates a crack-tip element, where the cracked and uncracked portions join, in which the generic loads are applied as already determined by a global beam analysis. In Fig. 2, Ni0, Qi0, and Mi0 (i = 1,2) are the applied axial forces, transverse shear forces, and bending moments at the crack tip in sub-layers (or substrates) 1 and 2, respectively; Ni, Qi, and Mi are the internal axial forces, transverse shear forces, and bending moments in sub-layers 1 and 2, respectively; NT, QT, MT are the total resultant applied axial

6

force, transverse shear force, and bending moment of the bi-layer beam system about the mid-plane of the layer 2. According to the Timoshenko beam theory, the stress resultants and displacements of each sub-layer can be expressed as [18,36]

Ni = Ai

dui dφ dw   , M i = Di i , Qi = Bi  φi + i  dx dx dx  

(1)

where ui , wi and φi are the displacements in the xi and zi directions and rotations of the sub-layer i of the middle plane, respectively; Ai, Bi and Di are the axial, transverse shear, and bending stiffness of layer i, respectively. P/2

P/2

pre-crack

b1

z x

1

h1

2

h2

d

b2

d a

L-a

Fig. 1. Schematic of general 4-AENF specimen Q1 z1 N10

M 10

Q 10

z z2

N 20

h1

x1

1 Q2

x x2

h2

M1

2 M2

M 20 Q 20

N1

QT

NT

N2 MT

Fig. 2. Crack-tip element of bi-layer beam joint model 2.1 Rigid joint model First, the classic composite bi-layer beam theory (rigid joint model), assuming that the cross-section at the crack-tip remain in one plane and is perpendicular to the mid-plane of the virgin beam, is adopted here. By substituting the corresponding expressions of moments and

7

shear forces into Eq. (1), the rotations and deflections in different portions of the cracked bi-layer beam are obtained as follows. for –a ≤ x ≤ –a+d:

φ1 = ∫

M1 αP dx = − x ( x + 2 a ) + c1 D1 4 D1

αP αP 2 Q w1 = ∫ ( 1 − φ )dx = − x+ x ( x + 3a ) − c1 x + c2 B1 2 B1 12 D1

(2)

for –a+d ≤ x ≤ 0:

φ1 = − w1 =

α Pd 2 D1

α Pd

x + c3

(3) 2

x − c3 x + c4

4 D1

for 0 ≤ x ≤ L–a–d:

φ1 = − w1 =

ϕ Pd 2 D1

ϕ Pd 4 D1

x + c5

(4) 2

x − c5 x + c6

for L–a–d ≤ x ≤ L–a:

φ1 = w1 =

ϕP

 x 2 +2(a − L) x  + c7 4 D1 

βP 2 B1

x−

ϕP 12 D1

(5)

x [ x+3(a − L)] − c7 x + c8 2

where ci (i = 1-8) are the integral constants determined by the continuity and boundary conditions; φ and β are the parameters based on the classical composite beam theory detailed in [37], which are given as

α=

η a D1 η h  a h D h  , β = −  + 1  M − 2 ,ϕ = 1 = −  M + 2  D1 + D2 DT  ξ 2  a0 2 D2ξ  ξ a0 2D2ξ 

where

ξ=

h1 h 1 1 (h + h ) h − 2 , η= + + 1 2 2, 2 D1 2 D2 A1 A2 4 D2 8

(6)

 1  1 ξ 1  h 1  h1 + h2  . ξ  , aM =  +  2 + a0 = −  + η + 2 D2   D1 D2  2 D2 D2  D1 D2 

It has to be mentioned that the derivation process of Eq. (6) is based on the displacement continuity condition along the interface as given in [18,37], which is different from the equivalent stiffness method of the composite beam. However, the two methods are equivalent as shown in Appendix 1 of [37]. The continuity and boundary conditions are

φ1

x =( − a + d )−

φ1

x = 0−

= φ1

x = 0+

φ1

x = ( L − a -d ) −

= φ1

w1

x =− a

= φ1

= 0, w1

x =( − a +d )+

, w1

, w1

= w1

x = 0−

x = ( L − a -d ) + x=L−a

, w1

x =( − a + d )− x = 0+

= w1

x =( − a +d )+

;

;

x = ( L − a -d ) −

(7) = w1

x = ( L − a -d )+

;

=0

From Eqs. (2) and (7), the eight unknown constants ci (i = 1-8) can be determined, with which the deflection along with the whole beam can be obtained. Strictly speaking, the beam compliance method is only valid for a linear elastic beam system characterized by a single applied load or a single applied displacement [38]. In this study, similar to the strategy adopted in the literature [13], the single applied load is defined as the total load P applied to the inner span, and the single applied displacement w is defined as the average displacement of the left and right loading points wL and wR. Then, the compliance C is defined as

C=−

w + wR w =− L P 2P

(8)

Once the compliance of the specimen is obtained, the energy release rate (G) of the specimen can be obtained by

9

G=

P 2 dC 2b da

(9)

where b is the width of the interface, i.e., the minimum of b1 and b2. Consequently, the compliance and ERR of cracked 4-AENF bi-layer beam specimen can be obtained, respectively, as: 2 d (α + β ) d 3Lϕ − 2d (α + ϕ ) + 3a (α − ϕ )  CC = + 4 B1 12 D1

GC =

P 2 dC P 2 d 2 = (α − ϕ ) 2b da 8D1b

(10)

(11)

where the subscript C refers to the solution based on the conventional composite bi-layer beam theory. It can be seen from Eq. (11) that G is independent of the crack length if the crack tip is between the two loading points, which shows that the 4-AENF specimen may be an effective way of determining the fracture toughness of bi-layer beam system, because there is no need to measure the crack length change or crack propagation during fracture test. 2.2 Flexible joint model

Although the rigid joint model described above is simple and widely used, it is fairly approximate due to omission of local deformation at the crack tip. To take into account the local deformation at the crack tip, an interface deformable bi-layer beam theory (flexible joint model), in which the local deformation due to interface normal and shear stress components is considered, was proposed by Qiao and Wang [36]. So the flexible joint model is further attempted to calculate the compliance and ERR. Here, the moment M1(x) and shear force Q1(x) in sub-beam 1 for 0 ≤ x ≤ L-a-d and L-a-d ≤ x ≤ L-a are modified as Case (a):

10

3

3

i =1

i =1

M1 ( x ) = M1C + ∑ fi Si e− Ri x , Q1 ( x ) = Q1C + ∑ f iTi e − Ri x

(12)

Case (b): M1 ( x ) = M 1C + f1S1e − R1x + e− R2 x  f 2 ( S2 cos( R3 x) − S3 sin( R3 x ) ) + f3 ( S3 cos( R3 x) + S2 sin( R3 x) )  Q1 ( x ) = Q1C + f1T1e− R1x

(13)

+ e− R2 x  f 2 ( T2 cos( R3 x ) − T3 sin( R3 x) ) + f 3 ( T3 cos( R3 x) + T2 sin( R3 x) )  where fi , Si and Ti are the parameters detailed in [36]. It is noted that extra terms are added in the expressions of moment and shear force for the reason that local deformation at the crack tip, which is caused by the stress singularity, is considered, and the detailed derivation process is given in [36]. Taking the same procedures of calculating the compliance and ERR described above based on the rigid joint model, the compliance by the flexible joint model is obtained as Case (a): d 2 (α − ϕ )  f12 S1 f 22 S 2 f 32 S3  C = CC + + +   4 D1 R2 R3   R1 a F

−

 f S f S  d 2  h2 f S + ξϕ   11 1 + 21 2 + 31 3   4 D1η  2 D2 R2 R3    R1 3

6

i =1

i =4

+ ∑ g i e− Ri ( a − L ) + ∑ g i e− Ri ( a + d − L )

Case (b):

11

(14)

 d 2 S1  f11  h2 + ξϕ    f12 (α − ϕ ) −  4 D1 R1  η  2 D2 

CFb = CC + +

d 2 (α − ϕ )  R2 ( f 22 S 2 + f 32 S3 ) + R3 ( f 32 S 2 − f 22 S3 )  4 D1 ( R22 + R32 )

 h2  d2 − + ξϕ   R2 ( f 21 S 2 + f 31 S3 ) + R3 ( f 31 S 2 − f 21S3 )  2 2  4 D1η ( R2 + R3 )  2 D2  + g1e

− R1 ( a − L )

+ g2e

− R1 ( a + d − L )

(15)

+

− R2 ( a − L )

 p1 sin ( R3 ( a − L ) ) + p2 cos ( R3 ( a − L ) )  − R a+d − L)  p3 sin ( R3 ( a + d − L ) ) + p4 cos ( R3 ( a + d − L ) )  + g4 e 2 ( + g3 e

where Ri (i = 1-3) are the roots of governing equations (11) in [36]. fi1 and fi2 (i = 1-3) are the coefficients of N and M as detailed in [36], gi (i = 1-6) and pi (i = 1-4) are the parameters with respect to the material properties and sectional dimensions of 4-AENF specimen. Considering that those expressions of gi (i = 1-6) and pi (i = 1-4) are rather complicated, they are not presented here for simplicity. M and N are the concentrated moment and axial force at the crack tip based on the rigid joint model, which have the following form for the considered 4-AENF specimen:

N =−

aM 1 Pd , M = (α − ϕ ) Pd 2a0 2

(16)

And the ERR based on the flexible joint model is then given as Case (a): 3

6

i =1

i =4

GFa = GC + ∑ ki e− Ri ( a − L ) + ∑ ki e − Ri ( a + d − L)

(17)

Case (b): GFb = GC + k1e

− R1 ( a − L)

+ k3 e

− R2 ( a − L )

+ k4 e

− R2 ( a + d − L )

+ k2 e

− R1 ( a + d − L )

+

 q1 sin ( R3 ( a − L ) ) + q2 cos ( R3 ( a − L ) )   

 q3 sin ( R3 ( a + d − L ) ) + q4 cos ( R3 ( a + d − L ) )  12

(18)

where ki (i = 1-6) and qi (i = 1-4) are the parameters dependent of material properties and sectional dimensions of 4-AENF specimen. Considering that those expressions are complicated, they are also not given here. It has to be mentioned that Cases (a) and (b) are classified by the form of the roots of the governing differential equation (11) in [36]. When the beam length L is several times the crack length a, the compliances for Cases (a) and (b) based on flexible joint model converge, respectively, as: Case (a): CFa = CC +

 f S d2 f S f S  (α − ϕ )  12 1 + 22 2 + 32 3  4 D1 R2 R3   R1

 f S f S  d 2  h2 f S − + ξϕ   11 1 + 21 2 + 31 3   4 D1η  2 D2 R2 R3    R1

(19)

Case (b): CFb = CC + + −

 d 2 S1  f11  h2 + ξϕ    f12 (α − ϕ ) −  4 D1R1  η  2 D2 

d 2 (α − ϕ )  R2 ( f 22 S 2 + f 32 S3 ) + R3 ( f 32 S 2 − f 22 S3 )  4 D1 ( R22 + R32 )

(20)

 h2  d2 + ξϕ   R2 ( f 21S 2 + f 31S3 ) + R3 ( f 31S 2 − f 21S3 )  2 2  4 D1η ( R2 + R3 )  2 D2 

In this special case, the compliance C Fa ( C Fb ) is a linear function of crack length a, and the coefficient in front of a is unchanged, so the expression of G Fa ( G Fb ) is the same as that of Eq. (11). Therefore, one has to design carefully the 4-AENF specimen in the actual experiment when characterizing the ERR if the rigid joint model is used. 2.3 Mode decomposition

Under mixed mode fracture conditions, the separation of the total ERR into individual mode-I and mode-II components is an important issue. There are two mode decomposition methods, i.e., the global and local methods. As revealed by Wang and Qiao [19], there exists 13

a certain relationship between the two decomposition methods, i.e., the local phase angle can be obtained by shifting a global phase angle. The global method is used here for its simplicity. The global method proposed by Wang and Qiao [19] assumed that the concentrated horizontal force Nc and concentrated peel force Qc only generate the respective mode II fracture and mode I fracture, and the global phase angle is expressed as

ψ G = arctan(

(GII ) GI

(21)

where

1 1 1 1 1 1 h12 h22 GI = ( + + )QC2 , GII = ( + ) NC2 . + 2 A1 A2 4 D1 4 D2 2 B1 B2 3. Design of the 4-AENF specimen

As presented in Section 2, once the compliance of the 4-AENF specimen is obtained, the fracture toughness (critical ERR) can be directly obtained if the critical load is known. Different from the FRP composites, the tensile strength of concrete is quite low. To ensure that the concrete sub-beam (substrate) has the enough load-carrying capacity and reduce the cracking tendency of concrete sub-beam, two strategies are considered: (1) steel bars with suitable diameter and concrete cover depth are placed in the tension zone of concrete sub-beam; and (2) the width of bonded concrete and GFRP/aluminum sub-layers is designed to be thinner as shown in Fig 3, so that it is prone to be debonded at the interface. To avoid the sudden change of cross-section and reduce the stress concentration near the GFRP-concrete bonded interface, the cross-section of the concrete sub-layer is wedged at the interface as shown in Fig. 3. To meet the applicability of beam theory, the interface length a and beam length L is relatively larger compared to the thickness of the whole beam h1 + h2 as 14

shown in Fig. 1. Also, the crack length is chosen so that the rigid joint model can be approximately used to calculate the ERR for its simplicity. The following step-by-step procedure summarizes the methodology for design of the proposed 4-AENF GFRP-concrete specimen: (1) Select an appropriate concrete height h c1 and hc2 with constant height of GFRP hf and variable height of aluminum h Al (Fig. 3), and calculate the effective stiffness coefficients Ai, Bi and Di (i = 1-2) and effective height hi according to [39]. (2) Select an appropriate width ratio of b 1/b2 so that the concrete sublayer has adequate load-carrying capacity. (3) Design the length of the specimen L, and select a proper initial crack length a and a proper distance d between loading points and support ends according to a parametric study as presented later in Sections 5.1 and 5.2. (4) Conduct a parametric study to obtain different fracture mode mixities with different aluminum heights. (5) Repeat Steps 1 through 4 to obtain feasible 4-AENF specimens for different mode mixities. Combined with the material properties shown later in Section 6.1, the longitudinal dimensions L, a and d of the specimen as shown in Fig. 1 are chosen as 750 mm, 300 mm, and 250 mm, respectively, and the sectional dimensions are b1 = 100 mm, b 2 = 50 mm, h c1 = 50 mm, hc2 = 15 mm, hf = 2.4 mm, and h Al = 20 (= 30 mm and 55 mm for different mode mixities) as shown in Fig. 3.

15

b1

hc

hc1

Concrete steel bar Φ

hc2 hAl

GFRP

Φ

hf

Aluminum b2

Fig. 3. Cross section of the 4-AENF specimen 4. Numerical finite element modeling

Geometrically linear finite element analysis (FEA) using commercial software ANSYS 16.0 is conducted to analyze the proposed 4-AENF fracture specimen of GFRP-concrete bonded interface and verify the analytical solution presented in Section 2. Sublayers 1 and 2 are both modeled by the four-node quadrilateral plane stress PLANE182 element with different real constants, which determine the width of layers. To prevent the mutual embedding of sublayers 1 and 2, the contact pairs consisting of TARGE169 and CONTA171 are created on the pre-crack interface between GFRP and concrete. To obtain a fine resolution of deformation and capture the crack tip deformation, the element size near the crack tip is chosen as 1.25 mm. Sub-layer 1 is concrete, while sub-layer 2 is composed of two layers (FRP and aluminum plates), in which the thickness of aluminum plate hAl varies to augment the stiffness of sub-layer 2 so as to obtain different fracture mixed mode ratios. The corresponding material properties of GFRP, concrete and aluminum are given in Table 1 in Section 6.1, and the total ERR is obtained by the virtual crack closure technique (VCCT) [40].

16

The global deformation of the cracked portion and crack tip deformation of the designed 4-AENF specimens is shown in Fig. 4. It can be seen from Fig. 4 that the top and bottom surfaces of the cracked portion are separated from each other for the reason that the stiffness values of sub-beams 1 and 2 are different from each other, which is consistent with the observation in [12]. Moreover, the contact area becomes larger with the increase of the stiffness of sub-beam 2. For the designed specimen with aluminum thickness of 55 mm, where the neutral axis is almost located at the bonded interface, sub-beams 1 and 2 of the cracked portion are almost completely in contact with each other except near the crack tip.

(a)

(b)

17

(c) Fig. 4. Global deformation of the cracked portion and crack tip deformation of the designed 4-AENF specimens with different thickness of aluminum layers: (a) hAl = 20 mm; (b) hAl = 30 mm; (c) hAl = 55 mm 5. Verification and parametric studies A parameter study is first performed to investigate influence of specimen length and aluminum thickness on the error of ERR based on the predictions of the rigid joint and flexible joint models, where the error is defined and expressed as ERROR = | GC − GF | /GC . Then, the influence of aluminum thickness on the mode mixities is conducted. Finally, based on the above parameter study, a verification study in comparison with the numerical finite element analysis is conducted to investigate influence of the crack length on the compliance and ERR of the designed 4-AENF specimen. 5.1 Effect of specimen length L The effect of varying specimen length L with a fixed loading point length d (Case a) and a fixed length L-2d (Case b) is presented in Fig. 5(a) and (b), respectively. As seen from Fig. 5(a), the difference (i.e., in the form of ERROR = | GC − GF | / GC ) of ERR between the rigid joint and flexible joint models is the same if the value of L-a-d (L-a) is the same for 18

specimens with different specimen length L, as expected in Eqs. (17) and (18). For Case (b), it can be seen from Fig. 5(b) that the difference of ERR between the rigid joint and flexible joint models becomes smaller with the increase of L-a-d. That is to say, the error of ERR of the rigid joint model from the flexible joint model becomes smaller with the reduction of pre-crack length for this case. What’s more, when the specimen length becomes larger, the error between the two joint models become smaller for all crack length, indicating the reduced local deformation effect as introduced in the flexible joint model. 0.12 0.10

L=0.60m L=0.70m L=0.75m L=0.80m L=0.90m L=1.00m

0.06 0.04

ERROR

ERROR

0.08

L=0.45m L=0.65m L=0.75m L=0.85m L=1.05m L=1.25m

0.2

0.1

hAl=0.02m, d=0.25m

hAl=0.02m, L-2d=0.25m

0.02 0.00

0.0 0

2

4

6

8

0.0

(L-a-d )/h1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

(L-a-d)/h1

(a)

(b)

Fig. 5. Effect of specimen length L on the difference of ERR between the rigid joint and flexible joint models with: (a) fixed length d; (b) fixed length L-2d 5.2 Effect of aluminum thickness

The effect of aluminum thickness with different crack lengths on ERROR is presented in Fig. 6. As presented in Fig. 6, for all the given crack lengths, the difference of ERR between the rigid joint and flexible joint models become larger with increase of aluminum thickness. It is also observed that the difference of ERR between the rigid joint and flexible joint models is almost the same for small thickness aluminum. That is to say, the ERR of the rigid joint model almost equals that of the flexible joint model, as expected in thin beams. Therefore, 19

there is a need to properly design the aluminum thickness in the actual design of 4-AENF specimen when the rigid joint model is considered, especially for the substrate with moderate aluminum thickness, where the local deformation is important in the prediction of ERR as given by the flexible joint model.

0.12

a=0.26m a=0.30m a=0.35m a=0.40m a=0.45m a=0.49m

0.10

ERROR

0.08 0.06 0.04

L=0.75m, d=0.25m

0.02 0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

hAl/h1

Fig. 6. Effect of aluminum thickness with repect to crack length on the difference of ERR between the rigid joint and flexible joint models 5.3 Effect of aluminum thickness on mode mixities

The variation of mode mixities with different thicknesses of aluminum plates is shown in Fig. 7, according to the global mode decomposition method presented in Section 2.3. As seen from Fig. 7, the mode ratios of GII/GT increase with the increase of aluminum thickness. Even though a wide range of mode mixities can be obtained as observed in Fig. 7, the beam theory will be no longer valid for an aluminum plate with great thickness. So the thickness of aluminum plate in the GFRP/aluminum sub-layer is chosen h Al = 20 mm, 30 mm and 55 mm with the mode ratios GII/GT of 54.2% ,75.2% and 99.85%, respectively.

20

1.0

GII/GT

0.8

0.6

0.4

0.2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

hAl/hc

Fig. 7. GII/GT with respect to the aluminum thickness hAl 5.2 Effect of crack length on the compliance and ERR

According to the design procedures presented in Section 3 and above parametric study, the dimensions (or sizes) of 4-AENF specimens for fracture test are obtained. The effect of crack tip deformation on the compliance and ERR of the designed specimens is hereby examined by comparing the solution based on the flexible joint model with those of the rigid joint model and numerical finite element analysis (FEA). A dimensionless parameter λ, which characterizes the relative location of crack tip, is defined as the ratio of the distances between a-d (the distance between the left loading point and crack tip) and L-2d (the distance between two loading points). The results of compliance and ERR of the designed 4-AENF specimens are presented in Figs. 8 and 9, respectively. It can be seen from Fig. 8 that almost all of these solutions predict a linear relationship between the compliance and crack length, indicating a constant compliance rate change (dC/da), as expected in the 4AENF specimen. By taking into account the relative rotation and deformation induced by the interface stresses of each sublayer of the virgin beam, the flexible joint model predicts a much closer solution of the

21

compliance than that of rigid joint model (see Fig. 8) when the FEA data is used as the benchmark. Compared with the rigid joint model, the flexible joint model gives almost the same ERR as that of rigid joint model when the crack tip is near the left loading point as shown in Fig. 9, in which the specimen length is much longer than the crack length. However, when the crack tip approaches to the right loading point, the ERR predicted by the rigid joint model begin to deviate from the results of the flexible joint model and VCCT based on FEA. These discrepancies may be caused by ignorance of local deformation caused by the interface stress components in the rigid joint model. Therefore, the rigid joint model can be used with reasonable confidence when the specimen is properly designed. It has to be mentioned that in the calculation of compliance using the flexible joint model, two interface compliance coefficients, Cni and Csi for sub-layer i must be determined properly. For the concrete sub-layer 1, the two coefficients are chosen in the same way as given by Qiao and Wang [36], and they are given as : Cn1 =

hc h , C s1 = c c c 10 E33 15G13

(22)

For sub-layer 2 which consists of the GFRP/aluminum composite beam, the two interface compliance coefficients, Cn2 and Cs2 are chosen similarly as given by Wang and Zhang [41] Cn 2 =

hf h h h + AlAl , Cs 2 = ff + AlAl f E33 10 E33 G13 15G13

22

(23)

8.5

9.5

8.5 8.0

hAl=0.02m

7.5 7.0 6.5 6.0

7.5 7.0

hAl=0.03m

6.5 6.0 5.5 5.0 4.5

5.5 5.0 0.0

Rigid joint model Flexible joint model FEA

8.0

Compliacne C (×10−8mN-1)

Compliacne C (×10−8mN-1)

9.0

Rigid joint model Flexible joint model FEA

0.2

0.4

0.6

0.8

4.0 0.0

1.0

0.2

0.4

(a)

0.8

1.0

(b) Rigid joint model Flexible joint model FEA

5.0

Compliacne C (×10−8mN-1)

0.6

λ

λ

4.5 4.0

hAl=0.055m

3.5 3.0 2.5 0.0

0.2

0.4

0.6

0.8

1.0

λ

(c) Fig. 8. Compliance of GFRP-concrete bonded 4-AENF specimen with respect to crack length: (a) hAl = 20 mm; (b) hAl = 30 mm; (c) hAl = 55 mm 1.60 1.52 1.56

1.44 1.40

G/P2 (×10−6m-1N-1)

G/P2 (×10−6m-1N-1)

1.48

hAl=0.02m Rigid joint model Flexible joint model FEA

1.36 1.32 1.28 0.0

1.52 1.48 1.44

hAl=0.03m Rigid joint model Flexible joint model FEA

1.40 1.36

0.2

0.4

0.6

0.8

1.32 0.0

1.0

λ

0.2

0.4

0.6

λ

(a)

(b)

23

0.8

1.0

1.10 1.08

G/P2 (×10−6m-1N-1)

1.06 1.04

hAl=0.055m

1.02

Rigid joint model Flexible joint model FEA

1.00 0.98 0.96 0.94 0.92 0.90 0.0

0.2

0.4

0.6

0.8

1.0

λ

(c) Fig. 9. The G/P2 values with respect to crack length: (a) hAl = 20 mm; (b) hAl = 30 mm; (c) hAl = 55 mm 6. Experimental characterization of GFRP-concrete bonded interface

Experimental programs using the designed 4-AENF specimen is conducted to characterize fracture behavior of GFRP-concrete bonded interface, and the critical loads for crack initiation and crack arrest as well as their corresponding critical energy release rates are obtained. As designed in Section 3, a reduced cross section scheme as shown in Fig. 3 is adopted with steel reinforcement bars in the tension zone of concrete beam so as to avoid the through-thickness cracking of concrete sub-beam (i.e., sub-layer 1) before the crack propagates along the GFRP-concrete bonded interface. The aluminum plate is also bonded to the GFRP thin layer and change the stiffness of sub-layer 2 (i.e., a composite beam made of aluminum and GFRP plates) so as to obtain different fracture mode mixities. 6.1 Materials

The designed C40 concrete sub-beams are cast with the same normal weight concrete patch produced by a professional manufacturer, and the maximum size of aggregates used is

24

10 mm. Two steel bars with a diameter of 10 mm are used to reinforce the concrete sub-beam. The glass fibers used are 3-ply fabrics formed by two-layer 0/90° cross stitched textile and one-layer random chopped fiber mat with a total weight 900 g/m2. Epoxy resin consists of two-part mixing components of resin and hardener, and it is used to impregnate or wet the glass fabrics to form a thin GFRP composite plate and then to act as adhesive to bond the GFRP composite plate to the concrete beam and aluminum plate. The mean compressive strength of concrete at 28 days is measured according to the requirements of GB/T 50081-2002 [42], and it is 40.1 MPa which is very close to the designed value. The other mechanical parameters of the concrete sub-beam are accordingly assumed to be these of C40 concrete and listed in Table 1. The material properties of GFRP plates are measured by the corresponding standards [43,44], and they are also listed in Table 1. The through-thickness elastic modulus of the GFRP plate is taken as 3.5 GPa. The common material properties of steel bars and aluminum plates are listed in Table 1, based on the data obtained from the literature. Table 1. Material properties of concrete, GFRP, aluminum and steel bar

Elastic modulus

Shear modulus

Compressive

Tensile strength

E (GPa)

G12 (GPa)

strength (MPa)

(MPa)

Concrete

32.5

13.5

26.8

2.4

GFRP

14.4

2.4

183.3

250.4

Aluminum

70.0

26.3

/

140.0

Steel bar

210.0

81.0

/

235.0

25

6.2 Specimen preparation

Three groups or a total of 18 4-AENF specimens are fabricated to evaluate the fracture toughness values of GFRP-concrete bonded interface. To achieve different fracture mode mixities, the thickness of aluminum plate in the GFRP/aluminum sublayer varies, of which Groups 1, 2 and 3 are with the aluminum plate thickness of h Al = 20 mm, 30 mm, and 55 mm, respectively, as elaborated in the aforementioned specimen design. First, the wedged concrete samples as shown in Fig. 10(a) are removed from the molds 24 hours after casting and cured at 25°C and 28 days before they are bonded with GFRP composites and subjected to the four-point bending test. Then, acetone is used to clear dust and oil of concrete sub-beam and aluminum plate before bonding to GFRP so as to achieve strong interface bond. Next, the epoxy resins are applied between the aluminum plate and GFRP plate as well as the GFRP plate and concrete, and assembly is compressed using steel clamps as shown in Fig. 10(b). A pre-crack is set between the GFRP plate and concrete through a plastic film. After 24 hours of clamping and another 24 hours of curing, the GFRP-concrete 4-AENF specimens are made. After final trimming and numbering, the specimens are ready for the fracture testing.

(a)

26

(b) Fig. 10. Fabrication of the 4-AENF specimen: (a) the wedged concrete sub-beam with reduced cross section; (b) bonding of the concrete sub-beam, GFRP and aluminum plates 6.3 Fracture testing

The fracture tests are performed on a Material Testing System (MTS) servo-hydraulic testing machine, and a four-point bend loading is applied through a steel-loading fixture as shown in Fig. 11. As designed in Section 3, the distances between the two loading rollers and two supports are set as 250 mm and 750 mm, respectively. In the test, all specimens are loaded to failure to measure the critical loads for crack initiation and arrest under displacement control with a loading rate of 0.25 mm/min. The machine load and displacement are automatically recorded by the MTS machine, and the displacements under the two loading rollers are synchronously and continuously recorded by two linear variable displacement transducers (LVDTs).

27

Fig. 11. Experimental setup and test configuration of GFRP-concrete bonded interface 4-AENF beam specimen 6.4 Results and discussion

Considering that crack propagation along the bonded interface or within the concrete sub-beam is of interest, the specimens with failure of concrete sub-beams (i.e., the crack develops/propagates in the concrete prior to the interface failure) are excluded in the further analysis. Typical total load vs. displacement relationships of the left (L) loading and right (R) loading rollers for specimens in Groups 1, 2 and 3 are shown in Fig. 12(a), 12(b) and 12(c), respectively. The crack propagation modes for Groups 1, 2 and 3 corresponding to three different mode mixities are presented in Fig. 13(a), 13(b) and 13(c), respectively. 40 30

Specimen S1-5 hAl=20mm

L

35

20 15

R

Specimen S2-4 hAl=30mm

30 Force (KN)

25

Force (KN)

R

L

25 20 15

10

10 5 0

5 0

2

4

6

8

0

10

Displacement (mm)

0

2

4

6

Displacement (mm)

28

8

(a)

(b) 70 60

Force (KN)

50

L

Specimen S3-6 hAl=55mm

R

40 30 20 10 0 0

1

2

3

4

5

Displacement (mm)

(c) Fig. 12. Typical load-displacement curve of GFRP-concrete 4-AENF specimens: (a) hAl = 20 mm; (b) hAl = 30 mm; (c) hAl = 55 mm

Crack tip

(a)

Crack tip

(b)

29

Crack tip

(c) Fig. 13. Typical failure modes of crack initiation and propagation of GFRP-concrete 4-AENF specimens: (a) hAl = 20 mm; (b) hAl = 30 mm; (c) hAl = 55 mm As revealed in Fig. 13, the crack first propagates along the bonded interface for a few centimeters and then into the concrete sub-beam for all specimens in Groups 1, 2 and 3. Considering that the interface debonding progress of GFRP-concrete specimens being studied here (i.e., closing form of crack at the cracked portion of beams) is more complicated than the case of opening crack as revealed in Figs. 4 (FEA) and 13 (experiment), only the first critical loads for crack initiation and crack propagation are approximately adopted in the data reduction analysis. According to the finite element analysis in Section 4, it can be seen that the top and bottom surfaces of the debonding interface are separated from each other for some distances before the propagation of debonding (see Fig. 4). However, the crack immediately closes after debonding happens for specimens in Group 3 for the reason that the neutral axis are almost located at the bonded interface. Therefore, the displacement at the left loading point varies intricately and trembles in Fig. 12(c) when the debonding happens as shown in Fig. 13(c). The crack arrest load for specimens in Group 3 is also hard to capture; therefore, only the crack initiation load is given here. The critical loads for crack initiation

30

and crack arrest are computed using the corresponding mean value as listed in Table 2. A statistical analysis of the critical loads for Group 1 yields a mean crack initiation load value of 16.85 KN with a COV of 6.3% and a mean crack arrest load value of 15.95 KN with a COV of 9.8%. While for Group 2, the mean value of the crack initiation load is 17.76 KN with a COV of 10.4%, and the mean value of the crack arrest load is 16.86 KN with a COV of 5.3%. For Group 3, the mean value of the crack initiation load is 30.16 KN with a COV of 2.9%, and the crack arrest load cannot be obtained. Table 2. Critical loads for the GFRP-concrete 4-AENF specimens in Groups 1, 2 and 3

Group

1

2

3

Specimen

Crack initiation load Pci (KN)

Crack arrest load Pca (KN)

S1-2

15.69

15.55

S1-5

17.09

14.62

S1-6

17.78

17.67

S2-2

19.74

19.66

S2-4

17.43

15.34

S2-6

16.10

15.59

S3-2

29.17

\

S3-3

30.82

\

S3-6

30.50

\

The fracture toughness values of the GFRP-concrete bonded interface for crack initiation and crack arrest defined as Gci and G ca are, respectively, determined by substituting the mean crack initial load Pci and mean crack arrest load Pca into Eq. (11). For Group 1, the fracture toughness values for crack initiation and crack arrest are 425.7 J/m2 and 381.4 J/m2, respectively. For Group 2, the fracture toughness values for crack initiation and crack arrest

31

are 494.7 J/m2 and 494.7 J/m2, respectively. While for Group 3, the fracture toughness for crack initiation are 981.3 J/m2, respectively. In addition, a brittle index I, which is the ratio of energy lost during crack growth to the energy required to initiate crack growth, is given as I=

Gci − Gca Gci

(24)

A large I value indicates catastrophic and unstable crack growth that is independent of the loading rate, and a small I value indicates slow tearing or growth in small increments. In the study of wood bonded interface conducted by River and Okkonen [45], a value of I = 0.43 was considered to represent a strong and moderately unstable crack growth, and an I value of 0.06 represented a moderately strong and stable crack growth. In the present study, the GFRP-concrete bonded interface for Groups 1 and 2 has I values of 0.104 and 0.099, respectively, which both indicate a moderately stable crack growth. 7. Conclusions

In this study, a combined analytical and experimental study is conducted to characterize the mode-II dominated fracture behavior of GFRP-concrete bonded interface using 4-AENF specimens. Compared with the rigid joint model, the flexible joint model which takes into account the crack tip deformation can give more accurate results of compliance and ERR in the 4-AENF specimen. Moreover, the calculated ERR by the flexible joint model can be reduced to that of rigid joint one when the specimen is properly sized. To ensure that the concrete sub-beam has enough tension load-carrying capacity and diminish the cracking in concrete, a reduced section scheme is adopted, i.e., the cross section width of the concrete 32

sub-beam and GFRP plate are designed differently. The aluminum plate is bonded to the GFRP plate so as to achieve different fracture mixed mode ratios. The fracture toughness values of GFRP-concrete bonded interface under three different fracture mixed mode ratios are obtained by the designed 4-AENF specimens. In conclusion, the 4-AENF specimens developed for GFRP-concrete bonded-interface in this study minimize the occurrence of potential tensile crack growth in concrete, and it is capable of measuring the mixed-mode fracture toughness values of FRP-concrete (or similar hybrid composite) bonded interfaces under different mode mixities. Acknowledgements

The authors would like to thank for the partial financial support from the National Natural Science Foundation of China (NSFC Grant No. 51478265) to this study. References

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Mixed Mode Fracture Characterization of GFRP-concrete Bonded Interface Using Four-point Asymmetric End-notched Flexure Test Qinghui Liu a, Pizhong Qiao a,b,∗ a

State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced

Ship and Deep-Sea Exploration, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China b

Department of Civil and Environmental Engineering, Washington State University, Sloan Hall 117, Pullman, WA 99164-2910, USA

Highlights:

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A four-point asymmetric end-notched flexure (4-AENF) specimen for mixed mode fracture of the glass fiber reinforced polymer (GFRP)-concrete bonded interface is proposed.

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Both the rigid joint and flexible joint models are used to calculate the compliance and energy release rate (ERR) of the 4-AENF specimens.

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The fracture toughness values of GFRP-concrete bonded interface under three different fracture mode ratios are experimentally obtained.

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The test method based on the 4-AENF specimen and data reduction procedures can be used to effectively characterize mode-II dominated fracture of hybrid material bonded interfaces.

∗

Corresponding author. E-mail addresses: [email protected] and [email protected] (P. Qiao). 39