concrete cohesive interface using digital image analysis

Composites: Part B 51 (2013) 35–43

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Investigation of sub-critical fatigue crack growth in FRP/concrete cohesive interface using digital image analysis Christian Carloni a,⇑, Kolluru V. Subramaniam b a b

Dept. of Architecture, University of Hartford, 200 Bloomfield Avenue, Hartford, CT 06119, USA Dept. of Civil Engineering, Indian Institute of Technology, Hyderabad, India

a r t i c l e

i n f o

Article history: Received 5 November 2012 Received in revised form 2 February 2013 Accepted 11 February 2013 Available online 26 February 2013 Keywords: A. Polymer–matrix composites (PMCs) B. Debonding B. Fatigue B. Stress transfer

a b s t r a c t The cohesive stress transfer during the sub-critical crack growth associated with the debonding of FRP from concrete under fatigue loading is experimentally investigated using the direct shear test set-up. The study focused on high-amplitude/low-cycle fatigue. The fatigue sub-critical crack growth occurs at a load that is smaller than the static bond capacity of the interface, obtained from monotonic quasi-static loading, and is also associated with a smaller value of the interfacial fracture energy. The strain distribution during debonding is obtained using digital image correlation. The results indicate that the strain distribution along the FRP during fatigue is similar to the strain distribution during debonding under monotonic quasi-static loading. The cohesive crack model and the shape of the strain distribution adopted for quasi-static monotonic loading is indirectly proven to be adequate to describe the stress transfer during fatigue loading. The length of the stress transfer zone during fatigue is observed to be smaller than the cohesive zone of the interfacial crack under quasi-static monotonic loading. The strain distribution across the width of the FRP sheet is not altered during and by fatigue loading. A new formulation to predict the debonding crack growth during fatigue is proposed. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Bonded fiber reinforced polymers (FRP) composites have been used for strengthening existing structures. Composite materials technology offers several advantages when applied to reinforced concrete (RC) structures, such as, excellent resistance to electrochemical corrosion and high strength-to-weight ratio. After more than two decades of application, the FRP technology has been proven to be economically advantageous and structurally effective for strengthening and repairing RC structures. Many issues related to the application of fiber reinforced polymers (FRPs) have been addressed and experience with bonded composites shows that the quality of the bond between the concrete substrate and the FRP is of critical importance in determining the level of strengthening achieved. Debonding is considered one of the most dangerous modes of failures for FRP-strengthened RC members. Debonding is produced by a crack which propagates in the interface region between concrete and FRP. Debonding mechanism has been studied within the framework of fracture mechanics [1–21]. Debonding crack initiation and propagation are modeled ⇑ Corresponding author. Address: College of Engineering, Technology, and Architecture, University of Hartford, 200 Bloomfield Avenue, West Hartford, CT 06117, United States. Tel.: +1 860 768 4857; fax: +1 860 768 5198. E-mail addresses: [email protected] (C. Carloni), [email protected] (K.V. Subramaniam). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.02.015

as Mode-II fracture problems. An interfacial cohesive material law, which relates the shear stress at the interface and the corresponding slip between the FRP and concrete has been proposed and calibrated against experimental data [11–13,18,19]. The interfacial fracture energy, which corresponds to the area under the cohesive material law curve, is associated with the maximum load that can be transferred through the interface before complete separation of the materials occurs [6,7,8]. Under monotonic quasi-static loading, the cohesive material law and the fracture energy have been measured experimentally and related to the properties of concrete and the type of composite [11–14,18,19,21]. The effect of environmental conditions on the fracture properties has also been studied [22–24]. Recent studies have highlighted that repeated or cyclic loading, which is a typical load condition in bridges, has a significant influence on the FRP-concrete interface properties [25–45]. From a general understanding of fatigue loading, it is known that fatigue produces crack growth at loads smaller than the critical load for quasi-static loading. Fatigue crack growth therefore is often called sub-critical crack growth. Most of the research studies on fatigueinduced debonding have been conducted using beam specimens [33–45]. The focus so far has been on degradation of stiffness, increase in deflections and design implication. An increase of the fatigue life of RC beams after strengthening is generally reported in literature and is attributed to the applied FRP sheets which are

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capable of reducing stresses in the steel reinforcement. When early debonding of the FRP sheet occurs, load-sharing between the concrete structure and the composite is prevented and the stress carried by the FRP is redistributed back to the internal reinforcing steel. Previous studies have shown that the fatigue response of FRP-strengthened beams depends on the range of the applied load: Pmax–Pmin [33]. In beam applications, Pmax and Pmin are defined in terms of the ‘‘ultimate load’’, which is not uniquely defined because it can be taken as the load corresponding to the ultimate capacity of the strengthened or un-strengthened [44] cross-section under static loading. Harries and Aidoo [41] pointed out that the degradation of the bond quality is observed even under very low-amplitude fatigue loading. Gheorghiu et al. [42] noticed that the fatigue response is independent of the number of cycles and the post-fatigue FRP–concrete interface behavior is altered by the previous cycling. Al-Rousan and Issa [44] tested six beams under fatigue loading. They considered several amplitudes, two different frequencies, and various bond arrangements. The authors concluded that the post-fatigue static behavior was not affected by the previous cycles even when the specimens were subjected to the same amplitude at different frequencies. Chaallal et al. [43] studied the fatigue performance of RC beams strengthened in shear and reported the effectiveness of the FRP system in extending the fatigue life under low-amplitude,/high-cycle fatigue. Dong et al. [45] presented an experimental work to study the fatigue response of RC beams strengthened with FRP in shear. The externally bonded composite is proven to increase the first crack load and the ultimate strength. Post-fatigue monotonic responses of beams with and without previous fatigue loading are very similar until the final failure stage. A survey of the literature reveals that while several observations have been made in relation to the fatigue crack growth and its influence on the load response of the structure, a clear understanding of the interfacial crack propagation during fatigue loading and the stress-transfer mechanism during fatigue loading is currently not available. Testing beams under fatigue loading does not allow for a full understanding of the debonding phenomenon during fatigue. The beam response is typically measured in terms of deflections of the beam or strain in the FRP. Several factors, such as, the presence of the steel reinforcement and how the load range is calculated with respect to the overall capacity of the beam, complicate the interpretation of the results. The applied loading cannot be directly related to the stress in the FRP–concrete interface. In addition, the available information from beam tests is hard to interpret since the observed change in deflection or stiffness can be attributed to several damage mechanisms and the contribution of FRP–concrete debonding cannot be isolated. Very few studies, which directly investigate the fatigue response of the FRP–concrete interface using direct shear tests of FRP–concrete joints, have been reported [25–32]. In direct shear tests, Pmax and Pmin are defined as a percentage of the ultimate load Pcrit, which corresponds to the debonding failure load of an equivalent specimen under monotonic quasi-static conditions. The available literature suggests that the fatigue response depends on the load range. Bizindavyi et al. [25] considered various parameters, such as, different amplitudes and mean values of the load ranges, and also different joint geometries (number of plies, bonded widths and length). The cyclic tests showed quasi-linear hysteretic behavior with very narrow loops. Higher maximum slip was observed for those joints with narrower width, even if the applied amplitude was lower than the ones with wider joints. Fatigue life of those joints tested with Pmin = 0, for the same load amplitude, were greater than those tested with Pmin – 0. Diab et al. [28] derived a theoretical solution of the fatigue debond growth and interfacial stress transfer of FRP–concrete joints. The study showed that the sub-critical fatigue crack growth rate (da/dN) of FRP–concrete interface diminished with fatigue cycles.

A threshold value for fatigue load amplitude equal to 30% of the static bond capacity of the FRP–concrete interface was suggested for preventing sub-critical crack growth. Ferrier et al. [30] published an extensive study using single-lap and double-lap shear tests to propose a methodology for the determination of the value of the allowable shear bond strength. S–N curves and the variation of the bond stiffness (the Young’s Modulus is used in their paper) against the number of cycles are obtained by fitting the experimental data using a Wohler’s law relationship. The experimental results reported by the authors [30] suggest a linear relationship between the maximum bond stress and the logarithm of the number of cycles. This approach however seems to point toward a strength approach rather than a fracture mechanics one. Experimental data which would provide an insight into the cohesive stress transfer during fatigue loading and its relation to the quasi-static behavior is currently not available. In this paper an experimental program focused on the high-amplitude/low-cycle fatigue behavior of the FRP–concrete interface is presented. The direct-shear test geometry which allows for directly determining the interfacial stress transfer during loading was used. Digital image correlation, which is a full field optical technique, was used to obtain strains in FRP and concrete. The test program provided a fundamental insight into the cohesive stress transfer during the sub-critical crack growth produced by fatigue loading. 2. Objectives The objectives of the present study are: (1) To study the cohesive stress transfer during the sub-critical crack growth under high-amplitude/low-cycle fatigue loading. (2) To compare the strain distribution in the FRP during and after fatigue loading with the monotonic quasi-static results. (3) To determine to what extent the Mode-II model could describe the crack propagation during fatigue loading. (4) To determine whether the fatigue loading could affect the strain distribution across the width of the FRP sheet which in turn would alter the stress-transfer mechanism. 3. Test set-up and materials In this experimental program, fatigue and monotonic quasistatic direct shear tests were performed using the classical pull– push configuration. The adopted test set-up was previously used by the authors to study the FRP–concrete interface behavior under monotonic quasi-static conditions. [11–13,18,19]. The composite sheet was bonded in the center on one side of the concrete block using the wet-layup procedure. The FRP composite comprised of continuous uni-directional carbon fiber sheet in a two part thermosetting epoxy matrix. The nominal diameter of the carbon fibers was equal to 0.167 mm (tf). The tensile strength and the Young’s Modulus (Ef) were equal to 3.83 and 230 GPa, respectively. The nominal width of the FRP sheet (b1) was 25 mm. The length (l) of the bonded area was 152 mm. In all specimens a notch was created by leaving 35 mm unbonded at the edge of the block. The dimensions of the concrete block were 330 mm (length, L), 125 mm (width, b) and 125 mm (thickness, h). The mixture proportions by weight of the constituents used in concrete were: cement (1.0): water (0.45): coarse aggregate (2.0): fine aggregate (2.0). River gravel with a maximum size of 10 mm was used as coarse aggregate. The average 28-day compressive strength of concrete was equal to 42 MPa [46]. Fig. 1 shows the specimen dimensions and the loading arrangement. The load was applied to the FRP sheet while the concrete block was restrained against movement. The

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37

Fig. 1. Test set-up.

restraints to the concrete specimen were placed at locations to minimize the effect of bending [11–13,18,19]. The strain components on the surface of the FRP and surrounding concrete during the fatigue and monotonic tests were determined from the displacement field, which was measured using a full-field optical technique known as digital image correlation (DIC) [47,48]. Details about DIC can be found in previous papers published by the authors [11–13,18,19] and in the book published by Sutton et al. [48]. The strain analysis reported in the next sections refers to the Cartesian system shown in Fig. 1. A photo of the test set-up is shown in Fig. 2. 4. Monotonic quasi-static tests Three specimens (DS-ST_1, DS-ST_2, and DS-ST_3) were tested under monotonic quasi-static conditions following the test procedure previously described. The quasi-static tests were performed under displacement control. The global slip, which is defined as the relative displacement between points on FRP and concrete initially located at the beginning of the bonded area, was increased at a constant rate equal to 0.00065 mm/s, up to failure. Global slip was measured using two LVDTs that were attached on the concrete surface on each side of the FRP sheet. The LVDTs reacted off of a thin aluminum plate, which was attached to the FRP surface at the edge of the bonded area as shown in Fig. 1. The load response of DS-ST_2 is reported in Fig. 3. The other load responses from all specimens were nominally similar. The modality of failure of all direct-shear-test specimens (including the ones under fatigue loading described in Section 5) was associated with progressive debonding of FRP. The load response plotted in Fig. 3 presents the same characteristics as the ones reported in previous papers by the authors [11–13,18,19]. The load response

Fig. 2. Photo of the test set-up.

Fig. 3. Load response of the monotonic quasi-static test DS-ST_2.

appears to level off with increasing global slip up to failure after an initial monotonic increase. The end of the monotonic load increase with global slip is marked by a sudden load drop (Points B and B0 ). In what follows, the portions of the load response before and after Point B are referred to as pre- and post-peak responses, respectively. The pre-peak response is comprised of a linear (OA) and non-linear (AB) part. In all tests the load drop corresponded to a value of the global slip within the range (0.25–0.30 mm). This range was consistent with previous quasi-static tests conducted by the authors [11–13,18,19]. In the post-peak response, the load is nominally constant and its value is identified as Pcrit. Pcrit is determined as the mean value of the load when the global slip varies between the values g1 and g2. Selection of the range for determining Pcrit is based on the results of the strain analysis which is presented later [11–13,18,19]. Table 1 reports the values of Pcrit, g1 and g2, and the values of the ultimate global slip at failure gu for the quasi-static monotonic tests. The longitudinal strain distribution along the direction of the fibers at point C in the load-global slip response of Fig. 3 is plotted in Fig. 4. For each value of y, the strain is computed as the average over the range 7 mm < x < 17 mm, where x = 12 mm corresponds approximately to the center of the FRP composite sheet. Fluctuations of the strain are observed along the length of the FRP sheet. They appeared to be at fixed location during the loading process and have previously been shown to be from local material inhomogeneities, variation in the thickness of the FRP and variations in local fracture toughness along the interface [11–13,18,19]. The nonlinear strain distribution is approximated by the following function [11]:

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Table 1 Monotonic quasi-static tests. Test #

Pcrit (kN)

(g1, g2) (mm)

gu (mm)

LSTZ (mm)

eyy (le)

smax (MPa)

s0(mm)

GF (N/mm)

bs (mm)

bd (mm)

DS-ST_1 DS-ST_2 DS-ST_3 Average

5.8 6.3 6.0 6.0

0.3;0.7 0.4;0.8 0.3;0.7 –

0.90 1.10 0.80 0.93

76 72 73 74

5500 5800 5700 5670

6.0 7.1 6.9 6.7

0.02 0.03 0.02 0.023

0.56 0.59 0.58 0.58

15.1 14.3 15.0 14.8

46.2 42.9 45.5 44.9

szy ¼ Ef tf

deyy dy

ð2Þ

Ef and tf are the Young’s Modulus and the thickness of the FRP sheet, respectively. The slip, s(y), between FRP and concrete, at a given location along the bonded length, is obtained by integrating the axial strain eyy in the FRP up to that point [11–14]. The interfacial fracture energy GF is obtained from the area under the entire szy – s curve [11–14]. The average values, determined from ten images for each test, of the maximum shear stress (smax), the corresponding slip (s0), and the fracture energy (GF) for the three monotonic quasi-static tests are reported in Table 1. 5. Fatigue tests Fig. 4. Axial strain at point C of the load response of DS-ST_2 (Fig. 3).

eyy ¼ e0 þ

a yy0 b

1 þ e

ð1Þ

where a, b, e0, y0 are determined using nonlinear regression analysis of the computed strains. The approximated strain distribution along the FRP obtained from Eq. (1) is also shown in Fig. 4. The observed strain distribution can be divided into three regions: (a) the unstressed region, corresponding to the unloaded end of the FRP where the strain is essentially zero. (b) The stress transfer zone (STZ), which is the intermediate region where the load is actually transferred from the FRP to the substrate. This region is characterized by variable gradients of the strain along the direction of the fibers. (c) The fully debonded zone, where the strains are essentially constant; the constant value of the strain in the fully debonded region is identified as eyy . The stress transfer zone (STZ) is fully established only for those points in the post-peak region where the load levels off at Pcrit [11–13,18,19]. Once the STZ is fully established, an increase of the global slip entails for a translation of the STZ further along the length of the FRP laminate, while its shape remains constant. The translation of the STZ indicates self-similar crack propagation along the interface. The values of the global slip [g1, g2], previously introduced, correspond to the range of the global slip where the STZ is fully established. The average values of the length of the STZ (LSTZ) and eyy for the three monotonic quasi-static tests are provided in Table 1. Ten images for each test were processed with DIC to calculate the average values reported in Table 1. The images corresponded to ten points in the load response within the range of the global slip [g1, g2]. The cohesive law, which relates the interfacial shear stress to the slip between corresponding points on the FRP and concrete substrate, is obtained using the strain distribution along the length of the FRP. The following assumptions were made in the analysis: (a) The FRP sheets are homogenous and linear elastic; (b) The thickness and the width of the FRP sheets are constant along the bonded length; (c) The interface is subjected only to shear loading; (d) The interface between the FRP and the concrete is assumed to be of infinitesimal thickness; and (e) The concrete substrate is rigid. From the measured strain eyy the interface shear stress szy is calculated as [49]:

Three constant amplitude fatigue tests (DS-FT_1, DS-FT_2, and DS-FT_3) were conducted using the direct-shear test configuration shown in Fig. 1. The global slip measured by the LVDTs was used during the cycles to estimate the interfacial crack propagation. The loading protocol was designed to investigate the residual load-carrying capacity of the interface at different stages of cycling and to understand the sub-critical crack growth during fatigue loading. Specimens were initially pre-loaded up to the mean value of the load range, and then a sinusoidal loading at 1 Hz was applied between fixed load levels, Pmax and Pmin, equal to 4.2 kN and 0.9 kN, respectively. Pmax and Pmin corresponded to 70% and 15% of the average critical load Pcrit obtained from the monotonic quasi-static tests, which was equal to 6.0 kN (see Table 1). Load was cycled between the two fixed load levels until a prescribed value of the global slip, gD, was measured. The value of gD for each test is provided in Table 2. Upon reaching the prescribed value of the global slip gD, ten slow cycles at 0.015 Hz were applied. The load was then decreased to 0.5 kN and monotonic quasi-static load was then applied up to failure by increasing the global slip at a constant rate equal to 0.00065 mm/s, similarly to the monotonic quasi-static tests. Digital images were taken during the slow cycles and the post-fatigue quasi-static loading. During the fatigue loading, data were recorded in ten-cycle blocks, every time the global slip increased by a threshold amount. Four ten-cycle blocks were recorded for each test during fatigue loading and were named A, B, C, D. Block A corresponded to the first ten-cycle block and block D corresponded to the slow-cycle block. gD was the global slip corresponding to the first cycle of block D. The load response of test DS-FT_1 is shown in Fig. 5. The load response of DS-ST_2 is plotted in Fig. 5 for comparison. Interestingly, the ultimate global slip at failure gu in the post-fatigue response of test DS-FT_1 is smaller than the average value of gu for the monotonic quasi-static tests (see Table 1). After recording the first ten cycles, ten-cycle blocks were recorded at 0.15 mm and 0.20 mm. Fatigue loading was then stopped when the global slip reached gD = 0.25 mm. The number of cycles at the beginning of each ten-cycle block is reported in Fig. 5. The post-fatigue load-carrying capacity, P PF crit , of DS-FT_1 is determined as the mean value of the load when the global slip varies between the values g1 = 0.45 mm and g2 = 0.70 mm in the postfatigue monotonic quasi-static response. Selection of the stated range for determining P PF crit is based on the same considerations

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C. Carloni, K.V. Subramaniam / Composites: Part B 51 (2013) 35–43 Table 2 Fatigue tests. Test #

P PF crit (kN)

gu (mm)

Pmax PPF crit

DS-FT_1 DS-FT_2 DS-FT_3 Average

5.7 5.3 5.6 5.5

0.85 0.80 0.85 0.83

74 79 75 –

(%)

Pmin P PF crit

16 17 16 –

(%)

DP PPF crit

58 62 59 –

(%)

(g1, g2) (mm)

gC (mm)

Cycles at C

gD (mm)

Cycles at D

AC =AA

0.45;0.75 0.45;0.70 0.65;0.80 –

0.20 0.25 0.40 –

7569 4735 8004 –

0.25 0.30 0.50 –

9367 5513 10,565 –

1.55 1.44 1.57 –

Fig. 5. Load response of DS-FT_1.

presented in the ‘‘monotonic quasi-static tests’’ section. The values of g1 and g2 can be found in Table 2 for the three tests. Table 2 summarizes the post-fatigue load-carrying capacity, PPF crit , and the range of the global slip (g1, g2) used to calculate PPF . The average value of crit P PF crit is in good agreement with the average quasi-static bond loadcarrying capacity Pcrit reported in Table 1. The actual ratios PF P max =PPF crit and P min =P crit are computed a posteriori for each test based on the post-fatigue response and are reported in Table 2 for the three specimens. Although the actual load ratios are similar, DS-FT_2 has higher amplitude. The percentage amplitude DP=PPF crit of each test is reported in Table 2. The hysteresis curves during fatigue loading are qualitatively similar (Fig. 5). For any cycle, the initial part of the response in the ascending branch is convex which then becomes concave [42]. The inflection point occurs approximately at 2.5 kN which corresponds to the mean value of the range [33]. The hysteretic loops increase in width as the global slip increases [42]. The residual slip at the bottom of the cycles also increases with the increasing number of cycles. The area enclosed by the loop represents the energy dissipated in the cycle [31–33,42]. In DS-FT_1, this area gradually increases with the number of cycles. The ratio of the area of block C to the area of block A, AC/AA, is 1.55 for DS-FT_1. Block D is not considered because it represents ten slow cycles at 0.015 Hz. Table 2 shows the ratio AC/AA and the number of cycles of the first cycle of blocks C and D, for the three tests. The increase of the energy dissipation with the number of cycles was observed in all specimens. It is important to note that the increase is relative to the initial block A of each test. Although the results presented in Table 2 are in good agreement with the recent work published by the authors [31,32], further research is needed to understand the role of the amplitude and mean value of the cycles, and the relationship between the number of cycles and the value of the global slip at which the energy dissipation is computed. Yun et al. [27] observed that the area enclosed by the loops increase gradually for high-amplitude cycling. This result is not confirmed by Gheorghiu et al. [42]. In their study the authors observed that the enclosed area does not increase with cycles both at low- and high-amplitude cycling. However, it should be taken into account

that the results presented by Gheorghiu et al. [42] refer to fatigue tests on beam specimens. The change in the energy dissipation could be potentially explained by the phenomenon of crack closure that typically occurs in cohesive materials during fatigue. The crack closure depends on the amplitude and mean value of the load range. Further research is needed to study the energy dissipation. The available data are not fully comparable because different frequencies have been adopted by researchers. Diab et al. [28] noticed that the load frequency has an influence on the bond degradation in terms of the interfacial constitutive material law. This observation is not confirmed by Al-Rousan and Issa [44] who tested beams under two different frequencies. The slope of the loops in test DS-FT_1 are found to be gradually decreasing with the number of cycles, indicating a progressive loss in the stiffness of the interfacial bond between the FRP composite and concrete substrate. Yun et al. [27] reported that under lowamplitude fatigue load, the slopes of the hysteresis loops remain unchanged, while it continuously decreases in high-amplitude cyclic tests. Similar results are reported by Al-Rousan and Issa [44] for beams tested under fatigue loading. Bizindavyi et al. [25] reported a decrease of the slope of the hysteresis. Gheorghiu et al. [42] reported a gradual decrease of the slope for low-amplitude cycling and a constant slope for high-amplitude cycling. The results by Gheorghiu et al. [37,40,42] are not easily comparable with the present study because refer to fatigue behavior of beams. The ten-cycle blocks of the three tests are compared in Fig. 6. DS-FT_2 cycles have a lower slope when compared to the equivalent cycles (same maximum global slip) of DS-FT_1, which in turn corresponds to a lower residual global slip. A similar trend is noticed when the ten-cycle blocks of DS-FT_2 and DS-FT_3 are compared. An explanation of this trend can be found in the actual value of the load amplitude DP, provided in Table 2. As previously discussed, the change in the slope depends on the amplitude. In particular, Bizindavyi et al. [25] reported that the rate of change of slope increases with increasing load amplitude. In this context, it is impossible to separate the effect of the amplitude and the mean value. In fact, the ratios of the amplitude of the cycles DP as well as the ratio of the mean value of the cycles to the static bond capacity

Fig. 6. Comparison of the ten-cycle blocks of the three fatigue tests.

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Fig. 7. Axial strain at Point E of the load response of DS-FT_1 (Fig. 5).

length of the bonded area is greater than LSTZ. A similar observation is also reported in the available literature [25,28,31,32].

Table 3 Fracture parameters of the post-fatigue response. Test # TestFT_1 TestFT_2 TestFT_3 Average

Fig. 8. Axial strain at point D (tenth slow cycle) of the load response of DS-FT_1 (Fig. 5).

LSTZ (mm)

eyy

smax

le

(MPa)

s0 (mm)

GF (N/ mm)

bs (mm)

bd (mm)

73

5400

5.8

0.03

0.48

15.5

45.0

65

5100

6.1

0.03

0.41

14.1

44.1

72

5300

5.5

0.03

0.47

15.1

45.6

70

5260

5.8

0.03

0.45

14.9

44.9

PPF crit vary. The limited number of blocks does not allow to draw a general trend and further research is needed to investigate this aspect. Because of the different amplitude, the number of cycles (3374) at the beginning of block B of DS-FT_2 is significantly lower than the number of cycles (7569) at the beginning of block C of DSFT_1. It could be interesting to compare cycle blocks of different tests corresponding to the same number of cycles.

6. Post-fatigue strain analysis The longitudinal strain distribution at point E of the post-fatigue response of test DS-FT_1 (Fig. 5) is plotted in Fig. 7, together with the approximated strain distribution by means of Eq. (1). Similarly to the monotonic quasi-static tests, the strain is computed as the average over the range 7 mm < x < 17 mm, where x = 12 mm corresponds approximately to the center of the FRP composite sheet. The strain distribution corresponds to a point of the portion of the monotonic quasi-static test, in which the load levels off (P PF crit ). The strain distribution clearly shows the same trend of the monotonic quasi-static tests. The interface cohesive law parameters and the fracture energy (GF) were derived from the strain distribution, following the procedure previously described [11–13,18,19] and are listed in Table 3. In Table 3, the average values of the strain at debonding (eyy ), calculated in the portion of the quasi-static response where the load levels off, are reported for the three fatigue tests. Five images, processed with DIC, were used to calculate the average values reported in Table 3. Fracture parameters reported in Table 3 are in good agreement with the results presented in Table 1. It appears that the quality of bond and the load–carrying capacity (see Tables 1 and 2) is not affected by a previous fatigue loading, provided that the interface has not completely debonded during cycles and the

7. Fracture parameters during fatigue loading The longitudinal strain distribution corresponding to the peak of the tenth slow cycle of block D of DS-FT_1 is plotted in Fig. 8. The strain was computed as the average along the central region of the FRP composite sheet, following the procedure reported in the ‘‘monotonic quasi-static tests’’ section. The general shape of the stress transfer region is similar to that observed in monotonic quasi-static loading. A distinctive stress-transfer region can be identified. Further, the strain at the loaded end appears to be constant, which suggests that debonding occurred. These observations indicate the presence of sub-critical crack growth during fatigue loading. Eq. (1) has been used to approximate the strain distribution of Fig. 8. By using Eq. (1), it is implicitly assumed that: (1) Debonding occurs during fatigue loading (constant strain at the loaded end); (2) The strain distribution during cyclic loading is similar to the one observed in the quasi-static tests. These assumptions are discussed at the end of this section. The strain levels off at a value (efatigue ) approximately equal to 3600 le. The length of the stress yy transfer zone (Lfatigue ) appears to be smaller than the one reported STZ in static tests. In Table 4, the average values of the fracture parameters and the value of the strain at debonding (efatigue ), calculated at yy the peak of the ten slow cycles, are reported for the three fatigue tests. Ten images, processed with DIC and corresponding to the peaks of the ten slow cycles, were used to calculate the average values reported in Table 4. The interface cohesive material law parameters and the fracture energy (Gfatigue ) are derived from the F strain distribution in a similar manner as described for the quasistatic tests. The cohesive crack length, a, during the cycles is determined as the distance from the loaded end to the point of inflection of the strain curve. The average value of the cohesive crack length at the end of the ten slow cycles, for the three tests, is also reported in Table 4. The reduced length of the stress transfer zone during fatigue can be attributed to the interfacial propagation in the epoxy layer rather than the FRP–concrete interface. This circumstance is supported by the visual analysis of the FRP sheets after failure and it was pointed out in a previous work by the authors [31,32]. A photo of the DS-FT_1 strip at failure is reported in Fig. 8. The debonded surface is smoother in the first 40 mm close to the loaded end, which approximately corresponds to the length of the cohesive crack (a) during fatigue. A similar observation is reported in Iwashita et al. [26]. Similar observations are reported also in Ferrier et al. [30]. The authors noticed that the fatigue performances of

41

C. Carloni, K.V. Subramaniam / Composites: Part B 51 (2013) 35–43 Table 4 Fracture parameters during fatigue loading. Test #

Lfatigue (mm) STZ

efatigue (le) yy

sfatigue (MPa) max

sfatigue (mm) 0

Gfatigue (N/mm) F

a (mm)

P Cycles (kN) crit

bs (mm)

bd (mm)

DS-FT_1 DS-FT_2 DS-FT_3 Average

50 50 56 52

3500 4000 4100 3870

6.0 6.7 3.9 5.5

0.01 0.02 0.02 0.017

0.22 0.27 0.29 0.26

40 68 100 –

3.7 3.6 3.3 3.5

15.2 13.9 15.4 14.8

44.6 44.9 45.2 44.9

FRPs are greatly influenced by the physical and mechanical properties of the epoxy. In this context, the authors want to focus the attention on the assumption that Eq. (1) can be used to approximate the strain distribution during the cycles. The assumption can be indirectly verified if one can prove that the fracture energy (Gfatigue ) during F fatigue can be used as an effective parameter to study the crack propagation. In quasi-static conditions, it has been proven [6,7,8] that the load-carrying capacity of the interface is related to the fracture energy:

Pcrit ¼ b1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GF  Ef  tf

ð3Þ Gfatigue F

The fracture energy (see Table 4), obtained from the ten slow cycles of block D, can be used to calculate the applied load during fatigue:

PCycles ¼ b1 crit

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Gfatigue  Ef  t f F

ð4Þ

PCycles crit

is provided in Table 4 for the three tests, and should be compared to the applied load at the peak of the cycles Pmax, which is equal to 4.2 kN. The calculated loads P Cycles are in good agreement crit with the applied load Pmax if one considers two factors: (1) The images used to calculate Gfatigue were taken during the cycles and F even though the frequency was 0.015 Hz the images might not correspond exactly to the peak load (4.2 kN). (2) The width effect [13,19], discussed in the next section, influences the value of the critical load and is not included in Eq. (3) [50]. Hence, these observations lead the authors to the conclusion that the crack propagation during fatigue can be treated still as a Mode-II problem within the fracture mechanics framework and Eq. (1) is suitable to approximate the strain distribution. 8. Width effect In the past, the authors studied the width effect in FRP–concrete debonding [13,19]. It was recognized that the longitudinal strain distribution across the width of the FRP sheet is not constant. The presence of a central region within the FRP width is associated with the Mode-II fracture propagation. The width of central region is constant during the debonding process and increases as the FRP width increases. The presence of the central region is confirmed by the shear strain distribution [13,19].The variation of the longitudinal strain across the FRP width is plotted in Fig. 9 for different locations along the bonded length. The strain distribution of Fig. 9 corresponds to point C of the load response of DS-ST_2 plotted in Fig. 3. The range of values of y has been determined from the strain distribution shown in Fig. 4. The STZ corresponds to the range 67 6 y 6 130 mm. The longitudinal strain is nominally constant in the central region of the FRP. bs and b1 are the width of the central region and the width of the FRP sheet, respectively. The total width bd defines the portion of concrete on each side of the FRP sheet that is involved in the stress transfer mechanism. The average values of bs and bd for the three monotonic quasi static tests are reported in Table 1. A similar strain distribution was observed for the post-fatigue quasi-static response and during the ten slow cycles of block D for tests DS-FT_1, DS-FT_2, and DS-FT_3. The values of bs and bd are reported in Table 3 and 4. A comparison

Fig. 9. Variation in the axial strain across the width of the FRP sheet at Point C of the load response of DS-ST_2 (Fig. 3).

of the values of bs and bd from Tables 3 and 4 with the ones of Table 1, indicates that the distribution of the longitudinal strain across the width of the FRP is not altered by and during fatigue loading. A Mode-II interface crack propagation can be assumed within the central region. In deriving Eq. (3) to predict the critical load, a Mode-II propagation is implicitly assumed across the entire width of the FRP sheet, without taking into account the more complex mixed mode in the edge regions of the composite sheet [13,19]. As observed before, this fact can partially justify the difference between the values of PCycles reported in Table 4 and Pmax . It is crit interesting to notice that even for the quasi-static loading condition Eq. (3) provides a load that is smaller than the values of Pcrit reported in Table 1. Cheng and Teng [3] proposed a design formula to compute the strain at failure when debonding occurs in FRP-to-concrete bonded joints:

sffiffiffiffiffiffiffiffi pffiffiffiffiffi fc0 eyy ¼ 0:427b b L P Ef tf

ð5Þ

where bL = 1 if the length of the bonded FRP is greater than LSTZ. bP takes into account the effect of the width ratio of the bonded plate to the concrete block b1/b [3]. For the monotonic quasi-static tests, Eq. (5) provides a value of the ultimate strain eyy equal to 6700 le; whereas if the coefficient bP is not considered, Eq. (5) provides a value of eyy equal to 5600 le, which is in good agreement with the average value reported in Table 1. The value of the strain calculated from Eq. (5), does not take into account that the axial strain is not constant across the width of the FRP sheet. Several formulations of the width coefficient bP have been proposed. It seems correct, however, to apply the width coefficient to the load–carrying capacity to account for the variation of the strain across the width, rather than to the strain, which is constant within the central region and is related to the interfacial fracture energy:

eyy ¼

Pcrit ¼ b1 Ef t f

sffiffiffiffiffiffiffiffi 2GF Ef t f

ð6Þ

42

C. Carloni, K.V. Subramaniam / Composites: Part B 51 (2013) 35–43

If the average fracture energy of Table 1 is used, Eq. (6) provides a value of eyy equal to 5500 le, which is in good agreement with the average value of the maximum strain at debonding reported in Table 1. 9. New fatigue life prediction Diab et al. [28] attempted a fracture mechanics approach to relate the rate of debonding growth to the strain energy release:

da Gfatigue ¼ m1 F dN GF

!n 1

b

ð7Þ

The coefficients m1, n1, b can be determined from experimental results. In particular, b takes into account that the crack propagation rate decreases as the debonded region increases. In Eq. (7) an additional coefficient, related to the effect of the frequency, has been considered equal to 1. An improvement of Eq. (7) is attempted in this section. It is important to notice that the fracture nature of Gfatigue and its relaF tionship with the amplitude and mean value of the load range have not been investigated in the available literature. However, the results presented in Sections 7 and 8 of this work indicate that: (1) The values of Gfatigue and GF are directly related to the applied load F during cycles and the load–carrying capacity of monotonic quasistatic tests, respectively; (2) The strain distribution across the width of the FRP sheet is not altered by and during fatigue loading. Hence, these observations suggest that the fracture energies of Eq. (7) can be substituted by the applied loads which are related to the corresponding fracture energies (see Eqs. (3) and (4)). In addition, Eq. (7) only considers the value of Gfatigue , without explicitly taking F into account the amplitude and the mean value of the cycle, which are proven to be determinant parameters in fatigue [51]. Eq. (8) provides an alternative formulation to Eq. (7):

pffiffiffiffiffiffiffiffiffiffiffiffiffi!n1 da a DP  P  1 b ¼m dN Pcrit

ð8Þ

DP ¼ Pmax  P min and P ¼ ðPmax þ Pmin Þ=2 are the amplitude and the mean value of the load range, respectively. Pcrit is the monotonic quasi-static load–carrying capacity of the interface. a is the frequency coefficient [28], which takes into account that the fracture properties during fatigue loading depend on the frequency. This approach requires further research to understand the role of Gfatigue . F  must be determined from the experimen 1; n 1 ; b The coefficients m tal data. The results herein obtained are not sufficient to characterize the coefficients and validate Eq. (8). 10. Conclusions In this paper, the authors attempted to gain an insight into the sub-critical crack growth during fatigue loading and the cohesive stress transfer during this crack growth. The following conclusions can be highlighted: 1. Debonding occurs during fatigue loading and is produced by an interfacial crack which appears to propagate at the epoxy/FRP interface rather than in the epoxy–concrete interface. 2. Fatigue crack growth occurs at a load which is smaller than the quasi-static critical load. 3. A cohesive stress transfer zone (STZ) of a finite length is established during the propagation of the fatigue-induced crack. The strain distribution along the STZ is similar to the strain distribution observed in quasi-static direct shear tests. 4. The strain profiles obtained at the peak of the cycles (see discussion in Section 7) show that the fatigue crack growth is associated with a cohesive stress-transfer zone of a smaller length than the LSTZ in quasi-static loading.

5. The cohesive material law is obtained from the strain distribution along the composite during fatigue and in the post-fatigue response. The fracture energy, which is the area under the cohesive law, associated with the sub-critical crack growth is smaller than that for the quasi-static loading. It was also observed in Section 7 that the cohesive crack model is indirectly proven to be adequate to describe the deboning mechanism during fatigue loading. 6. The strain distribution across the width of the FRP sheet is not altered by and during fatigue loading. The presence of a central region in the FRP sheet, allow to study the sub-critical stress transfer as a Mode-II fracture problem. Further work is needed to fully understand the cohesive stress transfer and debonding during fatigue loading. The role of the amplitude and mean value of the load range on the sub-critical crack growth during fatigue is still not clear. These are important factors to provide effective design guidelines. The effect of the frequency of the cyclic loading should be considered and investigated for a broad range of frequencies. Ferrier et al. [30] observed that conventional civil engineering structures are typically loaded at frequencies varying between 1 and 5 Hz. Therefore, it would be interesting to consider such a range to study the influence of the frequency. The use of Eq. (8) to predict the fatigue life of the FRP–concrete interface needs to be validated against a larger number of experimental data to include, for example, the effect of the amplitude and mean value of the load. Several FRP widths should be considered to support the conclusions reported in this paper. Finally, the decrease of the FRP–concrete interfacial stiffness and the energy dissipation during cycles require further research.

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