Modeling of fracture parameters for crack propagation in recycled aggregate concrete

Construction and Building Materials 106 (2016) 168–178

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Modeling of fracture parameters for crack propagation in recycled aggregate concrete Rajendra Kumar Choubey, Shailendra Kumar ⇑, M. Chakradhara Rao Department of Civil Engineering, Institute of Technology, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur (C.G.) 495009, India

h i g h l i g h t s
Fracture parameters of RAC with varying RCA content are determined using FCM and DKFM. un

ini


Pu and KIC , KIC

and Pini of RAC decrease linearly with varying RCA content as predicted by FCM and DKFM respectively. ini un are almost constant for NAC and RAC with varying RCA content.


The ratio Pini/Pu and KIC /KIC

a r t i c l e

i n f o

Article history: Received 29 June 2015 Received in revised form 14 November 2015 Accepted 15 December 2015

Keywords: Recycled aggregate concrete Recycled concrete aggregate Fracture energy Double-K fracture model Fictitious crack model

a b s t r a c t The paper presents the applications of fictitious crack model and double-K fracture model, to determine the fracture parameters for crack propagation in recycled aggregate concrete with varying content of recycled concrete coarse aggregate. The required material properties for modeling applications are derived using various empirical relations, for the known values of compressive strength. The fracture parameters of normal and recycled aggregate concrete are determined and a systematic comparative study has been carried out. The outcome has revealed an interesting finding and thus it is concluded that these two conventional concrete fracture models can be implemented for the study of fracture behavior of recycled aggregate concrete with varying substitution of natural aggregate by the recycled aggregates. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The global concrete industry uses approximately 10 billion tons of sand and rock annually, which makes it the largest consumer of natural resources in the world [1]. Use of the aggregate wreckage from old concrete structures is the need of these days to reuse them as recycled coarse aggregate for the production of new concrete. This practice will be conducive for the environmental preservation and sustainable development. Thus, it is now considered as an alternate concrete aggregate material source yielding a positive socio-economic impact. Researchers have been engaged in experimental investigations with regards to the mechanical properties and durability characteristics of recycled aggregate concrete (RAC), to ascertain the utilization of waste concrete aggregate by substituting the natural aggregate, in construction. Although significant research work has been carried out on the mechanical properties of RAC, but study on crack propagation and fracture parameters of RAC is meager in the literature. ⇑ Corresponding author. E-mail address: [email protected] (S. Kumar). http://dx.doi.org/10.1016/j.conbuildmat.2015.12.101 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.

Casuccio et al. [2] put forward from their experimental investigation, that recycled concrete aggregate (RCA) present in RAC slightly lowers the strengths (1–15%), lowers the modulus of elasticity (13–18%) and significantly reduces the fracture energy (27– 45%) and, consequently affects the fracture zone size, when it is compared with a concrete prepared with natural coarse aggregates. Reis and Jurumenh [3] have investigated the influence of replacement of fresh sand by recycled foundry sand on fracture properties of polymer concrete. The results showed that epoxy polymer concrete made with recycled foundry sand has higher fracture toughness and slight lower fracture energy when compared with fresh sand polymer concrete. A comprehensive review of the findings in China during the past decade, related to influences of the RCA on mechanical and fracture properties, has been presented by Xiao et al. [4]. In the study, it is reported that the compressive, tensile and shear strengths of RAC are generally lower than those for conventional concrete and these values decrease with the increase of RCA contents. The modulus of elasticity of RAC reduces with increase in RCA content; however the peak strain is larger than that of conventional concrete. The RCA content has a significant influence on the stress–strain curve

R.K. Choubey et al. / Construction and Building Materials 106 (2016) 168–178

of RAC in uni-axial compression, tension and shear. The RCA replacement percentage has nearly no influence on the bond strength between RAC and deformed rebar, while the bond strength between RAC and plain steel rebar decreases with the increase of RCA content. RAC behaves in a more brittle manner than conventional concrete in compression and the plastic deformation is less. Li et al. [5], reported in their study, that the mechanical properties of new mortar matrix and the corresponding new interfacial transition zone play a significant role in overall stress– strain response and fracture process of modeled recycled aggregate concrete. The finite element method (FEM) modeling is capable of simulating the complete stress–strain curve and also the overall fracture patterns including localization of deformation and the micro-crack pattern. Arezoumandi et al. [6], determined fracture energy from the experiments using five mixes with 0%, 30%, 50%, 70%, and 100% recycled concrete aggregate as a coarse aggregate replacement in concrete. The authors compared these results with different design codes and analytical equations and reported that, the values obtained from the CEB-FIP Model code-2010 [7] and Bazˇant equations [8] have a good agreement with the test results and the JSCE-07 [9] provision is found to underestimate the fracture energy for all specimens. The outcome of the study conducted by Riaz et al. [10], indicates that the mechanical properties of RAC starts degrading when recycled aggregates (RA) replacement level exceeds 25% and degradation further increases with the increase in the replacement level. The compressive strength and static modulus of elasticity is found to decrease with an increase in RCA content and a maximum decrease of 20% is observed corresponding to 100% replacement. A gradual decrease in value of fracture energy in tension is observed with increase in percentage of RA with a maximum drop of 38% when RCA replacement in concrete is 100%. However, fracture energy value in tension did not alter with concrete containing 25% RA, whereas fracture energy in compression was significantly reduced, i.e., 24% reduction in fracture energy was observed with concrete containing 25% RA. Maximum reduction of 45% in the value of fracture energy in compression was exhibited by the concrete made with 100% RA. The peak load value in flexure is noticed to be decreased by replacing the natural aggregates with RCA, however, the crack mouth opening displacement (CMOD) value at the peak load in flexure remains almost unchanged. The main reasons for reduction in strength, elastic property and fracture energy for RAC are increased concrete porosity, weak aggregate matrix interface bond in the new matrix due to residual adhered-mortar on RCA. The prime factor affecting the elastic modulus of RAC, is the modulus of elasticity of RCA itself, as it is more prone to deformation than the material aggregates. From the existing literature review it is observed that, extensive research work on mechanical and durability characteristics of RAC have been carried out. Also much of the experimental and numerical studies on the crack propagation and fracture properties for normal aggregate concrete have been carried out worldwide using various concrete fracture models. However, experimental and numerical studies on crack propagation and fracture properties of RAC are meager in the literature. It is also observed that, few experimental results are reported in the literature, on the studies of fracture behavior of RCA and no study has been reported on fracture behavior of RAC using conventional concrete fracture models. In order to determine the critical stress intensity factor of concrete, the concept of linear elastic fracture mechanics (LEFM) was first applied to concrete notched beam by Kaplan [11]. Thereafter, extensive research and studies for crack propagation and fracture properties of concrete like materials have been carried out. It is now clear that the concept of LEFM cannot be applied directly to concrete and concrete like materials, because a sizeable fracture process zone exists ahead of the initial crack tip. Then many non-

169

linear fracture models have been developed and applied to concrete structures by various researchers. These nonlinear models are: cohesive crack model (CCM) or fictitious crack model (FCM) [12–26]; crack band model (CBM) [27], two parameter fracture model (TPFM) [28], size effect model (SEM) [29], effective crack model (ECM) [30]; KR-curve method based on cohesive force distribution [31–34], double-K fracture model (DKFM) [26,32,35–50] and double-G fracture model (DGFM) [51]. Among these eight models, FCM and CBM are based on numerical approach, in which fracture energy is required as one of the input parameters. But, for the remaining models, either the analytical or semi-empirical and semi-analytical formulae in the modified form of LEFM are used in the form of stress intensity factor or fracture energy, to express the fracture toughness of concrete. Hillerborg et al. [12] initially applied fictitious crack model (FCM) as a suitable nonlinear model for mode-I fracture, to simulate the softening damage of concrete structures. The authors showed that, the analysis of crack formation, crack propagation and failure analysis can be done with fictitious crack model, even if coarse finite element is used, thereby eliminating the mesh sensitivity. This method is a simplified nonlinear fracture model, which can simulate satisfactorily the complex nonlinear phenomena in the fracture process zone of concrete and it predicts the localized real physical behavior in the vicinity of a crack and at the crack tip. With these advantages, the FCM has become more popular in its implementation. Extensive experimental and numerical studies based on fictitious crack model have been carried out by many researchers [12–26]. The model is based on the assumptions that: (i) the bulk of material behaves in a linear elastic and isotropic manner, (ii) the cohesive process zone begins to develop when the maximum principal stress becomes equal to the tensile strength and (iii) the material is in partial damaged condition and is still able to transfer the stress known as cohesive stress after the formation of cohesive fracture zone. The cohesive stress depends on the crack opening displacement. For development of the model three material parameters i.e., modulus of elasticity E, uni-axial tensile strength ft, and fracture energy GF along with stress-displacement softening relation are required. The three important stages of crack propagation in concrete i.e., crack initiation, stable crack propagation and unstable fracture in concrete can be well described by DKFM among all the concrete fracture models based on modified LEFM principle. The method does not require the closed loop testing system in the laboratory and is characterized by two material parameters viz initial cracking un toughness Kini IC and unstable fracture toughness KIC . The initiation toughness is defined as the inherent toughness of the materials, which holds for loading at crack initiation when material behaves elastically and micro cracking is concentrated to a small-scale in the absence of main crack growth. The total toughness at the critical condition is known as unstable toughness Kun IC which is regarded as one of the material fracture parameters at the onset of the unstable crack propagation. Extensive numerical and experimental studies have been carried out by the researchers around the world [26,32,35–50] on the behavior of double-K fracture parameters of concrete using varied tests configurations and material properties. It is obvious from existing literature that the strength and elastic properties of RAC gets influenced with the amount of RCA content and crack propagation study on fracture parameters of RAC is meager in the literature. Thus it is worth, to carry out a systematic study, on fracture parameters of RAC. Hence, the aim of the present paper is to investigate the application of concrete fracture models i.e., FCM and DKFM for predicting the fracture parameters of RAC with varying RCA percent replacement in conventional concrete. Also, the characteristics of fracture parameters predicted by DKFM

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in RAC have been studied in the paper. To determine the double K fracture parameter of RAC, critical crack mouth opening displacement (CMODc) and peak load are necessary from the fracture test results. However, in absence of these data, the input parameters are derived from FCM. Further, various material properties and softening function of RAC are required as input data for simulation of FCM. These data have been derived on the basis of empirical formula given in literature for the known value of compressive strength of RAC. 2. Material properties for determining the fracture parameters It is well known that, the material properties such as uni-axial tensile strength (ft), modulus of elasticity (Ec), fracture energy (GF) and softening function are required as input parameters for determining the fracture properties of concrete using FCM. For natural aggregate concrete (NAC), these properties can be determined from the test results or the empirical relation, if compressive strength of concrete is known. Similarly, for RAC, these material properties can also be determined. Xiao et al. [4] presented comprehensive review on the effect of content of RCA on various mechanical and strength properties of RAC. The authors presented that, with an RCA replacement of 100%, the reduction in compressive strength of RAC is between 10% and 30%. However, if the RCA content is less than 30%, the influence on compressive strength is not significant. The test results of cube compressive strength presented by Xiao et al. [4,52] are used in this investigation. These results are presented in Fig. 1 with legend fcu,exp, which shows that, the concrete compressive strength decreases linearly in general with increase of the RCA content. From the linear regression, the following equation can be written

f cu;RAC ¼ 1:0011  0:0026r f cu;NAC

ð1Þ

where fcu,RAC and fcu,NAC are the cube compressive strength of RAC and NAC respectively and r is% replacement of RCA. Based on the results of Xiao and Lan [53], it can be concluded that there is a reduction in the uni-axial tensile strength of concrete with increase in the RCA content. From the known values of compressive strength, the uni-axial tensile strength of RAC having 100% RCA is reduced by 31% as compared to that of NAC. The following equation was suggested by the authors for predicting the uni-axial tensile strength of RCA 2=3

f t ¼ ð0:24  arÞf cu

ð2Þ

where, f t and fcu are respectively, the uni-axial tensile strength and cube compressive strength of RAC. The value of coefficient ‘a’ is taken as 0.0006.The modulus of elasticity of NAC can be predicted by many empirical relations, if compressive strength of concrete is known. However, in present investigation, the empirical relation of TS500 [54], as given below, is used

qffiffiffiffi EC ¼ ð3:25 f c þ 14Þ GPa

ð3Þ

where fc is the cylinder compressive strength of NAC in MPa. Xiao et al. [52] put forward that average value of ratio of concrete compressive strengths, fc/fcu for RAC with different replacement percentages of RCA is found to be 0.81. In the present study, this ratio is taken as 0.80 for all practical purposes of computations. It has been also observed that the modulus of elasticity of RAC is always lower than that of the corresponding NAC. The following empirical equation proposed by Xiao et al. [55] is used in the present study for determining the modulus of elasticity of RAC

Ec ¼

105 2:8 þ 40:1 f cu

ð4Þ

where, f cu in Eq. (4) is the cube compressive strength (in MPa) of RAC. Based on the test results in the literature [4], the value of Poisson’s ratio of RAC is reported to be similar to that of NAC. Therefore, the Poisson’s ratio of 0.18 for NAC as well as RAC has been considered in the present study irrespective of the percent RCA replacement in concrete. Oh et al. [56], derived an empirical relation for fracture energy of NAC as given below

GF ¼ Cft

d Ec

ð5Þ

In which GF is in N/mm, ft and Ec in N/mm2,’d’ represents the maximum size of aggregate in mm and constant C = 56.24. Eq. (5) is used for the determination of GF of NAC in the present study. The GF (N/mm) for RAC has been expressed by Xiao et al. [4] as given below

GF ¼ b

 b  b2 fc d 1 t k t 30 EC d20

ð6Þ

where b = 85.93, b1 = 0.125, b2 = 0.211, d and t are respectively the maximum aggregate size (mm) and curing age (days), d20 = 20 mm, t30 = 30 days, parameter k = 1.0, fc and Ec are the compressive strength and elastic modulus of RAC respectively. In present study the values of d is taken as 20 mm in Eqs. (5) and (6) and t is taken as 28 days in Eq. (6).

1.2

Ratio of fcu,RAC/fcu,NAC

1 0.8 0.6 0.4

fcu,exp Linear (fcu,exp)

0.2 0 0

20

40

60

80

100

% of recycled aggregate Fig. 1. Influence of RCA content on compressive strength; Xiao et al. [4,52]

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Softening function of concrete is also known as constitutive law, which relates the cohesive stress r across the crack faces and the corresponding crack opening displacement w i.e. r = f(w). Out of many softening functions (such as linear, bilinear, exponential, quasi-exponential) the nonlinear softening function proposed by Reinhardt et al. [57] is used in the present study. It can be expressed as

("

)  3 #   c1 w c2 w w 3 exp  1þ ð1 þ c1 Þ expðc2 Þ wc wc wc

rðwÞ ¼ f t

The value of total fracture energy of concrete GF is expressed as

(

"  3 #    1 c1 3 6 6 expðc2 Þ GF ¼ wc f t 1þ6  1 þ c31 1 þ þ 2 þ 3 c2 c2 c2 c2 c2 c2    3 1 þ c1 ð8Þ  expðc2 Þ 2 in which c1 and c2 are the material constants. Also, the wc is the maximum crack opening displacement at the crack-tip at which the cohesive stress becomes to be zero. The value of wc is computed using Eq. (8) for a given set of values c1, c2 and GF. In the present investigation, the values of c1 and c2 in Eqs. (7) and (8) for concrete are determined using the following expression proposed by Xu [58]. These expressions are given as below:

d 0:75

8 d c2 ¼ 0:92  400 k h i0:7 f ck k ¼ 10  ð2f Þ

ð9Þ

cko

where fcko is 10 MPa and characteristic strength fck can be taken as fcu for all computation purposes. The values c2 for NAC and RAC can be determined by using the corresponding values of fcu. Considering the value of fcu = 35 MPa for NAC, the values of ft, Ec, GF and parameters of nonlinear softening function are computed using Eqs. (2), (3), (5) and (9) respectively. Now the% replacement by RCA is varied as 30, 50, 70 and 100 and these material constants such as fcu, ft, Ec, GF and parameters of nonlinear softening function are determined using Eqs (1), (2), (4), (6) and (9) respectively. The values of material constants as obtained for NAC and RAC are presented in Table 1 and Fig. 2. From the figure it can be seen that, a linear relationship between the material constants and the% replacement of RCA in concrete exists. In general the values of fcu, ft, Ec, GF decrease with increase in the RCA replacement. For RCA replacement in RAC by 30%, 50%, 70% and 100% there is a reduction in compressive strength of concrete by 7.7%, 12.9%, 18.1% and 25.9% respectively. Similarly, the reduction in tensile strength of RAC as compared to NAC is 12.5%, 20.2%, 28.02% and 38.5% for RCA replacement of 30%, 50%, 70% and 100% respectively. The decrease in elastic modulus of RAC as compared with NAC is 20.7%, 22.1%, 23.7% and 26.2% with RCA content of 30%, 50%, 70% and 100% respectively. There is linear

Table 1 Various material properties of NAC and RAC. RCA replacement (%)

fcu (MPa)

ft (MPa)

Ec (MPa  103)

GF (N/m)

0 30 50 70 100

35.00 32.31 30.49 28.67 25.94

2.57 2.25 2.05 1.85 1.58

31.197 24.746 24.300 23.817 23.010

92.66 88.46 85.01 81.56 76.38

reduction in the value of fracture energy of RAC as compared to NAC by 4.5%, 8.3%, 12.0% and 17.6% in case the RAC replacement is 30%, 50%, 70%, and 100%. From the figure it is also observed that there is sudden drop in the value of Ec of RAC for the RCA content of 30%. The similar observation can be seen elsewhere, Xiao et al.[4]. 3. Fictitious crack model simulation for three-point bend test

ð7Þ

c1 ¼

171

Nonlinear softening function parameters c1

c2

k

2 2 2 2 2

7.41 7.48 7.53 7.58 7.66

8.52 8.60 8.66 8.71 8.80

This paper is aimed for determination of double-K fracture parameters of RAC with varying content of RCA in concrete. For this purpose, the simplest test specimen i.e., three point bend test (TPBT) of standard dimensions (RILEM Technical Committee-50FMC) [59] is considered and are shown in Fig. 3. In the figure, the symbols B, D and S represent the width, depth and span respectively for S/D = 4. The governing equation of crack opening displacement (COD) along the potential fracture line is written for the development of the fictitious crack model. The influence coefficients of the COD equation are determined using linear elastic finite element method. Four node iso-parametric plane elements are used in finite element calculation. The COD vector is partitioned according to the enhanced algorithm introduced by Planas and Elices [15]. Finally, the system of nonlinear simultaneous equation is developed and solved using Newton–Raphson method. For standard TPBT with B = 100 mm having size range D = 200 mm and initial cracklength/depth (ao/D) ratio = 0.3, the finite element analysis is carried out for which the half of the specimens are discretized due to symmetry considering 80 numbers of equal iso-parametric plane elements along the dimension D. The discretization of the beam is shown in Fig. 4. Material properties such as modulus of elasticity E, uni-axial tensile strength ft and fracture energy GF as presented in Table 1 for NAC and RAC for different percentage replacement of RCA and nonlinear softening relation (Eq. (7)) with corresponding material constants given in Table 1 are used for simulation of FCM. Details of the formulation, development and model simulation of FCM for three point bend test can be seen elsewhere [25].

4. Determination of double-K fracture parameters Determination of double-K fracture parameters using linear elastic fracture mechanics (LEFM) is based on linear asymptotic superposition assumption [35]. Four different analytical methods: Gauss–Chebyshev integral method (GCIM) [35], simplified equivalent cohesive force method (SECFM) [37], weight function method (WFM) [41–42] and simplified Green’s function method (SGFM) [43] are available for determination of double-k fracture parameters of concrete. In this paper WFM with four terms of universal weight function is used for computing the double-K fracture parameters of NAC and RAC for the standard three point bend test. The input data required for obtaining the double-K fracture parameters (Pu and CMODc) are obtained from the developed fictitious crack model. Although, the detailed mathematical procedures are available in the literature [41–42], a summary for determining the double-K fracture parameters for TPBT with S/D = 4 is outlined below For standard TPBT geometry with S/D = 4 using Tada et al. [60] formulae, the stress intensity factor is expressed as

pffiffiffiffi K I ¼ rN DkðaÞ kðaÞ ¼

ð10Þ

pffiffiffi 1:99  að1  aÞð2:15  3:93a þ 2:7a2 Þ

a

ð1 þ 2aÞð1  aÞ3=2

ð11Þ

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Relative parameter made of RAC to NAC

1.2

1

0.8

0.6

0.4

fcu

ft

Ec

GF

0.2

0 0

20

40

60

80

100

120

RCA replacement (%) Fig. 2. Influence of RCA content on computed values of compressive strength, tensile strength, modulus of elasticity and fracture energy.

P B

D a S = 4D

(b) Mid span cross-section

(a) Longitudinal section

Fig. 3. Three point bending specimen geometry.

assumption using the following LEFM formulae for TPBT geometry, S/D = 4, Tada et al. [60]

P/2

CMOD ¼

6PSa BD2 Ec

VðaÞ

D

VðaÞ ¼ 0:76  2:28a þ 3:87a2  2:04a3 þ

0.75D

D/4

D

un C K ini IC ¼ K IC  K IC

Where k(a) is a geometric factor, a = a/D and rN is the nominal stress in the beam due to external load P and self weight of the structure which is given by

3S 2

4bD

½2P þ wg S

0:66 ð1  aÞ2

ð14Þ

In which, a is equal to ac equivalent-elastic crack length at maximum load, P equals Pu. Then, according to inverse analytical method, the following relation can be employed to determine the initial cracking toughness of the material.

Fig. 4. Finite element discretization of three point bend specimen.

rN ¼

ð13Þ

ð12Þ

In Eq. (12), wg is the self weight per unit length of the structure. Eqs. (10) and (12) can be used for calculation of initial cracking toughness Kini IC at the tip of initial crack length ao corresponding to initial cracking load Pini and unstable fracture toughness Kun IC at the tip of effective crack length ac corresponding to peak load Pu. In fact it is difficult to detect the crack initiation load from experimental approach, an inverse analytical method is used to calculate the value of Kini IC . The value of effective crack extension corresponding to peak load is determined using linear asymptotic superposition

ð15Þ

where, KCIC is known as cohesive toughness of the material. Once the value of KCIC is determined, the Kini IC can be obtained using Eq. (15). The value of KCIC using weight function method can be obtained as



 2 K CIC ¼ pffiffiffiffiffiffi A1 a 2s1=2 þ M 1 s þ 23 M2 s3=2 þ M23 s2 þ 25 M 4 s5=2 þ h 2pa io 4 4 A2 a2 43 s3=2 þ M21 s2 þ 15 M 2 s5=2 þ 35 M4 s7=2 þ M63 f1  ðao =aÞ3  3sao =ag ð16Þ s ðCTODc Þ where, A1 ¼ rs ðCTODc Þ; A2 ¼ f t raa and s ¼ ð1  ao =aÞ. At the o

critical effective crack extension, a is equal to ac corresponding to peak loading condition in Eq. (16). Also, crack tip opening displacement (CTOD) at initial crack-tip becomes its critical value denoted as CTODc at peak load and the corresponding value of cohesive stress, rs(CTODc) at the tip of initial notch, is determined using

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nonlinear softening functions as given in Eq. (7). The value of CTODc is computed using the following expression [28]

CTODc ¼ CMODc

( " 2  2 #)12 ao ac  ao ao 1 þ 1:081  1:149  ac D ac ac ð17Þ

In Eq. (16), M1, M2 and M3 are the weight function parameters of four terms universal weight function which can be represented as a function of a/D ratio. These parameters are expressed in the following form

Mi ¼

1 ð1  a=DÞ3=2

½ai þ bi a=D þ ci ða=DÞ2 þ di ða=DÞ3

þ ei ða=DÞ4 þ f i ða=DÞ5 

ð18Þ

for, i = 1 and 3 and

M i ¼ ½ai þ bi a=D for i ¼ 2

ð19Þ

The values of coefficients ai, bi, ci,..., fi are the constant and given in Table 2. 5. Results and discussion For given parameters of GF, ft, Ec and the nonlinear softening function for different% replacement of RCA in concrete the loadCMOD curves are obtained using FCM. These curves are shown in Fig. 5 in which legend represents the RAC with% replacement of RCA. It is clear from the figure that P-CMOD curve can be simulated for RAC similar to NAC. It can also be seen that terminals of PCMOD curves converge even though for different material properties of concrete with varying% replacement of RCA. This result is attributed to the basic assumptions of the FCM. The peak load Pu versus% replacement of RCA is shown in Fig. 6. From the figure it is interesting to note that the value of Pu decreases linearly with increase in content of RCA in concrete. There is reduction in load carrying capacity of the pre-cracked RAC beam specimen by 13.1%, 18.6%, 24.4% and 32.6% for% RCA replacement of 30, 50, 70 and 100 respectively. Fig. 7 shows the relationship between CMODc obtained using FCM and% RCA replacement. From the figure, it can be observed that no definite relationship exists between CMODc and the% RCA replacement in concrete. In general, the value of CMODc for RAC increases as compared to that of NAC. In order to determine the double-K fracture parameters, the values of effective crack length at peak load (ac) and the critical value of crack tip opening displacement (CTODc) are determined. The values ac/D ratio and CTODc are plotted with% RCA replacement in Figs. 8 and 9 respectively. From Fig. 8 it is observed that the value of ac/D ratio decreases up to 30% replacement of RCA and then it increases as the% replacement of RCA increases. The variation of CTODc with% replacement of RCA is almost similar to the variation of CMODc variation with% replacement of RCA. c ini The fracture parameters Kun IC , KIC and KIC and the initial cracking load Pini are plotted with% RCA replacement in Figs. 10 and 11, and respectively. It is interesting to observe that these parameters linearly decrease as the% RCA replacement increases in concrete. The values of Kun IC decreases by 14.9%, 18.6%, 20.4% and 27.5%, as the%

RCA replacement increases by 30, 50, 70 and 100 respectively. Similarly the values of KcIC decreases by 15.7%, 19.6%, 21.2% and 29.2% as the RCA replacement in concrete is 30%, 50%, 70% and 100% respectively. The value of initial cracking toughness of material also decreases by 13.5%, 16.4%, 18.8% and 24.4% as the% RCA increases by 30%, 50%, 70% and 100% in concrete. The values of initial cracking load decreases by 14.4%, 17.6%, 20.2% and 26.1% as the RCA replacement in concrete is 30%, 50%, 70% and 100% respectively. The modulus of elasticity of NAC has been determined using Eq. (3) as per empirical relation of TS500 [54] whereas this value for RAC has been obtained using empirical Eq. (4) for RAC with different RCA replacement varying between 30% and 100%. Graphical representation of Eq. (4) has been presented by Xiao and collaborators [55] which shows that the decrease in elastic modulus of RAC with RCA content of 30% as compared with the NAC is remarkably greater than that of RAC with different replacement percentage of RCA. As mentioned in the preceding section 2.0 of the paper, it is reiterated that the decrease in elastic modulus of RAC as compared with NAC is 20.7%, 22.1%, 23.7% and 26.2% with RCA content of 30%, 50%, 70% and 100% respectively. Generally, modulus of elasticity of RAC is lower than that of NAC but there is significant variation between studies as to how much the modulus is reduced. The most likely reason for this variation is the different properties of aggregates used in the study. This would further substantiate that the elastic modulus is controlled by the aggregates properties (e.g., aggregate elasticity) rather than the properties the concrete as a whole (e.g., compressive or flexural strength) [61]. This reason probably justifies the use of separate equations for NAC and RAC for determining the elastic modulus of two different materials. Further, the tensile strength of the concrete like material would be associated with its elasticity and the properties of the aggregates itself which directly influences the fracture behavior of concrete matrix. It obviously implies that fracture behavior of NAC and RAC is expected to be different. ini By and large, both the parameters Kun IC and KIC of double-K fracture model maintain a linear relationship with% RCA replacement in concrete. It can be observed that rate of decrease in unstable fracture toughness and initial cracking toughness of the RAC is high when the% RCA replacement is 30% and then is almost constant for further increase in RCA content in RAC. The trend of decrease of Kun IC and Kini IC is similar to that of modulus of elasticity of RAC. This may ini be so because the values of Kun IC and KIC are influenced of by elastic modulus of the material. Further, influence of RCA content on the ratio of Pini/Pu and Kini IC / Kun IC is shown in Fig. 14. From the figure an interesting phenomenon can be observed as the ratio of Pini/Pu slightly increases with the increase of RCA content in RAC. This ratio is almost 0.50 for the NAC and 0.54 for RAC having 100% RCA content. However, similar ratio in terms of un fracture parameters i.e., Kini IC /KIC is more interesting to observe. This value is almost constant for the NAC as well as RAC having varying percentage of RCA replacement which varies from 0.34 to 0.35. The results presented in this work will open up a new direction for the researcher to further carry out experimental and numerical investigations using conventional fracture models for material characterization of recycled aggregate concrete.

Table 2 Coefficients of four terms weight function parameters M1, M2 and M3. i

ai

bi

1 2 3

0.0572011 0.4935455 0.340417

0.8741603 4.43649375 3.9534104

ci 4.0465668 16.1903942

di 7.89441845 16.0958507

ei 7.8549703 14.6302472

fi 3.18832479 6.1306504

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6 5 RC-0 RC-50 RC-100

P (kN)

4

RC-30 RC-70

3 2 1 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

CMODc (mm) Fig. 5. Influence of RCA content on P-CMOD curves.

6

5

Pu (kN)

4

3

2

1

0 0

20

40

60

80

100

120

RCA replacement (%) Fig. 6. Influence of RCA content on peak load.

0.056 0.0555 0.055

CMODc (mm)

0.0545 0.054 0.0535 0.053 0.0525 0.052 0.0515 0.051 0

20

40

60

80

100

120

RCA replacement (%) Fig. 7. Influence of RCA content on the CMODc.

6. Concluding remarks For modeling fracture behavior of RAC many input parameters such as uniaxial tensile strength, fracture energy, modulus of elasticity, softening function of the material are required. These properties are determined in the present study using

various empirical formulas derived by previous researchers on the basis of their experimental results. Further, peak load and corresponding crack mouth opening displacement from fracture test viz. three point bend test are required for determining double-K fracture parameters of RAC. The values of peak load and corresponding crack mouth opening displacement in the

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0.48 0.475 0.47

ac /D

0.465 0.46 0.455 0.45 0.445 0.44 0

20

40

60

80

100

120

RCA replacement (%) Fig. 8. Influence of RCA content on ac/D values.

0.0285 0.028

CTODc (mm)

0.0275 0.027 0.0265 0.026 0.0255 0.025 0

20

40

60

80

100

120

RCA replacement (%) Fig. 9. Influence of RCA content on the CTODc.

40 35

KICun (MPa-mm1/2)

30 25 20 15 10 5 0 0

20

40

60

80

100

120

RCA replacement (%) Fig. 10. Influence of RCA content on the unstable fracture toughness.

present study are derived from the developed fictitious crack model. Thus, the fracture behavior of RAC has been studied in the present work. However, the input parameters required for

such innovative studies can also be obtained from the experiments. From the present study, the following concluding remarks can be highlighted:

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30

KICC (MPa-mm1/2)

25

20

15

10

5

0 0

20

40

60

80

100

120

RCA replacement (%) Fig. 11. Influence of RCA content on the cohesive toughness.

14

12

KICini (MPa-mm1/2)

10

8

6

4

2

0 0

20

40

60

80

100

120

RCA replacement (%) Fig. 12. Influence of RCA content on the initial cracking toughness.

3

2.5

Pini (kN)

2

1.5

1

0.5

0 0

20

40

60

80

RCA replacement (%) Fig. 13. Influence of RCA content on the initial cracking load.

100

120

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Ratio of Pini /Pu and KICini/KICun

0.6 0.5 0.4 0.3 0.2

Pini/Pu KICini/KICun

0.1 0 0

20

40

60

80

100

120

RCA replacement (%) un Fig. 14. Influence of RCA content on the ratio of Pini/Pu and Kini IC /KIC .


The material properties required to model fictitious crack model and double-K fracture model for RAC can be determined using the compiled equations presented in the paper. The fictitious crack model and double-K fracture model can be used to determine the fracture parameters of RAC with varying content of RCA.
The peak load of RAC predicted by fictitious crack model is found to be decreased from 13.1% to 32.6% for the RCA percent replacement from 30% to 100% respectively.
The values of unstable toughness, initial cracking toughness and initial cracking load of RAC decreases from 14.9% to 27.5%, 13.5% to 24.4% and 14.4% to 26.1% for RCA percent replacement from 30% to 100% respectively.
Interesting result is found from the present study that the ratio un of Pini/Pu and Kini IC /KIC are found to be almost constant for NAC and RAC with different percentage replacement of RCA. The un ratio Pini/Pu and Kini IC /KIC is observed to vary between 0.54–0.54 and 0.34–0.34 respectively, for natural aggregate concrete and recycled aggregate concrete.

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