Determination of DoubleK Fracture Parameters of Concrete Using SplitTension Cube: A Revised Procedure
International Journal of Concrete Structures and Materials volumeÂ 10,Â pages 163â€“175 (2016)
Abstract
This paper presents a revised procedure for computation of doubleK fracture parameters of concrete splittension cube specimen using weight function of the centrally cracked plate of finite strip with a finite width. This is an improvement over the previous work of the authors in which the determination of doubleK fracture parameters of concrete for splittension cube test using weight function of the centrally cracked plate of infinite strip with a finite width was presented. In a recent research, it was pointed out that there are great differences between a finite strip and an infinite strip regarding their weight function and the solution of infinite strip can be utilized in the splittension specimens when the notch size is very small. In the present work, improved version of LEFM formulas for stress intensity factor, crack mouth opening displacement and crack opening displacement profile presented in the recent research work are incorporated. The results of the doubleK fracture parameters obtained using revised procedure and the previous work of the authors is compared. The doubleK fracture parameters of splittension cube specimen are also compared with those obtained for standard three point bend test specimen. The input data required for determining doubleK fracture parameters for both the specimen geometries for laboratory size specimens are obtained using well known version of the Fictitious Crack Model.
1 Introduction
It is well known that fracture parameters of quasibrittle material like concrete cannot be determined by directly applying the concepts of linear elastic fracture mechanics (LEFM) because of the existence of large and variable size of fracture process zone (FPZ) ahead of a cracktip. In order to account for and characterize FPZ in the analysis, several nonlinear fracture mechanics models have been developed which primarily involve cohesive crack model (CCM) or fictitious crack model (FCM) (Hillerborg et al. 1976; Modeer 1979; Petersson 1981; Carpinteri 1989; Planas and Elices 1991; Zi and BaÅ¾ant 2003; Roesler et al. 2007; Park et al. 2008; Zhao et al. 2008; Kwon et al. 2008, Cusatis and Schauffert 2009, Elices et al. 2009; Kumar and Barai 2008b, 2009b) and crack band model (CBM) (BaÅ¾ant and Oh 1983), two parameter fracture model (TPFM) (Jenq and Shah 1985), size effect model (SEM) (BaÅ¾ant et al. 1986), effective crack model (ECM) (Nallathambi and Karihaloo 1986), K_{R}curve method based on cohesive force distribution (Xu and Reinhardt 1998, 1999a), doubleK fracture model (DKFM) (Xu and Reinhardt 1999a, b, c) and doubleG fracture model (DGFM) (Xu and Zhang 2008).
In recent time, much of research and studies (Xu and Reinhardt 1999a, b, c, 2000; Zhao and Xu 2002; Zhang et al. 2007; Xu and Zhu 2009; Kumar and Barai 2008a, 2009a, 2010; Kumar 2010; Zhang and Xu 2011; Kumar and Pandey 2012; Hu and Lu 2012; Murthy et al. 2012; Hu et al. 2012; Ince 2012; Kumar et al. 2013; Choubey et al. 2014; Kumar et al. 2014) have been carried out to determine and characterize the fracture parameters of concrete using doubleK fracture model for which the reasons are obvious (Kumar et al. 2013). The doubleK fracture model is characterized by two material parameters: initial cracking toughness K ^{ini}_{ IC } and unstable fracture toughness K ^{un}_{ IC } . 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 smallscale in the absence of main crack growth. It is directly calculated by knowing the initial cracking load and initial notch length using LEFM formula. The total toughness at the critical condition is known as unstable toughness K ^{un}_{ IC } which is regarded as one of the material fracture parameters at the onset of the unstable crack propagation and it can be obtained by knowing peak load and corresponding effective crack length using the same LEFM formula. Recently, Kumar and Pandey (2012) presented the formulation and determination of doubleK fracture parameters using splittension cube test specimen using weight function method in which the LEFM formulas for stress intensity factor (SIF), crack mouth opening displacement (CMOD) and crack opening displacement (COD) profile derived by Ince (2010) and the universal weight function of Wu et al. (2003) were adopted. The authors (Kumar and Pandey 2012) mentioned that there are several advantages of using splittension cube (STC) test specimen over the testing of other specimens like three point bend test (TPBT), compact tension (CT) and wedge splitting test (WST) specimens. However, there should be a limitation that the notch can be only produced at the time of casting of concrete cubes (precast notch) in the split tension cube specimen. The authors also presented the results of the initial cracking toughness, cohesive toughness and unstable fracture toughness obtained using split tension cube test specimen and they were compared with those obtained using standard compact tension specimen. From the study it was concluded that the doubleK fracture parameters as obtained using splittension cube test are in good agreement and consistent with those as calculated using standard compact tension specimen. However, the results of fracture parameters are influenced by the distributedload width during the loading of splittension cube specimen and it was observed that the values of unstable fracture toughness and cohesive toughness increase with increase in the distributedload width whereas the initial cracking toughness is not significantly affected by the distributedload width. In the formulation, the authors (Kumar and Pandey 2012) used the weight function of the centrally cracked infinite strip with a finite width specimen (Tada et al. 2000) and the equivalent four terms of universal weight function (Wu et al. 2003) for computing the value of cohesive toughness and consequently determining the initial cracking toughness. Later, Ince (2012) put forward a method for determination of doubleK fracture parameters using weight function for splitâ€”tension specimens such as splitting tests on cubical, cylindrical and diagonal cubic concrete samples. The author pointed out that there are great differences between a finite strip and an infinite strip regarding their weight function and the solution of infinite strip can be utilized in the splittension specimens when the notch size is very small. It was concluded that the central cracked plate can be considered as an infinite strip when the length/width (l/D) ratio of a plate is equal or greater than 3 (Isida 1971, Tada et al. 2000). In case of a cubesplit tension test specimen the value of the length/characteristic dimension (l/D) ratio is taken to be 1 for which Ince (2012) derived the four term universal weight function using boundary element method and finite element method. The author also presented the improved version of LEFM formulas for stress intensity factor, CMOD and COD profile over the previously derived LEFM equations by the same author (Ince 2010) for split tension cube test specimen. In view of the above development, it was felt necessary to carry out a comparative study on the doubleK fracture parameters computed using the procedure outlined by Kumar and Pandey (2012) and using the weight function of the centrally cracked plate of finite strip with a finite width incorporating the improved version of LEFM formulas for stress intensity factor, CMOD and COD profile derived by Ince (2012).
The paper presents the revised procedure for determination of doubleK fracture model using weight function method for the splittension cube specimen of concrete considering improved LEFM formulas for stress intensity factor, CMOD and COD profile and the weight function of the centrally cracked plate of finite strip with a finite width derived by Ince (2012). The results of the fracture parameters obtained using revised procedure and the previous work of Kumar and Pandey (2012) are compared. Further, the doubleK fracture parameters of splittension cube specimen are also compared with those obtained for standard three point bend test specimen. The input data required for determining for splittension cube test and three point bend test for laboratory size specimens are obtained using well known version of the fictitious crack model.
2 Dimensions of Test Specimens
For present investigation, the standard test geometries, dimensions and loading conditions for STC and TPBT specimens are considered as shown in Fig.Â 1. The symbols in Fig.Â 1(a): a _{ o }, D, h and t are half of the initial notchlength, characteristic dimension as specimen size (DÂ =Â h/2), height or total depth and half of the width of distributed load respectively for STC geometry. RILEM Technical Committee 50FMC (1985) has recommended the guidelines for determination of fracture energy of cementitous materials using standard threepoint bend test on notched beam. This method has been widely used for determination of fracture energy of concrete with certain modification in the experimental setup (Lee and Lopez 2014). In present study, standard threeâ€”point bend test (RILEM Technical Committee 50FMC 1985) is considered for which the symbols: B, D and S in Fig.Â 1b are the width, depth and span respectively with S/DÂ =Â 4.
3 Determination of DoubleK Fracture Parameters for STC Specimen
3.1 Assumptions
Linear asymptotic superposition assumption is considered to introduce LEFM for calculating the doubleK fracture parameters. The hypotheses of the assumption are given below:

1.
the nonlinear characteristic of the loadcrack mouth opening displacement (PCMOD) curve is caused by fictitious crack extension in front of a stressfree crack; and

2.
an effective crack consists of an equivalentelastic stressfree crack and equivalentelastic fictitious crack extension.
A detailed explanation of the hypotheses may be seen elsewhere (Xu and Reinhardt 1999b).
3.2 Effective Crack Extension
For the applied load (Fig.Â 1) on the STC specimen, the critical value of CMOD (CMOD_{c}) is measured across the crack faces at the centre of specimen. The PCMOD curve up to peak load for this test geometry should be known a priori for determining the value of effective crack extension during the crack propagation. Using linear asymptotic superposition assumption, the equivalentelastic crack length a _{ c } corresponding to maximum load P _{ u } is solved using the revised LEFM formulae (Ince 2012). Hence, the CMOD is expressed as:
In which Î±Â =Â a/D, Î² is the relative loaddistributed width and expressed as Î²Â =Â 2t/hÂ =Â t/D, V(Î±,Î²) is dimensionless geometric function, coefficients B _{ i } (iÂ =Â 0 to 5) are the function of Î² as given in TableÂ 1. EquationÂ (2) is valid for 0.1Â â‰¤Â Î±Â â‰¤Â 0.9 within 0.3Â % accuracy for 0Â â‰¤Â Î²Â â‰¤Â 0.2. The modulus of elasticity of concrete (E) obtained using cylinder test is taken as a constant value for a particular concrete mix. Ince (2012) used boundary element numerical method to improve the LEFM formulas over the previous LEFM formulas (Ince 2010) for the split tension cube specimens which was based on centrally cracked infinite strip with a finite width specimen. EquationÂ (2) and TableÂ 1 used in the present study are extracted by Ince (2012) from the numerical results based on centrally cracked finite strip with a finite width specimen. Since the values of coefficients B _{ i } (TableÂ 1) are given (Ince 2012) at discrete intervals, these coefficients can be determined by linear interpolation at any value of Î² for the given range 0Â â‰¤Â Î²Â â‰¤Â 0.2.
Also, the nominal stress for STC test specimen in Eq.Â (1) can be written using the following formula (Timoshenko and Goodier 1970).
At critical condition that is at maximum load P _{ u } the half of crack length a becomes equal to a _{ c } and Ïƒ _{ N } to Ïƒ _{ Nu } in which Ïƒ _{ Nu } is the maximum nominal stress. Karihaloo and Nallathambi (1991) concluded that almost the same value of E might be obtained from PCMOD curve, loadâ€“deflection curve and compressive cylinder test. Hence, in case that is not known the value of E determined using compressive cylinder tests may be used to obtain the critical crack length of the specimen.
3.3 Calculation of DoubleK Fracture Parameters
A linearly varying cohesive stress distribution is assumed in the fictitious crack zone, which gives rise to cohesion toughness as a part of total toughness of the cracked body. Superposition method is used in order to calculate the stress intensity factor (SIF) at the tip of effective crack length K _{ I }. According to this method, total stress intensity factor K _{ I } is taken as the summation of stress intensity factor caused due to external load K ^{P}_{ I } and stress intensity factor contributed by cohesive stress K ^{Câˆ’}_{ I } as shown in Fig.Â 2. The value of K _{ I } is expressed in the following expression:
After determining the critical effective crack extension at unstable condition of loading, the unstable fracture toughness K ^{un}_{ IC } is determined using the revised LEFM formulae (Ince 2012) for which the stress intensity factor is expressed as:
where k(Î±, Î²) is a geometric factor and coefficients A _{ i } (iÂ =Â 0â€“5) are the function of Î² as summarized in TableÂ 1. EquationÂ (6) yields results within 0.7Â % accuracy for 0.1Â â‰¤Â Î±Â â‰¤Â 0.9 and 0Â â‰¤Â Î²Â â‰¤Â 0.2. Within the range of 0Â â‰¤Â Î²Â â‰¤Â 0.2, any value of coefficients A _{ i } can be determined by linear interpolation. The unstable fracture toughness K ^{un}_{ IC } is calculated using Eq.Â (5) at maximum load P _{ u } when a becomes equal to a _{ c } and Ïƒ _{ N } to Ïƒ _{ Nu }.
If the crack initiation load P _{ ini } is known from experiment, the initiation toughness K ^{ini}_{ IC } is calculated using Eq.Â (5) in which P is equal to P _{ ini } and a is equal to a _{ o }. Alternatively, it can be determined analytically by applying the following relation.
EquationÂ (7) is known as inverse method for determining the initiation toughness.
4 Determination of SIF Due to Cohesive Stress
4.1 Cohesive Stress Distribution
The cohesive stress acting in the fracture process zone on STC test specimen is idealized as series of pair normal forces subjected symmetrically to central cracked specimen of finite strip and a finite width as shown in Fig.Â 3. The linearly varying distribution of cohesive stress is also shown in Fig.Â 4.
A centrally cracked specimen with finite strip of a finite width plate subjected to pair of normal forces as shown in Fig.Â 3 takes into consideration for a split tension test cube specimen where the value of the length/characteristic dimension (l/D) ratio becomes 1. The SIF due to cohesive stress distribution as shown in Fig.Â 4 becomes to cohesive toughness K ^{C}_{ IC } of the material at the critical loading condition with negative value because of closing stress in fictitious fracture zone. However, the absolute value of K ^{C}_{ IC } is taken as a contribution of the total fracture toughness (Xu and Reinhardt 1999b) at the critical condition.
At this loading condition, the cracktip opening displacement (CTOD) is termed as critical cracktip opening displacement (CTOD_{c}). In Fig.Â 4, the Ïƒ _{ s }(CTOD_{c}) is cohesive stress at the tip of initial notch where CTOD is equal to CTOD_{c} and then Ïƒ(x) can be expressed as:
The value of Ïƒ _{ s }(CTOD_{c}) is calculated using softening functions of concrete. In the present work, the nonlinear softening function (Reinhardt et al. 1986) is used for the computation which can be expressed as:
The value of total fracture energy of concrete G _{ F } is expressed as:
In which, Ïƒ(w) is the cohesive stress at crack opening displacement w at the cracktip and c _{1} and c _{2} are the material constants. Also, wÂ =Â w _{ c } for f _{ t }Â =Â 0, i.e., w _{ c } is the maximum crack opening displacement at the cracktip at which the cohesive stress becomes to be zero. The value of w _{ c } is computed using Eq.Â (10) for a given set of values c_{1}, c_{2} and G _{ F }. For normal concrete the value of c_{1} and c_{2} is taken as 3 and 7, respectively.
4.2 Determination of CTOD_{c}
For a given value of critical crack mouth opening displacement CMOD_{c}, the crack opening displacement within the crack length COD(x) is computed using the revised expression (Ince 2012) as given below.
The accuracy of Eq.Â (11) is greater than 4Â % for 0.1Â â‰¤Â Î±Â â‰¤Â 0.9 and any value of Î² and is greater than 2.5Â % for 0.1Â â‰¤Â Î±Â â‰¤Â 0.6 and any value of Î². The value of x is taken as a _{ o } and a as a _{ c } for evaluation of CTOD_{c} using Eq.Â (11).
4.3 Calculation of Cohesive Toughness Using Weight Function Approach
According to weight function approach (Bueckner 1970, Rice 1972), the SIF for mode â€“I loading is given by following expression.
where Ïƒ(x) is the distribution of stress along the crack line x in the uncracked body, the term m(x,a) is known as weight function, a is the crack length and dx is the infinitesimal length along the crack surface. The four term universal form of weight function (Glinka and Shen 1991, Kumar and Barai 2008a, 2009a, 2010) is written as:
For centrally through cracked specimen of infinite strip and a finite width subjected to pairs of normal forces symmetrically (Fig.Â 3), the weight function as given by Tada et al. (2000) is expressed as:
EquationÂ (14) as equivalently expressed in terms of universal weight function m(x,a) of Eq.Â (13) by Wu et al. (2003) was used by Kumar and Pandey (2012) in the previous formulation. In the present investigation the weight function parameters M _{1}, M _{2} and M _{3} derived by Ince (2012) for the split tension cube specimen are used. According to Ince (2012) the parameters of four term weight function for a centrally through cracked specimen of finite strip and a finite width subjected to pairs of normal forces (Fig.Â 3) can be obtained as:
where Î±Â =Â a/D and m _{ ij } (iÂ =Â 1â€“3 and jÂ =Â 0â€“7) are the coefficients of the polynomial Eq.Â (15) as presented in TableÂ 2. The sixth degree polynomial (m _{ i7}Â =Â 0) is used for M _{2}. EquationÂ (15) is valid for 0Â â‰¤Â a/DÂ â‰¤Â 0.9 and 0Â â‰¤Â x/a 1 (exactly 0.993). The accuracy of Eqs.Â (13) and (15) is greater than 3Â % for all the splitâ€”tension cube specimens.
Once the weight function parameters are determined, Eq.Â (12) is used to calculate the SIF at critical condition (cohesive toughness) due to trapezoidal cohesive stress distribution as shown in Fig.Â 4. The value of Ïƒ(x) in Eq.Â (12) is replaced by Eq.Â (8), hence the closed form expression of K ^{C}_{ IC } can be obtained in the following form.
where, \( A_{1} = \sigma_{s} (CTOD_{c} ),A_{2} = \frac{{f_{t}  \sigma_{s} (CTOD_{c} )}}{{a  a_{o} }} \) and \( s = (1  a_{o} /a) \), also aÂ =Â a _{ c } at PÂ =Â P _{ u }. After computing the value of K ^{C}_{ IC } using Eq.Â (16), initiation toughness can be evaluated using Eq.Â (7).
5 Fictitious Crack Model and Material Properties for DoubleK Fracture Model
The cohesive crack model (Modeer 1979; Petersson 1981; Carpinteri 1989; Planas and Elices 1991; Zi and BaÅ¾ant 2003; Roesler et al. 2007, Park et al. 2008, Zhao et al. 2008, Kwon et al. 2008; Cusatis and Schauffert 2009; Elices et al. 2009; Kumar and Barai 2008b, b) is developed for STC and TPBT specimens to determine the input data such as P _{ u } and CMOD_{c} for these specimens. Three material properties such as modulus of elasticity E, uniaxial tensile strength f _{ t }, and fracture energy G _{ F } are required to model FCM. In this method, the governing equation of COD along the potential fracture line is written. The influence coefficients of the COD equation are determined using linear elastic finite element method. Four noded isoparametric plane elements are used in finite element calculation. The COD vector is partitioned according to the enhanced algorithm introduced by Planas and Elices (1991). Finally, the system of nonlinear simultaneous equation is developed and solved using Newtonâ€“Raphson method. For standard STC and TPBT specimens with BÂ =Â 100Â mm having size range DÂ =Â 200500Â mm, the finite element analysis is carried out for which the onequarter of STC and half of TPBT specimens are discretized due to symmetry as shown in Fig.Â 5 considering 80 numbers of equal isoparametric plane elements along the characteristic dimension D. In the discretization, both the specimens are divided into three bands perpendicular to characteristic dimension D such as D/4, D/4 and D/2 in case of STC specimen and 0.25D, 0.75D and D in case of TPBT specimen as shown in Fig.Â 5. This arrangement facilitates to obtain finer mesh size near the potential fracture line. For STC specimen, the number of divisions is taken as 20, 5 and 5 in the bands D/4, D/4 and D/2 respectively whereas it is 20, 10 and 5 in the bands 0.25D, 0.75D and D respectively for TPBT specimen. Ten nodes from top along the potential fracture line are restrained against horizontal movement and all the nodes at the bottom perpendicular to fracture line are restrained against vertical movement in case of STC specimen. For the TPBT specimen, three nodes from top along the potential fracture line are restrained in horizontal direction. The concrete mix with material properties: Î½Â =Â 0.18, f _{ t }Â =Â 3.21Â MPa, EÂ =Â 30 GPa, and G _{ F }Â =Â 103Â N/m along with nonlinear stressdisplacement softening relation with c_{1}Â =Â 3 and c _{2}Â =Â 7 are used as the input parameters of FCM.
From simulation of FCM, the results of peak load P _{ u } versus CMOD_{c} for TPBT specimen at a constant a _{ o } /D ratio of 0.3 are presented in TableÂ 3. Similar results of peak load P _{ u } and the corresponding CMOD_{c} at different load distributed widths (Î²Â =Â 0.0, 0.05, 0.1 and 0.15) for STC specimens of varying sizes (200â€“500Â mm) at a constant a _{ o }/D ratio of 0.3 are also presented in TableÂ 3.
6 Results and Discussion
The input parameters such as P _{ u }, CMOD_{c}, E and softening function of concrete are required from the tests for determining doubleK fracture parameters of concrete using weight function analytical method. In the present study, the values of E, f _{ t }, nonlinear softening function (Eq.Â (9)) as mentioned in Sect.Â 5 and the values of P_{u}CMOD_{c} for STC and TPBT specimens obtained from FCM are used to determine doubleK fracture parameters. The weight function method with four terms is applied to calculate doubleK fracture parameters in which the value of critical crack extension a _{ c } is obtained using improved Eq.Â (1) for STC specimen. For given values of a _{ c } and CMOD_{c}, the values of CTOD_{c} are determined using revised Eq.Â (11). The values of a _{ c } and CTOD_{c} are also determined using corresponding equations presented in the previous work of Kumar and Pandey (2012) which were based on LEFM equations given by Ince (2010). The values of a _{ c } and CTOD_{c} for TPBT specimen are determined as mentioned elsewhere (Kumar and Barai 2008a, 2010). All the values of a _{ c } and CTOD_{c} determined as above are plotted in Figs.Â 6 andÂ 7, respectively. For determining the value of K ^{C}_{ IC } using weight function method, first of all the four parameters M _{1}, M _{2} and M _{3} of four terms weight function are computed using Eq.Â (15) and TableÂ 2, then closed form expression (Eq.Â (16)) is used to obtain the value of K ^{C}_{ IC } and finally the K ^{ini}_{ IC } is determined using inverse procedure (Eq.Â (7)). For TPBT specimen, doubleK fracture parameters are determined in a similar manner using four terms weight function method as mentioned elsewhere (Kumar and Barai 2008a, 2010). Thus the values of K ^{un}_{ IC } , K ^{C}_{ IC } and K ^{ini}_{ IC } as obtained for STC for different distributedload widths (0Â â‰¤Â Î²Â â‰¤Â 0.15) and TPBT specimens for specimen size 200Â â‰¤Â DÂ â‰¤Â 500Â mm at a _{ o }/D ratio of 0.3 are plotted in Figs.Â 8, 9 and 10 respectively. The legends in Figs.Â 6, 7, 8, 9 and 10 marked with star (*) indicate the fracture parameters of the specimens determined using revised equations presented in this work whereas those legends with no mark with star (*) show the respective parameters of the specimens determined using the equations presented by Kumar and Pandey (2012).
From Figs.Â 6 and 7 it can be seen that the revised formulae and the previous LEFM equations (Kumar and Pandey 2012) yield the same results of critical values of effective crack length and crack tip opening displacement. These values for split tension cube specimen and three point bend test specimen also depend upon the size of the specimens and show similar pattern. The value a _{ c }/D decreases with the increase in specimen size whereas CTOD_{c} increases with the increase in specimen size. From Fig.Â 6 it can be seen that for STC specimen these parameters also depend on distributedload width and the a _{ c }/D ratio shows maximum deviation for STC specimen with Î²Â =Â 0.15 from those obtained for TPBT for a given specimen size. This deviation is more for the lower specimen size and seems to be converging at higher specimen size. The a _{ c }/D values for STC specimen are on higher side as compared with those of TPBT specimen for all values of distributedload width (0Â â‰¤Â Î²Â â‰¤Â 0.15) considered in the study. On an average, these values for STC for all values of Î² (0Â â‰¤Â Î²Â â‰¤Â 0.15) are more than those for TPBT specimen by approximately 4.6Â % and 0.43Â % for DÂ =Â 200Â mm and DÂ =Â 500Â mm, respectively.
From Fig.Â 7 it can be observed that for STC specimen the value of CTOD_{c} depends on distributedload width and the value of CTOD_{c} shows maximum deviation for STC specimen having Î²Â =Â 0 from those obtained for TPBT for a given specimen size. The CTOD_{c} values for STC specimen are in lower side as compared with those of TPBT specimen for all values of distributedload width (0Â â‰¤Â Î²Â â‰¤Â 0.15). On an average, these values for STC for all values of Î² (0Â â‰¤Â Î²Â â‰¤Â 0.15) are less than those for TPBT specimen by approximately 19.9 and 18.3Â % for DÂ =Â 200Â mm and DÂ =Â 500Â mm respectively.
It can be observed from Fig.Â 8 that the values of K ^{un}_{ IC } determined using LEFM equations presented elsewhere (Kumar and Pandey 2012) and the revised LEFM equations in this work are the almost same for specimen sizes (DÂ =Â 200â€“500Â mm) for all values of Î² (0Â â‰¤Â Î²Â â‰¤Â 0.15). It is also seen from the figure that the unstable fracture toughness obtained from STC specimen is compatible with that of TPBT specimen. The value of K ^{un}_{ IC } for STC is the lowest for distributedload width Î²Â =Â 0 and is the highest for Î²Â =Â 0.15 which is in close agreement with that obtained from TPBT specimen for all sizes of specimens. The values of K ^{un}_{ IC } are 36.70 and 39.91Â MPa mm^{1/2} for STC with Î²Â =Â 0.15 and 38.16 and 42.10Â MPa mm^{1/2} for TPBT specimens for specimen size 200Â mm and 500Â mm respectively. It seems from Fig.Â 8 that there is relatively more difference in results of unstable fracture toughness between STC with Î²Â =Â 0 and TPBT. Therefore, in case STC specimen is adopted to replace TPBT to test unstable fracture toughness of concrete, the STC with Î²Â =Â 0.15 can be considered to be reasonable. That means the unstable fracture toughness of concrete can be determined using STC specimen.
The value of cohesive toughness obtained using equations presented elsewhere (Kumar and Pandey 2012) and the revised procedure in this work, varies with the value of Î² for STC specimen. The values of cohesive toughness for STC and TPBT specimens shown in Fig.Â 9 also show that these values either obtained using STC specimen or TPBT specimen are in consistent with each other.
The effect of finite strip in the present revised work over the infinite strip (previous work of Kumar and Pandey (2012)) of finite width cracked specimen on the cohesive toughness values for the 0Â â‰¤Â Î²Â â‰¤Â 0.15 is clearly observed from Fig.Â 9. It can be seen that for all values of distributed load width, the values of K ^{C}_{ IC } obtained considering the finite strip plate are slightly on higher side than those obtained considering the infinite strip plate.
For STC specimen with infinite strip and Î²Â =Â 0, the values of K ^{C}_{ IC } are found to be 29.36Â MPa mm^{1/2} and 23.87Â MPa mm^{1/2} for DÂ =Â 500Â mm and 200Â mm respectively whereas those values are obtained as 29.96Â MPa mm^{1/2} and 24.55Â MPa mm^{1/2} for finite strip for DÂ =Â 500Â mm and 200Â mm respectively. Similarly, for STC specimen with infinite strip and Î²Â =Â 0.15, the value of K ^{C}_{ IC } are found to be 30.64Â MPa mm^{1/2} and 26.93Â MPa mm^{1/2} for DÂ =Â 500Â mm and 200Â mm respectively whereas those values are obtained as 31.34Â MPa mm^{1/2} and 27.98Â MPa mm^{1/2} for finite strip for DÂ =Â 500Â mm and 200Â mm respectively. On an average for all values of Î², the K ^{C}_{ IC } as obtained using finite strip is 2.14 and 3.29Â % more than those obtained using infinite strip of plate for DÂ =Â 500Â mm and 200Â mm respectively. Also, the values of K ^{C}_{ IC } as determined using finite strip of STC is 4.82 and 1.86Â % less than those obtained using three point bend test specimen for DÂ =Â 500Â mm and 200Â mm respectively. It is also observed from Fig.Â 9 that the size effect on the K ^{C}_{ IC } values for STC specimen is less significant than that presented for three point bend test.
It can be observed from Fig.Â 10 that for STC specimen with infinite strip and Î²Â =Â 0, the values of K ^{ini}_{ IC } are found to be 9.46Â MPa mm^{1/2} and 8.32Â MPa mm^{1/2} for DÂ =Â 500Â mm and 200Â mm respectively whereas those values are obtained as 8.83Â MPa mm^{1/2} and 8.68Â MPa mm^{1/2} for finite strip for DÂ =Â 500Â mm and 200Â mm respectively. Similarly, for STC specimen with infinite strip and Î²Â =Â 0.15, the values of K ^{ini}_{ IC } are found to be 9.26Â MPa mm^{1/2} and 9.78Â MPa mm^{1/2} for DÂ =Â 500Â mm and 200Â mm respectively whereas those values are obtained as 8.71Â MPa mm^{1/2} and 8.74Â MPa mm^{1/2} for finite strip for DÂ =Â 500Â mm and 200Â mm respectively. On an average for all values of Î², the K ^{ini}_{ IC } obtained using finite strip is 6.21 and 5.96Â % lower than those obtained using infinite strip of plate for DÂ =Â 500Â mm and 200Â mm respectively. Also, the values of K ^{ini}_{ IC } as determined using finite strip of STC is 11.70 and 26.10Â % less than those obtained using three point bend test specimen for DÂ =Â 500Â mm and 200Â mm, respectively. According to the present trend, it seems that the difference in the value of K ^{ini}_{ IC } obtained between the STC and TPBT specimens may further increase for smaller size specimens such as 150Â mm or 100Â mm. As per the common convention, this difference should not be more thanÂ Â±25Â % in the fracture test which is a matter of further investigation. It is also seen from Fig.Â 10 that the size effect on the K ^{ini}_{ IC } values for STC specimen is less significant than that presented for three point bend test.
7 Conclusions
A revised formulation for determination of doubleK fracture parameters using weight function method for splittension cube test is presented in the paper. In the revised procedure, the weight function of the centrally cracked plate of finite strip with a finite width is used which is an improvement over the previous work of the authors. From the present study considering the specimen sizes (DÂ =Â 200â€“500Â mm) and distributedload width (0Â â‰¤Â Î²Â â‰¤Â 0.15) of splittension cube test the following conclusions can be drawn.

Use of weight function for splittension cube test considering a centrally cracked plate of finite width with the finite strip or the infinite strip yields the same results of critical values of effective crack length, critical value of crack tip opening displacement and unstable fracture toughness of concrete.

For all values of distributed load width (0Â â‰¤Â Î²Â â‰¤Â 0.15), the values of cohesive toughness obtained considering the finite strip plate is slightly higher than those obtained considering the infinite strip plate. On an average cohesive toughness obtained using finite strip is 2.14Â % and 3.29Â % more than those obtained using infinite strip of plate for DÂ =Â 500Â mm and 200Â mm, respectively

Consequently, on an average for all values distributed load width (0Â â‰¤Â Î²Â â‰¤Â 0.15), the initial cracking toughness determined using finite strip is 6.21 and 5.96Â % lower than those obtained using infinite strip of finite width plate for DÂ =Â 500Â mm and 200Â mm respectively.

The value of unstable fracture toughness determined using finite strip of splittension cube specimen is the lowest for distributedload width Î²Â =Â 0 and is the highest for Î²Â =Â 0.15 which is in close agreement with that obtained from three point bed test for all sizes of specimens. Also, on an average for all values of the distributedload width, the values of cohesive toughness determined using finite strip of splittension cube specimen is 4.82 and 1.86Â % less than those obtained using three point bend test specimen for DÂ =Â 500Â mm and 200Â mm respectively. Further, on an average for all values of distributedload width, the values of initial cracking toughness determined using finite strip of splittension cube specimen is 11.70 and 26.10Â % less than those obtained using three point bend test specimen for DÂ =Â 500Â mm and 200Â mm, respectively.
Abbreviations
 CBM:

Crack band model
 CCM:

Cohesive crack model
 CT:

Compact tension
 DGFM:

DoubleG fracture model
 DKFM:

DoubleK fracture model
 ECM:

Effective crack model
 FCM:

Fictitious crack model
 FPZ:

Fracture process zone
 LEFM:

Linear elastic fracture mechanics
 SEM:

Size effect model
 SIF:

Stress intensity factor
 STC:

Split tension cube
 TPBT:

Three point bend test
 TPFM:

Two parameter fracture model
 WST:

Wedge splitting test
 a _{ o } :

Initial crack length
 A _{ i } :

Regression coefficients
 a _{ c } :

Effective crack length at peak (critical) load
 B :

Width of beam
 B _{ i } :

Regression coefficients
 c _{1}, c _{2} :

Material constants for nonlinear softening function
 CMOD:

Crack mouth opening displacement
 CMOD _{ c } :

Crack mouth opening displacement at critical load
 CTOD:

Crack tip opening displacement
 CTOD_{c} :

Crack tip opening displacement at critical load
 D :

Depth or characteristic dimension of specimen
 E :

Modulus of elasticity of concrete
 f _{ t } :

Uniaxial tensile strength of concrete
 G _{ F } :

Fracture energy of concrete
 G(x,a):

Weight function
 H :

Height or total depth (2D) for split tension cube specimen
 k(Î±, Î²):

Nondimensional function for KI or geometry factor
 K _{ I } :

Stress intensity factor
 K ^{ini}_{ IC } :

Initial cracking toughness
 K ^{un}_{ IC } :

Unstable fracture toughness
 K ^{C}_{IC } :

Cohesive toughness
 m(x,a):

Universal weight function
 M _{1} , M _{2} , M _{3} :

Parameters of weight function
 P _{ u } :

Maximum applied load or critical load
 S :

Span of beam
 t :

Half of the width of distributed load for split tension cube specimen
 V(Î±,Î²):

Dimensionless function for CMOD
 w _{ c } :

Maximum crack opening displacement at the cracktip for which the cohesive stress becomes equals to zero
 Î± :

Ratio of crack length to depth of specimen (a/D)
 Î² :

Ratio of loaddistributed to height of specimen (2t/hÂ =Â t/D) for split tension cube specimen
 Ïƒ :

Cohesive stress
 Ï… :

The Poissonâ€™s ratio
 Ïƒ _{ s }(CTOD_{c}):

Cohesive stress at the tip of initial notch corresponding to CTOD_{c}
References
BaÅ¾ant, Z. P., Kim, J.K., & Pfeiffer, P. A. (1986). Determination of fracture properties from size effect tests. Journal of Structural Engineering ASCE, 112(2), 289â€“307.
BaÅ¾ant, Z. P., & Oh, B. H. (1983). Crack band theory for fracture of concrete. Materials and Structures, 16(93), 155â€“177.
Bueckner, H. F. (1970). A novel principle for the computation of stress intensity factors. Zeitschrift fÃ¼r Angewandte Mathematik und Mechanik, 50, 529â€“546.
Carpinteri, A. (1989). Cusp catastrophe interpretation of fracture instability. Journal of the Mechanics and Physics of Solids, 37(5), 567â€“582.
Choubey, R. K., Kumar, S., & Rao, M. C. (2014). Effect of shearspan/depth ratio on cohesive crack and doubleK fracture parameters. International Journal of Construction, 2(3), 229â€“247.
Cusatis, G., & Schauffert, E. A. (2009). Cohesive crack analysis of size effect. Engineering Fracture Mechanics, 76, 2163â€“2173.
Elices, M., Rocco, C., & RosellÃ³, C. (2009). Cohesive crack modeling of a simple concrete: Experimental and numerical results. Engineering Fracture Mechanics, 76, 1398â€“1410.
Glinka, G., & Shen, G. (1991). Universal features of weight functions for cracks in Mode I. Engineering Fracture Mechanics, 40, 1135â€“1146.
Hillerborg, A., Modeer, M., & Petersson, P. E. (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 6, 773â€“782.
Hu, S., & Lu, J. (2012). Experimental research and analysis on doubleK fracture parameters of concrete. Advanced Science Letters, 12(1), 192â€“195.
Hu, S., Mi, Z., & Lu, J. (2012). Effect of crackdepth ratio on doubleK fracture parameters of reinforced concrete. Applied Mechanics and Materials, 226â€“228, 937â€“941.
Ince, R. (2010). Determination of concrete fracture parameters based on twoparameter and size effect models using splittension cubes. Engineering Fracture Mechanics, 77, 2233â€“2250.
Ince, R. (2012). Determination of the fracture parameters of the DoubleK model using weight functions of splittension specimens. Engineering Fracture Mechanics, 96, 416â€“432.
Isida, M. (1971). Effect of width and length on stress intensity factor of internally cracked plates under various boundary conditions. International Journal of Fracture, 7, 301â€“316.
Jenq, Y. S., & Shah, S. P. (1985). Two parameter fracture model for concrete. Journal of Engineering Mechanics, 111(10), 1227â€“1241.
Karihaloo, B. L., & Nallathambi, P. (1991). Notched beam test: Mode I fracture toughness. In S. P. Shah & A. Carpinteri (Eds.), Fracture mechanics test methods for concrete, report of RILEM Technical Committee 89FMT (pp. 1â€“86). London, UK: Chamman & Hall.
Kumar, S. (2010). Behavoiur of fracture parameters for crack propagation in concrete. Ph.D. Thesis submitted to Indian Institute of Technology, Kharagpur, India.
Kumar, S., & Barai, S. V. (2008a). Influence of specimen geometry on determination of doubleK fracture parameters of concrete: A comparative study. International Journal of Fracture, 149, 47â€“66.
Kumar, S., & Barai, S. V. (2008b). Cohesive crack model for the study of nonlinear fracture behaviour of concrete. Journal of the Institution of Engineers (India), 89, 7â€“15.
Kumar, S., & Barai, S. V. (2009a). Determining doubleK fracture parameters of concrete for compact tension and wedge splitting tests using weight function. Engineering Fracture Mechanics, 76, 935â€“948.
Kumar, S., & Barai, S. V. (2009b). Effect of softening function on the cohesive crack fracture parameters of concrete CT specimen. SadhanaAcademy Proceedings in Engineering Sciences, 36(6), 987â€“1015.
Kumar, S., & Barai, S. V. (2010). Determining the doubleK fracture parameters for threepoint bending notched concrete beams using weight function. Fatigue & Fracture of Engineering Materials & Structures, 33(10), 645â€“660.
Kumar, S., & Pandey, S. R. (2012). Determination of doubleK fracture parameters of concrete using splittension cube test. Computers and Concrete, 9(1), 1â€“19.
Kumar, S., Pandey, S. R., & Srivastava, A. K. L. (2013). Analytical methods for determination of doubleK fracture parameters of concrete. Advances in Concrete Construction, 1(4), 319â€“340.
Kumar, S., Pandey, S. R., & Srivastava, A. K. L. (2014). Determination of doubleK fracture parameters of concrete using peak load method. Engineering Fracture Mechanics, 131, 471â€“484.
Kwon, S. H., Zhao, Z., & Shah, S. P. (2008). Effect of specimen size on fracture energy and softening curve of concrete: Part II. Inverse analysis and softening curve. Cement Concrete Res, 38, 1061â€“1069.
Lee, J., & Lopez, M. M. (2014). An experimental study on fracture energy of plain concrete. International Journal of Concrete Structures and Materials, 8(2), 129â€“139.
Modeer, M. (1979). A fracture mechanics approach to failure analyses of concrete materials. Report TVBM1001, Division of Building Materials. University of Lund, Sweden.
Murthy, A. R., Iyer, N. R., & Prasad, B. K. R. (2012). Evaluation of fracture parameters by DoubleG, DoubleK models and crack extension resistance for high strength and ultra high strength concrete beams. Computers Materials & Continua, 31(3), 229â€“252.
Nallathambi, P., & Karihaloo, B. L. (1986). Determination of specimensize independent fracture toughness of plain concrete. Magazine of Concrete Research, 135, 67â€“76.
Park, K., Paulino, G. H., & Roesler, J. R. (2008). Determination of the kink point in the bilinear softening model for concrete. Engineering Fracture Mechanics, 7, 3806â€“3818.
Petersson, P. E. (1981). Crack growth and development of fracture zone in plain concrete and similar materials. Report No. TVBM100, Lund Institute of Technology, Sweden.
Planas, J., & Elices, M. (1991). Nonlinear fracture of cohesive material. International Journal of Fracture, 51, 139â€“157.
Reinhardt, H. W., Cornelissen, H. A. W., & Hordijk, D. A. (1986). Tensile tests and failure analysis of concrete. Journal of Structural Engineering, 112(11), 2462â€“2477.
Rice, J. R. (1972). Some remarks on elastic cracktip stress fields. International Journal of Solids and Structures, 8, 751â€“758.
RILEM Draft Recommendation (TC50FMC). (1985). Determination of fracture energy of mortar and concrete by means of threepoint bend test on notched beams. Materials and Structures, 18(4), 287â€“290.
Roesler, J., Paulino, G. H., Park, K., & Gaedicke, C. (2007). Concrete fracture prediction using bilinear softening. Cement Concrete Composites, 29, 300â€“312.
Tada, H., Paris, P. C., & Irwin, G. R. (2000). Stress analysis of cracks handbook (3rd ed.). New York, NY: ASME Press.
Timoshenko, S. P., & Goodier, J. N. (1970). Theory of elasticity (3rd ed.). New York, NY: McGraw Hill.
Wu, Z., Jakubczak, H., Glinka, G., Molski, K., & Nilsson, L. (2003). Determination of stress intensity factors for cracks in complex stress fields. Archive of Mechanical Engineering, 50(1), s41â€“s67.
Xu, S., & Reinhardt, H. W. (1998). Crack extension resistance and fracture properties of quasibrittle materials like concrete based on the complete process of fracture. International Journal of Fracture, 92, 71â€“99.
Xu, S., & Reinhardt, H. W. (1999a). Determination of doubleK criterion for crack propagation in quasibrittle materials, Part I: Experimental investigation of crack propagation. International Journal of Fracture, 98, 111â€“149.
Xu, S., & Reinhardt, H. W. (1999b). Determination of doubleK criterion for crack propagation in quasibrittle materials, Part II: Analytical evaluating and practical measuring methods for threepoint bending notched beams. International Journal of Fracture, 98, 151â€“177.
Xu, S., & Reinhardt, H. W. (1999c). Determination of doubleK criterion for crack propagation in quasibrittle materials, Part III: Compact tension specimens and wedge splitting specimens. International Journal of Fracture, 98, 179â€“193.
Xu, S., & Reinhardt, H. W. (2000). A simplified method for determining doubleK fracture meter parameters for threepoint bending tests. International Journal of Fracture, 104, 181â€“209.
Xu, S., & Zhang, X. (2008). Determination of fracture parameters for crack propagation in concrete using an energy approach. Engineering Fracture Mechanics, 75, 4292â€“4308.
Xu, S., & Zhu, Y. (2009). Experimental determination of fracture parameters for crack propagation in hardening cement paste and mortar. International Journal of Fracture, 157, 33â€“43.
Zhang, X., & Xu, S. (2011). A comparative study on five approaches to evaluate doubleK fracture toughness parameters of concrete and size effect analysis. Engineering Fracture Mechanics, 78, 2115â€“2138.
Zhang, X., Xu, S., & Zheng, S. (2007). Experimental measurement of doubleK fracture parameters of concrete with smallsize aggregates. Frontiers of Architecture and Civil Engineering in China, 1(4), 448â€“457.
Zhao, Z., Kwon, S. H., & Shah, S. P. (2008). Effect of specimen size on fracture energy and softening curve of concrete: Part I. Experiments and fracture energy. Cement Concrete Res, 38, 1049â€“1060.
Zhao, Y., & Xu, S. (2002). The influence of span/depth ratio on the doubleK fracture parameters of concrete. Journal of China Three Gorges University (Natural Sciences), 24(1), 35â€“41.
Zi, G., & BaÅ¾ant, Z. P. (2003). Eignvalue method for computing size effect of cohesive cracks with residual stress, with application to kinkbands in composites. International Journal of Engineering Science, 41, 1519â€“1534.
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Pandey, S.R., Kumar, S. & Srivastava, A.K.L. Determination of DoubleK Fracture Parameters of Concrete Using SplitTension Cube: A Revised Procedure. Int J Concr Struct Mater 10, 163â€“175 (2016). https://doi.org/10.1007/s4006901601396
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DOI: https://doi.org/10.1007/s4006901601396