Determination of DoubleK Fracture Parameters of Concrete Using SplitTension Cube: A Revised Procedure
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.
Keywords
splittension cube test three point bend test concrete fracture doubleK fracture parameters weight function cohesive stress sizeeffectAbbreviations
 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
List of Notations
 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(α, β)
Nondimensional function for KI or geometry factor
 K_{I}
Stress intensity factor
 K_{IC}^{ini}
Initial cracking toughness
 K_{IC}^{un}
Unstable fracture toughness
 K_{IC}^{C}
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(α,β)
Dimensionless function for CMOD
 w_{c}
Maximum crack opening displacement at the cracktip for which the cohesive stress becomes equals to zero
 α
Ratio of crack length to depth of specimen (a/D)
 β
Ratio of loaddistributed to height of specimen (2t/h = t/D) for split tension cube specimen
 σ
Cohesive stress
 υ
The Poisson’s ratio
 σ_{s}(CTOD_{c})
Cohesive stress at the tip of initial notch corresponding to CTOD_{c}
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 Bažant 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) (Bažant and Oh 1983), two parameter fracture model (TPFM) (Jenq and Shah 1985), size effect model (SEM) (Bažant 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 _{ IC } ^{ ini } and unstable fracture toughness K _{ IC } ^{ un } . 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 _{ IC } ^{ un } 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 split—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
3 Determination of DoubleK Fracture Parameters for STC Specimen
3.1 Assumptions
 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
The values of coefficients A _{ i } and B _{ i } for split tension cube specimen (Ince 2012).
β = t/D  

Coefficient  0.0  0.067  0.1  0.133  0.167  0.2 
A _{0}  0.842  0.995  1.050  1.060  1.036  0.995 
A _{1}  2.861  −0.147  −1.366  −1.815  −1.655  −1.219 
A _{2}  −17.384  1.847  9.772  12.762  11.794  8.986 
A _{3}  53.695  −0.480  −23.296  −32.385  −30.268  −22.774 
A _{4}  −70.864  −1.908  27.794  40.275  38.365  29.263 
A _{5}  35.033  2.429  −12.082  −18.691  −18.479  −14.669 
B _{0}  1.159  1.192  1.211  1.216  1.208  1.188 
B _{1}  1.974  1.160  0.582  0.175  −0.047  −0.133 
B _{2}  −11.204  −5.970  −2.239  0.397  1.834  2.379 
B _{3}  37.233  22.364  11.650  3.942  −0.417  −2.252 
B _{4}  −48.035  −29.008  −15.160  −5.051  0.803  3.389 
B _{5}  23.823  14.741  8.015  2.972  −0.093  −1.597 
At critical condition that is at maximum load P _{ u } the half of crack length a becomes equal to a _{ c } and σ _{ N } to σ _{ Nu } in which σ _{ Nu } is the maximum nominal stress. Karihaloo and Nallathambi (1991) concluded that almost the same value of E might be obtained from PCMOD curve, load–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
4 Determination of SIF Due to Cohesive Stress
4.1 Cohesive Stress Distribution
A centrally cracked specimen with finite strip of a finite width plate subjected to pair of normal forces as shown in Fig. 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. 4 becomes to cohesive toughness K _{ IC } ^{ C } 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 _{ IC } ^{ C } is taken as a contribution of the total fracture toughness (Xu and Reinhardt 1999b) at the critical condition.
In which, σ(w) is the cohesive stress at crack opening displacement w at the cracktip and c _{1} and c _{2} are the material constants. Also, w = w _{ c } for f _{ t } = 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. (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}
The accuracy of Eq. (11) is greater than 4 % for 0.1 ≤ α ≤ 0.9 and any value of β and is greater than 2.5 % for 0.1 ≤ α ≤ 0.6 and any value of β. The value of x is taken as a _{ o } and a as a _{ c } for evaluation of CTOD_{c} using Eq. (11).
4.3 Calculation of Cohesive Toughness Using Weight Function Approach
Coefficients m _{ ij } (j = 0–7) of the four term universal weight function parameters M _{1}, M _{2} and M _{3} (Ince 2012).
M _{ i }  0  1  2  3  4  5  6  7 

1  0.070  0.407  −5.405  49.393  −199.837  384.617  −359.928  132.792 
2  −0.089  −2.017  24.839  −86.042  207.787  −243.596  114.431  
3  0.432  2.581  −31.022  134.511  329.531  437.642  −292.768  69.925 
5 Fictitious Crack Model and Material Properties for DoubleK Fracture Model
Values of P_{u} and CMOD_{c} obtained from FCM for TPBT and STC specimens for different specimen sizes.
D (mm)  a _{ o }/D  P_{u} (kN)  CMOD_{c} (mm)  

For TPBT  For STC specimen  For TPBT  For STC specimen  
Value of β for STC  Value of β for STC  
0.0  0.05  0.1  0.15  0.0  0.05  0.1  0.15  
500  0.3  10.73  20.66  20.81  21.24  21.96  0.0822  0.0426  0.0427  0.0442  0.0449 
400  0.3  9.47  17.56  17.69  18.074  18.72  0.0720  0.0379  0.0380  0.0385  0.0403 
300  0.3  7.94  14.15  14.27  14.604  15.18  0.0624  0.0316  0.0318  0.0323  0.0340 
200  0.3  6.05  10.33  10.43  10.724  11.23  0.0510  0.0243  0.0251  0.0259  0.0288 
6 Results and Discussion
From Figs. 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. 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 β = 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 (0 ≤ β ≤ 0.15) considered in the study. On an average, these values for STC for all values of β (0 ≤ β ≤ 0.15) are more than those for TPBT specimen by approximately 4.6 % and 0.43 % for D = 200 mm and D = 500 mm, respectively.
From Fig. 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 β = 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 (0 ≤ β ≤ 0.15). On an average, these values for STC for all values of β (0 ≤ β ≤ 0.15) are less than those for TPBT specimen by approximately 19.9 and 18.3 % for D = 200 mm and D = 500 mm respectively.
It can be observed from Fig. 8 that the values of K _{ IC } ^{ un } 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 (D = 200–500 mm) for all values of β (0 ≤ β ≤ 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 _{ IC } ^{ un } for STC is the lowest for distributedload width β = 0 and is the highest for β = 0.15 which is in close agreement with that obtained from TPBT specimen for all sizes of specimens. The values of K _{ IC } ^{ un } are 36.70 and 39.91 MPa mm^{1/2} for STC with β = 0.15 and 38.16 and 42.10 MPa mm^{1/2} for TPBT specimens for specimen size 200 mm and 500 mm respectively. It seems from Fig. 8 that there is relatively more difference in results of unstable fracture toughness between STC with β = 0 and TPBT. Therefore, in case STC specimen is adopted to replace TPBT to test unstable fracture toughness of concrete, the STC with β = 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 β for STC specimen. The values of cohesive toughness for STC and TPBT specimens shown in Fig. 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 0 ≤ β ≤ 0.15 is clearly observed from Fig. 9. It can be seen that for all values of distributed load width, the values of K _{ IC } ^{ C } 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 β = 0, the values of K _{ IC } ^{ C } are found to be 29.36 MPa mm^{1/2} and 23.87 MPa mm^{1/2} for D = 500 mm and 200 mm respectively whereas those values are obtained as 29.96 MPa mm^{1/2} and 24.55 MPa mm^{1/2} for finite strip for D = 500 mm and 200 mm respectively. Similarly, for STC specimen with infinite strip and β = 0.15, the value of K _{ IC } ^{ C } are found to be 30.64 MPa mm^{1/2} and 26.93 MPa mm^{1/2} for D = 500 mm and 200 mm respectively whereas those values are obtained as 31.34 MPa mm^{1/2} and 27.98 MPa mm^{1/2} for finite strip for D = 500 mm and 200 mm respectively. On an average for all values of β, the K _{ IC } ^{ C } as obtained using finite strip is 2.14 and 3.29 % more than those obtained using infinite strip of plate for D = 500 mm and 200 mm respectively. Also, the values of K _{ IC } ^{ C } as determined using finite strip of STC is 4.82 and 1.86 % less than those obtained using three point bend test specimen for D = 500 mm and 200 mm respectively. It is also observed from Fig. 9 that the size effect on the K _{ IC } ^{ C } values for STC specimen is less significant than that presented for three point bend test.
It can be observed from Fig. 10 that for STC specimen with infinite strip and β = 0, the values of K _{ IC } ^{ ini } are found to be 9.46 MPa mm^{1/2} and 8.32 MPa mm^{1/2} for D = 500 mm and 200 mm respectively whereas those values are obtained as 8.83 MPa mm^{1/2} and 8.68 MPa mm^{1/2} for finite strip for D = 500 mm and 200 mm respectively. Similarly, for STC specimen with infinite strip and β = 0.15, the values of K _{ IC } ^{ ini } are found to be 9.26 MPa mm^{1/2} and 9.78 MPa mm^{1/2} for D = 500 mm and 200 mm respectively whereas those values are obtained as 8.71 MPa mm^{1/2} and 8.74 MPa mm^{1/2} for finite strip for D = 500 mm and 200 mm respectively. On an average for all values of β, the K _{ IC } ^{ ini } obtained using finite strip is 6.21 and 5.96 % lower than those obtained using infinite strip of plate for D = 500 mm and 200 mm respectively. Also, the values of K _{ IC } ^{ ini } as determined using finite strip of STC is 11.70 and 26.10 % less than those obtained using three point bend test specimen for D = 500 mm and 200 mm, respectively. According to the present trend, it seems that the difference in the value of K _{ IC } ^{ ini } obtained between the STC and TPBT specimens may further increase for smaller size specimens such as 150 mm or 100 mm. As per the common convention, this difference should not be more than ±25 % in the fracture test which is a matter of further investigation. It is also seen from Fig. 10 that the size effect on the K _{ IC } ^{ ini } values for STC specimen is less significant than that presented for three point bend test.
7 Conclusions

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 (0 ≤ β ≤ 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 2.14 % and 3.29 % more than those obtained using infinite strip of plate for D = 500 mm and 200 mm, respectively

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

The value of unstable fracture toughness determined using finite strip of splittension cube specimen is the lowest for distributedload width β = 0 and is the highest for β = 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 1.86 % less than those obtained using three point bend test specimen for D = 500 mm and 200 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 26.10 % less than those obtained using three point bend test specimen for D = 500 mm and 200 mm, respectively.
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