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R. Ince et alii, Fracture and Structural Integrity, 75 (20YY) 435-462; DOI: 10.3221/IGF-ESIS.75.30 
 

435 
 

 

 
 
  
Utilizing cylindrical and cubical specimens with edge notch to 
determine size-independent fracture quantities of rock materials 
 
 
R. Ince, E. Eren 
Firat University, Faculty of Engineering, Department of Civil Engineering, Elazig, Turkey 
rince@firat.edu.tr, http://orcid.org/0000-0002-9837-8284 
eeren@firat.edu.tr, http://orcid.org/0000-0001-5282-398X 
 
 

  
KEYWORDS. Crack Closure Integral (CCI), Double-K model, Single notched 
cube bend specimen, Straight-notched disk bend specimen, Two-parameter 
model (TPM). 
 

 

 
 

Citation: Ince, R., Eren, E., Utilizing 
cylindrical and cubical specimens with edge 
notch to determine size-independent fracture 
quantities of rock materials, Fracture and 
Structural Integrity, 75 (2026) 435-462. 
 
Received: 04.10.2025  
Accepted: 11.12.2025 
Published: 11.12.2025  
Issue: 01.2026  
 
Copyright: © 2026 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided the 
original author and source are credited. 

 

 
 
INTRODUCTION  
 

n materials science, clay products such as bricks, natural stones, and concrete are generally classified as ceramic 
materials. From ancient times to the present day, bricks and natural stones such as limestone, granite, and sandstone, 
as well as Roman concrete, have been widely used in buildings, bridges, and retaining walls. These materials are 

categorized as quite brittle in materials science, meaning that while their compressive strength is reasonable, their tensile 
strength is poor. Therefore, in practice, ceramic materials are simulated using the Mohr-Coulomb criterion, one of the 
classical fracture theories, where materials are assumed to be flawless. Moreover, bricks, natural stones, and concrete contain 
many voids, both open and closed.  
In the 1920s, experimental research on brittle materials such as glass revealed that no material was perfect. Subsequently, 
the principle of linear elastic fracture mechanics (LEFM) was introduced. Irwin [1] improved this theory in the 1950s and 
proposed two equivalent material quantities, fracture toughness, and the critical strain energy release rate, to simulate 

I 

https://youtu.be/xdYUel_XbyE


 
 
 

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structures containing cracks. Irwin’s approach was applied to concrete beams of different sizes in 1961 and observed that 
Irwin’s material quantities were size-dependent for concrete. After the 1970s, experimental studies on the aforementioned 
ceramic materials revealed that these materials exhibit quasi-brittle failure, as a region containing a high density of 
microcracks, known as the fracture process zone (FPZ), can develop in the vicinity of the crack tip. In quasi-brittle materials, 
the initially sharp crack tip becomes blunted with increasing loading. As a result, failure in a cracked structure occurs only 
after the crack reaches a certain depth proportional to the FPZ, differing from truly brittle materials. Since 1976, various 
nonlinear fracture models, which employ at least two fracture quantities, have been developed to account for crack-tip 
blunting in quasi-brittle materials such as cementitious composites, rock, and asphalt [2–4]. 
The first applications of these fracture models focused on concrete and mortar materials. Until 1990, test results on rock 
material beams and compact tension (CT) specimens of various sizes revealed that the values of fracture toughness and the 
critical J-integral were independent of specimen width but dependent on specimen size. Considering this, a compliance 
technique commonly used in concrete fracture was first proposed for chevron bend specimens and short rod specimens to 
determine the nonlinear fracture toughness of rock materials by ISRM (International Society for Rock Mechanics) [5]. 
Bažant and Kazemi [3] successfully applied the size effect model (SEM) to experimental data obtained from beams and CT 
specimens of Hashida and Takahashi’s Iidate granite. Bažant et al. [6] simulated the results of fracture tests on Indiana 
limestone beams of four different sizes using the SEM. Subsequently, Ouyang et al. [7] modeled Bažant and co-workers’ [6] 
test results on Iidate granite using the two-parameter model (TPM). 
Although beams and CT specimens containing cracks are widely used to determine fracture quantities of quasi-brittle 
materials and metals, the use of compact specimens, illustrated in Fig. 1, has recently become more common for 
cementitious materials, rock, and asphalt concrete. The wedge-splitting (WS) specimens and compact compression 
specimens, shown in Figs. 1a and 1b, were developed as alternatives to CT specimens and have been widely employed to 
determine the nonlinear fracture quantities of cementitious composites such as mortar and concrete. While splitting 
specimens in cylindrical and cubical forms have been widely used to indirectly evaluate the tensile strength of quasi-brittle 
materials, their centrally notched forms, shown in Fig. 1c, have also been applied to determine the nonlinear fracture 
quantities of cementitious materials. Ince [8] modeled edge-notched mortar cubes, which behave as splitting specimens, 
using the TPM (Fig. 1d). The semi-circular bending (SCB) specimens illustrated in Fig. 1e are the most commonly used 
samples in fracture testing of rock and asphalt materials since they can easily be produced from cylindrical core samples. 
Many fracture tests on SCB specimens have been conducted to evaluate the critical J-integral value and the LEFM-based 
fracture toughness of asphalt composites. Zegeye et al. [9] modeled asphalt SCB specimens of various sizes at low 
temperatures using the SEM. Ince et al. [10] initially derived some LEFM relationships regarding crack blunting in SCB 
specimens and subsequently simulated various SCB tests on asphalt from the literature using the TPM in concrete fracture.  
While many studies on rock SCB specimens have been performed based on LEFM by several researchers, other investigators 
have modeled rock materials by considering the nonlinear fracture behavior of these materials. Wei et al. [11] modeled SCB 
specimens containing a notch and chevron-notched SCB specimens using the finite element method. They then conducted 
fracture tests on Changtai granite specimens of three different depths and determined the nonlinear fracture toughness 
parameters using their equivalent elastic fracture model. Guo et al. [12] determined fracture quantities based on the SEM 
for granite SCB specimens of various sizes at room and high temperatures. Recently, Ince [13] conducted two series of 
experimental studies with SCB and beam specimens made of concrete and mortar and discussed them based on the three 
most popular fracture models: the TPM, the SEM, and the double-K model. Subsequently, three sets of SCB rock tests from 
the literature were analyzed using the aforementioned concrete fracture models, emphasizing that the nonlinear fracture 
quantities of both rock and concrete materials can be effectively assessed using single-sized SCB specimens.  
Tutluoglu and Keles [14,15] derived the stress intensity factor (SIF) values for straight-notched disk bending (SNDB) 
specimens with different height-to-diameter ratios (Fig. 1f), a simplified form of the short rod specimen proposed by ISRM 
[5]. Subsequently, they conducted bending fracture tests on SNDB specimens, SCB specimens, and cracked chevron-
notched Brazilian disks composed of andesite and marble to comparatively determine mode I fracture toughness. They 
proposed a numerical model to compute the fracture process zone (FPZ) depth of these stone materials and concluded that 
the FPZ depth of the SNDB specimen is smaller than that of other specimens. As a result, deviations from the assumptions 
of linear elastic fracture mechanics (LEFM) are reduced. In another study, Tutluoglu and co-workers [16] performed 
extensive experiments on andesite using SNDB specimens, SCB specimens, and three-point bending plates. Their results 
revealed that the critical normal strain value could be considered a material property across all tested specimen types. The 
aforementioned studies primarily focused on compact specimens subjected exclusively to pure mode I loading. Ayatollahi 
and Aliha [17] modeled both SCB specimens and splitting cylinders under mixed-mode loading using the finite element 
method. 



 
 
 

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In this study, SNDB specimens and single-notch cube bending (SNCB) specimens, cubic forms of SNDB specimens that 
differ from the edge-notched cube shown in Fig. 1d only in boundary conditions, were investigated both experimentally 
and numerically. First, a two-dimensional crack analysis of SNCB specimens was conducted using the finite element method 
(FEM) in ANSYS, determining SIF values, crack blunting expressions, and Green’s functions. A three-dimensional crack 
analysis of SNCB specimens was also performed to obtain SIF values. Additionally, the existing SIF formulation for SNDB 
specimens with a diameter-to-height ratio of 2, originally developed by Tutluoglu and Keles [14], was refined based on this 
3D analysis. Other LEFM equations, including crack blunting expressions and Green’s functions, were also derived for 
SNDB specimens. Using these LEFM formulations, test data from Tutluoglu and Keles [14] were simulated using both the 
compliance method and the peak load method based on the two-parameter model (TPM), and the results were compared 
with the numerical model proposed by Tutluoglu and Keles [14]. A significant statistical correlation was observed between 
the three methods. Furthermore, fracture tests were conducted on Elazig limestone cubes with different edge crack depths 
but identical specimen sizes. The experimental results were analyzed using the peak load method with the SNCB specimen 
expressions. Consequently, this study emphasizes that, unlike the compliance approach, the nonlinear fracture properties of 
rock materials can be efficiently determined using the peak loads of single-sized cylindrical and cubic specimens with 
different edge-notch depths, eliminating the need for complex testing equipment. 
 

 
Figure 1: Notched compact specimens commonly used in testing of quasi-brittle materials a) WS specimen b) compact compression c) 
splitting specimens d) cube with edge notch e) SCB specimen f) SNDB specimen. 
 
 
NON-LINEAR FRACTURE MODELS EMPLOYED IN THE MODELING OF QUASI-BRITTLE MATERIALS  
 

uilding elements consisting of quasi-brittle materials, such as rock and cement-based materials, exhibit a three-stage 
behavior until failure occurs. As seen in a beam with initial crack length a0, whose load-crack mouth opening 
displacement relationship is presented in Fig. 2, these stages include: (1) crack initiation, corresponding to the range 

between the initial state and the first cracking load (Pi); (2) stable crack growth, corresponding to the range between the first 
cracking load and the peak load (Pc); and (3) unstable crack growth after the peak load. The beam, which behaves almost 
linearly elastic due to defects in the material until the first cracking load, exhibits non-linear behavior after this point due to 
the formation of a fracture process zone (FPZ) behind the free crack (Fig. 2c).  
 

B 



 
 
 

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Figure 2: a) A typical quasi-brittle beam b) P-CMOD curve c) FPZ formation behind the crack  

 
To fully characterize this multiphase behavior, cohesive crack models, which take into account the cohesive nature of the 
FPZ, simulate quasi-brittle structural members using the finite element method and the boundary element method. On the 
other hand, equivalent elastic fracture models, which consider only the behavior up to the peak load of the structural 
member, aim to predict the crack extension at the peak load (ac), which is proportional to the FPZ depth, as illustrated in 
Fig. 2c [2–4]. It is emphasized that none of the above-mentioned fracture models employ a single fracture parameter, such 
as fracture toughness or fracture energy, contrary to classical LEFM.  
Among the equivalent elastic fracture models, the two-parameter model (TPM) [2] predicts only the unstable crack growth 
at peak load in quasi-brittle structures by utilizing two critical quantities corresponding to the critical crack length 
(ac=a0+ac): the fracture toughness ( sKIc ) and the crack tip opening displacement value (CTODc), which describes the 

blunting at the crack tip well, as defined above: 
 

 
as cK a Yc cIc Nc d

     
 

         (1) 

  
4 0 ,1

a aa acNc c cCTOD V Mc c cE d a dc


 

      
   

      (2) 

 
Here,Nc refers to the strength value computed according to the classical strength of materials theory for unnotched 
specimens, and it is commonly referred to as nominal strength. E in Eqn. (2) represents Young’s modulus of the material. 
In Eqns. (1) and (2), the normalized functions Y, V1, and M are based on crack lengths (a0 and ac) and the specimen depth 
(d). These functions can be computed from crack analysis by employing numerical methods such as the finite element 
method (FEM) for the specimen considered or may be found in LEFM books for simple specimen types. As an example, 
the normalized functions employed for beam specimens with span/depth=2.5, which are also used in this study, can be 
given below: 
 

 
  

2 3 41.83 1.65 4.76 5.3 2.51
3

21 2 1
Y

   
  

   


 
        (3) 

 

 
 

0.682 30.65 1.88 3.02 2.691 21
V    


    


         (4) 

 

 
2 2

0 0 0 0, 1 1.081 1.149 c

a a a aacM ca d a a ac c c c
 

                        
    (5) 

 



 
 
 

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The double-K fracture model, based on equivalent elastic fracture models, simulates failure in quasi-brittle structures by 

utilizing two quantities: initiation and unstable fracture toughness ( unKIc  and iniKIc ). Therefore, contrary to the TPM, both 

crack initiation and unstable crack growth in a quasi-brittle structure can be determined using Eqn. (1) through this model, 
where the values of a₀ and the first cracking load (Pi) are utilized to determine unKIc . 

As pointed out above, the only unknown points in Eqns. (1) and (2) are the critical (or effective) crack depth (ac in the 
procedure for determining fracture quantities in the aforementioned fracture approaches. According to concrete fracture 
codes recommended by RILEM (International Union of Laboratories and Experts in Construction Materials, Systems and 
Structures), as illustrated in Fig. 2b, this critical quantity can be evaluated from the load–crack mouth opening displacement 
(P-CMOD) response of the test specimen (preferably a beam). However, to determine the P-CMOD response of the 
specimen, the use of a feedback-testing machine with CMOD control is necessary for the TPM, whereas no such 
requirement exists for the double-K model.  
In approaches based on RILEM, Young’s modulus (E) of the tested specimen is initially determined from the initial 
compliance value (Ci=CMODi/Pi), where Pi is commonly assumed to be Pc/2 due to difficulties in determining Pi (Fig. 2b). 
Subsequently, assuming that E remains constant for any pair of P and CMOD, the ac value of the notched specimen is 
computed using the secant compliance value (Cs) in the double-K model or the unloading compliance value (Cu) in the TPM, 
as indicated in Fig. 2b. Although both models employ different test equipment and compliance techniques, experimental 
studies on beams and wedge-splitting specimens by Xu et al. revealed that the fracture toughness and CTODc values obtained 
using the TPM were in good statistical agreement with those of the double-K model. 
An alternative method, which utilizes only the ultimate loads of at least three specimens with different notch depths, was 
developed for the TPM by Tang et al. [18]. In this so-called peak-load method in concrete fracture, both fracture parameters 
( sKIc and CTODc) are treated as unknowns instead of a𝑐 values for the specimens. Consequently, this system, which consists 

of multiple equations but only two unknowns, can only be solved using an optimization-based approach. To achieve this, 
Tang et al. proposed that sKIc  and CTODc curves corresponding to specific notch depths should first be determined for 

each specimen with a distinct a0. The values of the sample standard deviation (s) corresponding to the same sKIc  value are 

then calculated using Eqn. (6): 
 

    2 1, 1, 2,...,
1

nss K CTOD CTOD n i nc ciIc i
   


     (6) 

 

Here, CTODc  is the mean value of CTODci values. Consequently, the sKIc  and CTODc values, where  ss KIc is at its 

minimum, are accepted as the fracture quantities according to the peak-load method.  
 As stated above, determining the initial cracking load (Pi) of the specimens is difficult. Therefore, Xu and Reinhardt 
[4] recommended a simplified approach, known as the inverse method in concrete fracture, as follows: 
 

ini un cK K KIc Ic Ic            (7) 

 
Here, cKIc  refers to the cohesive fracture toughness. As depicted in Fig. 3a, cKIc corresponds to the toughness value 

consisting of cohesive stresses assumed to occur along the critical crack extension line (ac). According to the inverse 
method, this cohesive stress distribution is trapezoidal, such that the maximum value of this distribution is supposed to 
correspond to the tensile strength of the material (ft) while its lower value, s(CTODc), is the stress value determined 
according to the chosen stress–crack tip opening displacement function (-w), as indicated in Fig. 3a. In many applications 
up to the peak load in concrete fracture, using a linear function, as shown in Fig. 3b, provides an adequate approach for -
w [19]. The area under this function is equal to the fracture energy (Gf), which is converted to fracture toughness using 

Irwin’s G-K relation:  2unG K EIcf   [1,3]. On the other hand, the toughness value created by such a trapezoidal stress 

distribution at the crack tip can be calculated by integrating the toughness values created by the individual forces (P=dx) 
shown in Fig. 3c: 



 
 
 

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440 
 

 
0

2
,

ac x a axcc cK G dxIc a daa cc


 


 

    
 

        (8) 

 
Here, x refers to the vertical distance from the crack mouth of the specimen, and G is Green’s function. For instance, Tada 
et al. [20] derived the following Green’s function for an infinite strip as shown in Fig. 3c: 
 

   

     
 

1.53.52 1 4.35 5.28 1.3 0.3
, 0.83 1.76 1 11.5 0.5 0.52 211 1

G
      

 

 
   

          
   

 

    (9) 

 
The results of the experiments on compact quasi-brittle samples by Ince [21] indicated that the fracture quantities ( unKIc  

and iniKIc ) of the double-K model can also be evaluated using the peak-load method. 

 

 
Figure 3: Computing of c

IcK  according to the inverse method a) distribution of cohesive stress at peak load b) linear function for -w 
c) a body containing crack under wedge forces 
 
 
DERIVATION OF FRACTURE MECHANICS FORMULAS OF SNDB AND SNCB SPECIMENS  
 

wo prominent energy methods based on the FEM, such as the J-integral and the crack closure integral (CCI) 
technique[22], which can also be successfully utilized for three-dimensional crack problems, were employed in this 
study. Although conventional methods commonly used in FEM crack problems, such as extrapolation methods 

based on stress matching, displacement matching, and the hybrid method, require a large number of fine elements near a 
crack for accurate results, the aforementioned methods do not have such refined mesh requirements. The J-integral is a 
contour integral evaluated along a path that encircles the crack tip, beginning and concluding at the surfaces of the crack. 
Since this integral is path-independent for any of the contours, it has the same value for any similar contour. The use of the 
CCI technique is not as widespread as the J-integral approach and is not embedded in many commercial FEM programs, 
but the implementation of the CCI technique is very simple compared to the J-integral method, as will be detailed below.  
The basic principles of the crack closure integral (CCI) technique can be derived using the fundamental principle of LEFM 
[1]. To determine the relationship between the energy release rate G and the fracture toughness K, it is necessary to account 
for the work of the closing forces (P=dx), which corresponds to the initial state of the new crack profile when a crack of 
length a increases by a small amount (a), as shown in Fig. 4a. For a plate with a thickness of b, the energy released as a 
result of the small extension of the crack (Gba) must be equal to the work done by the internal forces that closed it. 
Accordingly, the following equation can be written: 
 

  
1

20

a
W vdx Gb a


                                                              (10) 

 
Here, v is the crack opening displacement (COD). The above work expression can easily be applied by running two-step 
computer numerical approaches, such as the finite element method, where the elastic body model is assumed to be 

T 



 
 
 

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continuous only at the selected nodes. To calculate the crack closure work in the body modeled with the linear plane finite 
elements, as shown in Fig. 4b, the nodal force at the crack tip (Pi) is determined for the original crack profile in the first 
step. Subsequently, the node at the crack tip is released, and the nodal displacement (vi) is determined for the extended crack 
profile in the second step. On the other hand, when sufficiently small linear elements are used, the simple work equation, 
which can also be computed by utilizing the displacement value at the node just before the crack tip (vi-1), could be obtained 
within acceptable limits by running a single step as follows (W(a)≈W(a+a)): 
 

 
Figure 4: a) Crack closure stresses occurring behind the crack tip b) CCI technique in FEM with linear elements c) crack-tip discretization 
by plane elements with 8 nodes d) singular crack-tip discretization with 8 nodes [22]. 

 
1 1

12 2
W P v P vi i i i             (11) 

 
Consequently, G and K can be determined for a cracked body modeled with linear plane finite elements by utilizing Eqns. 
(10) and (11) as follows: 
 

2
1

2

P vK i iG
E b a

 


          (12) 

 
Similarly, as depicted in Fig. 4c and Fig. 4d, in elastic crack problems where 8-node plane elements with or without singular 
are used, G (and hence K) can be calculated as follows, respectively. 
 

 1 1.5 412

viG P Pi ib a
     

         (13) 

 

2 1 1
2

P v P vi i i iG
b a

  


          (14) 

 
In this study, SNCB and SNDB specimens were simulated as deep beams subjected to three-point bending. While Tutluoglu 
and Keles [14] modeled SNDB specimens, as shown in Fig. 1f, they applied the external load to the specimen through rigid 
shell elements with an arch shape. However, Tang [23] revealed that the width of the distributed load has an insignificant 
effect on structures under three-point bending. Recently, Ince et al. [10] simulated semi-circular bending (SCB) specimens 
for asphalt concrete and concluded that this effect may also be ignored for SCB specimens. For instance, the relative error 



 
 
 

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between concentrated loading and distributed loading was smaller than 0.7% in the computations of the SIFs of SCB 
specimens.   
In this study, the SNCB specimen illustrated in Fig. 5a was initially analyzed using 8-noded plane elements. Because of the 
symmetry of the specimen, half of it, with a depth of d=100 mm and a width of b=1 mm, was considered in the modeling, 
as shown in Fig. 5b. The specimen was discretized into 100 finite elements to minimize computation errors along the notch 
line. Furthermore, a total of eight quarter-point elements with six nodes were employed to consider the stress singularity 
around the notch. Note that Ince [21] previously utilized the same discretization along the notch line to model split-tension 
cylinders containing central cracks, for which an analytical solution was available. Consequently, it was observed that this 
discretization provided an accuracy of 0.2%.  
 

 
Figure 5: FEM discretization of simulated specimens a) characteristics of SNCB specimen b) 2D-SNCB modelling c) 3D-SNCB 
modelling d) SNDB modelling. 
 
In the finite element analysis (FEA) by using ANSYS Parametric Design Language (APDL), the elastic constants of the 
material, elasticity modulus (E) and Poisson’s ratio (v), were selected as 1.0 MPa and 0.2, respectively, and the concentrated 
load (P) at the mid-span was set to 100 N. The relative notch depths (=a0/d) were varied between 0.1 and 0.9 with a step 
interval of 0.1, while span-to-depth (s/d) ratios were chosen as 0.5, 0.6, 0.7, and 0.8. The normalized function (Y), based on 
the SIF of the SNCB specimen, was computed using both the J-integral and the CCI technique based on Eqn. (13) for each 
 value. In this procedure, the value of the nominal strength (N) in Eqn. (1) was taken as P/(2bd), which was previously 
used for the SCB specimens and SNDB specimens by Tutluoglu and Keles [14]. In above 2D FEA, the type of PLANE183 
elements in ANSYS element library (8-noded isoparametric element) were utilized except around the crack, where 8 quarter-
point elements (6-noded singular elements) were used to account for the stress singularity at the crack tip. 
To simulate the SNDB specimen reliably, the SNCB specimen was also modeled in 3D. For this, a quarter of the SNCB 
specimen was considered due to the symmetry of the cubic geometry (b=100 mm), as indicated in Fig. 5c. The 3D SNCB 
specimen containing a notch was modeled using both 8-noded (SOLID185) and 20-noded (SOLID186) solid elements 
without singular finite elements. In FEA, the elastic constants of the material were chosen as in the 2D analysis, while the 
concentrated load (P=100 N) at the mid-span was distributed along the specimen width, as depicted in Fig. 5c. The Y values 



 
 
 

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of the 3D SNCB specimens were calculated using the J-integral and the CCI technique for each  and s/d ratio selected 
above. These values were determined for each finite element at the crack tip in the model with 20 nodes, whereas they were 
computed at the side surface of each finite element at the crack tip in the model with 8 nodes. In the crack analysis based 
on the CCI approach, Eqns. (12) and (14) were employed for the models with 8-noded and 20-noded solid elements, 
respectively.  
In Fig. 6a, the Y values computed for =0.5 are illustrated for s/d=0.5, 0.6, and 0.8 according to each solid element type. 
It is observed from Fig. 6a that there is a very close coincidence between both calculation methods, except for finite elements 
at the boundary of the specimen. The mean values, excluding the boundary values of the CCI technique, were also plotted 
according to both crack analyses in Fig. 6a. Based on this, it can be concluded that the solutions with 8-node elements are 
sufficiently compatible with those of 20-node elements.  
 

 
Figure 6: a) Y distribution along the section in 3D SNCB specimens with =0.5 b) Comparison of 2D and 3D FEA for SNCB specimen. 
 



 
 
 

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Fig. 6b shows the FEA results of both 2D SNCB specimens with 8 nodes and 3D SNCB specimens with 20 nodes for all 
and s/d values considered, comparatively. The equations of the regression line (Y=AX+C), the determination 
coefficients (R2) corresponding to these equations, and the maximum relative error (accuracy) computed for each data point 
were also reported in this figure. In an ideal model, the parameters A, C, and R2 should attain values of 1, 0, and 1, 
respectively. It can be seen from this figure that the results of the SNCB models based on 2D and 3D were satisfactorily 
consistent according to the J-integral way and the CCI technique.  
Based on the information above, the SNDB specimens with only a depth-to-diameter ratio (d/D) of 0.5 were simulated 
under the same conditions as the SNCB specimens for four different s/d ratios: 0.25, 0.30, 0.35, and 0.4. The FEM 
discretization of the quarter part of the SNDB specimen is illustrated in Fig. 5d. 
In this study, the results of 2D FEA were used while deriving the following LEFM formulas for the SNCB specimens. 
According to the FEM discretization in Fig. 5b, in addition to the SIF, the vertical displacement values at each node along 
the crack were recorded to determine the COD profile of the SNCB specimen. This procedure was also performed along 
the cylinder axis for the SNDB specimen. Note that the maximum COD value is equal to the CMOD value.   
Similar to the beam with span/depth=2.5 (Eqn. (3)), the following formulas were selected for the general form of Y for the 
SNDB specimen and the SNCB specimen, respectively. 
 

 

2 3 4
0 1 2 3 4

1.5(1 2 ) 1

A A A A Aa
Y

d

   


 

      
   

      (15) 

 

 

2 3 4 5
0 1 2 3 4 5

1.5(1 2 ) 1

A A A A A Aa
Y

d

    


 

       
   

     (16) 

 
The normalized equations for Y in Eqns. (15) and (16) were calculated by normalizing the FEM computations-based KI 
values with the values of (P√(a0)/2bd) for each specimen geometry, a/d, and span/depth ratio. In Tab. 1, the regression 
coefficients (Ai) computed using the least squares method are tabulated for each specimen geometry and span/depth ratio. 
The maximum relative error (accuracy) between the individual FEA analysis data and the above regression formulas was 
obtained as 0.2% and 0.3% for Eqns. (15) and (16), respectively. 
According to the mechanics of structures containing cracks, the displacement expressions of a cracked structure, such as 
CMOD, depend on both the elastic constants (E and v) of the material and the characteristic size of the structure. 
Consequently, a general expression of CMOD that satisfies boundary conditions was chosen as follows: 
 

 Pα
CMOD = V α12bE

          (17) 

 

 
 

B2 3 4V α = B + B α+ B α + B α +1 0 1 2 3 21- α
       (18) 

 
Similar to beam (Eqn. (4)), the following two formulas were selected for the dimensionless V1 function of the SNDB 
specimen and the SNCB specimen, respectively: 
 

 
 

B2 3 4 5V α = B + B α+ B α + B α + B α +1 0 1 2 3 4 21- α
      (19) 

 
The regression coefficients (Bi) corresponding to the above equations are reported for each specimen geometry and s/d 
ratio in Tab. 1. Eqns. (18) and (19) are valid for 0.1≤≤0.9 within 0.5% and 0.4% accuracy, respectively. The dimensionless 
functions derived for cylindrical and cubical specimens are represented by the curved line for 0.1≤≤0.8 in Fig. 7, as the 
values for =0.9 are too high. In the same figure, several values determined by Tutluoglu and Keles [14] are also plotted 
using + markers. It can be seen from this figure that the results of Eqn. (15) exhibit satisfactory accuracy. Note that 95% 
confidence interval (CI) values (CI-L lower and CI-U upper) were also presented for each coefficient in Tab. 1. 



 
 
 

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445 
 

 
 

Spec. s/D  s/d Coef. 0 1 2 3 4 5 
 

0.5 
 

0.25 
Ai 

CI-L 
CI-U 

3.516 
3.487 
3.545 

-7.146 
-7.464 
-6.829 

15.045 
13.972 
16.119 

-11.268 
-12,692 
-9.846 

3.139 
2.492 
3.787 

 

 
0.6 

 
0.30 

Ai 
CI-L 
CI-U 

3.936 
3.915 
3.956 

-5.304 
-5.526 
-5.083 

10.031 
9.282 
10.780 

-5.659 
-6.653 
-4.666 

0.927 
0.475 
1.379 

 

 
0.7 

 
0.35 

Ai 
CI-L 
CI-U 

4.408 
4.373 
4.442 

-4.119 
-4.495 
-3.744 

7.121 
5.852 
8.391 

-2.597 
-4.280 
-0.914 

-0.232 
-0.997 
0.534 

 

SNDB 
0.8 

 
0.40 

Ai 
CI-L 
CI-U 

4.905 
4.832 
4.979 

-3.182 
-3.978 
-2.385 

4.627 
1.934 
7.321 

0.392 
-3.178 
3.962 

-1.565 
-3.189 
0.059 

 

 
0.5 

 
0.25 

Bi 
CI-L 
CI-U 

10.819 
10.678 
10.960 

-28.272 
-29.608 
-26.936 

38.965 
35.491 
42.440 

-33.829 
-36.530 
-31.129 

7.464 
7.461 
7.466 

 

 
0.6 

 
0.30 

Bi 
CI-L 
CI-U 

12.559 
12.331 
12.788 

-27.789 
-29.951 
-25.628 

36.435 
30.815 
42.055 

-27.679 
-32.048 
-23.310 

9.029 
9.025 
9.033 

 

 
0.7 

 
0.35 

Bi 
CI-L 
CI-U 

15.105 
14.529 
15.681 

-30.183 
-35.637 
-24.729 

37.261 
23.080 
51.442 

-22.530 
-33.554 
-11.505 

10.595 
10.585 
10.605 

 

 
0.8 

 
0.40 

Bi 
CI-L 
CI-U 

18.059 
16.789 
19.329 

-31.357 
-43.382 
-19.332 

30.702 
-0.566 
61.970 

-8.206 
-32.514 
16.102 

12.069 
12.046 
12.091 

 

   
0.5 

Ai 
CI-L 
CI-U 

2.448 
2.186 
2.710 

-20.906 
-24.880 
-16.933 

82.673 
62.629 
102.717 

-146.001 
-190.470 
-101.533 

123.619 
78.705 
168.532 

-40.248 
-57.137 
-23.358 

   
0.6 

Ai 
CI-L 
CI-U 

2.854 
2.759 
2.951 

-16.967 
-18.442 
-15.492 

60.766 
53.325 
68.208 

-102.025 
-188.535 
-85.515 

84.581 
67.906 
101.256 

-27.245 
-33.515 
-20.974 

   
0.7 

Ai 
CI-L 
CI-U 

3.222 
3.183 
3.261 

-13.133 
-13.725 
-12.541 

40.903 
37.918 
43.889 

-64.107 
-70.731 
-57.484 

52.378 
45.688 
59.068 

-16.933 
-19.449 
-14.418 

SNCB   
0.8 

Ai 
CI-L 
CI-U 

3.672 
3.619 
3.725 

-10.172 
-10.978 
-9.366 

24.706 
20.640 
28.772 

-33.460 
-42.480 
-24.440 

26.953 
17.843 
36.064 

-9.015 
-12.441 
-5.589 

   
0.5 

Bi 
CI-L 
CI-U 

8.050 
7.031 
9.069 

-40.143 
-53.859 
-26.426 

28.182 
-30.674 
87.037 

56.025 
-42.544 
154.596 

-93.908 
-150.843 
-36.974 

3.674 
3.654 
3.694 

   
0.6 

Bi 
CI-L 
CI-U 

8.691 
7.744 
9.638 

-27.460 
-40.216 
-14.704 

5.715 
-49.017 
60.448 

74.589 
-17.077 
166.255 

-101.024 
-153.970 
-48.077 

4.476 
4.458 
4.495 

   
0.7 

Bi 
CI-L 
CI-U 

9.835 
8.833 
10.838 

-20.807 
-34.312 
-7.301 

4.416 
-53.534 
62.364 

63.861 
-33.190 
160.915 

-93.876 
-149.935 
-37.819 

5.276 
5.257 
5.296 

   
0.8 

Bi 
CI-L 
CI-U 

11.578 
10.480 
12.677 

-17.613 
-32.405 
-2.822 

13.057 
-50.409 
76.522 

40.423 
-65.868 
146.716 

-80.801 
-142.196 
-19.407 

6.075 
6.054 
6.097 

 

Table 1: The coefficients and confidence interval (CI) values of regression of SNDB and SNCB specimens. 
 



 
 
 

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446 
 

The COD/CMOD ratio depends on the structure's geometry and loading type but is independent of its size. When this 
ratio is known for any structure, the critical value of CTOD for the material can be easily determined. Consequently, Eqns. 
(20) and (21) were selected as the COD profiles for the SNDB specimen and the SNCB specimen, respectively: 
 

   
2 2

21 0.981 1.361 0.377
COD y y y y

CMOD a a a
 

             
     

     (20) 

 

   
2 2

21 1.007 2.747 2.018
COD y y y y

CMOD a a a
 

             
     

     (21) 

 
Consequently, the CTODc value for the material can be computed using the values y=a0, a=ac, and =ac/d. For all chosen 
values, Eqns. (20) and (21) achieve an accuracy better than 3.3%. 
The validity of the above simulations performed was demonstrated in Fig. 8. Two mesh convergence plots for both 2D and 
3D cases vs element size (1 mm, 2.5 mm, and 5 mm) were illustrated for α=0.1, 0.3, 0.5, 0.7, and 0.9 for SNCB specimens 
with s/d=0.8 in Fig. 8a. It is clearly seen from the figure that mesh structures with element sizes of 1 mm and 2.5 mm 
(corresponding to 100 and 40 elements along the crack line, respectively) gave very close results. In these comparisons, the 
maximum relative error calculated between 1 mm and 2.5 mm meshes was calculated as 1.3% for 2D mesh and 1.4% for 
3D mesh, respectively, while these errors were 11.7% and 7.9% for the comparison between 1 mm and 5 mm meshes. As 
a result, it can be concluded that, as mentioned above, using 100 finite elements along the notch line is sufficient to minimize 
calculation errors. In Fig. 8b, the distributed load effect was investigated for the SNDB specimen. It is clearly seen from the 
figure that the relative errors between concentrated loading and distributed loading were smaller than 0.2% and 1.6% for 
ratios of the width of the distributed load to the specimen depth b=0.05 and 0.10, respectively. As mentioned above, this 
effect may be ignored for bending specimens. Fig. 8c presents contour plots with various radii (r) and corresponding Y 
values computed to show path independence for the J-integral of the 2D-SNCB specimen with a/d=0.5. Since the 
differences for the Y values are very small here, it can be said that the calculated J-integrals are path independent. 
 

 
Figure 7: Dimensionless functions obtained from the FEA. 

 



 
 
 

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447 
 

 
Figure 8: Validation check of simulations conducted a) Mesh convergence of SNCB specimen for 2D and 3D computations  
b) Distributed load effect for SNDB specimen c) Contour plots for J-integral for various contours showing path independence for 2D-
SNCB specimen (a/d=0.5). 
 



 
 
 

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448 
 

Deriving Green’s functions for SNDB and SNCB specimens 
The most important application of Green’s function in Hookean structures, initially proposed by George Green in 1828 to 
model the response of a system to an impulse, is to estimate the stress field in response to a force acting at a point in the 
structure. For a cracked structure, the stress intensity factor (SIF, K) that occurs at the crack tip in response to a point force 
can be considered a special case of Green’s function. For a body with a notch of depth a, subjected to Mode-I loading, KI 
based on the Green's function, G(x,a), can be described as follows: 
 

   2
,

0

a
K x G x a dxI a




           (22) 

 
Here, (x), for 0 ≤ x ≤ a, represents the normal stress distribution function at the crack site for the structure without a 
crack due to external loads. According to Eqn. (22), a cracked structure can be analyzed as if it were the same structure 
devoid of external forces, where the relevant stresses are applied directly to the crack surface. In other words, once Green’s 
function is obtained for any structure, instead of modeling the cracked structure, it is sufficient to solve the above integral 
based on the stress distribution along the crack line. Green’s functions in the literature, primarily for two-dimensional elastic 
solids, have been derived analytically for strips, infinite sheets, and simple geometries such as disks. However, to address 
practical issues effectively, some form of approximation of the real problem requires numerical methods such as FEM.  
FEM analysis performed in this study revealed that not only the SIF, as shown in Fig. 6a, but also the stress distribution 
varies across the cross-section of SNDB specimens subjected to bending. Therefore, to obtain a uniform stress distribution 
across the cross-section of an SNDB specimen, the SNDB specimen subjected to the hoop (or circumferential) stress (0), 
as shown in Fig. 9a, was considered for determining the Green’s function. Consequently, by employing the FEM 
discretization in Fig. 5d, the SIFs of SNDB specimens were determined for 0.1<=a/d<0.9 using Eqn. (22). Similarly, 
SNCB specimens without a notch were initially modeled according to the FEM discretization in Fig. 5b under a uniform 
tensile stress (0) for 0.1<=a/d<0.9, as shown in Fig. 9b.  
The expression described in Eqn. (9), which is employed for bending members, was used as the basis for Green’s functions 
of the SNDB and the SNCB specimens because boundary conditions of the specimens examined in this study provide for 

  1.50 1.3 0.3G      and     21 3.52 1 1G       . Consequently, the remaining coefficients of Eqn. (9) 

(1.82, -1.39, -2.65, and 3.86) were adjusted to be suitable for each specimen geometry, such that Eqn. (22) was integrated 
with the stress function x obtained from the FEM analysis for the specimen without a crack, to determine the Green’s 
functions. The integration procedures in Eqn. (22) were performed using the Gauss-Chebyshev approach, while the 
Levenberg–Marquardt method was employed to compute the remaining coefficients. Therefore, for the SNDB and SNCB 
specimens, the derived Green’s functions were:  

   

     
 

1.53.52 1 0.73 0.86 1.3 0.3
, 7.35 12.78 1 11.5 0.5 0.52 211 1

G
      

 

 
   

          
   

 

  (23) 

 

   

     
 

1.53.52 1 3.43 6.7 1.3 0.3
, 0.8 5.55 1 11.5 0.5 0.52 211 1

G
      

 

 
   

          
   

 

   (24) 

 
For 0.1<=a/d<0.9 and any =x/a, the accuracies of the above regression expressions were computed as 2.7% (R2= 
0.9999932) and 3.8% (R2= 0.999778) for SNDB and SNCB, respectively, according to the individual FEM results. Fig. 9c 
illustrates the residuals as a percentage for the regression fits obtained for Green’s functions in Eqns. (23) and (24). 
 



 
 
 

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449 
 

 
Figure 9: a) SNDB specimen under hoop stress and stress distribution b) SNCB specimen under tensile stress c) Eqns. (23) and (24) of 
residual values according to a/d ratios 
 
 
IMPLEMENTATION OF THE COMPLIANCE METHOD AND THE PEAK LOAD METHOD TO SNDB SPECIMEN 

TESTS BY TUTLUOGLU AND KELES [14] 
 

xperimental studies on grained composites such as rock have revealed that their fracture properties are highly 
affected by the maximum grain size and the uniaxial compressive strength of the material (c). As a result, their 
behavior is characterized as being between brittle and ductile due to the internal structure of the material used. 

According to the TPM, the results of fracture test investigations performed on quasi-brittle materials have shown that there 
is a strong correlation between the uniaxial compressive strength and fracture toughness, while some properties related to 
the internal structure of the material, such as the maximum grain size, affect the CTODc parameter. 
Experimental and numerical studies on SNCB specimens were initially performed using Ankara andesite by Alkilicgil [15] 
and Tutluoglu and Keles [14]. For Ankara andesite, the elasticity modulus (E), the splitting tensile strength (fsp), and Poisson’s 
ratio were reported as 12334 ± 135 MPa, 7.00 ± 0.67 MPa and 0.15, respectively, by Tutluoglu and Keles [14]. The SNCB 
testing on cylindrical specimens with a diameter (D) of approximately 100 mm and a height (d) of approximately 50 mm 
was conducted for span/diameter ratios s/D = 0.6 and 0.7, and four different relative initiation notch lengths, = a0/d = 
0.1, 0.2, 0.3 and 0.4. Both P-displacement and P-CMOD curves of the specimens were recorded in the fracture tests.  
The geometrical properties (a0, D, r, d, and s), the peak loads, and the critical CMOD values (CMODc) at the peak load of 
successful test specimens are summarized in Tab. 2. In the specimen numbering in the first column of Tab. 2, the first 

E 



 
 
 

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450 
 

numbers (1, 2, 3, and 4) represent relative notch lengths corresponding to 0.1, 0.2, 0.3, and 0.4, while the next two-digit 
numbers (06 and 07) indicate the span/diameter ratios corresponding to 0.6 and 0.7. The last number refers to the number 
of successful test specimens for each notch length. A certain portion of the load-CMOD curves of SNDB specimens is 
illustrated separately for s/D = 0.6 and 0.7 in Fig. 10. Note that the P-CMOD curve of the specimen of 106-2, which was 
denoted with “*” in Tab. 2, is not presented in Fig. 10 since the curve was too noisy. 
Some characteristic values (0=a0/D, Nc=Pc/(2Dd), Ci, and Cs=CMODc/Pc) computed for each specimen in Tab. 2 are 
summarized in the 2nd to 5th columns of Tab. 3 to apply the compliance method to SNDB specimens. In this table, the 
values of Ci were computed from the slope of the linear regression equation, which was determined by considering pairs of 
P-CMOD values up to approximately Pc/2 in the initial straight portion of the P-CMOD plots, as stated above. Accordingly, 
Young’s modulus of the stone was determined from Eqns. (17) and (18) as follows: 
 

 0
1 02

E V
bCi


           (25) 

 
The relative critical notch depth of each specimen in Tab. 3 was computed using the Newton method, assuming that the E 
value of the specimen remains constant for any notch depth, in accordance with the equivalent elastic crack approach: 

 

 
 

1 0
0

1

C Vs
c C V ci


 


           (26) 

 
The fracture quantities of the TPM ( sKIc  and CTODc) were determined using Eqns. (1), (2), and (20) for each specimen in 

Tab. 3. As stated above, the unstable fracture toughness values of the TPM and the double-K are equivalent ( s unK KIc Ic ) 

and are therefore presented in a single column in Tab. 3. In Tab. 3, the mean values (), the standard deviation (), the 
coefficient of variation (C.V.), and the approximate 5% and 95% confidence limits (±) of the fracture quantities and 
Young’s modulus were also reported according to each span-to-diameter ratio. C.V. is beneficial when evaluating the 
variation among groups that exhibit notably different average mechanical properties, such as compressive strength while 
maintaining a consistent level of control. In many applications of concrete technology, a reasonable range for C.V. is 5% to 
20%. It can be seen from Tab. 3 that the C.V. values of the TPM quantities, including E, s unK KIc Ic and CTODc, are less 

than 20%. Furthermore, Tab. 3 shows that the value of E measured by Tutluoglu and Keles [14] falls within the confidence 
limits of that of the TPM. The crack extension values (ac) were also reported for each specimen in the last two columns of 
Tab. 3. Tutluoglu and Keles [14] stated by using the numerical model that the ac value was obtained as 3.92 ±0.86 mm for 
Ankara andesite, and this value also falls within the confidence limits of that of the TPM in Tab. 3. 

The initial fracture toughness values of the double-K ( iniKIc ) were computed using Eqns. (7), (8), and (23) and summarized 

in Tab. 3. In this procedure, to compute the lower limit stress of the cohesive stress distribution in Fig. 3a, the following 
linear expression proposed by Bažant [19], as depicted in Fig. 3b, was employed:  
 

 
2

1 , 0
0

GCTOD fcCTOD f ws c t w ft


 
    

 
       (27) 

 
According to concrete design codes, the splitting tensile strength of quasi-brittle materials such as concrete is 1.5 times the 
direct tensile strength (ft=fsp/1.5). Based on this, the upper limit stress in Fig. 3a was taken as 4.667 MPa, which is close to 
the lower limit of the range given by Arioglu et al. [24] for andesite (5-11 MPa).  
SNDB specimens were also analyzed using the peak-load method. The implementation of the peak-load method to a large 

number of specimens containing different notch depths  1 2, , ... na a ac c c  is based on the following two basic equations: 

 



 
 
 

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451 
 

 
 

, , 1, 2, ...,

,

i i i sK a K i ncI Nc Ic

i i iCTOD a CTODc cNc





 


      (28) 

 

Here, n is the number of sets, i is the ith set and i
Nc  is the mean critical nominal strength of the ith set, which corresponds 

to the mean critical notch depth ( i
ca ) of the ith set. By employing Eqns. (1) and (2), the sKIc  and CTODc curves of specimens 

are initially plotted for each different notch depth to apply the method to the specimens in sets. Subsequently, the mean 
sKIc  and CTODc curves, along with the standard deviation curve based on Eqn. (6), are plotted. Consequently, the fracture 

quantities of the material can be determined from the abscissa and ordinate of the point corresponding to the minimum 
standard deviation on the mean curve.  
Fig. 11 illustrates the implementation of the peak load method for SNDB specimens with Ankara andesite. In these figures, 
the sKIc  and CTODc plots of the SNDB specimens with s/D=0.6 are shown at the top, while those of the SNDB specimens 

with s/D=0.7 are at the bottom. The standard deviation plots based on Eqn. (6) are presented in the middle. In conclusion, 
the fracture quantities of the stone material were calculated from the smin values, which correspond to the minimum standard 
deviations, as depicted in Fig. 11. This figure demonstrates a strong correlation between both fracture quantities of Ankara 
andesite despite using SNDB specimens with two different s/D ratios. Note that the E value measured by Tutluoglu and 
Keles [14] was used in this analysis. In Tab. 4, the relative critical notch depths (c) and the crack extension values (ac) of 
the stone material are reported for each  and each s/D ratio. Using Eqns. (7), (8), and (23), the parameters of initiation 
fracture toughness of the double-K model, computed according to results of the peak load method, are also summarized in 
Tab. 4. 
 
 

No  
a0 

mm 
D 

mm 
r 

mm 
d 

mm 
s 

mm 
Pc 
N 

CMODc 
mm 

106-1 6.5 101.74 50.9 50.1 61.76 26020 0.0347 
*106-2 5.5 101.02 50.3 49.6 60.62 24549 0.0299 
106-3 5 100.98 50.5 50.1 61.02 21712 0.0334 
206-1 10 101.83 50.9 48 61.82 15882 0.035 
206-2 9.75 101.8 50.9 48.1 61.78 16346 0.038 
206-3 9.75 101.71 50.8 47.8 61.7 16535 0.0414 
306-1 15.63 101.68 50.9 50.1 61.72 15728 0.0471 
306-2 15.25 101.68 50.8 50 61.7 14456 0.0443 
406-1 20.5 101.65 50.8 50.5 61.6 10326 0.0488 
406-2 20.25 101.68 50.8 50.9 61.7 12250 0.0532 
406-3 20 101.43 50.7 50 61.32 12431 0.0513 
107-1 5.5 101.85 50.9 50.4 71.84 18337 0.0341 
107-2 5.5 101.48 50.8 50.2 71.52 20310 0.0352 
207-1 10.5 101.67 50.8 49.7 71.68 13649 0.0419 
207-2 10.75 101.81 50.9 50.2 71.8 15058 0.045 
207-3 9.75 101.8 50.9 50.4 71.8 15568 0.0398 
307-2 15.38 101.77 50.9 51.1 71.78 12917 0.0488 
307-3 15.5 101.68 50.8 50 71.66 12150 0.0548 
407-2 20 101.7 50.9 51.3 71.74 9596 0.048 
407-3 21 101.76 50.9 51.2 71.76 10377 0.0573 

 

Table 2: Test Results of SNDB specimens by Alkilicgil [15]. 
 



 
 
 

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Figure 10: Load-CMOD plots of SNDB tests by Alkilicgil [15]. 

 

 
 

Figure 11: Implementation of TPM to SNDB specimens experimented by Alkilicgil [15]. 
 
 
 
 
 



 
 
 

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No  0  
Nc 

MPa 
Ci×10-6 
mm/N 

Cs×10-6 
mm/N 

E 
MPa c  

s unK KIc Ic  

MPa√m 

CTODc 
mm 

iniKIc  

MPa√m 

ac 

mm 

106-1 0.130 2.552 0.687 1.334 19891 0.234 1.579 0.0210 0.640 5.21 
106-3 0.100 2.146 0.774 1.538 13581 0.190 1.201 0.0214 0.360 4.51 
206-1 0.208 1.625 1.278 2.204 18015 0.314 1.217 0.0173 0.340 5.09 
206-2 0.203 1.669 1.338 2.325 16665 0.309 1.201 0.0191 0.332 5.10 
206-3 0.204 1.701 1.717 2.504 13094 0.275 1.128 0.0180 0.411 3.39 
306-1 0.312 1.544 2.039 2.995 19313 0.392 1.396 0.0166 0.632 4.01 
306-2 0.305 1.422 2.192 3.064 17365 0.375 1.227 0.0149 0.528 3.50 
406-1 0.406 1.006 2.929 4.726 21144 0.501 1.263 0.0152 0.476 4.80 
406-2 0.398 1.183 3.282 4.343 18138 0.455 1.289 0.0134 0.682 2.90 
406-3 0.400 1.226 2.825 4.127 21346 0.476 1.414 0.0147 0.699 3.80 

mean=     17855  1.292 0.0172 0.510 4.23 

Std=     2828  0.134 0.0027  0.82 
C.V.     0.158  0.104 0.159  19.43 

     12199  1.023 0.0117  2.59 

     23511  1.560 0.0226  5.87 
107-1 0.109 1.786 0.885 1.860 15482 0.215 1.267 0.0221 0.362 5.34 
107-2 0.110 1.993 0.938 1.733 14720 0.193 1.337 0.0213 0.505 4.17 
207-1 0.211 1.351 1.644 3.070 17196 0.333 1.266 0.0216 0.323 6.06 
207-2 0.214 1.473 1.706 2.988 16820 0.324 1.355 0.0222 0.433 5.52 
207-3 0.193 1.517 1.419 2.556 17935 0.305 1.335 0.0208 0.403 5.64 
307-2 0.301 1.242 2.142 3.778 21031 0.418 1.475 0.0204 0.530 5.98 
307-3 0.310 1.195 3.349 4.510 14070 0.372 1.238 0.0174 0.583 3.10 
407-2 0.390 0.920 3.405 5.002 20336 0.468 1.264 0.0142 0.544 4.00 
407-3 0.410 0.996 3.557 5.522 21469 0.497 1.501 0.0170 0.719 4.45 

mean=     17673  1.338 0.0197 0.489 4.92 

Std=     2746  0.094 0.0028  1.03 
C.V.     0.155  0.071 0.142  20.87 

     12182  1.149 0.0141  2.87 

     23164  1.526 0.0253  6.97 
 

Table 3: Results of the compliance-based analysis for each data 
 

 

Group s/D 0 
Nc 

MPa 

s unK KIc Ic  

MPa√m 

CTODc 
mm c 

iniKIc  

MPa√m 

ac 

mm 

106 0.6 0.113 2.383 1.228 0.0195 0.174 0.512 3.06 
206 0.6 0.205 1.665   0.281 0.461 3.65 
306 0.6 0.308 1.483   0.368 0.589 3.00 
406 0.6 0.401 1.138   0.461 0.616 3.03 
107 0.7 0.109 1.890 1.221 0.0191 0.172 0.485 3.17 
207 0.7 0.206 1.447   0.271 0.504 3.26 
307 0.7 0.305 1.218   0.364 0.585 2.98 
407 0.7 0.400 0.958   0.456 0.628 2.87 

 

Table 4: Results of the TPM-based analysis for grouped data of SNDB specimen. 
 



 
 
 

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Statistical comparison of different fracture analysis methods used in SNDB specimens 
When comparing Tabs. 3 and 4, considering basic statistical parameters such as the mean value and the 5% and 95% 
confidence limits, it can be concluded that there is no significant difference between the compliance method and the peak 
load method, based on the fracture quantities of Ankara andesite. Similarly, when the critical crack extension values of TPM 
in Tab. 3 are compared with the values in the numerical model proposed by Tutluoglu and Keles [14], it can be concluded 
that there is no significant difference between both models, based on the FPZ depth.  
An important feature of statistics is its ability to facilitate the comparison of two data sets, allowing for the assessment of 
the likelihood that they differ significantly from one another. For this reason, two comparison tests are widely employed to 
determine whether they are potentially derived from the same sample population. These are the Student’s t-test, which 
compares the means of samples, and the F-test, which compares the standard deviations of samples. The expressions for 
these tests can be given for paired comparisons as follows:  
 

 
1 2

2
1

1

t ncalc n
d nii

 






 


         (29) 

 
2
1
2
2

Fcalc



            (30) 

 
In Eqn. (29), 1 and 2 are the means of the first and the second data sets for the number of data points n, respectively, 
while di and  represent the differences for each pair of data and the mean of di. In Eqn. (30), 1 and 2 are the standard 
deviations of the first and the second data sets, respectively, provided that 1>2. In addition to assuming that the 
populations are normally distributed in the two statistical methods defined above, the Student's t-test assumes that both 
populations have the same variance, while the F-test assumes that the populations have the same means. 
The aforementioned statistical test methods can also be employed to compare two different testing or analysis methods in 
any process. Moreover, the F-test can be utilized to assess the precision of a newly developed method based on a method 
whose accuracy is already known. In this study, a statistical comparison of SNDB specimens with different notch lengths 
was conducted between the compliance method proposed by RILEM and the peak load method, which is commonly 

employed in concrete fracture tests. For this comparison, the reciprocal iniKIc  parameters computed for each notch depth 

in Tab. 3 (based on compliance analysis) and Tab. 4 (based on the peak load method) were utilized, as this parameter is 
based on both s unK KIc Ic  and CTODc parameters.  

The comprehensive details of the statistical analysis are reported in Tab. 5. In this context, SNDB1, SNDB2, SNDB3, and 
SNDB4 represent specimens corresponding to relative notch lengths of 0.1, 0.2, 0.3, and 0.4, respectively. Prior to the 
comparative analysis, the normality of the dataset was verified using the Shapiro-Wilk test. Despite the small sample size 
(n=4), all p-values exceeded the 0.05 threshold, justifying the use of parametric tests. The values of tcalc and Fcalc were 
calculated using Eqns. (30) and (31), respectively, while their corresponding values from statistical tables (tn-1,0.025 and F0.05, n-

1, n-1, where n-1 is the degrees of freedom) were determined for the 5% significance level in this study. Note that since means 
are being compared, Student’s t-test was performed as a two-tailed test (0.05/2=0.025). To provide a robust comparison, 
95% Confidence Intervals for the means were calculated for each group. The overlap between the confidence intervals of 
the two methods suggests a strong similarity in the results. Furthermore, to address the practical significance of the results, 
the effect size (Hedges’ g) and post-hoc power were computed. For both geometries, the effect sizes were found to be 
negligible (g≤ 0.10), and the achieved power was low. These indicators demonstrate that the difference between the 
compliance and peak load methods is virtually non-existent for the tested material. Accordingly, it can be concluded from 
Tab. 5 that there is no significant difference between the compliance and peak load methods regarding initiation fracture 
toughness parameters, as the computed values are less than their corresponding values from statistical tables. Consequently, 
considering the discussion above, it can be concluded that the peak load method has the same level of precision as the 
compliance method proposed by RILEM. 
In this study, another statistical analysis was conducted to compare the compliance method and the numerical model 
proposed by Tutluoglu and Keles [14]. By using the FEA, Tutluoglu and Keles [14] simulated the SNDB specimens with 
d=(D/2)=50 mm and s/d= 0.3 and 0.35 for a0/d=0.1, 0.2, 0.3 and 0.4. In this analysis, the mean peak loads were utilized 



 
 
 

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for each a0/d and s/d. According to this, stress distributions along the notch line (xx) were determined. Subsequently, the 
FPZ depths (ac) in front of the notch were determined at Nc/xx=1. Consequently, the value of ac was determined as 
3.92±0.86 mm. 
For the statistical comparison of both approaches, by considering the above-mentioned numerical analysis, a total of 8 

fracture toughness values ( refKIc ) were computed based on Eqn. (1) using the values of ac=a0+3.92 mm (a0=5, 10, 15, and 

20 mm) and corresponding to the mean values of Nc for s/d= 0.3 and 0.35. In response, a total of 19 fracture toughness 
values ( s unK KIc Ic ) determined according to the compliance method were utilized for statistical comparison. 

The t-test based on Eqn. (29) is valid for paired comparisons. If the number of data to be compared is different, the 
following relation is used for the t-test: 

 

   
1 2

2 21 1 1 11 1 2 2
21 2 1 2

tcalc
n n

n n n n

 

 




  


 

       (31) 

 
 

s/D 0.6 0.7 

Method Compliance Peak Load di Compliance Peak Load di 
SNDB1 0.500 0.512 -0.012 0.434 0.485 -0.052 

SNDB2 0.361 0.461 -0.100 0.386 0.504 -0.118 

SNDB3 0.580 0.589 -0.009 0.557 0.585 -0.029 

SNDB4 0.619 0.616 0.003 0.632 0.628 0.003 

Descriptive Statistics 
n 4 4 

 0.515 0.545 -0.030 0.502 0.551 -0.049 

 0.114 0.071 0.048 0.112 0.067 0.051 
95% Confidence 
Interval for Mean [0.334, 0.696] [0.324, 0.680]  [0.432, 0.658] [0.444, 0.658]  

Normality Check 
Shapiro-Wilk (p) 0.608 0.761  0.516 0.596  
Result Normal Normal  Normal Normal  

Hypothesis Testing 

tcalc 1.24 (p>0.05) 1.89 (p>0.05) 
t3,0.025 2.35 2.35 
Result tcalc = 1.24 < t3, 0.025=2.35 tcalc = 1.89 < t3, 0.025 = 2.35 
Fcalc 2.58 (p>0.05) 2.77 (p>0.05) 

F0.05,3,3 9.28 9.28 

Result Fcalc = 2.58 < F0.05, 3, 3=9.28 Fcalc = 2.77 < F0.05, 3, 3 = 9.28 

Effect Size (Hedges' g) 0.10 (Negligible) 0.08 (Negligible) 

Achieved Power (1-) 0.056 0.054 

There is not a significant difference between initiation fracture toughness. 

Table 5:  Statistical comparison based on t-test and F-test for ini
IcK  in this study 

 
Here, n1 and n2 are the numbers of data in samples. Details of the second statistical comparison conducted are summarized 
in Tab. 6. Prior to applying the comparison formulas, the normality of the datasets was verified via the Shapiro-Wilk test. 
Despite the unbalanced sample sizes (n1=19 vs. n2=8), the data followed a normal distribution (p > 0.05), validating the 



 
 
 

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application of parametric tests. In this table, the values of tcalc and Fcalc were computed utilizing Eqns. (31) and (30), 
respectively, while their corresponding values from statistical tables (tn1+n2-2,0.025 and F0.05, n1-1, n2-1) were determined for the 5% 
significance level in this analysis. To further substantiate the comparison, 95% Confidence Intervals and the effect size 
(Hedges’ g) were calculated. Although a medium effect size (g = 0.66) was observed, the 95% Confidence Intervals exhibited 
overlap, suggesting consistency between the datasets. Accordingly, it can be concluded from Tab. 6 that there is no 
significant difference between the compliance method and the numerical method proposed by Tutluoglu and Keles [14] 
regarding non-linear fracture toughness parameters, as the determined test statistics remain below their corresponding 
tabular limits. 
 
 
IMPLEMENTATION OF THE PEAK LOAD METHOD FOR SNCB SPECIMEN TESTS ON ELAZIG LIMESTONE [14] 
 
Experimental program 

ased on the above statistical test analysis, it was concluded that there is no significant difference between the 
compliance method and the peak load method in determining the nonlinear fracture quantities of rock materials. 
Based on this conclusion, the fracture mechanics analysis of Elazig limestone was performed utilizing the peak load 

method. On the other hand, beams and SNCB specimens containing notch depths were fabricated from Elazig limestone 
to validate the results. To determine the mechanical properties of the stone, such as compressive strength, tensile strength, 
and Young’s modulus, cubes and beams without notches were also prepared. 
Compression cubes, as illustrated in Fig. 12a, were subjected to monotonic loading at a rate of 3.00 kN/s in a concrete 
hydraulic compression press with a 2000 kN actuator. The average compressive strength of three 70 mm cubes was 190.4 
MPa. Durmekova et al. [25] performed tests on many rock cylindrical and prismatic specimens and concluded that the ratio 
of cubical strength to cylindrical strength was 1.15. Accordingly, the cylindrical strength of the rock with a diameter of 70 
mm is 165.6 MPa. Consequently, when the cylindrical strength with a 70 mm diameter is transformed to the cylindrical 
strength with a 50 mm diameter, according to the size effect formula proposed by Hoek and Brown, where 

 0.18
50d dc c  , where andd

c c   represents the strengths of specimens with diameters d and 50 mm (standard), 

respectively, the standard strength value of Elazig Limestone was obtained as c=175.9 MPa. Note that the test data for 
limestone fit well with the formula by Hoek and Brown. The Young’s modulus of the rock (E) was determined by utilizing 
the following power formula proposed by Arioglu et al. [24]:   
 

0.720.76E c            (32) 
 

Here, c is in MPa, while E is in GPa. The above empirical formula, with a correlation coefficient of 0.812, was determined 
based on 119 different limestone test data. Accordingly, the value of E for the rock was evaluated as 52.7 GPa for the 
standard strength. 
The average bending strength (fr) of three beams (width × depth × length = 50 × 50 × 200 mm) tested under a loading 
span of 125 mm, whose failures are shown in Fig. 12a, was obtained as 11.468 MPa using a compression press with a 100 
kN actuator. Concrete design codes suggested that the direct tensile strength of quasi-brittle materials such as concrete 
should be taken as half of the bending tensile strength (ft=fr/2). According to this, in the following analysis, the tensile 
strength of the stone was assumed to be 5.734 MPa, which is within the range given by Arioglu et al. [24] for limestone (4-
7 MPa). 
The beams were loaded at three points with a span/depth=2.5 (Fig. 12b), while the SNCB specimens were subjected to 
three-point bending with a span-to-diameter ratio of 0.8 (Fig. 12d). All specimens containing a notch were fractured using 
the compression press with a 100 kN actuator, with the time to reach the peak load set to 3 minutes (± 30 seconds).  
Crack patterns at the failure of beams and SNCB specimens containing a notch are illustrated in Figs. 12c and 12e, 
respectively. The dimensions of section sizes (b and d) and the peak load values (Pc) of the notched beam and SNCB 
specimens were summarized in Tab. 7 according to the initial notch depths (a0). In this table, "B" and "SNCB" refer to the 
specimen types for beam and SNCB, respectively. In the specimen numbering in the first column of Tab. 6, the first numbers 
(1, 2, and 3) refer to relative notch lengths corresponding to 0.3, 0.4, and 0.5, while the last number illustrates the number 
of successful test specimens for each notch length. 
 
 

B 



 
 
 

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No  
s unK KIc Ic  

MPa√m 

refKIc  

MPa√m 
106-1 1.579 

1.265 
106-3 1.201 
206-1 1.217 

1.068 206-2 1.201 
206-3 1.128 
306-1 1.396 

1.269 
306-2 1.227 
406-1 1.263 

1.314 406-2 1.289 
406-3 1.414 
107-1 1.267 

1.201 
107-2 1.337 
207-1 1.266 

1.168 207-2 1.355 
207-3 1.335 
307-2 1.475 

1.276 
307-3 1.238 
407-2 1.264 

1.357 
407-3 1.501 

Descriptive Statistics 
n 19 8 

 1.313 1.240 

 0.116 0.091 
95% Confidence 
Interval for Mean [1.257, 1.369] [1.164, 1.316] 

Normality Check 
Shapiro-Wilk (p) 0.285 0.812 

Result Normal Normal 
Hypothesis Testing 

tcalc 1.58 (p>0.05) 

t25,0.025 1.71 

Result tcalc = 1.58 < t25, 0.025=1.71 
Fcalc 1.63 (p>0.05) 

F0.05,18,7 3.47 
Result Fcalc = 1.63 < F0.05, 18,7=3.47 

Effect Size (Hedges' g) 0.66 (Medium) 

Achieved Power (1-) 0.34 

There is not a significant difference between fracture toughness. 
 

Table 6:  Statistical comparison based on t-test and F-test between this study and Tutluoglu and Keles [14] for non-linear fracture 
toughness. 
 
 
 



 
 
 

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Figure 12: Test details of limestone specimens a) fractured flexural and compressive specimens b) notched beam test setup c) fractured 
notched beams d) SNCB specimen test setup e) fractured SNCB specimens. 

 
As stated above, the peak load method based on TPM is an optimization-based technique since the number of unknowns 
(three or more distinct specimens must be tested to ensure statistically valid results) is greater than the number of equations 
(Eqns. (1) and (2)). The objective function of the optimization problem is to minimize Eqn. (6), while the constraints of the 
problem are the critical crack lengths, which must be greater than initial crack lengths and smaller than specimen depth as 
detailed below. 
Fig. 13 indicates the implementation of the peak load method to beams and SNCB specimens with Elazig limestone. The 

sKIc  and CTODc plots of the beams are demonstrated at the top, and those of the SNCB specimens are at the bottom, while 

the s- sKIc  plots based on Eqn. (6) are illustrated in the middle in Fig. 13. For the dimensionless functions used in drawing 

the sKIc  and CTODc plots, Eqns. (3)-(5) were used for beams, while Eqns. (16), (19), and (21) derived in this study were 

employed for SNCB specimens. This figure shows that there is a strong correlation between the unstable fracture toughness 
of Elazig limestone for both specimen types. To compute the initiation fracture toughness based on the double-K, Eqns. 
(9) and (27) were employed for beams and SNCB specimens, respectively. The fracture quantities corresponding to each  
value were reported for two different bending specimens in Tab. 8.  
In the last column of Tab. 8, the relative crack extension values (c) are also presented for each specimen and each  
value. These values demonstrate that the effect of the FPZ on the fracture quantities of rocks cannot be ignored, as it is in 
other quasi-brittle civil engineering materials such as concrete and asphalt concrete. Although both specimen types are 
subjected to three-point bending loading, the c values of beams are smaller than those of SNCB specimens. The most 
important reason for this is that SNCB specimens behave like deep beams, where the depth of the compression zone is 
larger than that of the beam under three-point bending loading. On the other hand, it is well known that the depth of crack 
extension at the peak load is not a fracture parameter and varies depending on specimen type and specimen size in quasi-
brittle materials [2].  

 



 
 
 

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Specimen  
b 

mm 
d 

mm 
a0 

mm 
Pc 

kN 

B11 51.6 50.67 14.93 3.51 

B12 50.29 50.61 15.17 3.7 

B21 51.01 50.76 20.6 2.7 

B22 51.01 50.93 20.21 2.83 

B23 50.69 50.54 20.16 2.3 

B31 50.75 50.71 25 1.73 

B32 50.52 50.97 24.34 2.08 

B33 50.39 50.81 25.69 1.73 

SNCB11 70.91 71.05 20.75 29.49 

SNCB12 71.3 71.23 20.91 29.81 

SNCB13 70.68 71.37 20.98 29.2 

SNCB14 71.17 71.38 20.82 29.03 

SNCB21 71.34 71.61 31.7 18.36 

SNCB22 70.8 71.53 31.56 13.97 

SNCB23 71.71 71.36 31.45 14.45 

SNCB24 71.25 71.19 31.08 17.65 

SNCB25 71.27 71.45 31.4 16.48 

SNCB31 72.06 71.22 35.54 12.93 

SNCB32 71.48 70.99 35.5 12.71 

SNCB33 71.07 71.45 37.16 13.05 

SNCB34 71.15 71.29 35.76 15.31 

SNCB35 70.92 71.21 35.84 13.72 
 

Table 7: Test results of beams and SNCB specimens tested in this study 
 
 

Group s/d 0 
Nc 

MPa 

s unK KIc Ic  

MPa√m 

CTODc 
mm c 

iniKIc  

MPa√m 

c 

B1 2.5 0.297 2.383 2.260 0.0188 0.538 0.939 0.241 

B2 2.5 0.401 1.483   0.623 0.924 0.222 

B3 2.5 0.497 1.138   0.696 0.893 0.199 

SNCB1 0.8 0.293 1.890 2.244 0.0158 0.432 1.326 0.139 

SNCB2 0.8 0.440 1.447   0.605 0.916 0.165 

SNCB3 0.8 0.505 0.958   0.659 0.872 0.154 
 

Table 8:  Results of fracture analysis for grouped data of Elazig limestone. 
 



 
 
 

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Figure 13: Implementation of TPM to beams and SNCB specimens tested in this study. 

 
 
CONCLUSION 
 

onlinear fracture models, frequently applied in the examination of concrete fracture, were utilized in this study to 
evaluate the fracture quantities of quasi-brittle SNDB and SNCB specimens made from rock materials. The results 
of this investigation are summarized below: 

1) The examination of rock SNDB specimens with a depth/diameter ratio of 0.5, documented in the literature, utilized 
both the compliance method and the peak load method, revealing a significant correlation between the two 
approaches. This has previously been observed in other quasi-brittle materials such as concrete and asphalt 
concrete. It is emphasized that the most important advantage of the peak-load method is that it does not require 
sophisticated test equipment, unlike the compliance method, to determine the fracture quantities of the materials. 

2) Another statistical comparison based on the t-test and the F-test was conducted for the compliance method based 
on the TPM and the numerical method based on the FEA proposed by Tutluoglu and Keles [14]. It was revealed 
that there was a significant correlation between the two methods according to both the means and the variances. 

3) The fracture tests of comparative tests on beams and SNCB specimens reveal that the fracture quantities of rocks 
based on the TPM and the double-K model can easily be estimated using SNCB specimen tests, so specimens of 
differing sizes are unnecessary. On the other hand, the LEFM expressions derived for cubical specimens containing 
an edge notch can also be used for square prismatic specimens containing an edge notch subjected to a bending 
load. 

4) The primary benefits of cylindrical and cubical specimens containing an edge notch, examined in this study, are 
their lightness and compactness compared to beams. Furthermore, in the determination of the fracture quantities 
of rocks, the use of these specimens, facilitated by core drilling, enables the extraction of materials from rock 
formations, providing significant benefits.  

5) The CCI technique is not as commonly used as the J-integral approach and is not integrated into many commercial 
finite element method software. However, the implementation of the CCI technique is notably more 

N 



 
 
 

R. Ince et alii, Fracture and Structural Integrity, 75 (20YY) 435-462; DOI: 10.3221/IGF-ESIS.75.30 
 

461 
 

straightforward compared to the J-integral method. In the three-dimensional applications conducted in this study, 
the results of the CCI technique were satisfactorily consistent with the J-integral approach. So much so that even 
solutions with eight-node solid elements yielded sufficiently satisfactory results. Eight-node solid elements are the 
only element type used in commercial programs such as SAP2000, which is commonly used in civil engineering. 
 

 
ACKNOWLEDGMENTS 
 

his study was carried out within the scope of the Firat University Scientific Research Projects program (FUBAP) 
with the Research Project number of MF.24.08. We gratefully acknowledge the financial support provided by 
FUBAP. 

 
 
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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
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      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice