Mode I fracture toughness determination with straight notched disk bending method

International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

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Mode I fracture toughness determination with straight notched disk bending method Levent Tutluoglu n, Cigdem Keles Department of Mining Engineering, Middle East Technical University, Cankaya-Ankara 06800, Turkey

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 July 2010 Received in revised form 13 September 2011 Accepted 27 September 2011 Available online 28 October 2011

A new method called the straight notched disk bending method is developed for mode I fracture toughness determination using rock cores. Disk specimens of andesite and marble having a single straight edge notch were subjected to three-point bending loads. Dimensionless stress intensity factor estimations and fracture toughness tests were conducted for different notch lengths, span lengths, thicknesses and diameters of the cylindrical rock specimens. Stress intensity factors were computed by three-dimensional finite element modeling and the results were presented for a wide range of specimen geometrical parameters. Results of experiments were compared to the results of well-known mode I fracture toughness testing methods. For specimens having thickness equal to the radius, mode I fracture toughness was lower and close to the results obtained by semi-circular bending method. When thickness was increased and doubled, mode I fracture toughness increased and approached to the value found by the suggested cracked chevron notched Brazilian disk method. Advantages of the new method included easy specimen preparation and testing procedure, stiffer specimen geometry, smaller fracture process zone, and flexibility of the specimen geometry for the investigation of the size effect behavior. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Fracture mechanics Finite element analysis Stress intensity factor Toughness testing Rock

1. Introduction Rock fracture mechanics deals with the investigation of initiation and propagation of cracks in geological materials with fields of applications such as hydraulic fracturing, rock slope analysis, earthquake mechanics, blasting and rock fragmentation, and in many other practical problems in earth sciences. Fracture toughness is the resistance of a material to crack initiation and propagation. Since rock is usually a brittle material, it is weak under tension. Therefore, KIc, which is the fracture toughness under mode I loading (opening mode), is an important material property for fracture of rocks. Mode I loading condition in rock specimens can be generated by direct tensile loading, by three or four point bending, and by Brazilian type compressive loading. Due to the practical difficulties, direct tensile load application on rock specimens is not a common practice in fracture toughness tests. The short rod (SR) method [1] is one of the suggested methods by International Society for Rock Mechanics (ISRM) [2]. A tensile load is directly applied perpendicular to the initial chevron notch plane in SR method. This may result in bonding failures at the specimen-loading platen contacts, especially for hard rock types. Examples of direct tensile load application for KIc

n

Corresponding author. Tel.: þ90 312 210 5815; fax: þ90 312 210 5822. E-mail address: [email protected] (L. Tutluoglu).

1365-1609/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2011.09.019

determination can be found in [3] where compact tension type specimens of a granite rock were tested; the results were used by [4] in the verification of size effect law for rock-like materials. Specimen geometries under three-point bending or four-point bending and related testing techniques are attractive for KIc determination due to the easiness of specimen preparation and simplicity of testing configurations. For comparison of fracture toughness test results, wide availability of results with these geometries and testing techniques is another advantage of these methods. For cylindrical rock core specimens, common methods applying three-point bending to determine KIc include straight edge cracked round bar bend (SECRBB) method [5,6], semicircular bending (SCB) method [7,8], chevron bend (CB) test [2], and chevron notched semi-circular bending method [9]. For Brazilian type compressive loading of rock disks, various methods were proposed for KIc determination. Cracked straight through Brazilian disk (CSTBD) method [10], diametric compression test [11], cracked chevron notched Brazilian disk (CCNBD) method [12], modified ring test [13], Brazilian disk test [14], flattened Brazilian disk method [15], and hole-cracked flattened Brazilian disk method [16] are some of the methods used for fracture testing of rock cores under compressive upper and lower boundary loads. Among these methods, CCNBD method is one of the suggested methods of ISRM [17] for fracture toughness testing on rocks. Some methods such as SCB and SECRBB with straight edgenotched specimen geometry [18–21] under three-point bending

L. Tutluoglu, C. Keles / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

were reported to yield KIc values lower than the suggested methods by ISRM. Suggested methods by ISRM involve SR method [1,2], CB test [2], and CCNBD method [17]. Preparation of chevron-notched specimens requires special care in machining the specimens and chevron notches with desired precision and alignment considering the requirements regarding the geometrical constraints. Due to the plane strain assumption constraint in the development of CCNBD formulation to estimate mode I stress intensity factor (KI), typical CCNBD specimen geometry is subjected to some dimensional limitations. Limitations related to the two-dimensional (2D) estimations of KI were presented in [22]. Applying the geometrical constraints for CCNBD method, tip of the sharp chevron notch is positioned close to the upper free boundary and compressive concentrated load application point at the onset of unstable crack propagation point. These are the two high stress gradient regions. Crack tip at this point is assumed to be in plane strain condition for which computation of KIc is conducted. With CB specimens of different diameter values (D), importance of specimen size on KIc value was emphasized by [23]. KIc value increased about 20% with increasing specimen size with D values ranging from 32 to 76 mm. This is an indication of a possible specimen size effect in fracture toughness tests. KIc value found from CCNBD tests was reported to be lower compared to the results of tests with CB specimen geometry and related testing method in [24]. In [23,24], 1/OR term (R is the specimen radius) was proposed to be used in the basic formula for KIc determination with CCNBD method, instead of 1/OD. If 1/OD term was used in the formula for KIc determination the results would be 30% lower. This difference was attributed to the use of 1/OD term in the basic formula in the suggested version of CCNBD. In [25], mode I fracture toughness tests were conducted on SR and CCNBD core specimens. According to the comparisons of [25], for larger 68 and 74 mm D specimens, KIc values of CCNBD tests were approximately 8–13% lower than SR test results and 19–27% lower than the results of SR tests applying nonlinearity calibration. For smaller D groups like 50 mm D group, CCNBD tests produced very low KIc, (4.7 times lower KIc values compared to SR test results). Indications of specimen size effect were observed to exist in KIc determination compared to the other SR method results. Among the SR tests in [25], comparing the average KIc values of the smallest 50 mm D and the largest 74 mm D groups, KIc value for 50 mm D group was approximately 1.2 times lower than KIc value for 74 mm D group. In tests with rectangular concrete beams of different sizes, boundary influence factor was reported to affect KIc results significantly. [26] modeled specimen boundary influence, which may cause a size effect on specific fracture energy of concrete, and stated that boundary influence can be neglected only if a specimen is very large so that boundary region is only a small portion of the total fracture area. A fully developed fracture process zone (FPZ) in concrete can be quite large, e.g. around 50 mm or larger depending on the maximum aggregate size as discussed in [26,27]. This situation will be governed by the grain size in case of rocks. For relatively large specimens, the fully developed FPZ will maintain its size and shape during crack growth with a constant fracture energy distribution along the crack path. Experiments in [26] were carried out on edge-notched three-point-bend concrete beams with depths between 140–300 mm and crack-to-beam depth ratios between 0.2–0.6. Boundary influence decreased with increasing specimen size (beam depth), and fracture energy increased and eventually approached to the size-independent fracture energy. Depending on the specimen size and initial notch length/beam depth ratio (a/W), fracture energy values found were

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about three times lower for low-depth beams with longer notches. The true size-independent fracture energy was approached with a/W ratio of 0.1 for relatively large 300 mm deep rectangular concrete beam specimens. Another factor affecting the results of KIc determination is the estimation of KI values with linear beam material behavior assumption, which is often the case to estimate the KI of beam type specimen geometries with numerical modeling approach. Results of a linear analysis can be different from a nonlinear analysis with a larger beam deflection [28]. For rock fracture mechanics testing, large size beams like rectangular concrete beams in structural applications are not usually available for testing. Following a typical site investigation work in rock engineering, cylindrical core specimens of limited diameters are readily available for testing. It is still not well-known which method produces results close to the true fracture toughness, since there are different suggested methods for the determination of one single parameter KIc. Preparation of straight edge-notched disk type specimen geometry is relatively easy and less time consuming. Considering all these, a specimen geometry with a straight edge notch machined in a core disk specimen is proposed. This geometry is to be loaded under three-point bending to determine KIc of brittle rock-type materials. The proposed three-dimensional (3D) circular plate type specimen geometry is inherently stiffer than regular beam type specimen geometries. By adjusting the thickness of the disk specimens, size effect investigations can be extended to larger size specimens with this new specimen geometry. Specimen geometry and bending load application configuration for the new method, called straight notched disk bending (SNDB) method are illustrated in Fig. 1. In this method, a cylindrical plate type disk specimen is under three point bending load: one of the three rollers is used for load application along the central line of the upper boundary, and the other two rollers at the bottom boundary are used for supporting the circular plate type disk specimen. The SNDB specimen is a circular disk or cylindrical plate with a straight edge notch of length a machined perpendicular to the bottom surface of the disk throughout the diameter D all along the bottom boundary. Upper loading roller is in line with the initial notch machined at the bottom end of the disk. Support rollers at the bottom are positioned around the saw-cut notch with equal distances controlled by the desired span. Distance between the bottom support rollers is called span length and denoted with 2S. Vertical distance between the contact point of the upper loading roller and the bottom boundary surface where supporting rollers act is equal to the specimen or circular plate thickness t. Cylindrical plate thickness, and thus the size of the disk specimens can easily be changed by adjusting the disk thickness for desired specimen sizes.

Fig. 1. Straight notched disk bending (SNDB) specimen geometry and loading configuration.

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In a typical test, applied vertical load P over the top and bottom rollers is incremented till an unstable crack formation from the machined notch is initiated. This load is recorded as Pcr and is used for the computation of the mode I or opening mode fracture toughness of the rock specimen tested. Two different rock types, Ankara andesite and Afyon marble were used in the experimental work. Specimen dimensions and loading configuration can be adjusted by changing D, t, a and 2S of the specimens. SNDB method and the specimens used for this method involve 3D circular plate type core specimen geometry. 3D mode I stress intensity factor computations are required for this circular plate type geometry with different D, t, a, and 2S values. A finite element program named ABAQUS was used for stress intensity factor computations of SNDB specimen geometries of the method proposed here. Since the models are 3D disks under three-point bending, 3D specimen geometry models were generated and analyzed in the modeling work for KI estimation of SNDB method specimen geometries.

2. Modeling work for KI computation KI value computations were done by using ABAQUS finite element program. In the program, to evaluate stress intensity factor around the crack tip, a contour integral region is defined and KI values in this region are calculated by using J-Integral method. In fracture modeling, crack tips are regions of high stress gradients and high stress concentrations, and these concentrations result in theoretically infinite stresses at the crack tip. Therefore, to get accurate stresses and strains around the crack tip, finite element mesh must be refined in the vicinity of the crack tip. The final KI value at the crack tip is determined by averaging the KI values computed for a user-specified number of crack tip concentric mesh rings in the contour integral region. In order to take the advantage of specimen symmetry and symmetric boundary conditions in the modeling work, half specimen models of SNDB method were generated for all specimen geometries. For a more accurate simulation of the actual test loading conditions, the concentrated load P (P/2 for half models) was applied at a reference loading point over the top roller, which is defined as an analytical arc shaped rigid shell part in the models. Load application reference points in ABAQUS modeling process are used to transfer and distribute the applied loads over the contact parts such as the circular support rollers used in this modeling work. Applied load is transferred to the specimen through this analytical arc shaped rigid shell element. One of the support rollers at the bottom of three-point bending load configuration was again introduced to the half specimen model as an analytical arc shaped rigid shell element. Reference points were fixed against all rotation and displacement components. Analytical arc shaped rigid shell parts were fixed against rotations around all three coordinate axes. Displacement components of the rigid shell elements were fixed in all directions except the y-direction. Diameter of the support roller in the models was equal to the diameter of the circular support rollers used in the experiments. This diameter was 10 mm for each roller. To define the interaction between solid (specimen geometry model) and shell (support and loading rollers) contact pairs, a friction coefficient of 0.4 was assigned to the contact surfaces between the parts. For the specimen geometry model part, elastic modulus and Poisson’s ratio entries were introduced to the models as 12,334 MPa and 0.15, respectively, for andesite rock. These values are from the results of mechanical property determination tests on andesite rock material used in the experimental work. Results

of a number of model sensitivity analyses showed that neither variation in friction coefficient nor in material properties significantly affected KI value determinations. Therefore, separate model runs with different mechanical properties of marble were decided to be unnecessary. 2.1. Verification and assessment of accuracy for KI computations Before analyzing SNDB specimen model geometries in detail, the two well-known three-point bending specimen geometries, which are geometries of single edge notched bending (SENB) and SCB testing methods were modeled in order to assess the accuracy and applicability of the stress intensity factor computations with ABAQUS program and modeling procedures in this work. SENB and SCB methods and geometries were selected considering the similarities of load application configuration, boundary conditions, and relative crack position and geometry. Widely accepted analytically and numerically computed KI solutions are available for these two methods. KI values of both SENB and SCB testing method geometries can be determined by 2D plane strain modeling; however, SNDB model geometries are to be modeled as 3D, owing to the fact that basic geometry in this case is a circular plate type. Therefore, specimen geometries in the examples for verification and accuracy checks were modeled as 3D. Fig. 2 illustrates the typical mesh patterns generated for simulating the specimen geometries of SENB, SCB, and SNDB testing methods. For all three specimen model geometries, circular mesh region around the immediate notch front is defined as the contour integral region, and mesh is generated with concentric rings of special crack tip elements. For the remaining parts of the model geometries, ABAQUS continuum-brick elements coded as C3D8R and continuum-wedge elements coded as C3D6 were used in constructing the mesh for all specimen geometry models. Average KI value computed in the contour integral region increases as the specified contour integral region radius and the number of concentric rings increase. Reaching a convergence point, increasing contour integral region radius or number of rings do not further affect the quality of the average KI determination. After this point, average KI results converge to almost constant values. These values are accepted as KI values of the different specimen geometries considered. For the three specimen model geometries considered here, mesh was intensified at the crack front along the extent of the potential crack propagation paths. This path is governed by the beam depth W in case of SENB specimen geometry, by the radius R in case of SCB specimen geometry, and the disk or plate thickness t in case of SNDB specimen models. Proportioning and mesh intensity adjustments for contour integral region were applied in dimensions perpendicular to the initial notch planes for the three specimen model geometries. Related dimensional variables for this case are half-specimen length L/2 for SENB model geometry with a beam length L, and radii R for SCB and SNDB model geometries. In the modeling work, contour integral region radius was set to 3.75 mm for 75 mm diameter SCB and SNDB disk specimen geometries. Contour integral region radius over specimen radius R (for SCB and SNDB specimens), or contour integral region radius over half beam length L/2 (for SENB specimens) ratios were set to 0.1 for all specimen geometries. Within an acceptable tolerance, average KI values for 3D models of all specimen geometries considered were confirmed to converge to constant values with fourteen concentric rings of crack front mesh elements and a contour integral region of radius of 3.75 mm. These values were accepted as KI values of geometries modeled.

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Fig. 2. Refined mesh around crack tips of different three-point bend model geometries.

An example plot related to the convergence work is presented in Fig. 3 for a SENB modeling case; number of concentric rings is indicated in parentheses at model computation points with increasing size of the contour integral region. For mode I stress intensity factor computations, convergence trend of KI to a constant value in this figure with increasing size of the contour integral region radius and increase in the number of concentric element rings was typically observed for all SNDB models, and for the two calibration methods considered in the modeling work.

2.1.1. SENB specimen SENB specimen is an American Society for Testing and Materials E 399 standard fracture toughness testing specimen geometry. SENB specimen is a rectangular beam with a single edge notch under three-point bending load. Specimen geometry and loading configuration of the specimen are illustrated in Fig. 4. For KI computation of the SENB method, the following equation was developed by [29]. pffiffiffiffiffiffi pffiffiffi KIB W 3ð2S=WÞ a½1:99að1aÞð2:153:93a þ 2:7a2 Þ ¼ ð1Þ P 2ð1 þ2aÞð1aÞ1:5 where B, W, P, 2S, and a are beam specimen thickness, beam specimen depth, applied load, span length, and the dimensionless notch length (a ¼a/W), respectively. The formula was for a beam span of 2S¼4 W. Dimensionless stress intensity factor (FB) for SENB method was taken to be equal to FB ¼(1 a)1.5KIBW/(POa) in [29]. The dimensions used in the analysis of SENB specimen models are listed in Table 1. For 3D square plate type models of this geometry, average number of nodes and average number of elements were 15,888781 and 14,0437127, respectively, for a ¼0.20.7. With a unit 1 N concentrated load application on 3D model geometries, KI values were first determined directly in units of PaOm, and in order to compare the results with results of [29,30], KI values were normalized to a form in terms of a dimensionless stress intensity factor FB. FB values, and differences between the results of the 3D modeling work here and the results of [29,30] are listed in Table 2. According to [31], KI estimation within 5% is sufficient for KIc computations using this geometry. From Table 2, it is seen that difference between the numerically computed FB values in this work and the FB values from [29,30] stays below 5%. In fact, reason for this difference can be attributed to the fact that

Fig. 3. Stress intensity factor variation depending on contour integral region radius and number of rings.

Fig. 4. Single edge notched bending (SENB) specimen geometry and loading configuration.

original solutions are for plane strain condition, whereas 3D model geometry here is of a square plate type.

2.1.2. SCB specimen SCB method is one of the well-known and extensively used methods in fracture toughness testing of rock core type

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Table 1 Geometrical parameters of the 3D SENB specimen models.

Table 3 Geometrical parameters of the SCB specimen models.

Description

Values

Description

Values

Specimen depth or width, W (mm) Specimen thickness, B (mm) Specimen length, L (mm) Span length, 2S (mm) Notch length, a (mm) 2S/W a/W

15 75 75 60 3.0, 4.5, 6.0, 7.5, 9.0, 10.5 4 0.2, 0.3, 0.4, 0.5, 0.6, 0.7

Specimen radius, R (mm) Specimen thickness, B (mm) Span length, 2S (mm) Notch length, a (mm) S/R a/R

37.5 37.5 37.5, 60.0 3.750, 11.250, 18.750, 25.125, 30.000 0.5, 0.8 0.10, 0.30, 0.50, 0.67, 0.80

Table 2 Comparison of dimensionless FB values computed by the present work, and computed by [29,30].

a

0.2 0.3 0.4 0.5 0.6 0.7

FB

Table 4 Comparison of YI(SCB) values calculated by the present work and [8]. a/R

Difference (%)

Present work

[29]

[30]

Present work & [29]

Present work & [30]

7.244 6.309 5.630 5.130 4.757 4.417

7.519 6.506 5.825 5.325 4.927 4.596

7.513 6.518 5.834 5.317 4.919 4.602

3.65 3.03 3.35 3.65 3.44 3.90

3.58 3.21 3.50 3.51 3.28 4.03

Fig. 5. Semi-circular bending (SCB) specimen geometry and loading configuration.

0.1 0.3 0.5 0.67 0.8

Difference (%)

YI(SCB) Present work

[8]

2.724 2.538 3.550 6.209 12.665

2.760 2.573 3.603 6.315 12.930

1.29 1.36 1.48 1.67 2.05

Fig. 6. Variation of dimensionless stress intensity factors for SCB specimen geometries; YI(SCB) depending on a/R and S/R ratios.

ð2Þ

ratios of 0.5 and 0.8. In the modeling work for KI estimation of SCB geometry, KI values were first computed by 3D modeling of the SCB geometry, and then YI(SCB) values were computed using Eq. (2). For comparing KI results of 3D modeling here and the results of [8], Table 4 was prepared for a 75 mm diameter SCB specimen model geometries with initial notch lengths a/R¼0.10.8, and S/R¼0.5. Thickness of SCB model geometries was kept constant at B¼37.5 mm. Comparison of the overall results is illustrated in Fig. 6 for generalization of the verification efforts related to SCB geometry. YI(SCB) values estimated by 3D modeling work here were quite close to the YI(SCB) values computed by [8].

where YI(SCB) is dimensionless mode I stress intensity factor for SCB geometry, KI is mode I stress intensity factor, a is notch length; stress acting over the notch front is s0 ¼P/(2RB) where P, R, and B are applied load, specimen radius and specimen thickness, respectively. 3D model dimensions used in the analyses of SCB specimen geometry are listed in Table 3. Average number of nodes and elements used in 3D modeling of SCB geometry was 13,73471206 and 12,14371147 for a/R ratios between 0.1 and 0.8, and for S/R

2.1.3. General assessment of accuracy of the KI estimation by 3D modeling KI results from 3D modeling in this investigation were found to be in good agreement with the well-known KI solutions for both verification problems with beam type specimen geometries under three-point bending. The difference between KI results of the widely accepted solutions and the results in this work did not

specimens. SCB specimen is a semi-circular specimen with a single edge notch under three-point bending. SCB method specimen geometry and loading configuration are illustrated in Fig. 5. For KI computation of the SCB method, finite element method with quarter point displacement technique was used by [8]. KI values for different specimen geometrical parameters and loading configuration parameters were presented by [8] in a dimensionless form represented by YI. Dimensionless stress intensity factor YI in [8] is Y IðSCBÞ ¼

KI pffiffiffiffiffiffi

s0 pa

L. Tutluoglu, C. Keles / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

exceed 5% for 3D square plate SENB geometry models, and stayed below 2% for SCB geometry models with a/R ratios less than 0.67. In all, it is decided that 3D modeling procedure with ABAQUS program in this work yields acceptably consistent results for 3D fracture propagation modeling, and modeling results can be used for estimation of mode I stress intensity factors of SNDB specimen models with various geometrical parameters. 2.2. KI estimation for SNDB specimen SNDB method is introduced by [32] for KIc determination. SNDB specimen is a circular plate type disk geometry with a single straight edge notch under three-point bending load. To present 3D modeling results of KI estimations in dimensionless forms for specimens with different initial crack length/plate depth (a/t) and half-span/radius (S/R) ratios, mode I stress intensity factor KI is normalized by square root of the initial crack length a and the tensile stress s0 acting over the region involving the crack propagation path. By adopting the same normalization form, it is possible to compare KI values estimated by 3D modeling of SNDB geometry to the KI results of widely accepted SCB geometry given in [8]. YI is a dimensionless function that depends on the size and geometry of the initial notch, size, and geometry of the structural component, and the load application configuration. For estimation of dimensionless stress intensity factor YI(SNDB) of edge-notched circular plate type SNDB specimen geometry, KI values were normalized using the equation below: Y IðSNDBÞ ¼

KI pffiffiffiffiffiffi

s0 pa

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Table 5 Geometrical parameters of the SNDB specimen models. Description

Values

Specimen diameter, D (mm) Specimen thickness, t (mm) Notch length, a (mm) Span length, 2S (mm) t/R S/R a/t

75 18.75, 37.50, 56.25, 75.00, 93.75, 112.50 1.875–101.25 30.0, 37.5, 45.0, 52.5, 60.0 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 0.5, 0.6, 0.7, 0.8 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

Table 6 Average numbers of nodes and elements with various t of SNDB specimen model geometries. t (mm)

t/R

Nodes

Elements

18.75 37.50 56.25 75.00 93.75 112.50

0.5 1.0 1.5 2.0 2.5 3.0

20,0907 634 33,768 7 1100 37,101 7 1116 37,892 7 1472 44,768 7 1660 51,970 7 1991

17,902 7 615 30,596 7 1044 33,387 7 1075 34,739 7 1405 41,162 7 1569 47,909 7 1861

ð3Þ

where YI(SNDB) is dimensionless mode I stress intensity factor of SNDB geometry, s0 is the tensile stress over the crack propagation path, and a is the initial crack length of the model geometry. Indirectly generated tensile stress s0 by three-point bending acts perpendicular to the initial crack plane, and it causes crack propagation in mode I along the crack front. There are similarities in stress distributions along the beam depth (W for SENB and R for SCB) of beam type geometries, and depth or thickness (t) of plate type SNDB geometry. These specimen geometries are subjected to similar deflections due to three-point bending load applications. Upper part of the beam or plate geometry is under a compressive extreme fiber stress and the lower part is under a tensile extreme fiber stress. At the intersection of initial notch and lower boundary, there is an extreme tensile fiber stress directly proportional to the applied load P. However, towards the neutral axis of beam or plate, stress perpendicular to the region over the notch approaches zero. This means that, stress perpendicular to the crack plane at the lower boundary starts as tension proportional to the applied load P, and towards the neutral axis of the specimen it tends to zero. Then it becomes compressive, and maximum compressive extreme fiber stress is right under the upper boundary load P application point. It should be noted that neutral axis position changes and it moves up with increasing a. Assuming a linear distribution of load and stress perpendicular to the initial crack plane and its associated crack front along which a possible yield or fracture process zone is expected to develop, average tensile stress s0 will be proportional to average load P/2 in the lower part of the beam or plate geometry. Area perpendicular to the load P/2 along this section is equal to the specimen diameter multiplied with the specimen thickness (Dt) for the SNDB specimen geometry. Thus, tensile stress acting over the part involving crack plane and crack front is expressed as s0 ¼P/(2Dt). The dimensions used for models of SNDB specimens are listed in Table 5. D used in the KI analysis was constant, and number of nodes and elements was adjusted depending upon the variation

Fig. 7. Variation of dimensionless stress intensity factors for SNDB specimen geometries; YI(SNDB) depending on a/t and S/R ratios.

of t in the disk type specimen models. Average number of nodes and elements (with their standard deviations) used in 3D finite element modeling of SNDB geometry with various t is tabulated in Table 6. Similar to the KI and YI computations of SCB geometries, KI values for SNDB geometries were first computed in units of PaOm for an applied load of 1 N by using ABAQUS program, then YI(SNDB) values were determined using Eq. (3). Modeling results show that similar trends are observed for YI(SCB) and YI(SNDB) variations with dimensionless crack lengths (ratios a/R for SCB specimen geometries and ratios a/t for SNDB specimen geometries). The relationship between YI(SNDB) and a/t for specimens with t/R¼1 and S/R¼ {0.5, 0.6, 0.7, 0.8} is illustrated in Fig. 7. As observed in YI(SCB) results, for shorter a values (a/to 0.5) rate of change of YI(SNDB) is small and a gradual increase in dimensionless stress intensity factor is observed for a/t ratios between 0.3 and 0.5. For larger crack length ratios (a/t 40.5), YI(SNDB) increases rapidly. Similar YI(SNDB) versus a/t trends were found for specimen geometries with different thicknesses. For SNDB specimen geometries with different t/R ratios, linear relationship between YI(SNDB) values and S/R ratios were identified

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as in Fig. 8. As the specimen thickness increased, lines representing the variation of YI(SNDB) with S/R approached each other forming a group like behavior. For thicker specimens (t/R42.0), if the S/R ratio was decreased below 0.4, YI(SNDB) was observed to take negative values. Fig. 8 is for specimen geometries with a/t ¼0.5. For specimens having smaller a values (a/t o0.5), YI(SNDB) was again observed to take negative values when S/R o0.4. For possible future applications of SNDB testing method, two options will be analyzed to present the results of YI(SNDB) estimation work. First linear relationships observed in Fig. 8 for variation of YI(SNDB) with S/R will be fitted by lines, and the YI(SNDB) will be expressed as   S þn ð4Þ Y IðSNDBÞ ¼ m R

where slope is m and intercept is n for the fitted lines with R2 ¼1. Both m and n parameters depend on t/R and a/t ratios. The m and n parameters for various t/R and a/t ratios are tabulated in Table 7. Eq. (4) is recommended to be used for S/R ratios between 0.5 and 0.8. Second way to present the YI(SNDB) results is based on the variation of YI(SNDB) with t/R and a/t ratios as shown in Fig. 9. Fig. 9 is for specimen models of various a/t ratios of 0.3, 0.5, and 0.7 for a fixed ratio of S/R¼0.5. Regression analyses by fitting the results with polynomial functions in Fig. 9 result in R2 values being equal to one. A fifth order polynomial function of the form  5  4  3  2   t t t t t þC 6 ð5Þ Y IðSNDBÞ ¼ C 1 þC 2 þ C3 þ C4 þC 5 R R R R R is proposed in terms of various t/R ratios. Coefficients Ci of the polynomial fits are tabulated in Table 8. In this table, Ci values are presented for specimens having 0.5 rt/Rr3.0, 0.1 ra/t r0.9, and S/R¼0.5 and 0.8. For the other S/R ratios in the range 0.5–0.8,

Fig. 8. Variation of dimensionless stress intensity factors for SNDB specimen geometries; YI(SNDB) depending on S/R and t/R ratios.

Table 7 m and n parameters used for YI(SNDB) estimation. a/t

Fig. 9. Dimensionless stress intensity factor values of SNDB specimen geometries; YI(SNDB) depending on t/R and a/t ratios.

t/R 0.5

1.0

Table 8 Coefficients Ci used for YI(SNDB) estimation based on polynomial fits.

1.5

m

n

m

n

m

n

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

11.8930 12.0560 12.9210 14.5360 17.2960 22.0450 31.2320 52.5460 128.5000

 0.5768  0.7241  0.8442  0.9225  0.9789  1.0140  1.0969  1.3484  1.4060

5.2801 5.8903 6.5884 7.4728 8.8301 11.1730 15.8170 26.8770 68.6170

0.1421  0.4229  0.7633  0.9269  0.9966  1.0349  1.0957  1.2154  1.6095

4.8564 5.6252 5.8341 5.9777 6.4837 7.7662 10.6776 18.0195 46.7509

 0.4053  1.2537  1.4529  1.3607  1.2188  1.1242  1.1150  1.2300  1.6090

a/t

t/R 2.0 m

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

6.3261 6.7919 6.2760 5.7878 5.7162 6.3206 8.2422 13.5912 35.2833

2.5

3.0

n

m

n

m

n

 1.4995  2.3512  2.1945  1.8189  1.4915  1.2681  1.1777  1.2544  1.6939

8.1898 8.0491 6.9087 6.0466 5.6262 5.7759 6.9850 11.0152 28.2950

 2.5918  3.2258  2.7492  2.2277  1.8070  1.4846  1.2851  1.2776  1.6593

10.0567 9.0858 7.5028 6.4645 5.8488 5.6792 6.3506 9.3806 23.5940

 3.5972  3.8323  3.1345  2.5765  2.1323  1.7473  1.4429  1.3263  1.6242

a/t

S/R ¼0.5 C1

C2

C3

C4

C5

C6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

 0.2973  0.2917  0.2817  0.3062  0.3687  0.4919  0.6817  1.0547  2.6165

3.0347 2.9656 2.9118 3.2130 3.8845 5.1508 7.1378 11.0768 27.1915

 12.2232  11.8959  11.9440  13.4018  16.2597  21.4151  29.6798  46.2822  112.2499

24.6765 24.0699 24.8580 28.2971 34.3703 44.9179 62.3175 97.9476 234.9871

 25.9180  25.8703  27.4930  31.5108  38.1777  49.5204  68.9981  110.0070  261.8348

13.5145 13.5395 14.4730 16.5137 19.9682 25.9103 36.7181 60.5449 147.5423

a/t

S/R ¼0.8 C1

C2

C3

C4

C5

C6

 0.4323  0.4598  0.4897  0.5328  0.6166  0.7987  1.0882  1.7081  4.0531

4.6073 4.8526 5.1462 5.5929 6.4705 8.3406 11.3970 17.9442 42.3025

 19.6111  20.3864  21.4859  23.3203  26.9765  34.5795  47.4006  74.9644  175.4977

42.2449 43.1755 45.2005 49.0658 56.8168 72.4049 99.6026 158.5258 368.9376

 45.9929  46.5766  48.8896  53.5653  62.5400  79.6224  110.3646  177.7056  413.9705

23.5560 23.6807 25.0206 27.8074 32.9119 42.1584 59.4125 98.1946 235.5625

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

L. Tutluoglu, C. Keles / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

interpolation can be used to estimate YI(SNDB) of a particular geometry. Due to the boundary influence issue in the beam type specimen geometries under three-point bending, longer a values over a/t¼0.7 are not recommended practically.

3. Experimental work Mode I fracture toughness tests were performed on Ankara andesite and Afyon marble. Material properties of the rocks are tabulated in Table 9. Both materials are used in construction of pavements, roadways, stairs, etc. Therefore, it is important to determine KIc values of these rocks to have an idea about their resistance to crack initiation and propagation. Andesite is an igneous rock whereas marble is a metamorphic one. This difference gives us a chance to examine the variation of KIc value according to the rock types with the new testing method. Before testing the rocks with the SNDB method, KIc values of the rocks were determined with the two well-known methods, which were SCB and CCNBD methods. SCB method was the first method considered for KIc comparisons of existing methods and SNDB method, since the load application configuration and initial notch type of the specimens were similar for both methods. Then, CCNBD method was the next choice for comparison work, because of its simplicity in specimen preparation and testing procedure [17,22] compared to the other suggested methods by ISRM in [2]. For SCB and SNDB testing methods applied in this work, core specimens were loaded in a displacement controlled way with a rate of 0.005 mm/s and for CCNBD method tests, loading rate was equal to 0.002 mm/s; time elapsed till final fracturing is about a couple of minutes with these rates. 3.1. Preparation of notches YI estimations of SCB and SNDB specimen models are based on zero-thickness cracks in modeling work. Both SCB and SNDB method specimen geometries used in experimental work include initial saw-cut notches with finite widths and rounded notch tips. Controlling the roundness of the notch tip precisely may play important role in fracture toughness determination of very fine grained materials, such as glass and some ceramics with grain sizes measured in orders of magnitudes of mm. For rock-like materials with different mineral grains and binding matrices, size of grains and binding matrices shows large variations and reaches magnitudes in orders of mm. Notch tip sharpness factor on KIc determination is not expected to be as significant for such materials. In fact, a rounded notch tip may be advantageous in the sense that crack follows a path, which is the best representative path of the internal structure of the material. For rock-like materials, the important factor on KIc determination is to increase the statistical quality of the measurements by increasing the number of specimens used in test programs. At this point, there are three aspects that need to be discussed regarding the precracking, sharpness of the notch tip, and width of the machined notches for SCB and SNDB geometries. Table 9 Material properties of rock types used. Material property

Andesite

Marble

Elastic modulus (MPa) Poisson’s ratio Uniaxial compressive strength (MPa) Tensile strength (MPa)

12,334 7 135 0.157 0.01 82.84 7 4.14 7.007 0.67

34,294 7459 0.12 70.02 52.32 71.56 5.13 70.32

1255

According to [3] fatigue pre-cracking strongly effected specimen compliance used for fracture testing. Use of a very sharp crack with fatigue pre-cracking in [33] was decided to be unnecessary for SCB test specimens, considering the brittleness of rocks and possible uncontrolled early damage along the crack front. With similar arguments, [34] avoided fatigue pre-cracking in CSTBD specimens of soft friable sandstone. In fact, in [35] no significant differences were observed in fracture toughness results with the use of fatigue pre-cracking in a series of tests involving notch tip radii changing from 0.1 to 0.8 mm. In [36], the characteristic asymptotic behavior of crack tip stress fields at the tip of sharp and rounded semi-infinite cracks and notches were compared with corresponding behavior of selected finite bodies with finite size edge cracks and rounded notches. Stress intensity factor solutions with sharp crack and with rounded notch tip cases converge remotely but diverge as apex of crack or notch is approached for semi-infinite geometrical conditions. Closeness of the free boundaries to the tip may reduce the size of the process zone and change the nature of the expected asymptotic behavior of the crack or notch tip stress field. Provided that free boundaries of specimen geometries are sufficiently far from the crack or notch tip region, and a sufficiently large FPZ develops compared to the initial notch length and the radius of the saw-cut notch tip, stress intensity factor solutions for a notch and a sharp crack will not differ significantly. Rounding or sharpening of a notch tip locally changes the nature of the problem only around the crack or notch tip from one of a stress intensification (sharp tip) to stress concentration (rounded tip). Monitoring acoustic emission (AE) events and matching the results with numerical estimations, [37] investigated the FPZ size for SCB tests on asphalt mixtures. Initial notch width was around 2 mm, and notch lengths were changed between 5 and 30 mm for 150 mm diameter specimen geometries tested. Relatively large FPZ lengths were detected compared to the initial notch lengths applied. FPZ length covered almost 50% of the distance between notch tip and the upper free boundary where a concentrated load is applied parallel to the notch plane. Using real-time imaging and AE techniques, and direct optical analysis, existence of a significant FPZ surrounding the propagating crack was confirmed by [38] with CCNBD tests on granitic rock types. Size of FPZ was reported to be around 20 mm from the results of three-pointbend fracture tests on rectangular beam specimen geometries of high-strength concrete material in [39]. Based on this review on FPZ size, it can be concluded that a sufficiently large FPZ develops in notched-beam type fracture testing specimen geometries, and stress intensity factor solutions for a rounded notch and a sharp crack are not expected to differ significantly. It was concluded in [36] that FPZ size must be sufficiently small for the effects of the free boundaries of the specimen body to be small. This condition was investigated by [26] shown that, in order to achieve size-independent fracture toughness, a crack front with its fully developed FPZ in length must have enough frontal distance to propagate. This distance was proposed to be over 90% of the total fracture area until crack front was subjected to free boundary influence condition. If the initial FPZ length was too long covering about 60% of the distance between the initial notch tip and the upper free boundary zone, boundary influence dominated the results and fracture energy or KIc values were lower. A size-independent fracture toughness result could be reached with concrete beam specimens of beam depth approaching W¼ 300 mm with short initial notch lengths around a/W¼0.1. Considering the arguments above, fatigue pre-cracking and notch tip sharpness adjustments were not applied for initial notches of SCB and SNDB specimen geometries. Initial notch

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L. Tutluoglu, C. Keles / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

width was kept around 1 mm in the tests with these two specimen geometries. 3.2. SCB method tests SCB fracture toughness testing method is one of the simplest and extensively used methods in rock fracture toughness testing. This method was introduced for KIc testing by [7], and stress intensity factors in detail for various geometrical parameters of SCB specimens were presented in [8]. In SCB tests, 100 mm diameter core specimens were used for andesite. R and B of the andesite specimens were equal to 50 mm. Two different 2S values were applied as 60 and 70 mm with dimensionless ratio S/R varying between 0.6 and 0.7. a values were varied as 5, 10, 15, and 20 mm. For SCB tests on andesite, twenty one specimens were used. Diameters of the marble core specimens were 75 and 100 mm with constant 10 mm initial notch lengths for the two different diameters used. Total number of marble SCB specimens tested was seven. Five of the marble specimens had 75 mm D, and two of the specimens had 100 mm D. 2S values were applied as 45 mm (for D¼75 mm) and 60 mm (for D¼100 mm) with dimensionless ratio S/R¼0.6. For marble, 75 and 100 mm diameter SCB specimens had 37.5 and 50.0 mm radii (beam depths), respectively. Thickness B of the marble specimens was half of the D with a ratio B/R¼1. Initial notch length a was kept around 10 mm for marble specimens. Fracture toughness values were determined with the following expression: pffiffiffiffiffiffi K Ic ¼ Y IðSCBÞ scr pa ð6Þ where YI(SCB) is dimensionless stress intensity factor, and a is the initial notch length; critical stress at the critical load Pcr is scr ¼Pcr/(2RB), where Pcr, R, and B are load at fracture, specimen radius, and specimen thickness, respectively. Completing the bending type tests on SCB specimens and using YI(SCB) results of modeling work in Eq. (6), KIc values of andesite and marble were determined. No significant differences were observed among the results of tests with varying 2S and a. Therefore, test results were grouped together in computation of the average KIc results of each rock type. Average KIc was found as 0.9470.12 MPaOm for andesite with twenty one specimens and as 0.5670.06 MPaOm for marble with seven specimens. 3.3. CCNBD method tests CCNBD method was developed by [12] and became a suggested method of ISRM [17]. For CCNBD method, formulations for the estimation of KI values and determination of KIc with the use of 1/OR term in the basic formula were presented in [40]. This method was preferred among the other suggested methods, since it is the simplest suggested method in terms of the specimen preparation, test setup, and interpretation of the results. For CCNBD method tests, initial chevron notches were machined at the upper and lower boundaries of 125 mm diameter core disks by using a 110 mm circular diamond saw. Thickness of the disks was changed between 50 and 60 mm. To determine fracture load Pcr, disk specimens of andesite and marble were loaded under diametric compression till failure. In describing the CCNBD specimens, all the dimensions of the specimen geometry should be converted into dimensionless forms with respect to radius of the specimen. The parameters used in dimensionless forms are as follows (Fig. 10):

a0 ¼ a0 =R, a1 ¼ a1 =R, aB ¼ B=R, as ¼ Rs =R

ð7Þ

a0, a1, and Rs are geometric parameters related to the machining of chevron notch, and they are illustrated in Fig. 10. Valid ranges

Fig. 10. Cracked chevron notched Brazilian disk (CCNBD) specimen geometry and loading configuration [17].

and limitations were suggested for these parameters in [40]. The a0 values were between 0.056 and 0.225, the a1 values were between 0.728 and 0.782, and the aB values were between 0.742 and 0.948 for the andesite and marble CCNBD specimens tested in this work. With these ranges, requirements and conditions were satisfied in order to have a satisfactory plane strain state for practical fracture toughness measurements. In KIc computations of CCNBD specimens, the following formula was used [40]: P cr K Ic ¼ pffiffiffi Y nm B R

ð8Þ

where Y nm ¼ ueva1

ð9Þ

n

Y m is the critical dimensionless stress intensity factor at Pcr, which is the load at final fracture, B is the disk thickness, and R is the disk radius, u and v are geometrical constants determined based on geometrical parameters a0 and aB; u and v values are presented in tables of [17,22,40]. These constants have been updated recently in [41]. Using Eqs. (8) and (9) for evaluating the results of CCNBD method tests, KIc value of andesite was determined as 1.4570.06 MPaOm with five specimens, and KIc value of marble was obtained as 1.0970.12 MPaOm with four specimens. Using the constants updated in [41] for the computation of the dimensionless stress intensity factor of CCNBD specimen geometries, average KIc becomes 1.6270.07 MPaOm for andesite rock, and 1.2370.13 MPaOm for marble rock. 3.4. SNDB method tests The SNDB method was introduced in [32]. This new method was based on a circular plate type disk specimen under threepoint bending load application. Similar bending type of load application configuration is widely used for well-known mode I fracture toughness testing methods on beam type specimen geometries such as SENB, CB, and SCB method geometries. Mode I fracture toughness for SNDB method testing can be computed as pffiffiffiffiffiffi K Ic ¼ Y IðSNDBÞ scr pa ð10Þ where YI(SNDB) is the dimensionless stress intensity factor and a is the initial notch length for SNDB specimen geometries. Stress acting over the notch and crack front is expressed by scr ¼ Pcr/(2Dt), where Pcr, D, and t are load at fracture, specimen diameter, and thickness or plate depth, respectively. As presented before, YI(SNDB) values can be estimated using the results of 3D modeling work for SNDB specimen geometries.

L. Tutluoglu, C. Keles / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 1248–1261

To prepare straight edge-notched disk bending specimens, rock blocks were cored with two projected D values of approximately 100 and 75 mm. Using a rotary saw, core samples were sliced into disks, and to ensure parallel flat surfaces for disk specimens, surfaces were polished with a grinding machine. A straight edge notch of about 1 mm width was machined into the lower boundary of a specimen to the targeted length with a circular diamond saw. After an initial notch was machined, upper and lower boundary concentrated load application points were marked on the sample considering the targeted span length. Disk specimen was loaded under a three-point bending load configuration with an MTS 815 servo-controlled hydraulic testing machine. Loading was applied in a displacement controlled way with a rate of 0.005 mm/s. A typical test lasted about a couple of minutes until final fracture with this rate. Testing period varied depending on the specimen thickness and initial notch length. A series of SNDB method mode I fracture toughness tests were conducted to investigate the applicability of the proposed specimen geometry for KIc determination. In the experiments, effects of a, 2S, t, and D on KIc value were investigated. Effects of all these parameters on KIc determination were analyzed for Ankara andesite. Another rock type, namely Afyon marble was added to the experimental program to investigate the effect of plate depth or thickness t on KIc determination for a different rock type. In the first group of tests, D of the core specimens was set to 100 mm. a and 2S were changed to investigate the effects of these changes on KIc determination by using the SNDB method. t in this group was half of the D with ratio t/R¼ 1. a was varied as 5, 10, 15, and 20 mm resulting in dimensionless notch lengths between a/R¼ 0.1 and 0.4. Two different 2S were applied as 60 and 70 mm with dimensionless ratio S/R varying between 0.6 and 0.7. Twenty andesite specimens used in this group of tests. In the second group of tests, D and a of andesite and marble rock disks were set to 75 and 10 mm (a/R E0.27), respectively. The parameters t and 2S were varied for andesite specimens. After observing that the variation of 2S did not significantly affect the KIc determination for andesite, the 2S was fixed with S/R ratio¼ 0.6 and only thickness t of marble specimens was varied. Sixty four andesite specimens and twenty two marble specimens used in this group of tests.

4. Results and discussion Fracture toughness test results were analyzed and comparisons were made between the results of common test methods and SNDB testing method. Thickness of the SNDB specimen geometries can be adjusted, and thickness can be increased to up to the levels around the diameter of the disks. This way, a size effect investigation can be conducted by increasing the specimen thickness or circular plate depth. Results of this size effect investigation were analyzed, and compared to the KIc results obtained by well-known SCB and CCNBD testing methods. Stiffness values of SCB and SNDB geometries were compared in terms of slope of the load-crack mouth opening displacement (CMOD) data. Approximate yield zone sizes at the notch fronts of SCB and SNDB geometries were analyzed by applying a size and notch length dependent nominal strength to the stress distribution at the notch front. Results of FPZ size analyses were compared in terms of the estimated lengths of FPZ of both specimen geometries. 4.1. Fracture toughness tests Experimental and numerical data for the first group of SNDB tests on 100 mm diameter andesite disks are summarized in

1257

Table 10. In this table, initial notch lengths, span lengths, a/t and S/R ratios, YI values, individual Pcr and KIc results of the tests are presented. As in the case of SCB method test results, no significant differences in KIc results are observed with changing a/t and S/R ratios. Thus, combining all test results of this group together average KIc was evaluated as 1.00 70.09 MPaOm with twenty 100 mm diameter andesite disk specimens of t/R¼1. This result is very close to the result found as 0.9470.12 MPaOm in SCB method tests. Second group of tests were conducted on andesite and marble SNDB specimens with core disk diameters of 75 mm. Initially, a group of thirteen andesite specimens with t/R ¼1 were tested; for the disk specimens in this group thickness t was set equal to the radius R with t ¼R¼37.5 mm. 2S was varied with S/R ratios between 0.6-0.9. No significant differences in KIc results were observed with the reduction of diameter to D ¼75 mm, and varying loading span 2S. Thus, all results of tests with various S/R ratios were processed together and average result was found as KIc ¼0.9670.08 MPaOm for andesite. This result was again very close to the results of SCB tests and first group of tests. It was decided that reducing the D from 100 to 75 mm, and changing the loading span within the S/R ranges used here had no significant effects on KIc determination with SNDB testing method. Next step in the second group of SNDB tests was to conduct tests on andesite and marble with varying thickness or the circular plate depth t of the core disk type specimen geometries. A total of sixty four andesite and twenty two marble core specimens with diameter D ¼75 mm were tested for investigation of variation of KIc results with disk thickness t. Thickness t of the andesite SNDB disk specimens was changed between 18.8 and 90.4 mm at eight thickness group levels (t/R ¼0.5, 0.7, 1.0, 1.3, 1.5, 1.8, 2.0, and 2.4). S/R ratio was changed again between 0.6 and 0.9 in the tests on andesite; the purpose was to increase the validity of the results statistically, and to confirm that variation of 2S had no significant effect on KIc results of tests with a wider spectrum of 2S range. Then, mode I fracture toughness tests were conducted on a different rock type, namely Afyon Marble specimens of thicknesses ranging from 22.2 to 86.4 mm at seven thickness group levels (t/R ¼0.6, 0.9, 1.2, 1.5,

Table 10 Numerical and experimental data for the first group of SNDB andesite specimens. a mm

2S mm

a/t

S/R

YI

Pcr kN

KIc MPaOm

6.50 5.50 5.00

61.76 60.62 61.02

0.1

0.6

3.31

26.02 24.55 21.71

1.21 1.07 0.89

5.50 5.50

71.84 71.52

0.1

0.7

3.84

18.34 20.31

0.90 1.01

10.00 9.75 9.75

61.82 61.78 61.70

0.2

0.6

3.10

15.88 16.35 16.54

0.89 0.90 0.92

10.50 10.75 9.75

71.68 71.80 71.80

0.2

0.7

3.70

13.65 15.06 15.57

0.91 1.00 0.98

15.63 15.25

61.72 61.70

0.3

0.6

3.18

15.73 14.46

1.09 0.99

15.38 15.50

71.78 71.66

0.3

0.7

3.85

12.92 12.15

1.05 1.01

20.50 20.25 20.00 20.00 21.00

61.60 61.70 61.32 71.74 71.76

0.4

0.6

3.55

0.4

0.7

4.30

10.33 12.25 12.43 9.60 10.38

0.91 1.06 1.09 0.99 1.10

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2.0 and 2.3). 2S was kept constant at S/R¼0.6 for marble test groups. Interesting results are found regarding the effect of varying disk thickness or plate depth t on average KIc. For andesite, average KIc result is 0.63 MPaOm for the smallest thickness group (t/R¼0.5) with five specimens and 1.40 MPaOm for the largest thickness group (t/R¼2.4) with five specimens. Ratio of the KIc result of the largest thickness group to the result of the smallest is around 2.2. Similarly for marble, average KIc is 0.41 MPaOm for the smallest thickness group (t/R ¼0.6) with two specimens and 0.94 MPaOm for the largest thickness group (t/R ¼2.3) with two specimens. The ratio of the KIc results for the largest and the smallest thickness groups is around 2.3. These results clearly indicate the effect of beam or plate depth t on KIc value. Similar trends regarding the proximity of the free boundary and influence of this boundary on the FPZ size and size independent fracture energy of concrete are reported in [26]. In [26], boundary influence on fracture energy loses its effectiveness and a size independent fracture energy result can be asymptotically reached for rectangular concrete beam specimen geometries of a/W around 0.1 with beam depths around W¼300 mm. Even for maintaining a constant initial notch length ratio of a/W¼0.1, fracture energy decreases about 10% as the W drops down to levels of 140 mm or lower. Fracture energy or fracture toughness results are expected to be lower for the SNDB disk geometries with circular plate depth levels limited around 90 mm. Circular plate depth t for SNDB geometry is equivalent to the beam depth W of rectangular beam type specimen geometries. Maximum thickness of the SNDB disk specimens is to be kept about 90 mm in this work. Specimen geometries with thicknesses over this thickness result in invalid local fracturing underneath the concentrated load application locations. For tests on cylindrical core type specimen geometries 2S is limited to the core diameter, and disk thickness or plate depth is limited to the levels at which local failure under the concentrated load application points occur. Trends for the effect of t on KIc results in this work are illustrated in Fig. 11 for andesite and marble. Form of the fitted curves for variation of KIc versus t is approximately the same for both rock types. About the statistical quality of the fits, it should be noted that a larger number of test results for data processing is available in case of andesite. Slope of the logarithmic curves fitted is around 0.5 for both rocks. Intercept values for t/R ¼1, which

makes the logarithmic term zero in the fitted functional forms are approximately equal to the average KIc results of SCB tests for each rock type. Considering these trends and fitted expressions, KIc of the rock types tested with SNDB method can be estimated by using the expression below: K IcðSNDBÞ ¼ 0:5lnðt=RÞ þK IcðSCBÞ

ð11Þ

When t is increased and t/R ratio reaches values more than 2, KIc values determined with the SNDB method increase and reach values obtained by the results of CCNBD method tests. In estimations for a minimized boundary influence condition, taking a/t¼0.1 and t/R¼ 2.7 in Eq. (11), KIc result for andesite becomes equal to the average result of the CCNBD tests. A similar argument can be stated for the average results of the tests on marble. Remembering that plate depth t in SNDB work is limited to 90 mm compared to much larger beam depths in [26] and applying another factor of about 10% or 1.1 to the results, KIc values become higher and approach to the CCNBD results processed by using the constants updated in [41]. However, these results are still believed to stay lower compared to the results of the other testing methods such as CB method [24], SR method [25], and methods with Brazilian type compressive load application [42]. Further investigation is needed for the appropriate value of t/R to be applied in Eq. (11) as the size ratio to attain size independent KIc of different rock types with varying brittleness. SCB three-point bending type testing method with fixed beam depth around the radius R yields significantly lower KIc results compared to results of the other testing methods like CCNBD tests. The reason for this is believed to be due to the relatively lower beam depth of SCB geometry. As discussed before, a FPZ of significant length is presumed to form ahead of the notch front of SCB beam type specimens. FPZ size can be quite large for SCB geometry compared to the rectangular beams used commonly for fracture toughness testing of concrete. For fracture energy tests on rectangular concrete beams, there is flexibility to control the beam depth; beam depth can be increased and FPZ size can be kept proportionally smaller along the propagating crack front with respect to the beam depth. For fracture toughness testing of concrete, relatively large size rectangular concrete beams with flexibility to increase beam depth and 2S while keeping a values relatively small are readily available. Limited to the core type geometries, this is not the case for fracture testing of rocks. Practically, SCB method fracture toughness testing is relatively easier to conduct and to interpret compared to the other known KIc testing methods on core type specimens. By conducting SCB method tests and using Eq. (11), estimation of KIc for a larger beam or plate depth is possible for the beam or plate type geometries under three-point bending load. 4.2. Stiffness comparison of the SCB and SNDB specimen geometries

Fig. 11. Variation of average fracture toughness results of SNDB tests with t/R ratios for andesite and marble.

Stiffness of a beam type specimen under three-point bending is normally characterized by applied load and load point displacement or load-deflection data. However, for evaluation of fracture toughness results of tests with beam type specimens load-CMOD measurements with a clip gage are suggested to be preferred for nonlinearity corrections based on compliance calibration with loading-unloading cycles in the fracture toughness tests [2]. For some of the SCB and SNDB tests in the first group with S/R¼0.7, a clip gage was mounted at the crack tip and used to keep a record of CMOD with increasing load. CMOD values of different tests at peak loads (Pcr) were recorded as CMODcr. Stiffness values of SCB and SNDB specimens (kf ¼Pcr/CMODcr) were evaluated for the same diameter (D ¼100 mm) and beam depth values (t (for SNDB)¼ R (for SCB) ¼50 mm).

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Table 11 Average loads and scr values of SCB and SNDB fracture toughness tests on andesite to investigate and compare FPZ lengths. S/R

SCB a/R

Fig. 12. Variation of stiffness kf of andesite SCB and SNDB specimens depending on a/R or a/t ratios.

The kf values for SCB and SNDB tests were plotted for various initial notch length/radius (a/R for SCB geometry) or initial notch length/plate depth (a/t for SNDB geometry) ratios in Fig. 12. Initial rate of change of slope representing the kf of these geometries is high with increasing notch length; kf of both geometries falls down rapidly for relatively smaller notch lengths. Then the rate decreases for larger a. This behavior can be represented by the logarithmic trends illustrated in the figure. The value of kf of the SNDB method (kf (SNDB)) test specimen geometry is observed to be about twice as much the kf of SCB geometry (kf (SCB)). Higher stiffness state may be regarded as an advantage of SNDB test specimen geometry, considering the reduced differences of stress intensity factor estimations between linear and nonlinear beam modeling techniques with increasing stiffness; effect of beam stiffness factor on the determination of energy release rate of beam type geometries with linear and nonlinear analyses is reported in [28]. 4.3. Estimation and comparison of fracture process zone size In so called quasibrittle concrete and rock-like materials the plastic flow at fracture is next to nonexistent and the nonlinear zone is almost entirely filled by the FPZ, which is dominated by the softening damage. Softening damage includes zones with loss of cohesive forces transmitted across an almost formed crack and zones of micro cracking. Different from the metals, FPZ may occupy a much larger portion of the cross-section of the structure. For softening behavior, bilinear form of stressseparation law is generally used; the stress at slope change is generally considered to be between 0.15 and 0.33ft (ft is the tensile strength) [43]. Adopting the bilinear form of stress distribution ahead of the notch tip, stress normal to the notch tip rises from nearly zero to the yield strength at a distance that may vary depending on the degree of brittleness of the material. Location of peak and the slope change points controls the average fracture energy required for crack initiation and propagation at the notch front. Nominal tensile strength sN or flexural tensile strength of notched beams can be significantly lower than the tensile strength ft determined from the other testing methods like Brazilian splitting or direct tension testing methods. In [44], size dependent nominal tensile strength or sN/ft ratio of notched beams with initial notch/beam depth ratio of a/D is around 0.5 initially for beam depths around 50 mm, and this

SNDB No of tests

Pcr(avg) (kN)

rcr

a/t

(MPa)

No of tests

Pcr(avg) (kN)

rcr (MPa)

0.6

0.1 0.2 0.3 0.4

3 2 3 3

11.55 8.29 5.89 5.98

2.31 1.66 1.18 1.20

0.1 0.2 0.3 0.4

3 3 2 3

24.09 16.25 15.09 11.67

2.41 1.63 1.51 1.17

0.7

0.1 0.2 0.3 0.4

2 3 2 3

8.25 6.85 4.90 4.08

1.65 1.37 0.98 0.82

0.1 0.2 0.3 0.4

2 3 2 2

19.32 14.76 12.53 9.99

1.93 1.48 1.25 1.00

ratio decreases down to values around 0.3 for beam depths reaching 300 mm. Nominal strength is directly proportional to the average critical load normal to the notch plane and inversely proportional to the cross sectional area of the beam along this section. This area is represented by RB for SCB and by Dt for SNDB geometries. Based on these discussions, possible FPZ sizes for SCB and SNDB geometries were estimated by applying s0 ¼ scr as the nominal strength to the numerically computed stress distributions normal to the notch and crack section. Fracture loads for tests on 100 mm diameter SCB and SNDB specimens with 50 mm beam depth (B for SCB specimens and t for SNDB specimens) were processed. Average fracture loads (Pcr(avg)) of KIc tests were determined for groups with different a and 2S values. a was used as 5, 10, 15 and 20 mm. Two loading spans with S/R ratios of 0.6 and 0.7 were included in the analyses for both geometries. The Pcr(avg) and s0 ¼ scr values for various a/t (or a/R for SCB) and S/R groups used in the estimation of FPZ size are listed in Table 11. For both geometries with S/R ¼0.6, average scr approximately changes from 2.4 MPa to 1.2 MPa with notch length/beam or plate depth ratios varying from 0.1 to 0.4. To compare notch front stress distributions of both methods, stress component sxx normal to the notch plane was normalized by scr ¼Pcr(avg)/(2RB) for SCB specimens and scr ¼Pcr(avg)/(2Dt) for SNDB specimens. Length of FPZ (lFPZ) at the crack front of SCB and SNDB geometries was estimated by comparing dimensionless stress ratio sxx/scr to a value equal to one. Estimation was based on a criterion of sxx Z scr or sxx/scr Z1, and application of this criterion to the sxx/scr versus y/(t  a) (or y/(R  a) for SCB specimens) stress distribution plots. A typical sxx/scr versus y/(t  a) plot is illustrated in Fig. 13. In the figure, y is the distance change from crack tip to loading point of the specimen. y is equal to zero at the crack tip and t a (or R a for SCB specimens) at the upper boundary loading point. This example graph belongs to SCB and SNDB specimens having 100 mm D, 50 mm t (or B for SCB specimens), a/t¼0.2 (or a/R¼0.2 for SCB specimens) and S/R ¼0.6. Overall results of modeling and test data processing showed that for SNDB specimens having 100 m D, 50 mm t, 5 mm to 20 mm a range and 0.6 and 0.7 S/R ratios, average lFPZ was about 3.9270.86 mm. For SCB specimen geometries having the same D, t and a range, average lFPZ was estimated as 8.42 70.86 mm. Average lFPZ of SCB specimens was about 2.15 times greater than the average lFPZ of SNDB specimens. This difference was attributed to stronger boundary influence condition on crack front of threepoint bend SCB test specimen geometry.

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Fig. 13. Dimensionless crack front stress (sxx/scr) variation with dimensionless distance from the crack tip and FPZ length estimation procedure.

5. Conclusions A new test specimen called straight notched disk bending was proposed for mode I fracture toughness determination using rock. An advantage of this method over SCB method is the flexibility of testing specimens with different disk thicknesses for size effect investigations. Compared to SCB geometry, stiffness of this circular plate type specimen geometry is larger reducing the beam nonlinearity effects on the estimation of stress intensity factors, and fracture process zone length is estimated to be smaller. Results of 3D finite element modeling procedure used for modeling SNDB circular plate type geometry were verified by modeling well known SENB and SCB specimen geometries with approximately the same model and crack tip mesh intensities, and comparing the results of KI computations. Stress intensity factors of SNDB geometry were found to increase linearly with three-point bend loading span within a limited span range; computations were limited to a range due to diameter limitation of the core type specimens used in the experimental work for KIc determination. With increasing initial notch length, stress intensity factor of SNDB geometry increased in a nonlinear form with an increasing rate of change. Stress intensity factor was found to decrease nonlinearly with increasing plate depth. Mode I fracture toughness tests were conducted on two rock types, andesite and marble. No significant differences in KIc results were observed with the reduction of core specimen diameter from 100 to 75 mm for andesite rock. For andesite and marble, varying loading span 2S did not affect KIc results for a range of S/R¼0.6–0.9. SCB and CCNBD tests on both rock types were included in the test program for comparison purposes. For both rock types, KIc results demonstrated a nonlinear increasing trend with increasing thickness or plate depth of the SNDB disks. A parametric logarithmic expression was proposed for estimation of KIc in terms of thickness/radius ratio of the disk geometries. For thickness/radius ratio being equal to one, KIc results of the SNDB method tests were lower and very close to the results of SCB tests on both rock types. Increasing trend of the

logarithmic fit slowed down and rate decreased for larger thickness/radius ratios. For t/R ratio over two, KIc results were close to the results found from CCNBD tests. To determine size independent fracture energy of concrete fracture toughness testing, it is practically possible to prepare and test rectangular beams under three-point bending with larger beam depths around 300 mm. For fracture toughness testing of rocks, core type geometries available from the site investigation work are preferred, and there are limitations on the plate thickness and loading span that can be applied for the SNDB method and disk type specimen geometry. The value of 2S is limited to the core diameter, and disk thickness t is limited to the levels at which local failure under the concentrated load application points occur. However, estimations of the size independent fracture toughness can be made by applying larger t/R values in the logarithmic trend expression given in this work. Problem of how large a t/R ratio is to be applied to determine the size independent KIc needs further investigation on different rock types with varying degree of brittleness.

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