Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen

Engineering Fracture Mechanics 91 (2012) 133–150

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Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen Mahinda D. Kuruppu a,⇑,1, Ken P. Chong b,c,1 a

Curtin University, Western Australian School of Mines, Kalgoorlie, WA 6430, Australia George Washington University, Engineering, MAE, Washington, DC 20052, USA c National Institute of Standards and Technology, Engineering Lab., Stop 8615, Gaithersburg, MD 20899, USA b

a r t i c l e

i n f o

Article history: Received 17 September 2011 Received in revised form 1 January 2012 Accepted 2 January 2012

Keywords: Fracture mechanics Brittle fracture Toughness testing Stress intensity factor J-integral Finite element analysis Mixed mode fracture Dynamic fracture

a b s t r a c t The semi-circular bend (SCB) specimen was suggested in 1984 for testing mode I fracture toughness of rock and other geo or brittle materials. Since then SCB has been used worldwide, extended and improved for many other applications by various researchers. Formulations for mode I and mixed mode fracture of this specimen proposed by a number of researchers are presented. Methods to determine fracture toughness using both straightnotched and chevron-notched specimens have been proposed although the general consensus is that a specimen having a sharp straight notch should yield accurate fracture toughness. Other applications of SCB specimen include testing of rock subjected to in situ conditions such as elevated temperature, confining pressure and pore water pressure. Furthermore it has been proven that it is a suitable specimen to test fracture toughness of rock at very high strain rates. Areas requiring further research to improve the accuracy of formulations are identified. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fundamental research in deterioration and durability of structures and materials has shown great potential for increasing the life span of our infrastructure systems, as well as enhancing their functionality and serviceability. Brittle fracture mechanics is one of the main areas of research in deterioration of materials [1]. It is well known that linear elastic fracture mechanics is not strictly applicable for brittle materials like rock and other geomaterials due to the occurrence of relatively large fracture process zones as well as tension-softening behavior. In these materials, the process zone is made up of a region of microcracks that may be pre-existing and crack bridging may also occur in the process zone. Macrocrack initiation and propagation are associated with the activation of many microcracks in opening and/or shear modes, and in some cases, overcoming the resistance created by interlocking of relatively large and stiff grains of material. The sliding cracks that occur in compressive stress regimes, and the resulting wing cracks that occur due to local tensile stresses causing mode I microcracking, have been very helpful in explaining the initiation and growth of cracks in brittle materials subjected to global compressive stresses [2]. In order to account for these variations in material behavior, brittle or rock fracture mechanics uses

Abbreviations: BD, Brazilian disk specimen; BEM, boundary element method; CB, chevron bend specimen; CCNBD, cracked chevron notched Brazilian disk; COD, crack opening displacement; CMOD, crack mouth opening displacement; FEM, finite element method; GMTS, generalized maximum tangential stress fracture criterion; SCB, semi-circular bend specimen; SIF, stress intensity factor; SR, short rod specimen. ⇑ Corresponding author. Tel.: +1 618 9088 6173; fax: +1 618 9088 6151. E-mail address: [email protected] (M.D. Kuruppu). 1 Both authors Kuruppu and Chong are first authors. 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.01.013

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Nomenclature a C 0 C E Fmax G J K KI KIc KII KIIc P Pc rc R s t T T 0 YI, Y YII

a b d D

r rt rhh m (r,h)

crack length, mm compliance, mm/N normalized compliance modulus of elasticity, MPa maximum load, N energy release rate, N/m J-integral, N/m stress intensity factor, MPa mode-I stress intensity factor, MPa critical stress intensity factor in mode I, MPa mode-II stress intensity factor, MPa critical stress intensity factor in mode II, MPa force applied on specimen, N critical load, N crack tip damage zone, mm radius of specimen, mm half span length between supports, mm thickness of specimen, mm largest non-singular term of near-tip stress field (T-stress), MPa normalized T-stress normalized stress intensity factor in mode I normalized stress intensity factor in mode II crack angle, radian normalized crack length crack tip opening displacement, mm load point displacement, mm normal stress, MPa tensile strength, MPa tangential stress component, MPa Poisson’s ratio crack tip co-ordinate system

appropriate theories and test methods which are different from those applicable for metallic materials [3,4]. In the modeling of fracture propagation in brittle materials, Ingraffea is one of the pioneers [5]. Despite their relatively low strength, the crack initiation and growth in brittle rock and other geomaterials is associated with large dissipation of energy. Many researchers have found that energy-based fracture criteria, such as Rice’s J-integral [6] and specific work of fracture (R or G measured in J/m2), are more suitable for fracture characterization [7,8]. However, as the most widely-used fracture characterizing parameter, the critical stress intensity factor in mode I loading, also known asffi fracpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ture toughness KIc, is the preferable criterion for unstable fracture occurrence. Its dimensions are stress  cracklength (e.g. pffiffiffiffiffi MPa m). It has been found to be effective for geomaterials, such as rock, provided that certain minimum specimen-size criteria are met [9]. Some applications of rock fracture toughness are (a) a parameter for classification of rock (b) an index of fragmentation processes such as tunnel boring (c) a material property in the modeling of rock fragmentation such as rock cutting, hydraulic fracturing, explosive fracturing and crater blasting, and (d) stability analysis of mining and earthen structures (e.g. underground spaces). Plain strain fracture toughness that is derived using tests satisfying minimum-dimensional requirements has been proven to be a material property. A number of methods have been suggested to determine the fracture toughness of rock. These include the Short Rod (SR) specimen, Chevron Bend (CB) specimen and Cracked Chevron-notched Brazilian Disk (CCNBD) specimen [10–13]. These specimens have been incorporated into suggested standard testing methods for the fracture toughness measurement of rock by the International Society for Rock Mechanics (ISRM) [14,15]. The semi-circular bend (SCB) specimen [16] is complimentary to the standard methods as depicted in Fig. 1. It is a corebased specimen, and has certain inherent favorable properties such as its simplicity, minimal requirement of machining and the convenience of testing that can be accomplished by 3-point compressive loading using a standard test frame. 2. Fracture toughness tests The SCB specimen was proposed by Chong and Kuruppu [16], and many of the initial developments were done at the University of Wyoming [17–19]. The core-based SCB test specimen is shown in Fig. 1, along with some of the ISRM standard specimens. SCB can be used as a standalone or alternate specimen for standard specimens when determining the fracture toughness in orthogonal directions of brittle transversely isotropic materials or sedimentary rock, such as sandstone or

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Fig. 1. Core based fracture toughness test specimens. Crack orientation is in mutually perpendicular direction [17].

Fig. 2. Principal crack orientations with respect to bedding planes (left to right: arrester, divider and, short transverse configurations) [17].

Fig. 3. Mixed-mode fracture testing configuration of SCB specimen [23–25].

oil shale. This can be done using a single core and making use of the sampling procedure shown in Fig. 1 [17]. Besides rock, SCB has been applied to other core-based brittle materials, such as concrete, asphalt, sea ice and other materials. All sedimentary rock in general is layered in structure and can be treated as transversely isotropic [20]. Fig. 2 gives the orientation of cracks relative to bedding planes and illustrates the necessity of employing test specimens made in all three orientations. Studies done at the National Institute of Advanced Industrial Science and Technology in Japan [21] have shown that typical sedimentary rock exhibits a difference of up to 15% in fracture toughness measured in the three major orientations. Baek [22] reports that the fracture toughness of Elberton granite in one orientation is twice that of another. Another

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potential use of the SCB specimen is for mixed-mode fracture testing (Fig. 3). Such investigations have been reported by Chong and Kuruppu [23,24], Lim et al. [25] and Ayatollahi et al. [26–29]. 2.1. Mode I stress intensity factor determination using SCB specimen Normalized (i.e. non-dimensional) stress intensity factor (SIF) has been determined using the finite element method for SCB specimens with a flat notch [17]. Fig. 4 depicts the relationship between SIF and normalized crack length for a span to pffiffiffiffiffiffi radius ratio of 0.8. Note that the normalized SIF, Y0 , is expressed as K I =ðr0 paÞ where r0 ¼ P=ð2RtÞ. P is the load, a, R and t are the crack length, the radius and the thickness, respectively. Many investigators have since used s/R ratio of 0.8 as it is satisfactory for testing, except in the case of mixed-mode tests where a shorter span ratio is required to achieve pure mode II fracture [28]. The following formula gives the normalized SIF [19]:

Y 0 ¼ 5:6  22:2b þ 166:9b2  576:2b3 þ 928:8b4  505:9b5

ð1Þ

where b is the a/R ratio. Lim et al. [30] has found SIFs for the cases of other span ratios. Their results can be summarized by the following relation:

Y0 ¼

s ð2:91 þ 54:39b  391:4b2 þ 1210:6b3  1650b4 þ 875:9b5 Þ R

ð2Þ

More recently, Ayatollahi and Aliha [28] have published comprehensive data for crack tip SIFs KI and KII considering the crack angle, crack length ratio a/R and span ratio s/R. Alternatively, compliance can be used to determine the SIFs by numerical means (e.g. using FEM or BEM) or by experimental means. The non-dimensional elastic compliance is expressed as C0 = E’DC where C is the compliance, D is the diameter and E0 ¼ E=ð1  t2 Þ. E and m are Young’s modulus and Poisson’s ratio, respectively. For a SCB specimen having the span ratio s/R = 0.8, the relation between the non-dimensional compliance and the crack length is given as [31]:

C 0 ¼ 1366b3  867b2  51:9b þ 129:4

ð3Þ

The strain energy release rate is related to the compliance as:

G¼

P2 dC 2t da

ð4Þ

The SIF can then be related to the compliance using:

K¼



EG 1  t2

12 ð5Þ

and the resulting relationship is:

K¼

 0 12 P dC pffiffi 2R t da

ð6Þ

Compliance is also useful to measure the subcritical and post-critical crack growth that occurs during fracture testing. Kuruppu et al. [32] determined the fracture toughness of sandstone using the resistance curve method. They used the acous-

Fig. 4. Mode I stress intensity factor [17].

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137

tic emission method to determine the crack growth. The crack growth resistance plotted against crack extension yielded the fracture toughness. However, this method is difficult to use because of the need to measure the crack growth. The non-linear corrected level 2 fracture toughness [14] does not necessarily give representative fracture toughness if sub-sized specimens are used for testing. It has been shown that the resistance curve method, though cumbersome, yields correct fracture toughness using sub-sized specimens up to a certain limit. Adamson et al. [33], while using SCB specimen as a field test sample to determine the fracture properties of sea ice in the Arctic region, have made a significant contribution to its development as a test for fracture toughness measurement. They applied weight function method to the semi-circular specimen geometry and determined the SIFs and crack opening displacements. They considered 2 loading configurations: (a) the standard 3-point bend loading and, (b) a crack opening point load applied at the crack surface and showed that the weight function method gave an independent mathematical solution to determine SIF using the finite element stress distribution. They derived crack opening displacement (COD) and crack mouth opening displacement (CMOD) in addition to the SIF of the SCB specimen. Furthermore, they have proven that the SIFs and crack mouth opening displacements are comparable with those produced numerically using ABAQUS finite element analysis program. They provide results for the full range of non-dimensional crack length a/R from 0 to 1. 2.2. Application to mixed-mode fracture For mixed-mode fracture experiments, a suitable test configuration should have simple geometry and loading arrangement, inexpensive preparation procedure, convenience of testing and the ability of introducing complete combinations of mode I and mode II. The SCB specimen having an angled notch satisfies these requirements and enables the evaluation of the critical combinations of mixed-mode SIFs, as well as mode II fracture toughness. A number of investigations have been carried out to determine the mixed-mode SIFs using disk specimens, such as centre-cracked Brazilian and SCB specimens [23–28]. One of the earliest investigations reported was performed by Chong and Kuruppu [23] who determined the mode I and mode II SIFs in SCB specimen as a function of the angle of inclination of crack. Lim et al. [25] and Ayatollahi and Aliha [28] have subsequently determined more accurate values (Fig. 5). The mixed-mode fracture toughness combinations of KI and KII were determined using:

KI ¼ r

pffiffiffiffiffiffi paY I

and

pffiffiffiffiffiffi K II ¼ r paY II

ð7Þ

where r ¼ P c =2Rt and Pc is the applied load, R the specimen radius, t the specimen thickness, a the crack length and YI, YII are the normalized stress intensity factors determined using numerical stress analysis. Fig. 5 shows the values for a particular case of a/R = 0.5 and s/R = 0.5. Ayatollahi and Aliha [28] have further determined the value of the largest non-singular term of the near-tip elastic stress field, known as T stress. T stress is the stress component parallel to the crack and can be determined using finite element stress analysis, similar to deducing the SIFs. While its effect is insignificant for the case of mode I, its influence is significant when the crack angle is large [34]. The non-dimensional SIFs YI, YII and the non-dimensional T  stress are given as:

Fig. 5. Non-dimensional stress intensity factors (geometry factors) for angled crack in SCB specimen [25,28].

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K I 2Rt Y I ða; a=R; s=RÞ ¼ pffiffiffiffiffiffi pa P K II 2Rt Y II ða; a=R; s=RÞ ¼ pffiffiffiffiffiffi pa P T  ða; a=R; s=RÞ ¼ T

ð8Þ

2Rt P

where a is the crack angle, a is the crack length, R, s and t are the radius, the half span between supports and the thickness, respectively, and P is the applied load. An exhaustive range of the numerically-determined parameters are given in Ayatollahi and Aliha [28]. A particular note of interest is that the T  stress is positive for most of the range of a for the mixed-mode case. The case of YI = 0 results in pure mode II SIF. The combinations of a/R, s/R and a that yield mode II are useful for determining the mode II fracture toughness. The SIFs determined by the investigators reveal that s/R ratio should be less than 0.65 in order to yield practical pure mode II loading for crack angle a 6 50°. They used the generalized maximum tangent stress (GMTS) fracture criterion in order to determine the critical combinations of KI and KII at the onset of fracture in mixed-mode loading [35,36]. The criterion states that the fracture occurs when the tangential stress at a particular distance rc from the crack tip reaches a critical value denoted by rhhc. The elastic tangential stress inclusive of T stress is given as [37]:

1

h



h

3



rhh ¼ pffiffiffiffiffiffiffiffiffi cos K I cos2  K II sin h þ Tsin2 h þ Oðr1=2 Þ 2 2 2 2pr

ð9Þ

where h is the angle measured from the direction of crack in the crack tip co-ordinate system. When the tangent stress reaches its maximum value we get:

@ rhh ¼0 @h

ð10Þ

Using non-dimensional SIFs the solution for the direction of maximum tangential stress h0 can be given as:

Y I sin h0 þ Y II ð3cosh0  1Þ 

16T  3

rffiffiffiffiffiffiffi 2r c h0 cos h0 sin ¼ 0 a 2

ð11Þ

where the distance rc is considered equal to the crack tip damage zone. It is derived from the relationship [38]:

rc ¼

 2 1 K Ic 2p rt

ð12Þ

and rt is the tensile strength. Alternatively, this relation can be derived from Eq. (9) above for mode I case and by assuming rhh is equal to rt. For a given value of crack angle, the parameters YI, YII and T are known. Hence h0 can be determined from Eq. (11). For the case of mode I loading where KII = 0, h0 = 0 and KI = KIc, Eq. (9) and GMTS fracture criterion gives:

K

Ic rhhc ¼ pffiffiffiffiffiffiffiffiffiffi 2pr c

ð13Þ

Combining Eqs. (9) and (13) authors have given the following relation of the three crack parameters KI, KII and T that yield the critical combinations satisfying the onset of fracture:

  pffiffiffiffiffiffiffiffiffiffi 3 2 K Ic ¼ cos h=2 K I cos2 h0 =2  K II sin h0 þ 2prc T sin h0 2

ð14Þ

Dividing Eq. (14) by KI and KII, respectively, and finding the reciprocal gives:

" #1   rffiffiffiffiffiffiffi  KI h0 h0 3 Y II 2r c T 2 ¼ cos sin h0 þ sin h0 cos2  K Ic 2 2 2 YI a YI

and

" #1   rffiffiffiffiffiffiffi  K II h0 Y I 3 2rc T 2 2 h0 ¼ cos cos sin h0  sin h0 þ K Ic 2 Y II 2 2 a Y II

ð15Þ

ð16Þ

Since all the parameters in Eqs. (15) and (16) are known, including h0, for a given crack angle, the relationship between KI and KII can be found. Ayatollahi et al. [27] have given a comparison of theoretical values with experimental results for PMMA. For pure mode II where YI = 0 Eq. (16) gives KII as:

" !#1 rffiffiffiffiffiffiffi K II T  2rc 3 h0 sinh0  cos ¼ sin h0 2 K Ic Y II a 2

ð17Þ

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Ayatollahi and Aliha [26] give the relation of KKIIcIc and T  ¼ YT II as shown in Fig. 6. For the case of the SCB specimen, where the T stress is positive, KKIIcIc assumes a value less than 0.87. This suggests that the SCB specimen gives a conservative value for KIIc in comparison with other specimens that may have negative T stress associated with the crack tip stress field. Eq. (16) provides further support that positive T stress will decrease KII on the assumption of maximum tangent stress fracture criterion. More recently, Ayatollahi et al. [29] have demonstrated the possibility of using the SCB specimen with a symmetric notch in order to investigate the mixed-mode fracture by selecting an asymmetric loading configuration. 2.3. Chevron-notched SCB specimen In a loaded flat-notched specimen, it is possible that only some of the regions in the notch tip reaches the singular stress field, leaving the other regions of the crack-front subcritical [9]. In chevron-notched specimens, the crack is allowed to grow sub-critically, aided by the high stress concentration facilitated by its geometry. Crack propagation reaches the point of instability when the peak load is achieved. A limited SIF calibration of the chevron-notched SCB specimen was done by Kuruppu using a 3D finite element method [39–41]. The geometry is shown in Fig. 7. The analysis was performed taking a0/ R = 0.2, s/R = 0.8 and chevron notch angle of 90°. Fig. 8 illustrates the normalized SIF, defined as:

K nd ¼

pffiffiffi KIt R P

ð18Þ

where KI is the SIF and P is the applied load. Fig. 8 shows the characteristic minimum value of Knd, equal to 7.4, associated with the critical point of fracture. Further numerical work is required to determine the characteristic curves for a range of a0/R values, and to develop a more general mathematical relation that can be used to evaluate the fracture toughness. Eq. (18) can be used to determine the level 1 fracture toughness as:

7:4P c K Ic ¼ pffiffiffi t R

ð19Þ

Satoshi [42] has determined a formula for level 1 fracture toughness evaluation using a different chevron-notch geometry of the SCB specimen. The relationship given as:

K I ¼ 15:22

F max

ð20Þ

R1:5

is applicable to the geometry shown in Fig. 9. An appropriate non-linearity correction is required to determine a representative value for level 2 fracture toughness in both the preceding cases. Wang et al. [43] have determined the SIFs of cracked chevron-notched Brazilian disk specimen using a method easier than the conventional methods. They used the thin slice approximation to simulate the CCNBD specimen, where only the SIF for straight cracked specimen is required to deduce the SIF of chevron notched specimen. Wang et al. [44] used a sub-model for convenience in the discretization of their numerical FEA model. These developments would be useful for similar work with the SCB specimen. For example Dai et al. [68] have derived the stress intensity factor of a cracked chevron-notched SCB specimen and shown that it assumes a minimum value with increasing crack growth beyond which fracture becomes unstable.

Fig. 6. The mode II critical stress intensity factor variation with T stress [26].

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Fig. 7. Chevron-notched semi-circular bend test specimen [39].

Fig. 8. Mode I stress intensity factor for chevron-notched SCB specimen [40].

Fig. 9. Chevron-notched SCB geometry used by Satoshi where W/R = 0.826, s/R = 0.8, h = 54.6°, a0 = 0.2R [42].

2.4. Tension softening method for determining mode I fracture toughness The theory of tension softening makes use of the energy release rate during the post-peak part of the load and load-pointdisplacement curve. A number of investigators have examined the post-failure behavior associated with brittle rock material, where the failure process continues under decreasing load until the separation occurs [18,45–47]. Using the definition of the J integral as the specific area under the post-peak tension softening curve (i.e. crack tip opening stress versus crack opening displacement (COD) curve), the critical value of Jc can be determined considering the complete unloading portion of the load and load-point displacement curve. The peak value of the J integral, or alternatively, the area under the tension softening curve, is a measure of the critical energy release rate during the crack tip stress relaxation. Hashida [45] describes it as a reliable method of fracture toughness determination. This method also has the potential to yield correct fracture toughness using relatively small specimens. However, it is likely that applications are somewhat limited for rocks since not many rock materials exhibit tension softening behavior. Application of the method to determine the fracture toughness of basalt rock and cement mortar using SCB specimen has been done by the University of Wyoming group [18]. However, in view of the ability of the tension-softening curve to characterize fracture associated with large process zones, it is worthy to re-visit this method with the intention of incorporating

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141

Fig. 10. Load–displacement behavior of SCB specimens with slightly different crack lengths [47].

700

J-Integral (N/m)

600 500 400 300 200 100 0 0

20

40

60

80

100

120

140

160

Load line displacement (Micrometer) Fig. 11. J-integral versus load line displacement curve [47].

into a standard method. Furthermore, the experimental procedure is relatively simple (e.g. no partial unloading) and requires only normal laboratory equipment. The method can be used to determine important fracture parameters for tension-softening materials. Furthermore, a circular disk cut from a core can be sawed into two identical semi-circular specimens. Two slightly different crack lengths (a1, a2) are introduced in these two otherwise identical specimens [47], see Fig. 10. These two specimens with slightly different crack lengths were tested while measuring load, load-point displacement (D) and crack tip opening displacement (d). The following parameters were determined:

JðDÞ ¼

   A1 A2 1  a2  a1 B1 B2

ð21Þ

where A1, A2 = area under load versus load-point displacement curves for specimens with crack lengths a1, a2 respectively. B1, B2 = net thickness of specimens having crack lengths a1, a2 respectively. Fig. 11 shows J versus D curve. Considering the near tip zone the critical value of J derived from the r  d curve is:

J c ðdÞ ¼

Z

dc

rðdÞdd

ð22Þ

0

where r is the tensile stress at or near the crack tip and dc is the crack tip opening displacement corresponding to r = 0 (i.e. specimen is fully cracked). Eq. (22) gives:

r¼

dJ dd

ð23Þ

Using the experimentally determined relations of J, D and d the J  d curve can be constructed. Hence using Eq. (23), the tension softening curve r  d can be determined as shown in Fig. 12. The area under the curve gives the critical crack tip energy release rate Jc (analogous to critical energy release rate Gc), which can be converted to fracture toughness as

Jc ¼

ð1  m2 ÞK 21c E

ð24Þ

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10 9

Stress (MPa)

8 7 6 5 4 3 2 1 0 0

10

20

30

40

50

60

70

80

Crack tip separation (Micrometer) Fig. 12. Tension softening curve of Basalt [47].

3. Size effects It is well recognized that owing to the formation of a proportionately large nonlinear process zone the theory of liner elastic fracture mechanics (LEFMs) is not strictly applicable for the fracture of ‘small’ specimens of rock [8]. The nonlinear process zone in small specimens which do not satisfy the LEFM minimum size requirements causes a size effect. Earlier work investigating specimen size effect gave quite conservative size requirements [48,49]. Bazant and Planas [50] presented a comprehensive book on the size effects in the fracture of quasi-brittle materials. Chong et al. [17] suggested the following size requirement for the flat-notched SCB specimen:

D P 2:0

 2 K Ic

ð25Þ

rt

where rt is the tensile strength. A number of investigations (e.g. [21]) have confirmed the above size requirement. Subsequent developments that used chevron-notched specimens with non-linear correction [10] and J-based evaluation methods gave somewhat relaxed size requirements for various specimen geometries [51]. It may be possible to relax the size requirement for chevron-notched specimens when J-integral based methods are used. However, it is essential that the size requirements are defined under such circumstances (e.g. [52]). This has been somewhat neglected in many of the proposed methods for fracture toughness evaluation. For the SCB specimen having straight notch, Kuruppu et al. [32] have shown that the size requirements can be relaxed to:

D P 1:6

 2 K Ic

ð26Þ

rt

when the nonlinear correction method is used. Also the amount of crack growth required to give fully developed crack growth resistance is:

 2 K Ic a  a0 P 0:2

ð27Þ

rt

In the SCB geometry, the compressive load applied at the top central loading pin causes compressive stresses ahead of the mode I fracture process zone. It is plausible that the occurrence of compressive stresses restricts the size of the process zone, therefore making it easier to use relatively small specimens to achieve valid fracture toughness results. We further note that when using nonlinear correction method, Ouchterlony [51] gives the following size requirements for the Chevron Bend specimen and the Short Rod specimen, respectively:

DSR P 0:6  0:9

 2 K Ic

rt

 2 K Ic and DCB P 1:3  2:0

rt

ð28Þ

where DSR and DCB are the diameters of Short Rod and Chevron Bend specimens, respectively. The subcritical crack growth is given as:

 2 K Ic a  a0 P 0:2  0:3

rt

ð29Þ

These results also show that the minimum specimen size requirement for the SCB specimen is at least in the same order, if not smaller, than that of standard specimens. Horii et al. [3] investigated the effect of the second term of the crack tip stress field and determined that it is the dominant factor which influences the specimen size-dependency of fracture toughness.

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Bazant and Kazemi [8] have shown that geometrically similar specimens made of a rock material do not necessarily fail at loads determined by LEFM theory. They proposed a size effect law that can be used to determine the fracture energy as well as the effective length of the fracture process zone of the material by measuring the maximum loads of geometrically similar specimens. They define the nominal stress at the peak failure load of a test specimen (or a substructure) as:

rN ¼ C n

Pu bd

ð30Þ

where Pu is the ultimate (maximum) load, b is specimen thickness and d is the depth or any other characteristic length of the specimen. Cn is a constant that is determined considering the load and the stress relation for the specimen geometry. For example the elastic bending stress formula for the SCB specimen considering a half span length of s and uncracked ligament length l is given by:

rN ¼ 3sP=bl2 ¼ cn P=bl which is equivalent to Eq. (30) with Cn = 3s/l and Cn is a constant for geometrically similar specimens. Assuming that the energy dissipated at failure is a smooth function of the specimen size and the fracture process zone width Bazant [53] showed that rn can be given in the following form:

Bf

u ffi where b ¼ rN ¼ pffiffiffiffiffiffiffiffiffiffiffi

1þb

d d0

ð31Þ

fu is a measure of material strength such as the tensile strength. B and d0 are constants. Eq. (31) can be given in the parametric form Y = AX + C in which:

Y ¼ ðfu =rN Þ2 ;

X ¼ d;

1 B ¼ pffiffiffi ; C

d0 ¼ C=A

By plotting Y versus X experimentally the unknowns B and d0 are determined. The fracture energy release rate G is given as:

G¼

P2 gðaÞ 2

Eb d

ð32Þ

where P is the applied load, g(a) is a geometric factor and a = a/d. For the SCB specimen Eq. (32) is analogous to Eq. (4) and hence g(a) can be related to compliance given in Eq. (3). For plane strain E is replaced by E/(1  m2) where m is the Poisson’s ratio. The fracture energy Gf may be uniquely defined as that required for crack growth in a specimen satisfying LEFM requirements [54]. For the limit d ? 1 Eq. (32) yields:

B2 fu2

Gf ¼

C 2n E

d0 gða0 Þ

ð33Þ

where a0 is the initial notch length. Gf is then related to KIc as:

K Ic ¼

qffiffiffiffiffiffiffiffi Bf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d0 gða0 Þ EGf ¼ Cn

ð34Þ

which gives the fracture toughness in relation to the size effect parameters. The crack length cf corresponding to Gf is related to the size effect parameters as:

cf ¼

d0 gða0 Þ g 0 ða0 Þ

ð35Þ

where cf is defined as the crack growth beyond the initial notch of a specimen satisfying LEFM conditions. Bazant et al. [55] has shown that the crack growth resistance curve widely known as R-curve can be determined using the size effect law. R-curve is considered a material property and is independent of the specimen size. If the crack grows by c = (a  a0) where a0 and a are the initial notch length and the crack length after growth respectively in a specimen having a characteristic dimension d and if the elastic energy release is in balance with the increase in crack growth resistance then:

G da þ R dc ¼ 0 or RðcÞ ¼ Gða; dÞ

ð36Þ

Based on the assumption that R-curve is size independent oR(c)/od = 0, which gives:

@Gða; dÞ ¼0 @d Eqs. (32) and (33) yield:

ð37Þ

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Gða; dÞ ¼ RðcÞ ¼ Gf

gðaÞ d gða0 Þ d þ d0

ð38Þ

Substituting Eq. (38) into Eq. (37) and differentiating we get

d þ d0 gðaÞ ¼ ða  a0 Þg 0 ðaÞ d0

ð39Þ

Substituting Eq. (39) into Eq. (38) and combining with Eq. (35) we get the crack growth resistance:

RðcÞ ¼ Gf

g 0 ðaÞ c g 0 ða0 Þ cf

ð40Þ

Furthermore Eqs. (38) and (40) give the relationship of crack length c of the fracture process zone at maximum load at size d and cf as:

  c g 0 ða0 Þ gðaÞ ¼ a þ a  0 cf gða0 Þ g 0 ðaÞ

ð41Þ

Eqs. (40) and (41) along with Eq. (33) can be used to construct the R-curve parametrically. 4. Applications Due to the inherent favorable properties of the SCB specimen, such as its ease of preparation from standard core samples and ease of loading, it has been utilized by numerous investigators since its inception. Applications have been as diverse as fracture mechanics studies of rock, pavement asphalt and arctic sea ice. An attempt is made here to cover some of these applications. Applications have been prompted from its advantages listed below.  Simple, core-based geometry. It is relatively easy to make and involves very little machining. Therefore, it is very attractive from the practical point of view.  Suitable to be used as a specimen for testing sedimentary (layered) rock. This enables the evaluation of fracture toughness from a single block of core, by cutting specimens in such a way that the notch is oriented in three principal directions of material properties.  Suitable for testing materials that behave non-linearly. A number of researchers have used it successfully for testing soft rock and other materials.  The geometry is especially suitable for mixed-mode fracture studies. It is likely that more researchers will use it as the knowledge base of mixed-mode fracture and the demand for such studies increase. A number of investigators have used the SCB specimen for the fracture toughness evaluation of rock. The University of Wyoming group [17,19] used this method to determine the fracture toughness of oil shale. They developed most of the theoretical basis for using the specimen, as described in the previous sections. Karfakis et al. [56] have investigated the effect of chemical additives on the fracture toughness. Chong et al. [18] determined the fracture toughness of rock and cement mortar by making use of the tension softening material behavior. Lim et al. [25,30,57] used the specimen for fracture toughness evaluations of a soft rock and proved that its geometry and loading configuration are good for testing materials that behave nonlinearly. Their work proved that the water saturation is a dominant factor affecting the fracture toughness. Furthermore, they reported an increasing trend of fracture toughness with strain rate. Singh and Sun [58–61] determined fracture toughness of Welsh limestone, coal measures sandstone and Newhurst granite. In addition, they determined the effect of parameters such as water content. Obara et al. [62–64] further examined the effect of pore water on fracture toughness using the SCB specimen. Baek [22] investigated the effects of specimen size, crack orientation with respect to bedding plane and water content on fracture toughness using Chevron Bend (CB), Single Edge Cracked Round Bar in Bending (SECRBB) and SCB specimens. He used three different rock materials and found that:  Using Elberton granite, the CB specimen gave the highest value and SCB specimen gave the lowest value for level 1 fracture toughness.  For Comanche Peak limestone, CB and SCB specimens gave similar values, while SECRBB specimen gave lower values.  In general, the fracture toughness measured by the SCB specimen was lower than that of the CB specimen, showing that the SCB specimen gives somewhat conservative results for the fracture toughness. These results show that one needs to exercise caution when comparing fracture toughness values derived using different types of specimens. There are differences in the development of fracture process zones in different specimens and this can lead to variations in fracture. Baek also measured fracture toughness of Elberton granite in three perpendicular directions with respect to the material structure using the SCB specimen. The values were very different to each other, which reflects

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Fig. 13. Fracture toughness of oil shale as a function of specific gravity (i.e. shale grade) in the divider orientation [17].

the anisotropy of Elberton granite. The investigator further determined that water-saturated Elberton granite had its fracture toughness reduced by 18%, compared with un-saturated rock. The case of anisotropy must be dealt with special care as anisotropic rock exhibit direction-dependent properties. A classic case is sedimentary rock having bedding planes; properties of this rock can be treated as transversely isotropic. Chong et al. [17] have evaluated the fracture toughness of oil shale and shown its dependence on crack orientation with respect to the bedding planes. Fig. 13 gives fracture toughness dependence on shale grade when measured in one orientation using SCB specimen. In case fracture energy based method is used to find the fracture toughness the conversion of J to K involves the elastic modulus and tensile strength, both of which are direction-dependent. Japanese researchers (e.g. [42]) have regularly used the SCB specimen, although most of the publications are written in Japanese. Kuruppu and Seto [31,32] used the SCB specimen to measure the fracture toughness of rock at elevated temperatures and high confining pressures simulating the in situ conditions of deep buried rock. They used the resistance curve method to determine representative fracture toughness values from small SCB specimens. The crack growth was measured using compliance and an acoustic emission source location method. They found that the resistance curve method gave comparable results with level 2 fracture toughness measured using the ISRM standard method. Figs. 14 and 15 show the variation of Kimachi sandstone at elevated temperatures and confining pressures. A particular note of interest was the observation of the material reaching ‘limiting’ fracture toughness when subjected to high confining pressure. The investigators attributed this to closing of microcracks and other discontinuities within the material under such conditions. Al-Shayea et al. [65] who used Brazilian disk for testing also reported large increases of fracture toughness of a limestone with increasing confining pressure. They also determined that the fracture toughness of most rock increased up to about 200 °C and then started to decrease. Chang et al. [66] performed a detailed study of fracture toughness of granite and marble using a number of different specimens including CB, CCNBD, straight notched SCB and chevron notched SCB. Their results showed that straight cracked SCB gave lower values of KIc compared to those given by CB and CCNBD specimens. However, they reported that chevron notched SCB results were comparable with those produced by CCNBD tests.

4.1. Dynamic fracture toughness Chen et al. [67] and subsequently Dai et al. [68] used the SCB specimen in a Split Hopkinson Pressure Bar (SHPB) apparatus to measure dynamic rock fracture toughness. In this research, Laurentian granite which is an isotropic fine-grained rock was tested. A strain gauge was mounted near the crack tip to determine the fracture initiation time-on-arrival of the incoming stress wave. A laser gauge was used to determine the crack opening displacement. Additional strain gauges were mounted away from the specimen on the incident and transmitted bar to monitor stress waves. Hence the forces on the test specimen could be determined using the incident, transmitted and the reflected strains, as well as the material properties and geometry of the bar. Both dynamic and quasi-static finite element analyses were used to determine the evolution of SIFs. Dai et al. [68] investigated the suitability of both cracked chevron-notched SCB specimen (CCNSCB) and cracked chevronnotched Brazilian Disk (CCNBD) for dynamic fracture tests. They found that the CCNSCB specimen is better as the CCNBD specimen has inherent drawbacks such as the high energy dissipation at loading pins under the dynamic loading conditions. A chevron-notch was introduced to initiate a natural crack. They have given (a) stress intensity factor calibration of CCNSCB specimen that is equally applicable for both static and dynamic fracture toughness measurements and (b) dynamic initiation fracture toughness and dynamic propagation fracture toughness with respect to the loading rate.

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Fig. 14. Fracture toughness of Kimachi sandstone at elevated temperatures [31].

Fig. 15. Variation of Kimachi sandstone with confining pressure. Upper and lower graphs are for short transverse and divider orientations, respectively [31].

In dynamic fracture testing, static force balance on the boundary does not necessarily ensure the stress equilibrium of the entire sample. Therefore, it is necessary to determine the maximum dynamic load that causes fracture initiation. However, Chen et al. [67] have shown that the dynamic fracture toughness can be calculated by substituting the peak far-field load, as measured by the transmitted load, to the quasi-static equation if the dynamic force balance is achieved. They tested Laurenpffiffiffiffiffi pffiffiffiffiffi tian granite and found that the dynamic fracture toughness was 3.47 MPa m with a loading rate of 79.7 GPa m=s. The stapffiffiffiffiffi tic fracture toughness of this material is 1.5 MPa m. This shows that fracture toughness increases substantially under dynamic loading conditions. The investigators have shown that the SCB specimen can be conveniently adapted for testing dynamic fracture properties at very high strain rates. 4.2. Recent efforts and the significance placed on the application of SCB specimen for fracture testing of asphalt mixtures Improving the fracture resistance is paramount as cracking is the main cause of failure of asphalt pavements. Many researchers have used fracture toughness as a measure of fracture resistance, and in an attempt to find a readily adaptable and convenient test method, they have resorted to the SCB test [69,70]. This decision has been supported by the practical consideration that pavement field samples are commonly collected by circular coring methods. The literature that has been published from 2000-present reveals that the United States research community working in the development of stronger asphalt mixtures for highway applications almost unanimously use the SCB specimen for fracture toughness testing. Some of the investigations are briefly summarized below:  Mull et al. [71] used the strain energy release rate method to measure the critical fracture resistance of three asphalt mixtures. Specifically, they considered the fracture energy up to the peak load of the load–deflection plot. The SCB specimen was chosen for testing due to the rigidity that is required when testing visco-elastic and/or visco-plastic materials, in comparison with bend bar, which it replaced as a commonly used test specimen. Furthermore, the SCB specimen was preferable as (a) it can be made from the gyratory compacted cylindrical specimens or as core obtained from the field and, (b) multiple specimens can be obtained from one core. The investigators found that the test gave consistent and reproducible results for Jc, the specific fracture energy, up to the maximum load. However, the fracture resistance was temperature-dependent owing to the visco-elasticity of the mixtures. The test allowed for examination of the cohesive strength of the binder as well as the interfacial strength between the binder and the aggregate. The authors also suggest that it is important to further this research in order to develop a standardized technique for fracture toughness determination of unmodified and modified asphalts. Moreover, durability and fatigue life are both important for the choice of asphalt mixtures, which led to the investigation of fatigue crack propagation behavior using the SCB specimen [72]. This research will help to understand the fracture of asphalt mixtures under cyclic loading caused by vehicular traffic.  Wu et al. [73] also highlighted the advantages of the SCB specimen for fracture mechanics testing of asphalt mixtures. In line with the method used by Mull et al. [71], they used the strain energy release rate method to determine the critical fracture resistance up to crack initiation Jc, by considering the stored energy in the specimen up to the peak load. They

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Fig. 16. Modeling of crack pattern for the asphalt mixture during the SCB test (a) finite element model, (b) crack pattern at point of fracture initiation, (c) crack pattern at visible macrocrack and (d) crack at final load step [77].

reported that larger aggregates, as well as softer binders, improved the fracture resistance. The study suggested that the SCB test is a valuable correlative tool in the evaluation of fracture resistance of asphalt mixtures.  Li and Marasteanu [74] used the SCB specimen to determine the fracture resistance of six asphalt mixtures at low environment temperatures. They also investigated the effect of loading rate on fracture energy. The investigators used the specific fracture energy as the total area under load deflection plot (converted to energy units) divided by the uncracked ligament area at the start of the test. The specimen size used was 150 mm diameter and 25 mm thickness, in comparison with the nominal maximum aggregate size of 12.5 mm. The notch lengths employed were 5 mm, 15 mm and 30 mm, all of which seem to be relatively small in comparison with other specimen dimensions. The span length used was 120 mm making s/r ratio of 0.8. The fracture energy determined in the tests was dependent on the initial notch lengths used; higher fracture energy was shown for specimens having short notches, except for those tested at very low sub-zero temperatures. Experimental results showed a strong dependence of fracture resistance on the test temperature, with about fourfold increase from 30 to 6°C. Also, the specific fracture energy was found to decrease with the increase in loading rate at all three temperatures, and that variation was particularly pronounced at the higher temperature of 6°. The type of binder also had an effect on the fracture energy. Post-failure observation of the specimens revealed that a significant portion of the fracture passed through the aggregate particles in the mixture made with limestone, whereas most fractures in mixture made with granite passed through the interface between the mastic.  Aglan et al. [75] studied the effect of the addition of short polypropylene fiber on the fracture resistance and indirect tensile strength of nanostructured [76] perlite-cementitious super-compounds. The SCB specimen was chosen for the fracture tests in anticipation that its geometry and loading configuration can reveal the degree of interaction between the nanostructured cement binder, perlite particulates and the fiber reinforcement, as well as the cohesion of the binder itself. Short fibers of about 15 mm in length and 45 lm in diameter were used along with the nano-clay binder to make the material. SCB specimens of diameter 152 mm and thickness 25 mm were molded directly along with notch of length a = 12.7, 25.4 and 38.1 mm in three sets of specimens. Fracture energy up to the point of peak load in specimens with various crack lengths was measured and used to determine the critical strain energy release rate. A fourfold increase in fracture resistance was shown to be achieved with a fiber loading (i.e. content) of 3%-by-weight of the mixture. Micrographs revealed failure mechanisms of fiber/matrix interaction, such as fiber bridging, pullout and fiber rupture. These mechanisms aided to dissipate a portion of the crack driving force, leading to a slow fracture process and enhancing the toughness of the composite compounds.  Birgisson et al. [77] presented a comparison between predicted and measured crack pattern development in hot mix asphalt mixtures during common fracture tests. The fracture behavior is predicted using a displacement discontinuity boundary element method to model the microstructure of asphalt mixture. The results were found to be consistent with observed cracking behavior. The fracture tests were done using the SCB specimen without a pre-notch. During testing, a crack initiated at the center-bottom surface of the 3-point loaded specimen, which corresponds to the area of highest bending moment. The crack grew upwards making a macrocrack similar to that of the standard SCB specimen. Furthermore, their numerical model predicted that the highest strain occurs in a restricted zone located at the bottom edge of the specimen where the crack initiated. The numerical simulation gave a glimpse of how the crack growth occurs in a standard SCB specimen made of a composite material such as asphalt (Fig. 16).

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Bitumen materials have a natural tendency to lose strength with crack growth. It is suggested that the fracture mechanics community should investigate the application of tension softening behavior to these materials. While the calculations may be somewhat more tedious, it may prove to yield more consistent results for fracture properties.

5. Conclusions The SCB specimen has been widely used by many researchers throughout the world for fracture toughness testing of mainly rock and asphalt, and to a lesser extent, of other brittle materials. The straight notched SCB specimen has been well developed to measure level 1 and level 2 fracture toughness. The latter includes nonlinearity correction. However, the bulk of its applications in the past have been for the determination of level 1 fracture toughness. This is due to the fact that only a small percentage of applications require it to be accurate to level 2. Most users found it a satisfactory test method with the added advantage of the convenience of specimen preparation and testing. It has also been used for testing the dynamic fracture toughness. While the complete range of SIFs for a straight edge notched SCB specimen in modes I and II were derived during the early stages of specimen development, SIF accuracy was subsequently improved by considering the T stress effect in the case of mode II fracture testing. The SIFs of modes I and II for a comprehensive range of the crack length ratio a/R and the span ratio s/R are now available. Conversely, limited data of SIFs of chevron notched SCB specimens is available, and future work is needed to make that more accurate and reliable. However, the applications of the specimen have not been hindered by the non-availability of SIFs of chevron notched SCB specimen. The SCB specimen, along with the cracked Brazilian disk specimen, has been found by many users to be adaptable to mixed-mode fracture toughness testing. This is achieved either by adjustment of the loading configuration (BD specimen) or by introducing an angled crack (SCB specimen). In mixed-mode fracture testing, the SCB specimen has been found to provide conservative values of fracture toughness due to its T stress being inherently positive as opposed to negative T stress found in most other fracture toughness test specimens. Researchers have used the R-curve method and the tension softening curve as the alternative methods of determining the fracture toughness and these methods are well adaptable to non-linearly behaving rock and other geomaterials. The SCB specimen was developed for testing brittle rock materials and this will continue to be its main application. Over the past decade there has been extensive use of the specimen to determine fracture properties of asphalt used for road pavements. Its applications encompass the determination of fracture toughness, fracture energy and fatigue resistance. 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