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Determination of fracture toughness KIC by using the flattened Brazilian disk specimen for rocks
Engineering Fracture Mechanics 64 (1999) 193±201
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Determination of fracture toughness KIC by using the ¯attened Brazilian disk specimen for rocks Qi-Zhi Wang a,*, Lei Xing b a
Department of Civil Engineering and Applied Mechanics, Sichuan University, Chengdu, Sichuan 610065, People's Republic of China b Hainan Airlines, Hainan, People's Republic of China Received 23 December 1998; accepted 18 June 1999
Abstract The ¯attened Brazilian disk specimen subjected to compression is proposed for testing fracture toughness KIC of quasi-brittle materials such as rocks. Neither crack nor notch is needed for such specimens, the two parallel ¯attens of the specimen are introduced speci®cally for ease of loading. The important condition for crack initiation at the center of the disk during testing is outlined, which is closely linked to the load angle determined by the width of the ¯atten. The key parameter in the formula for calculating fracture toughness is determined by using boundary element method. Tentative test result for a rock is given, the recorded load-displacement plot is used to illustrate the test principle, and the critical load is the local minimum load which corresponds to the unique turning point immediately succeeding the peak load. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Flattened Brazilian disk specimen; Fracture toughness; Rock
1. Introduction Fracture toughness KIC is an important parameter for rocks and other geomaterials; it is especially useful for stability and fragmentation analysis in rock engineering. However, it is rather dicult to prepare rock specimens for fracture toughness testing according to the procedures usually adopted for metals. For example, the electricity discharge machine can not * Corresponding author. E-mail address: [email protected] (Q.-Z. Wang) 0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 6 5 - X
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be used to produce a thin slot in a rock specimen and the fatigue-cracking method is hard to apply for brittle rocks. Recently the International Society for Rock Mechanics (ISRM) has proposed two suggested methods for testing rock fracture toughness KIC using a core-based specimen with a chevron notch [1,2]. For such chevron-notched specimens, the dicult precracking for rock specimens is waived, however, the chevron notch should be introduced, and this alternative con®guration is still not convenient for its preparation, which hinders its wide application in rock engineering practice. To cope with this situation, Guo et al. [3] proposed a new method to test rock fracture toughness using the diametrically compressed Brazilian disk specimens [3], for which the diametrical load is assumed to be uniformly distributed on an arc of angle 2a shown in Fig. 1.The outstanding merit of the method is its convenience in specimen preparation: the specimen does not need any crack or notch in its con®guration, the crack required by the specimen is to be generated automatically during testing. Another advantage of the method is that the critical point for determining fracture toughness is easy to be identi®ed from a test record, and measurement of crack length is unnecessary. Guo's method may be considered as the simplest and most economic one for the determination of fracture toughness of rocks and other brittle materials [3]. However, there are still some setbacks and unresolved points in Guo's method which deserve further research. Firstly, Guo's method neglected the crucial problems: how to guarantee the crack initiation at the center of the disk specimen and how the load angle (2a in Fig. 1) aects location of the crack initiation. Secondly, Guo's solution of the stress intensity factor (SIF) for center cracked disk was inappropriate, they used Green's function for in®nite plate to treat the ®nite domain problem in the disk. Thirdly, in practice, the arc loading of the disk specimen (Fig. 1) is hard to satisfy exactly, and the stress distribution on the load arc can not be uniform; however, it was assumed to be uniform in Guo's analysis. In this paper we propose a modi®ed Brazilian disk specimen for rock fracture toughness testing, the modi®cation is that two parallel planes of equal width are introduced to the
Fig. 1. The Brazilian disk specimen under uniformly distributed diametrical load [3] (The dashed line represents the crack formed during loading, P is the summation of distributed load.)
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195
Brazilian disk specimen (Fig. 2), the ¯attens facilitate easy loading of the specimen. Numerical results of SIF for ¯attened Brazilian disk specimens using the two-dimensional boundary element method (BEM) are presented. The condition for guarantee of crack initiation at the center of the disk is considered, this condition is closely linked to the load angle. Tentative experimental result in testing a typical lime stone by using the ¯attened Brazilian disk specimen is given, the result is in adequate agreement with those by using the ISRM-suggested method [2]. 2. Determination of the load angle for the crack initiation at the center of the Brazilian disk specimen The precondition for rock fracture toughness testing using the Brazilian disk specimen under diametrical compressive loading (Fig. 1) is that the crack should initiate at the disk center during loading and then propagate along the diameter. On the contrary, if the crack initiates at a place other than the center of the loading diameter, e.g. at somewhere near the load arc, the case will become very complicated, since calculation of the SIF and determination of the route of crack propagation all become dicult, hence losing ground for its analysis. In their analysis, Guo et al. assumed a uniformly distributed load along the load arc, this uniformity is not realistic, more than that, they did not consider the important condition for center crack initiation, they did not consider the eect of load angle (2a ) on the location of crack initiation, they chose arbitrarily an angle 2a=108 for their specimen. These unresolved problems will be addressed in the following. According to the analysis by Satoh [4], whose work was based on the fracture criterion by Grith [5] and stress solution for the Brazilian disk by Hondoros [6], the angle sustained by the load arc strongly aects the position of crack initiation (see Appendix A), it will be illustrated in Appendix A that only when the load angle satis®es the condition 2a > 19.58, can
Fig. 2. The ¯attened Brazilian disk specimen (The dashed line represents the crack formed during loading, P is the summation of distributed load.)
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we guarantee center crack initiation for the Brazilian disk specimen under uniform distributed diametrical loading (Fig. 1), this condition should be satis®ed for loading the Brazilian disk specimen in rock fracture toughness test. 3. The ¯attened Brazilian disk specimen and its numerical analysis The arc loading device for the achievement of uniform load distribution on the Brazilian disk (Fig. 1) is only an approximate approach; it is hard to realize actually. According to the theory of contact mechanics [7], the contact force between the circular boundary of specimen and the arc-loading device is not uniformly distributed; instead, it is parabolically distributed. Furthermore, in practice the load angle for the arc-loading device is hard to determine accurately, the load angle is aected by several factors, especially by the mismatch of curvature between the load jig and the specimen. Another disadvantage for the original Brazilian disk is the waste in design and manufacture of loading devices, as for disk specimens of dierent sizes, various arc-loading devices with dierent curvature should be designed and manufactured. To overcome these diculties and setbacks, the ¯attened Brazilian disk specimen (Fig. 2) is proposed, the specimen has two equal-width parallel planes, which are prepared speci®cally for load appliance. The merits for this specimen con®guration over the original one are as follows: no special loading device is required, the load angle is proportional to the width of the ¯atten and hence its determination and adjustment is easy, the traction applied on the plane of the ¯attened Brazilian disk tends to be more uniform than that on the arc of the original disk. Numerical analysis for the ¯attened Brazilian disk specimen is performed by using twodimensional BEM. The analysis was in two aspects: stress analysis and SIF analysis. First, a comparative stress analysis is carried out between ¯attened Brazilian disk and original Brazilian disk, the result is shown in Table 1, from which it can be seen that the stress
Table 1 A comparison of stress distribution on the loading diameter between arc loading (Fig. 1) and plane loading (Fig. 2), the load angle is 2a=208 r/R
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
sy/(P/Rt )
sr/(P/Rt )
arc loading
plane loading
arc loading
plane loading
ÿ3.0857 ÿ0.4573 0.4032 0.7092 0.8385 0.8996 0.9310 0.9476 0.9565 0.9628
ÿ3.0755 ÿ0.4539 0.4043 0.7134 0.8427 0.9039 0.9353 0.9521 0.9602 0.9673
ÿ8.8636 ÿ7.3359 ÿ5.8388 ÿ4.8005 ÿ4.1008 ÿ3.6269 ÿ3.3088 ÿ3.1072 ÿ2.9966 ÿ2.9614
ÿ8.8460 ÿ7.3574 ÿ5.8591 ÿ4.8188 ÿ4.1172 ÿ3.6421 ÿ3.3239 ÿ3.1209 ÿ3.0108 ÿ2.9746
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197
distribution along the loading diameter for the two cases is almost identical, thus the condition for crack initiation at the center of the specimen, i.e. 2a > 19.58, which was derived for the original Brazilian disk (see the Appendix A), can also be adopted to the ¯attened Brazilian disk specimen. Then the SIF of the ¯attened Brazilian disk with a center through crack is also determined by using two-dimensional BEM. In Ref. [3] Guo et al. gave an analytical solution for the SIF of cracked Brazilian disk, they took a certain SIF solution in an in®nite plate as the Green's function, this approach may not be adequate and hence was questioned by other scholars [8]. Guo's solution is only valid when the relative crack length (a/R ) is small, otherwise the error caused by the ®nite boundary cannot be neglected. As for our ¯attened Brazilian disk, it is not possible to obtain an analytical solution, thus we have to resort to numerical method. As stated above the load angle should satisfy the condition 2a > 19.58, so two load angles (2a equals to 208 and 308, respectively) are chosen for present analysis. The SIF solution is put into a dimensionless form and ®tted into a polynormial form as follows. For the load angle 2a=208, the dimensionless SIF of center cracked ¯attened Brazilian disk (Fig. 2, the dotted line represents the crack) is: p f a=R KI Rt=P ÿ4:2892 a=R7 ÿ 26:6765 a=R6 84:9054 a=R5 ÿ 93:0870 a=R4 50:7763 a=R3 ÿ 14:3776 a=R2 2:7408 a=R:
is:
1
For the load angle 2a=308, the dimensionless SIF of center cracked ¯attened Brazilian disk p f a=R KI Rt=P ÿ33:9811 a=R7 ÿ 128:5613 a=R6 189:8983 a=R5 ÿ 146:3809 a=R4 64:0804 a=R3 ÿ 15:7996 a=R2 2:7115 a=R,
2
where f is dimensionless SIF, a/R is the relative crack length, a half the crack length, R the radius of the disk, t thickness, P total load (summation of distributed load), KI is mode-I SIF. The error of curve ®tting for these two formulas is less than 0.5%. The SIF solutions for the two load angles were plotted in two ®gures respectively, it was found that basically these ®gures are similar to the shape of a bell, the numerical trend is ascending±maximum± descending. Since only the maximum values of the dimensionless SIF solutions are of our main concern, which will be used in the formula for the fracture toughness test, only a schematic presentation of these two ®gures is shown in Fig. 3. It can be derived by the numerical calculation that for 2a=208, when a/R = 0.8, f gets its maximum value fmax=0.78; and for 2a=308, when a/R = 0.73, fmax=0.58. These maximum dimensionless SIF values are essential parameters for the rock fracture toughness test using the ¯attened Brazilian disk specimen.
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Fig. 3. A schematic presentation for the dimensionless SIF f vs dimensionless crack length a/R.
4. The principle of rock fracture toughness test using ¯attened Brazilian disk specimen When 2a > 19.58, it can be expected that a crack can be initiated at the center of the specimen, then the crack extends along the diameter. As shown in Fig. 3, the value of the SIF gradually increases from zero, represented by point a, which corresponds to the crack initiation, to the maximum, represented by point b, where fmax is obtained, after that f decreases until ®nal breakage of the disk, represented by point c. In the ®rst region(ab ), dimensionless SIF f increases with increment of relative crack length a/R, this region has unstable crack growth, because the crack will go on extension even when the load is held constant. In the second region (bc ), after achievement of fmax at point b, f decreases with increment of relative crack length a/R, this region has stable crack growth, since the crack will stop extension if the load is not increased, only when the applied load is increased can the crack extend further. Point b is the turning point between unstable and stable regions, this point b corresponds to the local minimum load immediately succeeding the peak load, as shown in the test record (Fig. 4), this point is chosen as the critical point, which can be identi®ed very easily in a test record. The reason for this choice can be explained brie¯y as follows. For brittle materials such as rock, fracture toughness KIC can be considered to be a material constant, so basically any point during crack extension can be used to determine fracture toughness if the current load and the crack length are known, however it is better to choose point b as the critical point, since this point has maximum value fmax which can be determined beforehand for any prescribed specimen geometry, there is no necessity to measure the critical crack length, also convenient is the determination of the critical load, which is the local minimum load occurred right after the peak load (see Fig. 4). The crack initially extends unstably, it turns to arrest at the critical point, and subsequent development of the crack is stable. This kind of specimen behavior is very unique, and is advantageous for fracture toughness test. The formula to calculate fracture toughness KIC using local minimum load as the critical point is as follows
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199
Fig. 4. A typical test record of P±v curve by using the ¯attened Brazilian disk specimen.
Pmin KIC p fmax , Rt
3
where Pmin is the local minimum load which can be identi®ed directly from the test record, R and t are radius and thickness of the specimen, respectively, fmax is the maximum dimensionless SIF, which is given in the previous section for two load angles, respectively. 5. Experimental results A MTS 810 model test machine is used to conduct fracture toughness measurement for a Chongqing lime stone, the elastic modulus for this rock is 6.7 GPa, the Poisson's ratio is 0.236. The displacement-controlled testing manner is adopted, the loading rate is 0.18 mm/min. Specimen diameter is 100 mm, the load angle is 308, and hence Eq. (3) becomes Pmin KIC p 0:58 Rt
2a 308
4
Figure 4 is a typical test record of the P±v plot, where P is the total applied load and v the total diametrical displacement. It can be seen from this plot that the loading process can be divided into three stages: stage 1(oa ) represents elastic deformation of the disk, this stage ends with the peak load, i.e. point a; stage 2 (ab ) is for unstable crack extension, at the peak load, the crack initiate at the center of the specimen, then the crack develops unstably until the load drops to the local minimum load, i.e. point b, point b is the turning point, which separates the unstable and stable region; this point b is chosen as the critical point in the test. Stage 3 (bc ) is for stable crack extension, beginning from point b, the load must increase for further crack growth until complete breakage of the specimen. The ups and downs towards the test end may
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be caused by subcrack formation near the contact area, these irregularities do not aect test results, since the critical point b is in advance of their happening. p The fracture toughness of a Chongqing lime stone is measured to be KIC=1.25 MPa m using the ¯attened pBrazilian disk specimen, this test result is in good agreement with the value(1.26 MPa m) tested with chevron-notched Brazilian disk specimen according to the ISRM suggested method [2].
6. Conclusion The ¯attened Brazilian disk specimen (Fig. 2) for rock fracture toughness test is a modi®cation of Brazilian disk specimen proposed by Guo et al. [3]. Neither crack nor notch is required for the specimen. According to the analysis, when the load angle 2a > 19.58, the crack is to be initiated at the center of the disk, which occurs at the peak load, the initiated crack will propagate along the diameter, the propagation is ®rst unstably and then stably, the turning point between unstable and stable crack propagation is chosen as the critical point to calculate fracture toughness, this critical point corresponds to the local minimum load, which is immediately succeeding the peak load. The introduction of two parallel planes in our specimen waives the unreliable arc-loading device, which is required by the un¯attened specimen, and makes loading simpler and more uniform. The local minimum load Pmin occurs right after the peak load and is very easy to be identi®ed from the test record (point b in Fig. 4), it is used in Eq. (3) to calculate fracture toughness KIC, Pmin is matched with maximum dimensionless SIF fmax, which is obtained by numerical analysis using BEM. fmax=0.78 for load angle 2a=208, and fmax=0.58 for 2a=308. Test result of fracture toughness for a Chongqing lime stone using ¯attened Brazilian disk specimen is in adequate agreement with that obtained by using the ISRM suggested method [2], however the con®guration of the ¯attened Brazilian disk specimen is much simpler than the chevron-notched specimen, the eectiveness and reliability of the new test method for rock fracture test is demonstrated.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Project No. 19872046). Appendix A. The condition for guarantee of crack initiation at the center of the Brazilian disk under uniform arc loading [4] Grith [5] considered that in a two-dimensional elastic body there are a lot of very ¯at elliptical cavities lying in all directions, the ratio of minor to major axis of the elliptical hole is nearly zero. Speci®cally we think this situation is true for rocks and there are small elliptical cavities lying in all radial directions in a Brazilian disk of rock. When an elliptical hole is
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201
under the maximum principal stress s1 along the minor axis and minimum principal stress s3 along the major axis, the maximum tensile stress sG ' at the edge of the elliptical hole is determined by Grith [5], it was classi®ed in two conditions: (1) 3s1+s3 r0, (2) 3s1+s3 < 0, using s1 and s3 two sG ' expressions were given, respectively, in Ref. [4]. Hondros [6] gave out the formulas for the principal stress sy and sr in a Brazilian disk under uniform arc loading, sy and sr are the maximum principal stress and the minimum principal stress, respectively. It is easy to show that for this loading condition we have 3s1+s3 < 0. Combining the theoretical results of Grith [5] and Hondros [6], the maximum tensile stress sG ' at the edge of the elliptical hole, which is both very small and ¯at and exists in the Brazilian disk, is given by Satoh [4], this most signi®cant result(i.e. Eq. (13) of Ref [4]) is now reproduced as below " #2 " ( )#ÿ1 2 2 P 1 f1 ÿ r=R gsin2a 1 r=R arctan tana 5 sG0 pRt 4sina 1 ÿ 2 r=R2 cos2a r=R4 1 ÿ r=R2 Where r denotes the radial location, R the radius of the Brazilian disk, t thickness, 2a the load angle and P the total load. In order to determine the location for crack initiation in a Brazilian disk, the location of maximum value of sG ' should be known. It is shown by numerical calculation with Eq. (5) that when the load angle 2a r0.32, (sG ')max occurs at the disk center, and when 2a < 0.32, (sG ')max does not occur at the disk center, it takes place near the outer boundary(r/R > 0.65), where r is the location of (sG ')max. So, 2a=0.32 (i.e. 19.58) is the critical value, across which there is a discontinuous and abrupt change for the location of (sG ')max, a change from the disk center to certain place near its boundary. References [1] ISRM. Suggested methods for determining the fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1988;25:71±96 [co-ordinator: F Ouchterlony]. [2] ISRM. Suggested methods for determining mode I fracture toughness using cracked chevron notched Brazilian disk (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 1995;32:57±64 [co-ordinator: RJ Fowell]. [3] Guo H, Aziz NI, Schmidt LC. Rock fracture toughness determination by the Brazilian test. Eng Geol 1993;33:177±88. [4] Satoh Y. Position and load of failure in Brazilian test, a numerical analysis by Grith criterion. J Soc Mater Sci Jpn 1987;36:1219±24. [5] Grith AA. The theory of rupture. In: Proc. First Int. Congr. Appl. Mech., 1924. p. 56±63. [6] Hondros G. The evaluation of poisson's ratio and the modulus of materials of a low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Aust J Appl Sci 1959;10:243±68. [7] Johnson KL. Contact mechanics. Cambridge University Press, 1985. [8] Zhao XL, Fowell RJ, Roegiers J-C, Xu C. Discussion: Rock fracture toughness determination by the Brazilian test, by H. Guo, N.I. Aziz and L. Schmidt. Eng Geol 1993;38:181±4.
www.elsevier.com/locate/engfracmech
Determination of fracture toughness KIC by using the ¯attened Brazilian disk specimen for rocks Qi-Zhi Wang a,*, Lei Xing b a
Department of Civil Engineering and Applied Mechanics, Sichuan University, Chengdu, Sichuan 610065, People's Republic of China b Hainan Airlines, Hainan, People's Republic of China Received 23 December 1998; accepted 18 June 1999
Abstract The ¯attened Brazilian disk specimen subjected to compression is proposed for testing fracture toughness KIC of quasi-brittle materials such as rocks. Neither crack nor notch is needed for such specimens, the two parallel ¯attens of the specimen are introduced speci®cally for ease of loading. The important condition for crack initiation at the center of the disk during testing is outlined, which is closely linked to the load angle determined by the width of the ¯atten. The key parameter in the formula for calculating fracture toughness is determined by using boundary element method. Tentative test result for a rock is given, the recorded load-displacement plot is used to illustrate the test principle, and the critical load is the local minimum load which corresponds to the unique turning point immediately succeeding the peak load. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Flattened Brazilian disk specimen; Fracture toughness; Rock
1. Introduction Fracture toughness KIC is an important parameter for rocks and other geomaterials; it is especially useful for stability and fragmentation analysis in rock engineering. However, it is rather dicult to prepare rock specimens for fracture toughness testing according to the procedures usually adopted for metals. For example, the electricity discharge machine can not * Corresponding author. E-mail address: [email protected] (Q.-Z. Wang) 0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 6 5 - X
194
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be used to produce a thin slot in a rock specimen and the fatigue-cracking method is hard to apply for brittle rocks. Recently the International Society for Rock Mechanics (ISRM) has proposed two suggested methods for testing rock fracture toughness KIC using a core-based specimen with a chevron notch [1,2]. For such chevron-notched specimens, the dicult precracking for rock specimens is waived, however, the chevron notch should be introduced, and this alternative con®guration is still not convenient for its preparation, which hinders its wide application in rock engineering practice. To cope with this situation, Guo et al. [3] proposed a new method to test rock fracture toughness using the diametrically compressed Brazilian disk specimens [3], for which the diametrical load is assumed to be uniformly distributed on an arc of angle 2a shown in Fig. 1.The outstanding merit of the method is its convenience in specimen preparation: the specimen does not need any crack or notch in its con®guration, the crack required by the specimen is to be generated automatically during testing. Another advantage of the method is that the critical point for determining fracture toughness is easy to be identi®ed from a test record, and measurement of crack length is unnecessary. Guo's method may be considered as the simplest and most economic one for the determination of fracture toughness of rocks and other brittle materials [3]. However, there are still some setbacks and unresolved points in Guo's method which deserve further research. Firstly, Guo's method neglected the crucial problems: how to guarantee the crack initiation at the center of the disk specimen and how the load angle (2a in Fig. 1) aects location of the crack initiation. Secondly, Guo's solution of the stress intensity factor (SIF) for center cracked disk was inappropriate, they used Green's function for in®nite plate to treat the ®nite domain problem in the disk. Thirdly, in practice, the arc loading of the disk specimen (Fig. 1) is hard to satisfy exactly, and the stress distribution on the load arc can not be uniform; however, it was assumed to be uniform in Guo's analysis. In this paper we propose a modi®ed Brazilian disk specimen for rock fracture toughness testing, the modi®cation is that two parallel planes of equal width are introduced to the
Fig. 1. The Brazilian disk specimen under uniformly distributed diametrical load [3] (The dashed line represents the crack formed during loading, P is the summation of distributed load.)
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195
Brazilian disk specimen (Fig. 2), the ¯attens facilitate easy loading of the specimen. Numerical results of SIF for ¯attened Brazilian disk specimens using the two-dimensional boundary element method (BEM) are presented. The condition for guarantee of crack initiation at the center of the disk is considered, this condition is closely linked to the load angle. Tentative experimental result in testing a typical lime stone by using the ¯attened Brazilian disk specimen is given, the result is in adequate agreement with those by using the ISRM-suggested method [2]. 2. Determination of the load angle for the crack initiation at the center of the Brazilian disk specimen The precondition for rock fracture toughness testing using the Brazilian disk specimen under diametrical compressive loading (Fig. 1) is that the crack should initiate at the disk center during loading and then propagate along the diameter. On the contrary, if the crack initiates at a place other than the center of the loading diameter, e.g. at somewhere near the load arc, the case will become very complicated, since calculation of the SIF and determination of the route of crack propagation all become dicult, hence losing ground for its analysis. In their analysis, Guo et al. assumed a uniformly distributed load along the load arc, this uniformity is not realistic, more than that, they did not consider the important condition for center crack initiation, they did not consider the eect of load angle (2a ) on the location of crack initiation, they chose arbitrarily an angle 2a=108 for their specimen. These unresolved problems will be addressed in the following. According to the analysis by Satoh [4], whose work was based on the fracture criterion by Grith [5] and stress solution for the Brazilian disk by Hondoros [6], the angle sustained by the load arc strongly aects the position of crack initiation (see Appendix A), it will be illustrated in Appendix A that only when the load angle satis®es the condition 2a > 19.58, can
Fig. 2. The ¯attened Brazilian disk specimen (The dashed line represents the crack formed during loading, P is the summation of distributed load.)
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we guarantee center crack initiation for the Brazilian disk specimen under uniform distributed diametrical loading (Fig. 1), this condition should be satis®ed for loading the Brazilian disk specimen in rock fracture toughness test. 3. The ¯attened Brazilian disk specimen and its numerical analysis The arc loading device for the achievement of uniform load distribution on the Brazilian disk (Fig. 1) is only an approximate approach; it is hard to realize actually. According to the theory of contact mechanics [7], the contact force between the circular boundary of specimen and the arc-loading device is not uniformly distributed; instead, it is parabolically distributed. Furthermore, in practice the load angle for the arc-loading device is hard to determine accurately, the load angle is aected by several factors, especially by the mismatch of curvature between the load jig and the specimen. Another disadvantage for the original Brazilian disk is the waste in design and manufacture of loading devices, as for disk specimens of dierent sizes, various arc-loading devices with dierent curvature should be designed and manufactured. To overcome these diculties and setbacks, the ¯attened Brazilian disk specimen (Fig. 2) is proposed, the specimen has two equal-width parallel planes, which are prepared speci®cally for load appliance. The merits for this specimen con®guration over the original one are as follows: no special loading device is required, the load angle is proportional to the width of the ¯atten and hence its determination and adjustment is easy, the traction applied on the plane of the ¯attened Brazilian disk tends to be more uniform than that on the arc of the original disk. Numerical analysis for the ¯attened Brazilian disk specimen is performed by using twodimensional BEM. The analysis was in two aspects: stress analysis and SIF analysis. First, a comparative stress analysis is carried out between ¯attened Brazilian disk and original Brazilian disk, the result is shown in Table 1, from which it can be seen that the stress
Table 1 A comparison of stress distribution on the loading diameter between arc loading (Fig. 1) and plane loading (Fig. 2), the load angle is 2a=208 r/R
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
sy/(P/Rt )
sr/(P/Rt )
arc loading
plane loading
arc loading
plane loading
ÿ3.0857 ÿ0.4573 0.4032 0.7092 0.8385 0.8996 0.9310 0.9476 0.9565 0.9628
ÿ3.0755 ÿ0.4539 0.4043 0.7134 0.8427 0.9039 0.9353 0.9521 0.9602 0.9673
ÿ8.8636 ÿ7.3359 ÿ5.8388 ÿ4.8005 ÿ4.1008 ÿ3.6269 ÿ3.3088 ÿ3.1072 ÿ2.9966 ÿ2.9614
ÿ8.8460 ÿ7.3574 ÿ5.8591 ÿ4.8188 ÿ4.1172 ÿ3.6421 ÿ3.3239 ÿ3.1209 ÿ3.0108 ÿ2.9746
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distribution along the loading diameter for the two cases is almost identical, thus the condition for crack initiation at the center of the specimen, i.e. 2a > 19.58, which was derived for the original Brazilian disk (see the Appendix A), can also be adopted to the ¯attened Brazilian disk specimen. Then the SIF of the ¯attened Brazilian disk with a center through crack is also determined by using two-dimensional BEM. In Ref. [3] Guo et al. gave an analytical solution for the SIF of cracked Brazilian disk, they took a certain SIF solution in an in®nite plate as the Green's function, this approach may not be adequate and hence was questioned by other scholars [8]. Guo's solution is only valid when the relative crack length (a/R ) is small, otherwise the error caused by the ®nite boundary cannot be neglected. As for our ¯attened Brazilian disk, it is not possible to obtain an analytical solution, thus we have to resort to numerical method. As stated above the load angle should satisfy the condition 2a > 19.58, so two load angles (2a equals to 208 and 308, respectively) are chosen for present analysis. The SIF solution is put into a dimensionless form and ®tted into a polynormial form as follows. For the load angle 2a=208, the dimensionless SIF of center cracked ¯attened Brazilian disk (Fig. 2, the dotted line represents the crack) is: p f a=R KI Rt=P ÿ4:2892 a=R7 ÿ 26:6765 a=R6 84:9054 a=R5 ÿ 93:0870 a=R4 50:7763 a=R3 ÿ 14:3776 a=R2 2:7408 a=R:
is:
1
For the load angle 2a=308, the dimensionless SIF of center cracked ¯attened Brazilian disk p f a=R KI Rt=P ÿ33:9811 a=R7 ÿ 128:5613 a=R6 189:8983 a=R5 ÿ 146:3809 a=R4 64:0804 a=R3 ÿ 15:7996 a=R2 2:7115 a=R,
2
where f is dimensionless SIF, a/R is the relative crack length, a half the crack length, R the radius of the disk, t thickness, P total load (summation of distributed load), KI is mode-I SIF. The error of curve ®tting for these two formulas is less than 0.5%. The SIF solutions for the two load angles were plotted in two ®gures respectively, it was found that basically these ®gures are similar to the shape of a bell, the numerical trend is ascending±maximum± descending. Since only the maximum values of the dimensionless SIF solutions are of our main concern, which will be used in the formula for the fracture toughness test, only a schematic presentation of these two ®gures is shown in Fig. 3. It can be derived by the numerical calculation that for 2a=208, when a/R = 0.8, f gets its maximum value fmax=0.78; and for 2a=308, when a/R = 0.73, fmax=0.58. These maximum dimensionless SIF values are essential parameters for the rock fracture toughness test using the ¯attened Brazilian disk specimen.
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Fig. 3. A schematic presentation for the dimensionless SIF f vs dimensionless crack length a/R.
4. The principle of rock fracture toughness test using ¯attened Brazilian disk specimen When 2a > 19.58, it can be expected that a crack can be initiated at the center of the specimen, then the crack extends along the diameter. As shown in Fig. 3, the value of the SIF gradually increases from zero, represented by point a, which corresponds to the crack initiation, to the maximum, represented by point b, where fmax is obtained, after that f decreases until ®nal breakage of the disk, represented by point c. In the ®rst region(ab ), dimensionless SIF f increases with increment of relative crack length a/R, this region has unstable crack growth, because the crack will go on extension even when the load is held constant. In the second region (bc ), after achievement of fmax at point b, f decreases with increment of relative crack length a/R, this region has stable crack growth, since the crack will stop extension if the load is not increased, only when the applied load is increased can the crack extend further. Point b is the turning point between unstable and stable regions, this point b corresponds to the local minimum load immediately succeeding the peak load, as shown in the test record (Fig. 4), this point is chosen as the critical point, which can be identi®ed very easily in a test record. The reason for this choice can be explained brie¯y as follows. For brittle materials such as rock, fracture toughness KIC can be considered to be a material constant, so basically any point during crack extension can be used to determine fracture toughness if the current load and the crack length are known, however it is better to choose point b as the critical point, since this point has maximum value fmax which can be determined beforehand for any prescribed specimen geometry, there is no necessity to measure the critical crack length, also convenient is the determination of the critical load, which is the local minimum load occurred right after the peak load (see Fig. 4). The crack initially extends unstably, it turns to arrest at the critical point, and subsequent development of the crack is stable. This kind of specimen behavior is very unique, and is advantageous for fracture toughness test. The formula to calculate fracture toughness KIC using local minimum load as the critical point is as follows
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Fig. 4. A typical test record of P±v curve by using the ¯attened Brazilian disk specimen.
Pmin KIC p fmax , Rt
3
where Pmin is the local minimum load which can be identi®ed directly from the test record, R and t are radius and thickness of the specimen, respectively, fmax is the maximum dimensionless SIF, which is given in the previous section for two load angles, respectively. 5. Experimental results A MTS 810 model test machine is used to conduct fracture toughness measurement for a Chongqing lime stone, the elastic modulus for this rock is 6.7 GPa, the Poisson's ratio is 0.236. The displacement-controlled testing manner is adopted, the loading rate is 0.18 mm/min. Specimen diameter is 100 mm, the load angle is 308, and hence Eq. (3) becomes Pmin KIC p 0:58 Rt
2a 308
4
Figure 4 is a typical test record of the P±v plot, where P is the total applied load and v the total diametrical displacement. It can be seen from this plot that the loading process can be divided into three stages: stage 1(oa ) represents elastic deformation of the disk, this stage ends with the peak load, i.e. point a; stage 2 (ab ) is for unstable crack extension, at the peak load, the crack initiate at the center of the specimen, then the crack develops unstably until the load drops to the local minimum load, i.e. point b, point b is the turning point, which separates the unstable and stable region; this point b is chosen as the critical point in the test. Stage 3 (bc ) is for stable crack extension, beginning from point b, the load must increase for further crack growth until complete breakage of the specimen. The ups and downs towards the test end may
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be caused by subcrack formation near the contact area, these irregularities do not aect test results, since the critical point b is in advance of their happening. p The fracture toughness of a Chongqing lime stone is measured to be KIC=1.25 MPa m using the ¯attened pBrazilian disk specimen, this test result is in good agreement with the value(1.26 MPa m) tested with chevron-notched Brazilian disk specimen according to the ISRM suggested method [2].
6. Conclusion The ¯attened Brazilian disk specimen (Fig. 2) for rock fracture toughness test is a modi®cation of Brazilian disk specimen proposed by Guo et al. [3]. Neither crack nor notch is required for the specimen. According to the analysis, when the load angle 2a > 19.58, the crack is to be initiated at the center of the disk, which occurs at the peak load, the initiated crack will propagate along the diameter, the propagation is ®rst unstably and then stably, the turning point between unstable and stable crack propagation is chosen as the critical point to calculate fracture toughness, this critical point corresponds to the local minimum load, which is immediately succeeding the peak load. The introduction of two parallel planes in our specimen waives the unreliable arc-loading device, which is required by the un¯attened specimen, and makes loading simpler and more uniform. The local minimum load Pmin occurs right after the peak load and is very easy to be identi®ed from the test record (point b in Fig. 4), it is used in Eq. (3) to calculate fracture toughness KIC, Pmin is matched with maximum dimensionless SIF fmax, which is obtained by numerical analysis using BEM. fmax=0.78 for load angle 2a=208, and fmax=0.58 for 2a=308. Test result of fracture toughness for a Chongqing lime stone using ¯attened Brazilian disk specimen is in adequate agreement with that obtained by using the ISRM suggested method [2], however the con®guration of the ¯attened Brazilian disk specimen is much simpler than the chevron-notched specimen, the eectiveness and reliability of the new test method for rock fracture test is demonstrated.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Project No. 19872046). Appendix A. The condition for guarantee of crack initiation at the center of the Brazilian disk under uniform arc loading [4] Grith [5] considered that in a two-dimensional elastic body there are a lot of very ¯at elliptical cavities lying in all directions, the ratio of minor to major axis of the elliptical hole is nearly zero. Speci®cally we think this situation is true for rocks and there are small elliptical cavities lying in all radial directions in a Brazilian disk of rock. When an elliptical hole is
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under the maximum principal stress s1 along the minor axis and minimum principal stress s3 along the major axis, the maximum tensile stress sG ' at the edge of the elliptical hole is determined by Grith [5], it was classi®ed in two conditions: (1) 3s1+s3 r0, (2) 3s1+s3 < 0, using s1 and s3 two sG ' expressions were given, respectively, in Ref. [4]. Hondros [6] gave out the formulas for the principal stress sy and sr in a Brazilian disk under uniform arc loading, sy and sr are the maximum principal stress and the minimum principal stress, respectively. It is easy to show that for this loading condition we have 3s1+s3 < 0. Combining the theoretical results of Grith [5] and Hondros [6], the maximum tensile stress sG ' at the edge of the elliptical hole, which is both very small and ¯at and exists in the Brazilian disk, is given by Satoh [4], this most signi®cant result(i.e. Eq. (13) of Ref [4]) is now reproduced as below " #2 " ( )#ÿ1 2 2 P 1 f1 ÿ r=R gsin2a 1 r=R arctan tana 5 sG0 pRt 4sina 1 ÿ 2 r=R2 cos2a r=R4 1 ÿ r=R2 Where r denotes the radial location, R the radius of the Brazilian disk, t thickness, 2a the load angle and P the total load. In order to determine the location for crack initiation in a Brazilian disk, the location of maximum value of sG ' should be known. It is shown by numerical calculation with Eq. (5) that when the load angle 2a r0.32, (sG ')max occurs at the disk center, and when 2a < 0.32, (sG ')max does not occur at the disk center, it takes place near the outer boundary(r/R > 0.65), where r is the location of (sG ')max. So, 2a=0.32 (i.e. 19.58) is the critical value, across which there is a discontinuous and abrupt change for the location of (sG ')max, a change from the disk center to certain place near its boundary. References [1] ISRM. Suggested methods for determining the fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1988;25:71±96 [co-ordinator: F Ouchterlony]. [2] ISRM. Suggested methods for determining mode I fracture toughness using cracked chevron notched Brazilian disk (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 1995;32:57±64 [co-ordinator: RJ Fowell]. [3] Guo H, Aziz NI, Schmidt LC. Rock fracture toughness determination by the Brazilian test. Eng Geol 1993;33:177±88. [4] Satoh Y. Position and load of failure in Brazilian test, a numerical analysis by Grith criterion. J Soc Mater Sci Jpn 1987;36:1219±24. [5] Grith AA. The theory of rupture. In: Proc. First Int. Congr. Appl. Mech., 1924. p. 56±63. [6] Hondros G. The evaluation of poisson's ratio and the modulus of materials of a low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Aust J Appl Sci 1959;10:243±68. [7] Johnson KL. Contact mechanics. Cambridge University Press, 1985. [8] Zhao XL, Fowell RJ, Roegiers J-C, Xu C. Discussion: Rock fracture toughness determination by the Brazilian test, by H. Guo, N.I. Aziz and L. Schmidt. Eng Geol 1993;38:181±4.