Effect of Tensile Strength of Rock on Tensile...

ISSN 1062-7391, Journal of Mining Science, 2016, Vol. 52, No. 4, pp. 647–661. © Pleiades Publishing, Ltd., 2016.

647

_________________________________ GEOMECHANICS _______________________________ ____________________________________________________________________________________________________________________________________ ___________________________________________________________________________________________________________________________

Effect of Tensile Strength of Rock on Tensile Fracture Toughness Using Experimental Test and PFC2D Simulation1

H. Haeria*, V. Sarfarazib, A. Hedayatc, and A. Tabaroeid aYoung Researchers and Elite Club, Bafgh Branch, Islamic Azad University, Bafgh, Iran

*e-mail: [email protected]; [email protected] bHamedan University of Technology, Hamedan

cDepartment of Civil and Environmental Engineering, Colorado School of Mines, Golden, CO 80401 USA dDepartment of Civil, Science and Research Branch, Islamic Azad University, Tehran, Iran

University, Tehran, Iran

Received December 8, 2016

Abstract—The effect of tensile strength on the tensile fracture toughness of rock like specimen was studied in this paper. Brazilian test was done to determine tensile strength of material. A compression to tensile load transforming (CTT) device was developed for determination of mode I fracture toughness of concrete. Also particle flow code (PFC) was used for validation of the experimental outputs. Three concrete slabs with different tensile strength were prepared for investigation of the effects of tensile strength on the fracture toughness. The samples were made from a mixture of water, fine sand and cement with different ratio. These samples were installed in CTT device. A 30-tons hydraulic load cell applied compressive loading to CTT end plates with a constant pressure of 0.02 MPa per second. Compressive loading was converted to tensile stress on the sample because of the overall test design. The results show Fracture toughness has a close relationship with tensile strength of concrete so it increases with increasing the tensile strength. In constant join length, the angle of crack growth related to normal load was decreased with increasing the grain size. Numerical simulation shows that failure pattern and fracture toughness was nearly similar to experimental results. Finally, it can be concluded that CTT device was capable for determination of fracture toughness of concrete.

Keywords: PFC2D, mode I fracture toughness, compression to tensile load transforming device.

DOI: 10.1134/S1062739116041046

INTRODUCTION Linear elastic fracture mechanics has been developed to describe crack growth and fracture within

a material under essentially linear elastic conditions. It is based on the assumption that the influence of applied loads upon crack extension can be represented in terms of certain parameters that characterize the stress – strain intensity near the crack tip. The introduction of the fracture mechanics approach to engineering geology and rock engineering has led to the development of rock fracture mechanics, which mainly refers to the discrete initiation and propagation of an individual crack or cracks in geological materials subjected to a particular stress field [1]. The explosion in rock fracture mechanics research has touched many diverse areas including blasting , hydraulic fracturing and in situ stress determination, mechanical fragmentation, rock slope analysis, earthquake mechanics, earthquake prediction, plate tectonics, magmatic intrusions, hot dry rock geo-thermal energy extraction, fluid trans port proper ties of fracturing rock masses, propagating oceanic rifts, crevasse penetration and other glaciological problems, the development of steeply dipping extension fractures that are nearly ubiquitous at the earth’s surface and are formed through folding, up warping and rifting and the modeling of time-dependent rock failure [2, 3].

A fundamental fundamental feature of rock fracture mechanics lies in its ability to establish the relations hip between rock fracture strength to the geometry of a crack or cracks and the fracture toughness, the most fundamental parameter in fracture mechanics describing resistance of a material

1The article is published in the original.

648 HAERI et al.

JOURNAL OF MINING SCIENCE Vol. 52 No. 4 2016

to crack propagation. It follows that for quasi -brittle geological materials, crack propagation is the major cause of material failure in many cases. Thus, assessment of fracture toughness is important to the understanding of behavior of structures involving geological materials. In addition, rock fracture toughness has been applied as a parameter for classification of rock materials, an index for rock fragmentation process and a material property in the interpretation of geological features and in stability analysis of rock structure s, as well as in modeling of fracturing in rock [4].

According to the applied stress condition, a crack propagates under the three basic failure modes or the mixed-mode condition. Mode I is the tensile opening mode, in which the crack faces separate in a direction norm al to the plane of the crack. Mode II is the in-plane sliding or shear mode, in which the crack faces are mutually sheared in the direction norm al to the crack front. Mode III is the tearing or out of plane mode, in which the crack faces are sheared parallel to the crack front (Fig. 1).

There are a number of methods and test specimens for obtaining the fracture toughness of brittle materials for all mode mixtures from pure mode I to pure mode II [5–51]. Among many different testing methods for rock fracture toughness, the International Society for Rock Mechanics (ISRM) suggested the chevron bend (CB) and the short-rod (SR) specimens in 1988 and cracked chevron -notched Brazilian disc (CCNBD) specimen in 1995. But it should be noted that most studies are relevant to mode I (opening mode) with some studies on mode II (in-plane shear mode) or the mixed-m ode. Considering specimen geometries, tensile (mode I) cracks are induce during CB and SR tests. In addition, it has been reported that they are not appropriate for testing the fracture toughness of rock under mode II or mixed-mode cases [52–55].

Compact tension test [56] is other method for determination of fracture toughness of material. The fracture toughness of material is determined by applying the tensile load on the crack surface (Fig. 2).

Fig. 1. Three basic modes of crack propagation.

Fig. 2. Specimen consisted of an edge crack under tensile loading.

EFFECT OF TENSILE STRENGTH OF ROCK ON TENSILE FRACTURE TOUGHNESS 649


The fracture toughness was determined by follow equation:

,aKI πσ= (1)

where KI is the mode -I stress intensity factor, σ is the far field stress at failure, a is the crack length. There are also some numerical simulations for investigation of fracture toughness of brittle

material [57–62]. In this paper a new test method was developed experimentally/numerically for determination of

fracture toughness of concrete. Also the effect of tensile strength on the fracture toughness of rock like specimen was determined.

1. EXPERIMENTAL SET-UP

1.1. Modeling Material and Its Physical Properties

For investigation of the effect of tensile strength on the fracture toughness of concrete, three concrete samples with different tensile strength were prepared. The samples were prepared from a mixture of the water, fine sand and cement. Uniaxial compression and indirect tensile strengths of the intact material was tested in order to control the variability of material. The uniaxial compressive strength (UCS) of the model material is measured on fabricated cylindrical specimens with 56 mm in diameter and 112 mm in length. The indirect tensile strength of the material is determined by the Brazilian test using fabricated solid discs 56 mm in diameter and 28 mm in thickness. The testing procedure of uniaxial compressive strength test and the Brazilian test complies with the ASTM D2938-86 [63] and ASTM C496-71 [64], codes respectively. The specification of grain size, mixture ratio and their mechanical properties has been depicted in Table 1. Figure 3 shows the failure pattern occurred in Brazilian test. As can be seen, the curvature of tensile failure surface was decreased by decreasing the particles size. The maximum curvature occurs in sample 1 which has smaller grain size. Table 1. Specification of grain size, mixture ratio and their mechanical properties

Sample no. Grain size, mm Cement/grain/

water Density, kg/m3

Wave velocity,

(m/s

Uniaxial

strength,

MPa)

Tensile strength,

MPa

1 0.3–2.5 1/0.5/1 2100 3075 43.7 4.1

2 0.05–0.5 0.75/0.5/1 1660 2786 29.2 2.6

3 0.005–0.2 0.5/1/1 1900 3500 53.3 4.6

Fig. 3. Failure patterns in samples with grains size of a) 0.005–0.2 mm, b) 0.05–0.5 mm and c) 0.3–2.5 mm.

650 HAERI et al.


1.2. The Technique in Preparing the Jointed Specimens The material mixture is prepared by mixing fine sand and cement in a blender; the mixture is then

poured into a Plexiglasbox with internal dimension of 19×19×6 cm. The main box consists of four sheets bolted together and of two rectangle blocks with internal dimension of 2×6×6 cm were fixed at the left and right sides of the box (Fig. 4a). A PMMA plates with 1/6 inch thick, having slitare placed at the bottom of the main box. The width of slits is 0.5 mm. The positions of the shims are predetermined to give a desired non-persistent joint. Through these slits, greased metallic shims are inserted through the thickness of the mold before pouring the mixture. The box with the fresh mixture is vibrated and then stored at room temperature for 8 h afterward. The specimens un-molded and the metallic shim pulled out of the specimens; the grease on the shims prevents adhesion with the mixture and facilitates the removal of the shims.

As the mixture seated and hardened, shim leaves in the specimen an open joint through the thickness and perpendicular to the front and back of the specimen (Fig 4b).

Immediately after removing the shim, the front and back faces o.f the specimens are polished and the specimen is stored in laboratory for 4 days. At the end of the curing process, the specimens are tested. It does not appear that the pull out of the shim produces any damage through the joints. The aperture of joints is 0.5 mm, therefore the joint surface has not any effect on the failure mechanism. The temperature of experimental room was 22°C.

Fig. 4. (a) Plexiglas box with internal dimension of 19×19×6 cm; (b) cement specimens with edge joint length of 6 cm.

Fig. 5. The components of compression to tension convertor load device.



1.3. Compression to Tensile Load Transformer Device A compression-to-tension load transformer device (CTT) was developed to determine the tensile

strengths of specimens with a hole in the middle. This device converts the compression load to tensile load. The compression-to-tension load transformer device comprises of four parts made from hardened stainless (Fig. 5). Part No. 1 is composed of two pieces as shown in Fig. 2a. The front view of both pieces is “n” shaped and the side views look like “I” and “L” on the left and right, respectively (Fig. 5a). Part No. 2 is one piece and its front view is U shaped (Fig. 5b) and the side views look like “П”. The dimensions of the pieces are shown on this figure.

1.4. Installation of CCT Device on the Specimen The set up procedure of CTT device is consisted of three stages, as shown in Fig. 6. Firstly, the

“L” shape segment of part number 1 is situated in left side of the specimen so its edge is in contact with upper sample edge (Fig. 6a). Secondly, the “П” shape segment is situated in right side of the specimen so its edge is in contact with lower sample edge (Fig. 6b). Thirdly, the “l” shape segment is screwed to the “L” shape segment of part number 1 and the apparatus set up is completed. Therefore the upper cast is in contact to the lower sample edge and the lower cast is in contact to upper sample edge (Fig. 6c). When this set up is situated between the uniaxial loading frames, the upper loading frame compress the upper cast so the lower part of the joint is compressed. In similar condition, the lower loading frame compresses the lower cast so the upper part of the joint is compressed. Compression of upper and lower parts of the join brings the joint to tensile loading (Fig. 6d). A 30-ton hydraulic load cell applies compressive load to the CTT end plates. An electronic load cell is used to measure the increase of the applied load. To isolate the effect of loading rate from the results a constant loading of 0.02 MPa/s was applied for all specimen. This rate is within the range recommended for the Brazilian tensile strength testing by ASTM (D 3967-08).

Fig. 6. The set up procedure of CTT device.

652 HAERI et al.


Fig. 7. The tensile failure pattern in samples.

1.5. Failure Mechanism of Sample Consisting Edge Joint In all tested samples, Failure surface is tensile because it is varnish and no pulverized material and

no trace of shear displacement was occurred. The failure of sample 1 with fine grain size (Fig. 7a) shows that tensile crack initiates from joint tip and propagates diagonally with 40 degree related to tensile load direction and coalesces with the edge of the sample.

The failure of sample 2 with medium grain size (Fig. 7b) shows that tensile crack initiates from joint tip and propagates diagonally with 65 degree related to tensile load direction and coalesces with the edge of the sample. The failure of sample 3 with large grain size (Fig. 7c) shows that tensile crack initiates from joint tip and propagates diagonally with 85 degree related to tensile load direction and coalesces with the edge of the sample.

Totally, in constant join length, the crack growth angle related to normal load was decreased by increasing the grain size. This is comparable by failure pattern occurred in Brazilian test (Fig. 3).

1.6. Determination of Fracture Toughness of Samples with Different Tensile Strength The fracture toughness of three different samples with similar joint length was measured by

equation 1 (Table 2). By comparison between fracture toughness and tensile strength of concrete specimens (Fig. 8), it can be concluded that the fracture toughness has a close relationship with tensile strength. It increases with increasing the tensile strength.

Table 2. Fracture toughness of samples with different grain size

Sample no. Fracture toughness, MPa

1 2.72

2 1.35

3 1.96

Fig. 8. Variation of fracture toughness based on joint length.



2. NUMERICAL SIMULATION

Particle flow code in two dimensions (PFC2D) was used for investigation of the effect of the joint length and particle dimension on the fracture toughness of concrete. This also performed to study the effect of tensile strength of model material on the fracture toughness.

2.1. Particle Flow Code

Particle flow code (PFC) (Itasca [65]) developed by Itasca consulting group, is a distinct element method program, which is used to model physical problems that are concerned with the movement and interaction of disc particles. It is also possible to create particles of arbitrary shape by attaching two or more particles together, such that each group of particles acts as an autonomous object. The particle assembly is created at a given uniform size-distribution or Gaussian size-distribution with radii in the range of the minimum radius to maximum radius set by the user. The particle flow code model is composed of distinct particles that displace independent of one another, and interact only at contacts or interfaces between particles. A bonded particle model should satisfy the following assumptions. 1) The particles are treated as rigid bodies; 2) The contacts occur over a vanishingly small area; 3) Behavior at the contacts uses a soft-contact approach where the rigid particles are allowed to overlap one another at contact points; 4) The magnitude of the overlap is related to the contact force via the force–displacement law, and all overlaps are small in relation to particle sizes; 5) Bonds can exist at contacts between particles; 6) All particles are spherical. However, the clump logic supports the creation of super-particles of arbitrary shape. Each clump consists of a set of overlapping particles thatacts as a rigid body with a deformable boundary. Two types of walls including infinite walls and finite walls are provided in the PFC2D program. The walls are usually used to apply boundary conditions or as loading platens. Two bonding behaviors are embodied in contact bonds and parallel bonds, both of which can be envisioned as a kind of glue joining the two neighboring particles (Fig. 9). A contact bond also can be envisioned as a pair of elastic springs with constant normal and shear stiffness acting at the contact point. The two springs located between two neighboring particles have specified shear and normal strength, and control the micro-mechanical behavior of a contact bond (Fig. 10a). If the magnitude of the tensile normal contact force equals or exceeds the normal contact bond strength, the bond breaks, and both the normal and shear contact forces are set to zero. If the magnitude of the shear contact force equals or exceeds the shear contact bond strength, the bond breaks, but the contact forces are not altered, it is provided that the shear force does not exceed the friction limit, and the normal force is compressive (Fig. 10b). A parallel bond model was used in the work to model the micro-mechanical behavior of a rock-like material. The Parallel-bond model has eight micro-mechanical parameters listed in Tables 3, 4 and 5. The micro-mechanical parameters are related to the macro- mechanical behavior of the intact material. Even though some equations are given in the PFC2D manual, no solid theory exists to calibrate micro-mechanical parameters by knowing the macro-mechanical properties. The calibration has to be done through a trial and error procedure. Generally, the contact elastic modulus is directly related to the elastic modulus of bonded particle model, and the shear strength and normal strength of contact-bond are directly related to the strength of bonded particle model.

2.1.1. Preparing and calibrating the numerical model

The standard process of generation of a PFC2D assembly to represent a test model involves four steps: (a) particle generation and packing the particles, (b) isotropic stress installation, (c) floating particle elimination, and (d) bond installation.

654 HAERI et al.


Fig. 9. Two bonding behaviors, a)contact bonds and b) parallel bonds, Potyondy [66].

Fig. 10. (a) Two springs located between two neighboring particles with shear and normal strength, () bond break under external

loading, Potyondy [66].

Table 3. Micro properties used to represent the model with tensile strength of 4.6 MPa

Property Value Property Value Number of clump particle per 2290 mm2 / clump diameter, mm

50/1.5 Parallel bond radius muliplier 1.4

Densiy, kg/m3 3000 Youngs modulus of parallel bond, GPa 1.7 Minimum radius, mm 0.27 Parallel bond stifness ratio, pb_kn/pb_ks 3 Size ratio 1.56 Particle friction coefficient 0.5 Porosity ratio 0.05 Parallel normal strength, mean, MPa 50 Local damping coefficient 0.7 Parallel normal strength, SD, MP) 5 Contact young modulus, GPa 12 Parallel shear strength, mean, MPa 50 Stiffness ratio, kn/ks 1.7 Parallel shear strength, SD, MPa 5

Table 4. Micro properties used to represent the model with tensile strength of 2.6 MPa

Property Value Property Value Number of clump particle per 2290 mm2 / clump diameter, mm

96/3 Parallel bond radius muliplier 1.4

Densiy, kg/m3 2200 Youngs modulus of parallel bond, GPa 1.7 Minimum radius, mm 0.27 Parallel bond stifness ratio, pb_kn/pb_ks 3 Size ratio 1.56 Particle friction coefficient 0.5 Porosity ratio 0.05 Parallel normal strength, mean, MPa 25 Local damping coefficient 0.7 Parallel normal strength, SD, MP) 5 Contact young modulus, GPa 12 Parallel shear strength, mean, MPa 25 Stiffness ratio, kn/ks 1.7 Parallel shear strength, SD, MPa 5



Table 5. Micro properties used to represent model with tensile strength of 4.1 MPa

Property Value Property Value

Number of clump particle per

2290 mm2 / clump diameter, mm 240/4.5 Parallel bond radius muliplier 1.4

Densiy, kg/m3 2600 Youngs modulus of parallel bond, GPa 1.7

Minimum radius, mm 0.27 Parallel bond stifness ratio, pb_kn/pb_ks 3

Size ratio 1.56 Particle friction coefficient 0.5

Porosity ratio 0.05 Parallel normal strength, mean, MPa 42

Local damping coefficient 0.7 Parallel normal strength, SD, MP) 5

Contact young modulus, GPa 12 Parallel shear strength, mean, MPa 42

Stiffness ratio, kn/ks 1.7 Parallel shear strength, SD, MPa 5

2.1.2. Brazilian test

Brazilian test was used to calibrate the tensile strength of three different models in PFC2D. Adopting the micro-properties listed in Table 3, 4 and 5 with the standard calibration procedures [66], three calibrated PFC particle assembly was created. The diameter of the Brazilian disk considered in the numerical tests was 54 mm. The specimens were made of 5615 particles with different clump particle distributed in itto gain the best results. The disk was crushed by the lateral walls moved toward each other with a low speed of 0.016 m/s. Figures 11–13 illustrate the failure patterns of the numerical and experimental tested samples, respectively. The failure planes experienced in numerical and laboratory tests are well matching. The numerical tensile strength and a comparison of its experimental measurements are presented in Table 6. This table shows a good accordance between numerical and experimental results.

Fig. 11. (a) Failure pattern in a physical sample; (b) PFC2D model.


656 HAERI et al.



Table 6. Brazilian tensile strength of physical and numerical samples

Physical tensile strength, MPa 4.6 2.6 4.1

Numerical tensile strength, MPa 4.35 2.7 4

2.2. Numerical Simulation of CTT Test

2.2.1. Preparing the model

After calibration of PFC2D, numerical simulation of CTT test was simulated by creating a box model in the PFC2D (Fig. 14). The PFC specimen had dimensions of 75×100 mm. A total of 11,179 disks with a minimum radius of 0.27 cm were used to make up initial model condition. Two thickness bands with dimension of 10×20 mm were removed from both side of the model to build the proper geometry similar to physical sample.

For creating the joint in the model, a narrow band of particle was removed from the model. The opening of join was 2 mm. The joint length “a” varies in three different lengths; 1.1, 2.2 and 3.3 mm. the ratio of joint length to model width in numerical model and experimental sample was similar. Four loading wall were situated in the four edges of the model. Wall id 1 and 3 moves in Y direction. Wall id 2 and 4 moves in opposite side of Y direction with a low speed of 0.016 m/s.

Fig. 14. A box model in the PFC2D.



Fig. 15. The tensile failure pattern in samples with tensile strength of (a) 4.35 MPa, (b) 2.7 MPa and (c) 4 MPa.

2.2.2. Failure mechanism of sample consisting edge joint Figure 15 shows progress of cracks in the models. Black lines represent the tensile cracks in the

mode. It is clear that under this condition, tensile cracks develop in all models. The failure of sample 1 with fine grain size (Fig. 15a) shows that tensile crack initiates from joint

tip and propagates diagonally with 40 degree related to tensile load direction and coalesces with the edge of the sample. This failure was similar to the failure occurred in the physical sample (Fig. 7a).

The failure of sample 2 with medium grain size (Fig. 15b) shows that tensile crack initiates from joint tip and propagates diagonally with 65 degree related to tensile load direction and coalesces with the edge of the sample. The failure was similar to the failure in the physical sample (Fig. 7b).

The failure of sample 3 with large grain size (Fig. 15c) shows that tensile crack initiates from joint tip and propagates diagonally with 85 degree related to tensile load direction and coalesces with the edge of the sample. This failure was similar to the failure occurred in the physical sample (Fig. 7c).

Totally, in constant join length, the crack growth angle related to normal load was decreased by increasing the grain size. This is comparable by failure pattern occurred in numerical Brazilian test (Fig. 13).

Table 7. Fracture toughness of samples with different grain size

Sample no. Fracture toughness, MPa

1 2.92

2 1.55

3 2.08

Fig. 16. Variation of fracture toughness based on joint length.

658 HAERI et al.


2.3. Determination of Fracture Toughness of Samples within Different Tensile Strength The fracture toughness of three different samples with similar joint length was measured by

equation 1 (Table 7). By comparison between fracture toughness and tensile strength of concrete specimens (Fig. 16), it can be concluded that the fracture toughness has a close relationship with tensile strength. It increases with increasing the tensile strength.

By comparison between Figs8 and 16 it can be concluded that fracture toughness measured by experimental test was similar to the numerical one. Also the relationship between fracture toughness and tensile strength in numerical model was similar to the experimental one.

CONCLUSIONS

The CTT device was designed to obtain themode I fracture toughnessof concrete. Also the effect of tensile strength on the fracture toughness of concrete was cleared.For this purpose, concrete samples with constant edge joint length and different ensile strength wereprepared and tested using CTT device. The results show that:

• In constant join length, the crack growth angle related to normal load was decreased by increasing the grain size.

• Fracture toughness has a close relationship with tensile strength. It increases with increasing the tensile strength.

• The failure pattern and fracture toughness determined by numerical simulation were similar to experimental results.

CTT device is capable for determination of fracture toughness of concrete. The proposed device was designed and fabricated for the applications with commercially available compression loading machines. It is durable, inexpensive and easy to use.

REFERENCES

1. Lajtai, E.Z., Strength of Discontinuous Rock in Direct Shearing, Geotechnique, 1969,

vol. 19, pp. 218–233.

2. Atkinson, B.K., Fracture Mechanics of Rock, Academician Press, London, 1987.

3. Whittaker, B.N., Singh, R.N., and Sun, G., Rock Fracture Mechanics—Principles, Design and

Applications, Elsevier, Amsterdam, 1992.

4. Ouchterlony, F., Suggested Methods for Determining the Fracture Toughness of rock, Int. J. Rock Mech.

Mineral Sci. Geo-Mech. Abstr., 1988, vol. 25, no. 1, pp. 71–96.

5. Lim, I.L., Johnston, I.W., Choi, S.K., and Boland, J.N., Fracture Testing of a Soft Rock with Semi-

Circular Specimens under Three-Point Bending. Part 2: Mixed Mode, Int. J. Rock Mech. Min. Sci.

Geomech. Abstr., 1994, 31(3), pp. 199–212.

6. Atkinson, C., Smelser, R.E., and Sanchez, J., Combined Mode Fracture via the Cracked Brazilian Disc

Test, Int. J. Fract., 1982, 18, pp. 279–91.

7. Chang, S.H., Lee, C.I., and Jeon, S., Measurement of Rock Fracture Toughness under Modes I and II and

Mixed-Mode Conditions by Using Disc-Type Specimen, Eng. Geol., 2002, 66, pp. 9–97.

8. Shetty, D.K., Rosenfield, A.R., and Duckworth, W.H., Mixed-Mode Fracture in Biaxial Stress State:

Application of the Diametral-Compression (Brazilian Disk) Test, Eng. Fract. Mech., 1987, vol. 26,

no. 6, pp. 825–40.

9. Khan, K. and Al-Shayea, N.A., Effect of Specimen Geometry and Testing Method on Mixed I–II Fracture

Toughness of a Limestone Rock from Saudi Arabia, Rock Mech. Rock Eng., 2000, 33(3), pp. 179–206.



10. Maccagno, T.M. and Knott, J.F., The Fracture Behavior of PMMA in Mixed Modes I and II, Eng. Fract. Mech., 1989, 34(1), pp. 65–86.

11. He, M.Y., Cao, H.C., and Evans, A.G., Mixed-Mode Fracture: The Four Point Shear Specimen, Acta Metal Mater., 1990, 38, pp. 839–46.

12. Suresh, S, Shih, C.F., Morrone, A., and O’Dowd, N.P., Mixed-Mode Fracture Toughness of Ceramic Materials, J. Am. Ceram. Soc., 1990, 73, pp. 1257–67.

13. Huang, J. and Wang, S., An Experimental Investigation Concerning the Comprehensive Fracture Toughness of Some Brittle Rocks, Int. J. Rock Mech. Min. Sci., Geomech. Abstr., 1985, 22(2), pp. 99–104.

14. Buchholz, F.G., Pirro, P.J.M., Richard, H.A., and Dreyer, K.H., Numerical and Experimental Mixed-Mode Analysis of a Compact Tension-Shear–Specimen, Proc. 4th Int. Conf. Numerical Methods in Fracture Mechanics, Swansea: Pineridge Press, 1987, pp. 641–56.

15. Mahajan, R.V. and Ravi-Chandar, K., An Experimental Investigation of Mixed-Mode Fracture, Int. J. Fract., 1989, 41, pp. 235–52.

16. Banks-Sills, L. and Bortman, Y., A Mixed-Mode Fracture Specimen, Analysis and Testing, Int. J. Fract., 1986, 30, pp. 181–201.

17. Chong, K.P. and Kuruppu, M.D., New Specimen for Fracture Toughness Determination for Rock and Other Materials, Int. J. Fract., 1984, 26, pp. R59–62.

18. Chong, K.P. and Kuruppu, M.D., Fracture Toughness Determination of Layered Materials, Eng. Fract. Mech., 1987, 28(1), pp. 43–54.

19. Singh, R.N. and Sun. G.X., A Numerical and Experimental Investigation for Determining Fracture Toughness of Welsh Limestone, Min. Sci. Tech., 1990, 1 pp. 61–70.

20. Lim, I.L., Johnston, I.W., Choi, S.K., and Boland, J.N., Fracture Testing of a Soft Rock with Semi-Circular Specimens under Three-Point Bending. Parts 1, 2, Int. J. Rock Mech. Mineral Sci. Geomech. Abstr., 1994, 31(3), pp. 185–212.

21. Ouchterlony, F., A Core Bend Specimen with Chevron Edge Notch for Fracture Toughness Measurement, Proc. 27th US Symp. Rock Mechanics, 1986, pp. 177–84.

22. Barker, L.M., A Simplified Method for Measuring Plane Strain Fracture Toughness, Eng. Fract. Mech., 1977, 9, pp. 361–9.

23. Sarfarazi, V., Ghazvinian, A., Schubert, W., Blumel, M., and Nejati, H.R., Numerical Simulation of the Process of Fracture of Echelon Rock Joints, Rock Mechanics and Rock Engineering, 2014, 47(4), pp. 1355–1371.

24. Bobet, A., Fracture Coalescence in Rock Materials: Experimental Observations and Numerical Predictions, Sc.D. Thesis, MIT, Cambridge, USA, 1997.

25. Ouchterlony, F., A New Core Specimen for the Fracture Toughness Testing of Rock, Swedish Detonic Research Foundation Report, Stockholm, Sweden, DS, 1980.

26. Dai, F., Chen, R., Iqbal, M.J., and Xia, K., Dynamic Cracked Chevron Notched Brazilian Disc Method for Measuring Rock Fracture Parameters, Int. J. Rock Mech. Min. Sci., 2010, 47(4), pp. 606–613.

27. Kataoka, M., Obara, Y., and Kuruppu, M., Estimation of Fracture Toughness of Anisotropic Rocks by SCB Test and Visualization of Fracture by Means of X-Ray CT, Proc. 12th ISRM International Congress, Beijing, China, 2011, pp. 667–670.

28. Tutluoglu, L. and Keles, C., Mode I Fracture Toughness Determination with Straight Notched Disk Bending Method, Int. J. Rock Mech. Min. Sci., 2011, 48, pp. 1248–1261.

29. Amrollahi, H., Baghbanan, A., and Hashemolhosseini. H., Measuring Fracture Toughness of Crystalline Marbles under Modes I and II and Mixed Mode I–II Loading Conditions Using CCNBD and HCCD Specimens, Int. J. Rock Mech. Min. Sci., 2011, 48(7), pp. 1123–1134.

660 HAERI et al.


30. Aliha, M.R.M., Sistaninia, M., Smith, D.J., Pavier, M.J., and Ayatollahi, M.R., Geometry Effects and Statistical Analysis of Mode I Fracture in Guiting Limestone, Int. J. Rock Mech. Min. Sci., 2012, 51, pp. 128–135.

31. Zhou, Y.X., Xia, K., Li, X.B., Li, H.B., Ma, G.W., Zhao, J., Zhou, Z.L., and Dai, F., Suggested Methods for Determining the Dynamic Strength Parameters and Mode-I Fracture Youghness of Rock Materials, Int. J. Rock Mech. Min. Sci., 2012, 49, pp. 105–112.

32. Kataoka, M., Hashimoto, A., Sato, A., and Obara, Y., Fracture Toughness of Anisotropic Rocks by Semi-Circular Bend (SCB) Test under Water Vapor Pressure, Proc. 7th ARMS, Seoul, Korea, 2012, pp. 458–465.

33 Kataoka, M. and Obara, Y., Estimation of Fracture Toughness of Different Kinds of Rocks under Water Vapor Pressure by SCB Test, J MMIJ, 2013, 129, pp. 425–432.

34. Ramadoss, P. and Nagamani, K., Stress–Strain Behavior and Toughness of High-Performance Steel Fiber Reinforced Concrete in Compression, Computers and Concrete, An Int’l Journal, 2013, 11(2), pp. 55–65.

35. Ayatollahi, M.R. and Alborzi, M.J., Rock Fracture Toughness Testing Using SCB Specimen, Proc. 13th Int. Conf. Fracture, Beijing, China, 2013, pp. 1–7.

36. Kuruppu, M.D., Obara, Y., Ayatollahi, M.R., Chong, K.P., and Funatsu, T., ISRM-Suggested Method for Determining the Mode I Static Fracture Toughness Using Semi-Circular Bend Specimen, Rock Mech. Rock Eng., 2014, 47, pp. 267–274.

37. Pan, B., Gao, Y., and Zhong, Y., Theoretical Analysis of Overlay Resisting Crack Propagation in Old Cement Concrete Pavement, Structural Engineering and Mechanics, An Int’l Journal, 2014, 52(4), pp. 167–181.

38. Wei, M.D., Dai, F., Xu, N.W., Xu, Y., and Xia, K., Three-Dimensional Numerical Evaluation of the Progressive Fracture Mechanism of Cracked Chevron Notched Semi-Circular Bend Rock Specimens, Eng. Fract. Mech., 2015, 134, pp. 286–303.

39. Kequan, Y.U. and Zhoudao, L.U., Influence of Softening Curves on the Residual Fracture Toughness of Post-Fire Normal-Strength Concrete, Computers and Concrete, An Int’l Journal, 2015, 15(2), pp. 102–111.

40. Lee, S. and Chang, Y., Evaluation of RPV According to Alternative Fracture Toughness Requirements, Structural Engineering and Mechanics, An Int’l Journal, 2015, vol. 53, no. 6.

41. Haeri, H., Shahriar, K., Marji, M.F., and Moarefvand, P., Investigating the Fracturing Process of Rock-Like Brazilian Discs Containing Three Parallel Cracks under Compressive Line Loading, Strength of Materials, 2014, 46(3), pp. 133–148.

42. Haeri, H., Marji, M.F., and Shahriar, K., Simulating the Effect of Disc Erosion in TBM Disc Cutters by a Semi-Infinite DDM, Arabian Journal of Geosciences, 2015, 8(6), pp. 3915–3927.

43. Haeri, H., Khaloo, K., and Marji, M.F., Experimental and Numerical Analysis of Brazilian Discs with Multiple Parallel Cracks, Arabian Journal of Geosciences, 2015, 8(8), pp. 5897–5908.

44. Haeri, H., Shahriar, K., Marji, M.F., and Moarefvand, P., The HDD Analysis of Micro Cracks Initiation, Propagation and Coalescence in Brittle Substances, Arabian Journal of Geosciences, 2015, 8, pp. 2841–2852.

45. Haeri, H., Propagation Mechanism of Neighboring Cracks in Rock-Like Cylindrical Specimens under Uniaxial Compression, J. Min. Sci., 2015, vol. 51, no. 3, pp. 487–496.

46. Haeri, H., Influence of the Inclined Edge Notches on the Shear-Fracture Behavior in Edge-Notched Beam Specimens, Comput. Concrete, 2015, 16(4), pp. 605-623.

47. Haeri, H., Experimental Crack Analysis of Rock-Like CSCBD Specimens Using a Higher Order DDM, Comput. Concrete, 2015, 16(6), pp. 881-896.



48. Akbas, S., Analytical Solutions for Static Bending of Edge Cracked Micro Beams, Structural Engineering and Mechanics, An Int’l Journal, 2016, 59(3), pp. 66–78.

49. Rajabi, M., Soltani, N., and Eshraghi, I., Effects of Temperature Dependent Material Properties on Mixed Mode Crack Tip Parameters of Functionally Graded Materials, Structural Engineering and Mechanics, An Int’l Journal, 2016, 58(2), pp. 144–156.

50. Mohammad, A., Statistical Flexural Toughness Modeling of Ultra High Performance Concrete Using Response Surface Method, Computers and Concrete, An Int’l Journal, 2016, 17(4), pp. 33–39.

51. Haeri, H. and Sarfarazi, V., The Effect of Micro Pore on the Characteristics of Crack Tip Plastic Zone in Concrete, Computers and Concrete, 2016, 17(1), pp. 107–12.

52. Fowell, R.J. and Chen, J.F., The Third Chevron-Notch Rock Fracture Specimen—The Cracked Chevron-Notched Brazilian Disk, Proc. 31st U.S. Symp. Rock, Balkema, Rotterdam, 1990, pp. 295–302.

53. Fowell, R.J. and Xu, C., The Cracked Chevron Notched Brazilian Disc Test—Geometrical Considerations for Practical Rock Fracture Toughness Measurement, Int. J. Rock Mech. Mineral Sci. Geomech. Abstr., 1993, 30(7), pp. 821–824.

54. Lim, I.L., Johnston, I.W., and Choi, S.K., Assessment of Mixed-Mode Fracture Toughness Testing Methods for Rock, Int. J. Rock Mech. Mineral Sci. Geomech. Abstr., 1994, 31(3), pp. 265–272.

55. Zhang, Xiao-ping and Wong, L.N.Y., Cracking Process in Rock-Like Material Containing a Single Flaw under Uniaxial Compression: A Numerical Study Based on Parallel Bonded-Particle Model Approach, Rock Mechanics and Rock Engineering, 2012, vol. 45(5), pp. 711−737.

56. Sato K., Fracture Toughness Evaluation Based on Tension-softening Model and its Application to Hydraulic Fracturing, Pure Appl. Geophys., 2006, 163, pp. 1073–1089.

57. Wang, Q., Jia, X., Kou, S., Zhang, Z., and Lindqvist, P.A., More Accurate Stress Intensity Factor Derived by Finite Element Analysis for the ISRM Suggested Rock Fracture Toughness Specimen—CCNBD, Int. J. Rock Mech. Min. Sci., 2003, 40, pp. 233−241.

58. Wang, Q., Jias, X., Kou, S., Zang, Z., and Lindqvist, P.A., The Flattened Brazilian Disc Specimen Used for Testing Elastic Modulus, Tensile Strength and Fracture Toughness of Brittle Rocks: Analytical and Numerical Results, Int. J. Rock Mech. Min. Sci., 2004, 41 pp. 245−253.

59. Ke, C.C., Chen, C.S., and Tu, C.H., Determination of Fracture Toughness of Anisotropic Rocks by Boundary Element Method, Rock Mechanics and Rock Engineering, 2008, 41, pp. 509−538.

60. Dai, F., Wei, M.D., Xu, N.W., Zhao, T., and Xu, Y., Numerical Investigation of the Progressive Fracture Mechanisms of Four ISRM-Suggested Specimens for Determining the Mode I Fracture Toughness of Rocks, Comput. Geotech., 2015, 69, pp. 424–441.

61. Xu, N.W., Dai, F., Wei, M.D., Xu, Y., and Zhao, T., Numerical Observation of Three Dimensional Wing-Cracking of Cracked Chevron Notched Brazilian Disc Rock Specimen Sbjected to Mixed Mode Loading, Rock Mech. Rock Eng., 2015, vol. 49, pp. 79–96.

62. Yaylac, M., The Investigation of Crack Problem through Numerical Analysis, Structural Engineering and Mechanics, An Int’l Journal, 2016, vol. 57, no. 6.

63. ASTM, Test Method for Unconfined Compressive Resistance of Intact Rock Core Specimens, ASTM Designation, D2938-86, 1986.

64. ASTM, Standard Method of Test for Splitting Tensile Resistance of Cylindrical Concrete Specimens, ASTM Designation, C496-71, 1971.

65. Itasca PFC2D, Particle Flow Code in 2 Dimensions, Theory and Background, Itasca Consulting Group, Minneapolis, MN, 1999, pp. 1–124.

66. Potyondy, D.O. and Cundall, P.A., A bonded-particle model for rock, Int. J. Rock Mech. Min. Sci., 2004, 41, pp. 1329–1364.

Effect of Tensile Strength of Rock on Tensile...

Documents

Transcript of Effect of Tensile Strength of Rock on Tensile...