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Research Papers

A Probabilistic Approach to Investigate the Size and Boundary Condition Effects on the Fracture Response of Brittle Materials Loaded in Diametral Compression

[+] Author and Article Information
Mahdi Saadati

Atlas Copco,
Örebro SE-70225, Sweden;
Department of Solid Mechanics,
KTH Royal Institute of Technology,
Stockholm SE-10044, Sweden
e-mail: msaadati@kth.se

Kenneth Weddfelt

Atlas Copco,
Sweden SE-70225, Örebro
e-mail: kenneth.weddfelt@se.atlascopco.com

Per-Lennart Larsson

Department of Solid Mechanics,
KTH Royal Institute of Technology,
Stockholm SE-10044, Sweden
e-mail: plla@kth.se

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received May 12, 2017; final manuscript received December 15, 2017; published online March 14, 2018. Assoc. Editor: Vadim V. Silberschmidt.

J. Eng. Mater. Technol 140(3), 031004 (Mar 14, 2018) (13 pages) Paper No: MATS-17-1136; doi: 10.1115/1.4039290 History: Received May 12, 2017; Revised December 15, 2017

The focus in this work is toward an investigation of the fracture response of brittle materials with different specimen size loaded in diametral compression using different boundary conditions. The compacted zone underneath the loading points is assumed to be limited and only responsible for the load transition to the rest of the material. Therefore, the theory of elasticity is used to define the stress state within a circular specimen. A tensile failure criterion is used, and the final load capacity is related to the formation of a subsurface crack initiated in a probabilistic manner in a region in the vicinity of the loaded diameter of the specimen. This process is described by Weibull theory, and it is assumed here that the growth of the subsurface crack occurs in an unstable manner. Therefore, the assumption in Weibull theory that the final failure occurs as soon as a macroscopic fracture initiates from a microcrack is fulfilled. The concept of disk effective volume used in Weibull size effect is presented in a convenient way that facilitates the application of the model to transfer the tensile strength obtained from different methods such as three point bending and Brazilian test. The experimental results for Brazilian test on a selected hard rock are taken from the literature and a fairly close agreement is obtained with the model predictions.

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References

Hondros, G. , 1959, “ 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., 10(3), pp. 243–268.
Fairhurst, C. , 1964, “ On the Validity of the ‘Brazilian’ Test for Brittle Materials,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 1(4), pp. 535–546. [CrossRef]
Jaeger, J. C. , 1967, “ Failure of Rocks Under Tensile Conditions,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 4(2), pp. 219–227. [CrossRef]
Mellor, M. , and Hawkes, I. , 1971, “ Measurement of Tensile Strength by Diametral Compression of Discs and Annuli,” Eng. Geol., 5(3), pp. 173–225. [CrossRef]
Hudson, J. A. , Brown, E. T. , and Rummel, F. , 1972, “ The Controlled Failure of Rock Discs and Rings Loaded in Diametral Compression,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 9(2), pp. 241–244. [CrossRef]
Wijk, G. , 1978, “ Some New Theoretical Aspects of Indirect Measurements of the Tensile Strength of Rocks,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 15(4), pp. 149–160. [CrossRef]
Andreev, G. E. , 1991, “ A Review of the Brazilian Test Rock Tensile Strength Determination—Part I: Calculation Formula,” Min. Sci. Technol., 13(3), pp. 445–456. [CrossRef]
Bazant, Z. P. , Kazemi, M. T. , Hasegawa, T. , and Mazars, J. , 1991, “ Size Effect in Brazilian Split-Cylinder Tests: Measurements and Fracture Analysis,” ACI Mater. J., 88(3), pp. 325–332.
Andreev, G. E. , 1991, “ A Review of the Brazilian Test Rock Tensile Strength Determination—Part II: Contact Conditions,” Min. Sci. Technol., 13(3), pp. 457–465. [CrossRef]
Rocco, C. , Guineae, G. V. , and Elices, M. , 1999, “ Size Effect and Boundary Conditions in the Brazilian Test: Theoretical Analysis,” Mater. Struct., 32(6), pp. 437–444. [CrossRef]
Rocco, C. , Guinea, G. V. , and Elices, M. , 1999, “ Size Effect and Boundary Conditions in the Brazilian Test: Experimental Verification,” Mater. Struct., 32(3), pp. 210–217. [CrossRef]
Bazant, Z. P. , 1999, “ Size Effect on Structural Strength: A Review,” Arch. Appl. Mech., 69(9–10), pp. 703–725. [CrossRef]
Weibull, W. , 1939, “A Statistical Theory of the Strength of Materials,” Royal Institute of Technology, Stockholm, Sweden, Report No. 151.
Weibull, W. , 1951, “ A Statistical Distribution Function of Wide Applicability,” ASME J. Appl. Mech., 18(3), pp. 293–297.
Vardar, Ö. , and Finnie, I. , 1975, “ An Analysis of the Brazilian Disk Fracture Test Using the Weibull Probabilistic Treatment of Brittle Strength,” Int. J. Fract., 11(3), pp. 495–508. https://link.springer.com/article/10.1007/BF00033536
Saadati, M. , 2015, “On the Mechanical Behavior of Granite: Constitutive Modeling and Application to Percussive Drilling,” Ph.D. thesis, KTH Royal Institute of Technology, Stockholm, Sweden. http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A787683&dswid=-117
Saadati, M. , Forquin, P. , Weddfelt, K. , Larsson, P. L. , and Hild, F. , 2018, “ On the Mechanical Behavior of Granite Material With Particular Emphasis on the Influence From Pre-Existing Cracks and Defects,” J. Test. Eval., 46(1), pp. 33–45.
Saadati, M. , Forquin, P. , Weddfelt, K. , and Larsson, P. L. , 2016, “ On the Tensile Strength of Granite at High Strain Rates Considering the Influence From Preexisting Cracks,” Adv. Mater. Sci. Eng., 2016, p. 6279571. [CrossRef]
Jaeger, J. C. , Cook, N. G. W. , and Zimmerman, R. , 2009, Fundamentals of Rock Mechanics, Wiley, Hoboken, NJ.
Lundborg, N. , 1967, “ The Strength-Size Relation of Granite,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 4(3), pp. 269–272. [CrossRef]
ANSYS, 2013, “ANSYS 15.0,” ANSYS Inc., Canonsburg, PA.
Japaridze, L. , 2015, “ Stress-Deformed State of Cylindrical Specimens During Indirect Tensile Strength Testing,” J. Rock Mech. Geotech. Eng., 7(5), pp. 509–518. [CrossRef]
Davies, D. G. S. , 1973, “ The Statistical Approach to Engineering Design in Ceramics,” Proc. Brit. Ceram. Soc., 22, pp. 429–452.
Forquin, P. , and Hild, F. , 2010, “ A Probabilistic Damage Model of the Dynamic Fragmentation Process in Brittle Materials,” Adv. Appl. Mech., 44, pp. 1–72. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

A cylinder with radius a and length t is loaded diametrically along its length. The length t is usually chosen equal to the radius a. In practice, the loading force will be distributed circumferentially over an angle 2θ0.

Grahic Jump Location
Fig. 2

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, and (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen with a point load boundary condition using the analytical solution by Jaeger [3]. Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 3

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, and (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen with a point load boundary condition using FE solution (all the stress components are normalized by p·θ0). Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 4

Different elastic stress components: (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen loaded with a 15 deg uniformly distributed pressure boundary condition using the analytical solution Jaeger [3]. Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 5

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen loaded with a 15 deg uniformly distributed pressure boundary condition using FE solution (all the stress components are normalized by p·θ0). Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 6

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen loaded with a 48 deg uniformly distributed pressure boundary condition using the analytical solution Jaeger [3]. Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 7

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen loaded with a 48 deg uniformly distributed pressure boundary condition using FE solution (all the stress components are normalized by p·θ0). Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 8

The FE discretization of one quarter of the specimen loaded with rigid walls boundary condition including 74,078 plane stress eight-noded quadratic elements and 223,082 nodes

Grahic Jump Location
Fig. 9

Different elastic stress components (a) σr, (b) σx, (c) σθ, (d) σy, (e) τrθ, (f) τxy, contour plots in cylindrical and Cartesian coordinate systems within a Brazilian specimen with rigid walls boundary condition using FE solution (all the stress components are normalized by p·θ0). Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 10

First principal stress contour plots within a Brazilian specimen with (a) and (b) point load, (c) and (d) 15 deg (e) and (f) 48 deg uniformly distributed pressure and (g) flat rigid walls boundary conditions using FE and analytical solutions (the first principal stress is normalized by p·θ0 in all cases). Plots show one quarter of the disk where both axes are axes of symmetry (see Fig. 1 for directions).

Grahic Jump Location
Fig. 11

Fraction of the total volume that is effectively active in tensile failure

Grahic Jump Location
Fig. 12

Comparison of the Brazilian experimental results versus the model prediction based on three point bending test

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