PSI - Issue 25

Ch.F. Markides et al. / Procedia Structural Integrity 25 (2020) 214–225

215

2

Ch. F. Markides et al., Structural Integrity Procedia 00 (2019) 000 – 000

2 f

P Dh

  t

 BD

(1)





where P f is the compressive load in the direction of the loaded disc’s diameter D at the moment of fracture and h is the disc’s thickness . Eq.(1) is in agreement with analytic solutions for a circular disc under diametral line compressive forces (Muskhelishvili, 1933; Hondros, 1959) as well as with recent solutions (Markides et al., 2010; Markides & Kourkoulis, 2012) for distributed compressive forces assuming that the width of distribution tends to zero.

(c)

(a)

(b)

Fig. 1. (a) Brazilian-disc test; (b) Circular Ring test; (c) Circular Semi Ring compression/tension test.

The validity of the results of the BD test, however, was criticized due to the fact that failure at the disc center is the result of a biaxial rather than a uniaxial stress field (Fairhurst, 1964), and because stress concentration in the disc under the loading platens causes often premature fracture at the disc-platen contact region instead of the disc center (Hobbs, 1964; Mellor & Hawkes, 1971; Hooper 1971). Thus, soon after the introduction of the BD, Ripperger & Davis (1947) introduced the Circular Ring test (CR) (Fig.1b), in an attempt to remove some of the inherent problems of the BD. Indeed, the critical stress field in CR, at point C , is uniaxial and the fracture load required is less than that of the BD test. Along the same lines, Hobbs (1964; 1965) arrived at the following expression for the tensile strength of the material as provided from CR ( D =2 R 2 , R 2 : the outer radius of the ring, h : the ring thickness): where k is a function of the ratio 1/ ρ = R 1 / R 2 (where R 1 is the inner radius of the ring). Eq.(2) is in agreement with ana lytic solutions for a ring under diametral line compressive forces P (Timoshenko, 1910; Jaeger & Hoskins, 1966) and, also, with recent solutions (Kourkoulis & Markides, 2012) for distributed compressive forces assuming that the width of distribution tends to zero. Thus it seemed that the tensile strength as obtained from the CR test was a fraction of that obtained from the BD test making Hudson (1969) and Hudson et al. (1972) to raise concerns about whether the tensile strength obtained by the CR test was a material property or an experimental result depending on the specimen’s geometry. Moreover, it was concluded that, in spite of the reduction of the fracture load in the CR test with respect to that of the BD test, it was still high causing again premature fracture at the CR-loading platen contact region. In this direction, Mellor & Hawkes (1971) carried out a thorough study of the parameters influencing the BD- and CR-tests concluding that to avoid premature fracture curved jaws instead of flat platens should be used, leading to the standardization of the BD test by ISRM. It is to be mentioned, that besides the BD- and CR-tests, alternative con figurations for obtaining the tensile strength of brittle materials were suggested, even recently, as, for example, the flattened BD test (Wang, 2014), the semi-circular bend test (Kuruppu et al., 2014), each one with its pro and cons. In this study, a configuration recently suggested as possible alternative for obtaining the tensile strength of brittle materials (Kourkoulis et al., 2018; Markides et al.; 2018), is analytically studied, simultaneously with another closely related alternative test, in an attempt to eliminate, as far as possible, some of the drawbacks of existing tests. Namely,       CR BD (1)  2 f t t P k k Dh   (2)