PSI - Issue 41

Christos F. Markides et al. / Procedia Structural Integrity 41 (2022) 351–360 Christos F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

352

2

1. Introduction The Flattened Brazilian Disc Test (FBDT) was introduced almost twenty years ago (Wang et al. 2004, Wang and Wu 2004) as an alternative of the familiar Brazilian Disc Test (BDT) (Carneiro 1943, Akazawa 1943), which is used for the determination of the tensile strength of very brittle materials. Its introduction was dictated by the need to con front the premature fracturing, which is quite often observed during the BDT in the immediate vicinity of the contact region of the disc with either the loading platen (ASTM 2008) or the jaw (ISRM 1978). Obviously, this premature crack ing far from the disc‟s center undermines the validity of the BDT results, as it was long ago indicated by many researchers (Fairhurst 1964, Hobbs 1965, Jaeger & Hoskins 1966, Hudson 1969, Hooper 1971, Mellor & Hawkes 1971). After it was introduced the FBDT is used worldwide by many researchers. The field of its applications covers a broad variety of protocols, including both static (Kaklis et al. 2005, Ren et al. 2022) and dynamic (Wang et al. 2009, Chen et al. 2013, Liu et al. 2018, Zhang et al. 2021) loading schemes. Moreover, and besides its application for the determination of the tensile strength, the FBDT is used to determine the fracture toughness of brittle materials (Wang et al. 2010, Wang et al. 2011, Elghazel et al. 2015, Yagin et al. 2021, Yu et al. 2021). It is here recalled, that Wang and Xing (1999) suggested the use of FBDT for the determination of fracture toughness well before it was introduced as an alternative for the indirect determination of the tensile strength. The theoretical background of the FBDT solution is even today assessed further, either experimentally (Yan et al. 2021) or numerically (Wang and Cao 2016, Wu et al. 2018) or even analytically, although the analytical studies are rather scarce (Huang et al. 2015, Markides and Kourkoulis 2020), due to the extremely complicated nature of them. Indeed, Wang et al. (2004) accepted that “… for the flattened Brazilian disc a similar (to that of the BDT proposed by Hondros (1959)) exact elasticity solution cannot be obtained ”. As a result, Wang‟s scientific te am (Wang et al. 2004) suggested a hybrid numerical/analytic approach, which, in spite of its efficiency, renders detailed parametric studies somehow complicated and time consuming. In this direction, an attempt was recently presented by Markides and Kourkoulis (2020) for an alternative analytic approach to the solution of the 1 st fundamental problem of the FBD (Fig.1a) under the usual assumption of a uniform pressure applied to its flat edges (Fig.1b), based on complex potentials method (Muskhelishvili 1963). As it was shown, the assumption of uniform pressure on the flat edges of the FBD leads to an unnatural bending of them during loading. In this context, an attempt is here described aiming to improve the existing approaches, at least as far as it concerns the assumption of uniform pressure on the flat edge of the FBD. Alternative, more realistic distributions are considered that could remove the unnatural bending of the flat edge during loading, leading to a more accurate formula for the tensile strength of the material tested. It is to be mentioned here that the present Part I of this ongoing research project deals with an FBD that is squeezed between two flat plates of high rigidity (stamps) and of length equal to that of the flat edge of the FBD, in the presence of friction (Fig.2).

y

P frame

P frame

P =const.

0.02 0.04 0.06

Platen

α α

t

R

t

Bent flat edge after deformation

FBD

-0.06 -0.04 -0.02 0

Ο

x

y [m]

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

P frame

x [m]

P frame

(a) (c) Fig. 1. (a) FBDT; (b) FBD under uniform pressure; (c) Deformed FBD with the unnatural bending of its flat edges. (b)