PSI - Issue 28

Christos F. Markides et al. / Procedia Structural Integrity 28 (2020) 710–719 Christos F. Markides and Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2019) 000–000

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In an attempt to cure the above mentioned drawback of the BDT, an alternative configuration was proposed, some fifteen years ago, by Wang and his scientific team, widely known as the Flattened Brazilian Disc (FBD) (Wang et al. 2004, Wang and Wu 2004). The respective test is known as the Flattened Brazilian Disc Test (FBDT). According to this approach, two cyclic segments, corresponding to two parallel chords, are cut from the circular disc, as it is schematically shown in Fig.1d. The external load is applied normally to the two flat edges of the as above obtained truncated disc. It was proven by Wang et al. (2004) that the stress concentration is significantly relieved and, assuming that the central angle corresponding to the flat edges of the FBDT exceeds a specific limit (equal to about 20 o ), the fracture starts always from the center of the truncated disc. Since then the FBDT has been widely used by many researchers worldwide in a wide variety of experimental protocols, including both static (Kaklis et al. 2005) and dynamic (Wang et al. 2009, Chen et al. 2013, Liu et al. 2018) loading schemes, not only for the determination of the tensile strength of brittle materials but also for the determination of their fracture toughness (Wang et al. 2010, Wang et al. 2011, Elghazel et al. 2015). It is here mentioned, for reasons of historical accuracy, that Wang and Xing (1999) proposed the determination of fracture toughness by means of the FBD configuration well before it was proposed for the determination of tensile strength. Concerning the stress field developed in a FBD, the majority of studies dealing with this issue are of numerical nature (Wang and Cao 2016, Wu et al. 2018), while the respective analytic ones are very scarce (Huang et al. 2015). Indeed, as it was explicitly stated by Wang et al. (2004) “ … 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 et al. (2004) were forced to adopt a hybrid numerical/analytic approach to confront the problem, rendering detailed parametric studies somehow complicated and time consuming. Driven by this observation, an attempt is described in this study to obtain flexible closed-form expressions for the stress field components all over the FBD, adopting the complex potentials technique. 2. Mathematical formulation and solution of the problem 2.1 Theoretical preliminaries The equilibrium of a linear elastic, homogeneous, isotropic flattened disc, under an overall compressive force P frame , uniformly distributed along its flat edges, is to be studied analytically (Fig.2a). Due to difficulties of a straightforward analytic approach by means of conformal mapping (related to the shape of the boundary of the flattened disc), the problem is here confronted indirectly by considering an alternative, equivalent configuration. Namely, instead of the flattened, an intact disc is considered, subjected to a large number of vertical point forces of magnitude P , acting along two symmetric parallel chords z 1 z 2 , z 3 z 4 of it, assuming that these chords coincide with the flat edges of the relevant flattened disc. Actually, a single point force of magnitude P is to act at each point z 1 , z 2 , z 3 , z 4 and at a large number n of equaly spaced internal points , k k z z (over-bar denotes the conjugate complex and 1≤ k ≤ n ) on the straight segments z 1 z 2 , z 3 z 4 respectively (Fig.2b), resulting to the same overall load P frame , acting on the flat edges of the respective flattened disc.

P frame

P frame

P frame

p= const.

P= const.

P= const.

z 2

z 2

z 1

z 2

z 1

z 1

R

R

R

t

≈

z 3

z 4

z 3

z 4

z 3

z 4

P frame

P frame

P frame

(a) (c) Fig. 2. (a) The flattened-disc configuration; (b) The uniform circular disc under normal point forces P along the chords z 1 z 2 and z 3 z 4 ; (c) The approximate equivalency between cases (a) and (b). (b)