PSI - Issue 10

Ch.F. Markides / Procedia Structural Integrity 10 (2018) 163–170

164

Ch.F. Markides / Structural Integrity Procedia 00 (2018) 000 – 000

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for transmitted caustics and next extended by Theocaris (1970) to the reflected ones, the method has constituted the grounds of many researches on various engineering problems. It is recalled that Theocaris was the first one to solve the problem analytically using complex potentials (Theocaris and Gdoutos (1974)). Namely, caustics have been used in stress singularity- (Theocaris (1981)) and stress concentration-problems (Pazis et al. (2011)), in the evaluation of elastic constants (Ioakimidis and Theocaris (1979)) and stress-optical constants (Theocaris and Ioakimidis (1979); Younis and Zachary (1989); Badaluka and Papadopoulos (2011)) of materials, in dynamic problems (Papadopoulos (1990); Chun yang et al. (1999); Xuefeng et al. (2005; 2011)), in plasticity problems (Rosakis and Freund (1982)), in the evaluation of the J-integral (Kikuchi and Hamanaka (1990)) and in many others. The method has also applied to various ma terials, as birefringent ones (Papadopoulos and Pazis (2003)), anisotropic ones (Kezhuang and Zheng (2008)), graded ones (Yao Xuefeng et al. (2008)) and rock-like ones (Yang et al. (2009)). In contact problems, caustics were used in several cases (Theocaris (1978; 1979); Theocaris and Stassinakis (1978); Semenski (2003); Raptis et al. (2011); Pa padopoulos (2004; 2005)). Of particular interest for the present study is the paper by Theocaris and Stassinakis (1978), where formulae for the contact length between two discs in contact were derived. Those formulae are further exploited here to assess the theoretic contact length using existing analytic solutions (Kourkoulis et al. (2012)) in case of a trans parent circular disc (or ring), squeezed between the curved jaws of the ISRM (1978) device for the implementation of the Brazilian-disc test. In this context, the equations of reflected caustics are here reproduced assuming zero light refraction within the specimen. Under this assumption, simple conditions for reflected caustics from the front and rear faces are revealed and two ways are proposed to locate the y ΄ =0 line on photos of experimental caustics. Then the contact length between the disc (or ring with a small whole) and the ISRM’s jaw is obtained with the aid of a closed-form formula, not used until now (Theocaris and Stassinakis (1978)). Increasing the accuracy of the measurement of the contact length aims to contribute to the open discussion on the reliability of the Brazilian-disc test and the influential role of the contact region in failure onset (Lanaro et al. (2009)). Let a disc of radius R and thickness t , of homogeneous, isotropic and linearly elastic material, be smoothly compressed against the ISRM’s meta llic curved jaw of radius R j =1.5 R , by an overall load P frame (Fig.1a). Increasing P frame the contact between them, starting from a single point, is eventually realized along a finite arc. Assume that the cross-sections of the disc and the upper jaw lie in the ζ -complex plane ( ζ = x +i y = r e i θ ) and the origin of the Cartesian reference is at the mid-point of their common contact arc, denoted by - ℓℓ (Fig.1b). If – ℓℓ is small compared to R , it can be assumed that the disc and upper jaw occupy the lower and upper half plane, respectively, whence particularizing Muskhelishvili’s (1963) solution for a mixed fundamental problem (of linear relationship), the complex potential Φ ( ζ ) for the disc, the distribution p ( τ ) of P frame ( τ = ζ : – ℓ ≤ ζ≤ℓ ) and the magnitude of ℓ , for the Brazilian disc- ISRM’s jaw complex, are (Kourk oulis et al. (2012)): 2. The analytic solution of the contact problem and the theoretic contact length

P frame

Half ball bearing

Upper jaw

y

(a)

(b)

P frame

O y

Disc

x

O

– ℓ

Upper jaw

+ ℓ

τ

z

R

t

x

Guide pin

θ 2

θ

r 2

r

r 1

θ 1

R j =1.5 R

R

Disc

ζ = x +i y = r e – i θ ( y <0)

R j =1.5 R

Lower jaw

Fig. 1. (a) Brazilian-disc test according to ISRM; (b) The mathematical configuration.