Issue 50

Ch. F. Markides, Frattura ed Integrità Strutturale, 50 (2019) 451-470; DOI: 10.3221/IGF-ESIS.50.38 452 elastostatic problems [8–10], the mechanical behaviour and crack propagation under dynamic loading conditions [11–17], the measurement of strain concentration in plasticity problems [18], the determination of the J-integral [19], the determination of stress-optical as well as elastic constants of materials [20–22], the location of the crack-tip position [23], the description of load distribution and the definition of the contact length in contact problems [24–29]. What is more, the method has been successfully applied to a wide range of materials, including birefringent ones [30], rock-like [31] and anisotropic ones [32–35], viscoelastic ones [36] and even composite [37] or graded ones [38]. In parallel, several studies have concerned with the improvement of the method of caustics [39–43] while an experiment on the method of Caustics for educational purposes was recently proposed by Younis [44]. In the present study, further to a work of Theocaris and Stassinakis [26], general formulae for double initial curves and their corresponding reflected or/and transmitted caustic curves are provided in the case two cylindrical elastic bodies of arbitrary radii are compressed against each other in the absence of friction. In this direction and under the usual assumptions made in the method of caustics, the conditions for the development of double initial and contact caustic curves are established. As it is seen, the generation or not of double initial and caustic curves depends on six independent parameters which are related to the material properties, the relative dimensions of the two bodies in contact, as well as to the characteristics of the optical set-up. In light of those formulae, expressions for the contact length given in [26] are revisited and easy-to-use closed-form expressions are given for obtaining the contact length based on the well-known Muskhelishvili’s solution for the contact problem [45] particularised in [46]. Then an experimental protocol was implemented concerning a divergent laser light beam incident on a thin cylindrical specimen made of a transparent optically isotropic, linearly elastic material, squeezed between the curved jaws of the International Society for Rock Mechanics (ISRM) suggested device for the standardized Brazilian-disc test [47]. As it is shown, the experimental method of caustics can provide the contact length even in the case of double caustic curves (either reflected or transmitted) which seem that can occur not only in the case of wide contact regions, but also in the case of relatively small ones, under certain conditions concerning, among others, the optical set-up, thus satisfying the small contact length assumption in the theoretic solution employed. T HEORETICAL CONSIDERATIONS The contact problem et a circular disc of radius R and thickness t be in equilibrium upon compressed against a curved jaw of arbitrary radius R J ( R ≤ R J <∞), by an overall load P frame . Assuming that both the disc and jaw are made of homogeneous, isotropic and linearly elastic materials and their cross-sections lie in the ζ = x +i y = r e i θ plane, the expressions for Muskhelishvili’s complex potential Φ ( ζ ) for the disc, and the corresponding half contact length ℓ realized between the disc and the jaw, read in the Oxy coordinate system (Fig.1) as [45, 46]:   frame          2 2 +1 2 1 +1 ( ) i , , + 2 π (1 ) 4 4 J J ζ κ KRP ρ κ Φ K t ρ ζ ζ KR μ μ (1) In Eqs. (1), ρ = R / R J , κ is Muskhelishvili’s constant, equaling (3– ν )/(1+ ν ) for plane strain and (3–4 ν ) for plane stress, and μ = Ε /[2(1+ ν )] is the shear modulus (with E and ν denoting Young’s modulus and Poisson’s ratio, respectively). Figure 1: The contact region (- ℓ , ℓ ) and the definition of symbols. L