PSI - Issue 3

Christos F. Markides et al. / Procedia Structural Integrity 3 (2017) 334–345 Christos F. Markides, Stavros K. Kourkoulis / Structural Integrity Procedia 00 (2017) 000–000

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are strongly violated when the degree of anisotropy is not negligible and therefore the familiar Hondros’ (1959) for mula, widely used in engineering praxis, is not applicable. As a result, the determination of the state of stresses or dis placements in an anisotropic disc under diametral compression becomes imperative. The mathematical problem is complicated, especially for finite domains. In most cases it is confronted experimentally (Barla & Inaurato 1973; Ver voort et al. 2014) while, recently, interesting numerical approaches appeared also (Tan et al. 2015). The number of analytical approaches is rather limited (Amadei 1996; Exadaktylos & Kaklis 2001; Markides & Kourkoulis 2016). In this context, an attempt is described here to obtain an analytic solution for the displacement-field developed in a circular disc made of an orthotropic material, assuming that the disc is submitted to diametral compression. A main innovation of the present approach is related to the simulation of the disc’s loading scheme: The diametral compression of the disc is here realized by imposing a parabolic distribution of radial stresses, which act along two finite arcs of the disc’s periphery (antisymmetric with respect to the disc’s center), the length of which is approximated by the solution of the respective contact problem. It could be anticipated of course that, from a quantitative point of view, the specific assumption for the arcs’ length is a first approximation. However, it is definitely closer to experimental reality, independently of whether the test is implemented according to the standardized procedure suggested by ISRM (1978) or by the respective one suggested by ASTM (2014). Indeed, the actual distribution of contact stresses in a cylinder compressed between either plane or circular jaws is of cyclic nature (Timoshenko & Goodier 1970), which is obviously approached better by a parabolic distribution rather than by a uniform one or by a pair of point forces. The solution is achieved by taking advantage of the complex potential technique for rectilinear anisotropic materials as it was formulated by Lekhnitskii (1968; 1981). Initially the complex potentials and the general expressions for the displacements are obtained at any point of a disc made of an orthotropic material. As a second step, attention is focused on discs made of transversely isotropic materials, considering their importance in engineering praxis. The simplified formulae obtained are used to study the variation of the components of the displacement field along some strategic loci of the disc enlightening, among others, the crucial role of the degree of anisotropy. 2. Theoretical considerations 2.1. Assumptions and theoretical formulation Consider a linearly elastic, orthotropic disc of radius R and thickness d, the cross-section of which is parallel to one of the three mutually perpendicular planes of elastic symmetry passing from each point of its volume. The disc is in equilibrium while squeezed smoothly between the jaws of the device suggested by ISRM for the standardized implementation of the Brazilian-disc test. The overall load externally exerted to the disc, P frame , forms an arbitrary angle ϕ o with respect to one of the two planes of elastic symmetry normally settled to its cross-section as it can be seen in Fig.1. For the specific configuration, the displacement field is to be defined at any point of the disc. The cross- section of the disc lies in the z=x+iy=re iθ complex plane. The origin of the Cartesian reference is the disc’s center and

P frame

ϕ o

Half-ball bearing

Guiding pin

1.5R

Steel jaws of the ISRM device

R

Traces of planes of elastic symmetry normally to the disc’s cross-section

Cylindrical specimen

Fig. 1. An orthotropic disc compressed between the jaws of the ISRM device for the standardized implementation of the Brazilian-disc test.