This paper attempts to provide a theoretical framework for the determination of the stress state induced in a hollow cylinder (ring) under Hertzian compressions on its outer boundary. The solution is obtained using Airy stress functions, Michell-Fourier series expansions and Fourier-Bessel series expressions in polar coordinates. Parametric studies are also performed to investigate the influence of the loading angle and the geometrical aspect ratio of the ring on the induced stress tensor.
Among the various mechanical parameters needed in rock mechanics, accurate estimation of the tensile strength is critical since rock-like geo- and construction materials are much weaker in tension than compression or shear. The direct uniaxial pulling test (DPT), the split cylinder test also refereed as the Brazilian test (BT), the point load strength test (PLST), and lateral compression of a thin disc with concentric holes (also known as the ring test), are among the well-accepted techniques used to obtain a measure of the tensile strength of brittle rocks and pavement materials (Jaeger 1966). The DPT is theoretically the simplest test method, but is difficult to carry out in practice with geo-materials (Liao et al. 1997). The Brazilian test involves splitting a disc loaded by a compressive line load that generates a tensile stress inside the disc causing failure. However, it has long been evident that the concentration of shear stresses that develop in the vicinity of the contacts can often interfere with the tensile breakage of the disc, through the formation of inverse shear conical plugs and multiple cracking (Hobbs 1964). In harder materials, it has also been shown that following strictly the standard Brazilian test methodology is almost impractical (Serati et al. 2015). Deviations from the standard test methodology with hard and stiff materials arise mainly from the violation of the accepted boundary conditions in the Brazilian test, e.g. three-dimensional stresses are developed in the contact regions (Serati 2014). In the case of the PLST, the available relationships correlating the strength index measured in this test with the tensile strength of the rock are generally either empirical or controlled by the characteristics of the rock, and thus cannot be accepted as universal and fundamental (Broch & Franklin 1972). The ring test, initially suggested by Ripperger and Davids (1947), has attracted more recent attention by investigators mainly for its ease of sample preparation and its unique breakage mechanism in pure tension. Unlike the Brazilian test, in which the tensile rupture initiates at the centre of the solid disc in a biaxial stress field condition, failure in pure tension initiates away from the loading platens at the boundaries of the inner hole in the ring test (points m in Fig. 1). Unwanted shear effects and premature local fractures commonly reported in other mentioned techniques can also be controlled in the ring test by suitably choosing the hole size (Hudson 1969). Among many contributions to the theoretical study of the stresses induced inside the ring, are the work of Hobbs (1965), Jaeger & Hoskins (1966), Timoshenko & Goodier (1969), Mellor & Ivor (1971), Ma (1994) and Kourkoulis & Markides (2014). Other works includes that of Pilkey (2008) and Vullo (2014), covering both theoretical and experimental results. However, despite it being proven that the actual loading condition transmitted at the ring-platen interface follows an ellipsoidal (Hertzian) distribution, the available theoretical solutions (including the author's previous research) are substantially limited by the contact stresses being simplified as either point (line) forces, uniform radial stresses or a parabolic distribution (Johnson 1987, Kourkoulis & Markides 2014, Serati & Williams 2015). This study attempts to eliminate this limitation by investigating the induced stress field when the ring is under the influence of Hertzian radial compressive stresses at the contacts. The assumed Hertzian loading on the perimeter of the ring is expected to provide a more realistic boundary condition than other contact distributions adopted in previous studies (Fig. 2). To pursue these objectives, boundary conditions are first introduced by means of a Fourier-Bessel expansion, and Michell's stress functions in the form of a Fourier series are then employed to derive the stress tensor.