The 3-D exact analytic solution by Wei & Chau (2001) was used to analyze the stress distributions within cylinders under the Brazilian test. By comparing the present solution with the classical 2-D solution by Hondros (1959), we found that, if the height-to-diameter ratio of the cylinder is smaller than 0.25, the tensile stress at the center of the cylinder obtained by the present 3-D solution tends to that by the classical 2-D solution, and the difference does not exceed 1%. However, if the height of the cylinder Increases, the difference will increase. The larger the Poisson's ratio, the larger the difference between the two solutions. In particular, the largest difference, which is as high as 29%, is found when the height-to-diameter ratio of the cylinder equals 1 and Poisson's ratio is 0.5. Even for the standard Brazilian test, i. e., the height-to-diameter ratio is 0.5, the difference can be 12%. More importantly, contrast to the 2-D analytic solution, our calculations show that the tensile stress at the center of the cylinder is affected's ratio, and the tensile stress distribution along the thickness of the cylinder is not uniform. The tensile stress at the center of the cylinder is not the largest tensile stress within the cylinder, but the tensile stress near the end surfaces along the axis of symmetry of the cylinder is the largest.
The Brazilian test is one of the most popular indirect tensile strength tests for rocks. Solid cylinders cut from rock cores are usually used as specimens for the Brazilian test. Many investigations have been done for the Brazilian test. For example, the size effect of the specimen has been studied by Newman & Bennett (1990), Bazant et al (1991),Kafka et al (1996) and Rocco et al (1999a-c). The deformability, the tensile strength and the evaluation properties of anisotropic rock have been investigated by Vancauwelaert & Eckmann (1994). Chen et al (1996, 1998), and Bohloli & Ronge (1999). The extensively used formula for estimating the tensile strength of rocks under the Brazilian test is that by Hondros (1959). However, it should be emphasized that the formula is based on the 2-D elastic theory and only suitable for very long or very short cylinders. Note that as suggested by the International Society for Rock Mechanics, the height-to-diameter ratio of the cylinder is approximately 0.5 (Bieniawski & Hawkes, 1978). Recently Wei & Chau (2001) derived a 3-D exact analytic solution for finite solid cylinders under the Marshall test, which is the most commonly used indirect tensile test for asphalt.
Our theoretical model for a finite solid cylinder of radius R0(or diameter D) and length H=2h under the Brazilian test is shown in Figure 1. The external load is simulated by uniform stress P0 acting on two small strip areas on the curved surface of the specimen. Symbol b represents the half-width of the loading areas.