## 1 Introduction  
Mechanical strength is deemed as a significant feature for brittle materials such as concrete, rock, and ceramic [1, 2, 3, 4]. Numerical research [5, 6, 7] has suggested that the instability of structures composed of these brittle materials generally arises from a tensile failure. It is thus of vital importance to accurately quantify the tensile mechanical behaviours of brittle materials [8]. Measuring the tensile strength of brittle materials is mainly concentrated on direct and indirect tensile-testing methods [9, 10, 11]. In the direct tensile test, a bone-shaped specimen is subjected to a uniaxial tensile load directly [12, 13, 14, 15]. Theoretically, the direct tensile method aims to measure the uniaxial tensile strength of materials. However, the local stress concentration and eccentricity are generally inevitable, leading to unexpected crushing and an apparent dispersion of the tensile strength [16, 17, 18]. The Brazilian test is the most widely used indirect tensile method which translates a compressive load to a local tensile stress concentrated at the centre of the disc specimen [19, 20, 21, 22, 23]. In a Brazilian test, a cylindrical sample is submitted to a compressive force along its diameter through diametrical compression between two flat platens, thus inducing tensile stresses perpendicular to the direction of compressive force. Currently, the Brazilian test has been recommended as the acknowledged approach to measure the tensile strength of brittle materials by the ISRM [24], ASME(American Society of Mechanical Engineers) [25], RILEM (International Union of Laboratories and Experts in Construction Materials, Systems and Structures) [26], and Chinese standards [27]. Generally, the Brazilian test supposes an elastic disc in a 2D-plane stress state and deduces the maximum tensile stress located at the centre of discs using the elastic theory [28]. Considering failure criteria based on the maximum tensile stress, the tensile strength is determined by the peak force which causes the failure [29]. To obtain reliable and scientific experimental outputs, the Brazilian test requires two preconditions as follows: (1) the crack initiates at the centre of the disc; (2) the in-plane stress/strain fields follow the elastic theory. However, it is hard to satisfy the condition for the initiation of cracks at the centre of specimens, due to incorrect failure patterns caused by the concentrated force near the loading area. The tiny contact area between platens and cylinder promotes the development of stress concentration. This indicates that a large enough contracting stress can result in contacting areas before the initiation of the central crack. To avoid shear failure by contacting, studies have adapted the Brazilian test to reduce the stress concentration caused by linear loads. These adaptions involve changing the contact geometry between the loading platens and specimens (such as curved loading platens) [30; 31], flat loading platens with strips [32; 33; 34; 35], and flattened Brazilian discs [36; 37; 38; 39]. However, changing the contact configurations means the transition of boundary conditions from concentrated loads to distributed loads. Therefore, setting concentrated boundary conditions may bring about a great inaccuracy in the measured tensile strength of materials with distributed boundary conditions. Previous works have conducted numerical, analytical/semianalytical, and experimental investigations on elastic solutions of tensile strength considering different boundary conditions. However, their research focus largely remains at the theoretical level, which is in the absence of real-time experimental investigations. Consequently, it is necessary to discuss the real-time failure process of cracks before evaluating the tensile strength of brittle materials for Brazilian tests with different boundary conditions.