# Acoustic Emission Analysis of Fracture and Size Effect in Cementitious Mortars
## 3 Analysis of Mechanical Behavior  
In this section, the resulting tensile splitting strength (\(f_{\text{ct}}\)) and effects of the sample size (diameter D) and the boundary condition (strip width B) are discussed. This is achieved by conducting Brazilian splitting tests on samples with varying strip widths and sizes. [PERSON] et al. [24] reported that \(f_{\text{ct}}\) is not affected by one of these parameters individually (size or strip width), but by their ratio \(\beta\) (\(\beta=B/D\)). However, EN 12390-6 [27] recommends a fixed strip width of 15 mm without referencing a specific sample size. In addition, theequation used in EN 12390-6 [27] for calculating the \(f_{\rm ct}\) does not take into account the strip width. In the literature, various equations are found for calculating \(f_{\rm ct}\) as a function of \(\beta\). They are summarized in Table 2.  
The initial development of the equations shown in Table 2 began in the 1890s [42, 43]. [PERSON] [44] developed an exact mathematical solution for the stress state of an elastic homogeneous isotropic disk under a diametrical compression line load. This solution was later modified by [PERSON] [23] for a load distributed over an arc of contact [22, 43, 45, 46]. By superimposing the elastic solutions for opposite compression point loads and circumferential tension load on a circular disk, [PERSON] et al. [47] arrived at a similar conclusion as [PERSON] [44]. [PERSON] [41] modified this solution to account for a load distributed over a strip. However, unlike [PERSON] [23], the distribution was not over an arc but instead on a horizontal projection plane of the arc (the distributed load on the strip is assumed to be only vertical). The equation used in EN 12390-6 [27] is derived similarly, but with the assumption that the load is applied as a line load (\(\beta=0\)). All of the equations in Table 2 are under the assumption that failure occurs at the point of maximum tensile stress (at the center of the disk). Figure 3 shows the resulting \(f_{\rm ct}\) as a function of \(\beta\). The tensile splitting strength is calculated from the experimentally obtained maximum load (P) according to EN 12390-6 [27]. It is observed that the \(f_{\rm ct}\) computed with EN 12390-6 [27] increases with the strip width. This is because, as the strip width increases, the failure load increases, and the maximum tensile stress decreases. However, the equation in EN 12390-6 [27] only considers the increment in load and not the contribution of \(\beta\) in stress reduction. This results in an overestimation of \(f_{\rm ct}\). However, as seen in Table 2, the equations proposed by both [PERSON] [41] and [PERSON] [23] account for this stress reduction. Accordingly, a more accurate estimation of \(f_{\rm ct}\) that allows it to be considered as a material property can be obtained with these two equations. In Figure 4, a comparison is made between the different equations by calculating the ratio F between the average tensile splitting strength (\(f_{\rm ct}\)) for varying \(\beta\) values. An F ratio of one hence indicates no effect of \(\beta\) on the calculated \(f_{\rm ct}\). As seen in Figure 4, the change in \(f_{\rm ct}\) with \(\beta\) raised to 25% for the medium and 75% for the small samples for the EN 12390-6 [27] equation; however, this change is significantly reduced to 11% and 9% (close to one) for the medium and the small samples, respectively, when applying [PERSON] [23] and Tang [41] equations. When investigating the role of the strip width in the Brazilian splitting test of brittle materials, [PERSON] et al. [24] have found the strip width to affect not only the \(f_{\rm ct}\) but also the extent of the size effect. The research concluded that as this width approaches zero (from \(\beta=0.16\) to \(\beta=0.04\)), the size effect disappears, and \(f_{\rm ct}\) will become an intrinsic material property. However, as described in Section 2.2, distributing the applied load over a small width is essential to avoid local compression failure [48].