## 2 Development of a Combined Size Effect Model for Rocks under Uniaxial Compression  
### Size Effect Model of Surface Effects  
The ASE in rocks under uniaxial compression has been widely reported by researchers [14; 61; 67]. [PERSON] and [PERSON] [62] and [PERSON] et al. [63] argued that the ASE of small samples is related to surface flaws or imperfections that are introduced during sample preparation. In fact, as the sample size increases, the volume fraction of surface flaws or imperfections decreases, resulting in the well-known anomalous size effect (ASE). If this idea of surface flaws holds true, then polishing the surfaces of the sample may remove or decrease surface flaw effects. [PERSON] and colleagues [61] performed compressive tests on unpolished sandstone samples with diameters of 19, 25, 31, 50, 65, 96, 118, and 146 mm, obtaining the highest nominal uniaxial compressive strength (UCS) for the samples with a diameter of 65 mm. They also conducted compression tests on two polished samples of sandstone, each with a diameter of 25 mm [61]. It is noted that the nominal UCS of polished samples with a 25 mm diameter is higher than that of unpolished samples with the same diameter, but lower than that of unpolished samples with a 65 mm diameter. Therefore, [PERSON] et al. [61] argued that the fractal characteristics proposed by [PERSON] [58] seem to be the main cause of the ASE observed in the nominal UCS, while surface flaws play a secondary role. Yet, the physical mechanism of the fractal characteristics of the ASE remains ambiguous. Existing studies have shown that as the height-diameter ratio increases, the nominal UCS of rocks decreases, and the macroscopic failure characteristics gradually transition from splitting failure to shear failure [74; 75]. This is due to the generation of confining stress in the sample caused by the platen-sample interface friction coefficient [76]. As the height-diameter ratio increases, the percentage of the confined zone decreases due to the interface friction coefficient, leading to a decrease in the nominal UCS. Moreover, as the friction coefficient increases, the simulation results indicate that the influence of the height-diameter ratio on the nominal UCS is more significant [75], as shown in Figure 1. This means that the platen-sample interface friction coefficient is not the cause of the ASE. As is known, flaws present on a free surface can reduce the strength of a sample's surface material, and the material around these flaws is more prone to cracking. More importantly, the free surfaces of samples cannot act as obstacles to internal crack propagation, and internal cracks can easily reach these free surfaces, resulting in a reduction in the nominal strength of the samples. For example, failure process simulation [64; 65] indicates that  
Figure 1: The relationship between height–diameter ratio and nominal UCS under different friction coefficients [75]. smaller samples can quickly slide along a flaw on the surface, resulting in a lower nominal UCS. [PERSON] et al. [66], [PERSON] et al. [67], and [PERSON] [68] observed that regarding the uniaxial compression of basalt, sandstone, high-strength gypsum, limestone, and granite, cracks originating on one surface of smaller samples can promptly propagate to the opposite surface. The above analysis suggests that the flaws on a free surface, as well as the free surface itself, have a substantial influence on the mechanical behavior of rocks under uniaxial compression, and eventually result in the ASE of nominal UCS. Therefore, considering a sample under uniaxial compressive stress \(\sigma_{\rm n}\), the sample is composed of an interior zone covered by a free surface layer, as illustrated in Figure 2. The diameter and height of the sample are \(D\) and \(H\), respectively. The thickness of the surface layer is \(t\). Compressive stresses are taken to be positive in this investigation.