## 1 Introduction  
It is crucial to comprehend how size impacts mechanical properties in structures and materials for the secure and dependable construction of engineering structures [1, 2, 3, 4, 5, 6]. In rocks, the nominal strength tends to rise as the sample size decreases [7, 8, 9], demonstrating the normal size effect (NSE) of nominal strength. However, studies investigating the size effects of rocks have demonstrated that the nominal strength decreases with decreasing sample sizes, a phenomenon referred to as the anomalous size effect (ASE) [10, 11, 12, 13, 14, 15, 16]. Other more complex size effect laws have also been reported by researchers [17, 18, 19, 20, 21]. Hence, it is crucial to explore the effect of sample size on nominal strength in rocks and construct a model that can predict the NSE and ASE reliably. To date, researchers have put considerable work into investigating how the mechanical properties of rocks vary with sample size, as well as understanding the reasons for the NSE and ASE [22, 23]. [PERSON] initially discovered the inverse relationship between the strength of a cord and its length in the 15 th century. Fifty years later, [PERSON] argued that there could be a defect in the long cord which causes the short cord to be stronger, thus exemplifying the statistical size effect. Following this, [PERSON] argued that the low strength and size dependence in isotropic solids stem from discontinuities and flaws, and introduced fracture mechanics [24]. [PERSON] established the statistical size effect based on the weakest-link model [22]. Following this, numerous enhanced models were suggested to depict the statistical size effect of mean strength and the variation, and this research has continued to date [8; 25; 26; 27; 28; 29; 30; 31]. There is a general consensus that the size effect observed afterward is related to statistics and can be elucidated using a statistical size effect model. For example, [PERSON] and colleagues [32; 33] created a statistical size effect model that can effectively capture the mixed type ASE-NSE of nominal strength. Nevertheless, these models fail to capture the deterministic size effect, which involves the fracture process zone (FPZ) [34]. [PERSON] [35] is a pioneer in the study of the size effect by assuming that the stress on a specific region (referred to as an FPZ) does not change around the notch root. [PERSON] introduced the dimensional analysis method to address the size-dependent mechanical property [23; 36; 37]. [PERSON] and colleagues [34] proposed the size effect law (SEL) based on the balance of energy release. The multifractal scaling law (MFSL), pioneered by [PERSON] et al. [38; 39; 40], forecasts a limited strength in large intact samples; this phenomenon cannot be accounted for by the SEL [41]. [PERSON] et al. [42; 43] later formulated a universal SEL to include the observed finite strength of large samples. [PERSON] and [PERSON] [44] proposed and improved the boundary effect model (BEM) to clarify the NSE and predict the strength and fracture toughness [2; 45; 46; 47]. [PERSON] [48] presented a model that incorporates size effects by utilizing fracture mechanics. The theory of critical distances (TCD) has been utilized to develop and enhance a commonly accepted size effect model [49; 50]. In addition to these theoretical models, simulated approaches have been devised to study the size dependence of nominal strength [51; 52; 53; 54; 55]. Nevertheless, the theories and simulations mentioned above cannot capture and explain the ASE. In addition, the BEM has been proven to effectively capture the mixed type ASE-NSE of nominal strength, but it fails to forecast size-dependent unnotched samples [56; 57]. For the ASE, the concept of fractal fracture energy is integrated into the SEL to explain this phenomenon. When the fractal exponent of fractal fracture energy exceeds 1, the integrated SEL can represent the ASE [58].