## 3 Optimal Softening Law  
A typical softening law was often assumed for convenience in many studies [39; 43; 50]. However, it is well known that quasi-brittle materials are usually heterogeneous, especially concrete-like materials. It should be noted that we have confined our investigation mainly to concrete in this work due to the complexity of other quasi-brittle materials, such as fiber-reinforced concrete, which deserve additional analyses. Due to the excellent properties of the PF-CZM, such as no mesh bias, accuracy and insensitivity to the length scale parameter, it is possible to focus only on the effects of softening laws on the mechanical responses of different concretes [1; 25; 26]. In this section, we propose and validate a simple new approach to determining the optimal softening law in the framework of the PF-CZM. It could be found in the following section that the geometry of the specimen would not affect the softening properties significantly, which means the approach would be suitable for engineering analysis directly once the optimal softening law is determined. It should be noted that complex, large-scale materials and structures are not involved in this work.  
A set of notched concrete beams under three-point bending is modeled, as tested by [43] and shown in Figure 3. Only two groups of the test are considered in this work, i.e., C60 and C100. Additionally, there are three specimens in each group. The material properties are shown in Table 1. The maximum tensile stress failure criterion is adopted. It can be easily found in Figure 4 that the predicted curves using the concrete damage plasticity (CDP) model with [PERSON]'s [49] softening law are lower than the experimental results, and the error increases when the strength of the specimen increases. The same tendency could be observed in the P-CMOD curves using the PF-CZM. It is well known that the softening law can reflect the stochastic distribution of mesoscale mixtures in FPZ implicitly. For instance, we only compared the results with three typical softening curves in this paper to determine the optimal softening law. It should be noted that the consideration of other softening laws is naturally based on this approach [26; 51]. In addition, the length scale parameter is \(\ell_{0}=2.5\) mm, and the plane stress assumption is adopted for 2D problems in this work. Displacement-controlled is used for the acquisition of mechanical response curves, and the prescribed displacement \(u=0.8\) mm is applied in 80 incremental steps, with each incremental step \(\Delta u\) fixed at 0.01 mm.  
Figure 4 depicts the P-CMOD curves predicted using the PF-CZM with linear, exponential and Petersson's softening curves, together with the experimental results. Since the crack path was not given, here we mainly analyzed the P-CMOD curves. It can easily be found in Figure 4 for both C60 and C100 concrete samples that the P-CMOD curves obtained with the linear softening law are higher than with other softening laws, and the results using the exponential softening law are the minimum. In Figure 4a, all predicted curves are higher than the experimental results. The numerical results are consistent with the experimental data on the whole, except when using the linear softening law. The specific predicted peak loads are listed in Table 2. By comparing the experimental mean peak load and the predicted peak load, we can assume that the exponential softening curve is the optimal softening law for C60 concrete, while for C100 concrete, it is the Petersson softening curve. Furthermore, it can also be found that for C60 concrete, the predicted results are higher than in the experiments in general; this deviation can be modified by adjusting the softening law. A more accurate method of obtaining an accurate softening