## 1 Introduction
Multiscale methods are also adopted by combining the above approaches for the accurate analysis of quasi-brittle materials [30; 31]. [PERSON] et al. [32] developed a two-scale phase field fracture framework and discussed computational homogenization. However, the representative volume element (RVE) concept is not completely in accordance with actual quasi-brittle materials [2; 4]. A more popular strategy is considering the local multiscale, i.e., only the potential cracking zone is considered with multiphase at the mesoscale [31]. In this way, both indirect and direct approaches are often adopted. In the indirect approaches, the critical fracture parameters of the materials, such as tensile strength and fracture energy, are modeled as stochastic variables with random fields so that different mixtures at the mesoscale are modeled implicitly. Recently, [PERSON] et al. [33] developed a method combining the PF-CZM and random fields for quasi-brittle materials in a 2D setting; only tensile strength is modeled by random fields. [PERSON] et al. [34] analyzed the size effect at the mesoscale with the same method. [PERSON] et al. [35] adopted statistical reconstruction algorithms and the Karhunen-Loeve expansion technique with microscale X-ray computed tomography (CT) images in order to get the random fields of tensile strength. In addition, [PERSON] et al. [36] generated a random field with a stochastic harmonic function (SHF). On the other hand, various phases at the mesoscale are generally taken into account explicitly in the direct approaches. [PERSON] et al. [37] investigated the mechanical behaviors of concrete in a 2D mesoscale setting, with the aggregates generated using a random sequential addition (RSA) method. [PERSON] et al. [38] developed a 3D mesoscale method for the damage and fracture simulation of quasi-brittle materials with the PF-CZM; the aggregates were built using a random generating and packing algorithm. [PERSON] et al. [39] proposed a new mesoscale modeling method in which the PF-CZM is combined with the cell-based smoothed finite element method (CSFEM) to improve computational efficiency. However, it is not easy to extend these methods to engineering applications directly. First of all, most studies focus on small specimens, and the distribution of both aggregates and random fields is assumed to be ideal. Moreover, the extensive Monte Carlo simulations (MCS) cost a lot, even for the RVEs. Additionally, the determination of the micromechanical parameters usually needs extra discussion, and all these factors may lead to an unreal deviation. Another promising strategy is offered by peridynamics (PD), which substitutes integrodifferential motion equations for partial differential equations (PDEs). These equations are always applicable whether the displacement field is continuous or discontinuous. PD has been drawing a lot of attention for its simple and powerful ability to capture complicated failure processes. [PERSON] et al. [40] proposed a mesoscale PD for modeling the cracking process in concrete. To properly predict cohesive crack propagation, [PERSON] et al. [41] proposed a PD-CZM and an approach combining PD and the finite element method (FEM). See [42] for a comparative review between PD and PF models. In this work, the comparison between the PF-CZM and the modified PDs will be provided as the second numerical example. It can be seen that most of the existing studies focus mainly on the realization of the failure simulation of quasi-brittle materials. However, there have been a few studies on the effects of softening properties in numerical simulations. One key point is the inherent drawbacks of the various models. Inspired by the work by [PERSON] [43] and [PERSON] [44] and the excellent properties (e.g., no mesh bias, accuracy and insensitivity to the length scale parameter, etc.) of the PF-CZM, it is thus possible to focus on the softening laws in the accurate analysis of quasi-brittle materials.