## 1 Introduction  
Peridynamics (PD) is a nonlocal theory originally developed to address discontinuity issues [1], which discretizes an object into a number of material points connected by bonds. Different from conventional continuum mechanics (CCM), PD theory adopts the integral form of a governing equation without any spatial derivative [2]. The macroscopic cracking is strongly associated with spontaneously progressive breaking of bonds within the model [3]. Therefore, PD can predict damage development without additional assumptions as in the CCM, for example, crack extension criteria and node enrichment functions. PD provides a powerful tool to analyze material and structural failure processes [4; 5], and has already been successfully applied to various problems, such as deformation and fracture simulation for brittle materials [6; 7; 8; 9; 10; 11], asphalt [12], ferrite and pearlite wheel materials [13] and composite beams [14], etc. Currently, generic variations of PD fall into two branches: bond-based peridynamics (BBPD) and state-based peridynamics (SBPD) [15]. The SBPD is a reformulation of the BBPD relations, which is divided into ordinary state-based PD and nonordinary state-based PD. The SBPD introduces the force vector state \(\underline{T}\) which maps the deformation state \(\underline{M}\) to the force state \(\underline{T}\) at all points within the affect region, and the bond force depends on the deformation of all bonds in the affected region. Nevertheless, the BBPD model is more simple to comprehend and carry out, which is well suited in multiscale analyses of composite materials, involving deformation and brittle fracture analysis [16; 17; 18; 19; 20; 21; 22]. The prototype microelastic brittle (PMB) model [23] is the most widely employed bond force constitutive model in BBPD, which is a linear microelastic model firstly proposed to simulate brittle failure process [24] in materials such as concrete [25; 26; 27; 28; 29] and cement [30; 31]. The bond force in PMB can be regarded as the spring force, and the micromodulus represents the bond stiffness. A micromodulus can be calculated based on both the energy equivalence principle and force intensity. For instance, [PERSON] [32] developed the expression of the one-dimensional (1D) numerical micromodulus by introducing the concept of PD stress for linear elastic solids, which provides a new inspiration for the implementation of PD. However, there is a lack of research on the numerical micromodulus for the two-dimensional (2D) model. Some practical engineering problems (e.g., thin plate tension and compression) can be simplified into plane problems with the purpose of low computational expense and high precision. Therefore, it is necessary to address this knowledge gap. In this work, the numerical micromodulus for the 2D plane stress problem is explored first. It is important to note that any numerical method will have errors. Technical aspects such as the loading algorithm and long-range force effect must be considered to achieve higher computational accuracy while minimizing expenses in BBPD. In practical application, the micromodulus is typically a constant when the interaction range is determined, that is, the bond force does not vary with the distance between two points [16]. As a matter of fact, it does not conform to the laws of physics due to the nonlocal feature in PD. Essentially, bond force owns the long-range force effect, which is similar to the atomic potential, that is, the bond force between two points decreases as their distance increases. The previous studies have revealed that long-range force effects can be represented by the influence function [33; 34]. Therefore, in order to improve the applicability of the numerical micromodulus, it is necessary to introduce the influence function into the numerical micromodulus and explore the effect of the influence function types on the simulation precision.