# An Improved Contact Force Model of Polyhedral Elements for the Discrete Element Method
## 1 Introduction  
Granular materials refer to a collection of discrete solid particles characterized by individual shapes, sizes, and interaction features. Various substances constitute granular materials, including sand, soil, powders, grains, and even small stones. They are among the most commonly encountered materials in natural environments, engineering applications, and daily life [1]. Therefore, granular materials are closely related to our lives and production. Studying the dynamic behavior of granular materials has always been an important research topic [2, 3]. For example, the stability of earth-rock dams under external loads [4], the problem of slope hazards [5], and the fragmentation of railway ballast under the pulsation of trains all involve the dynamic behavior of granular materials [6]. Compared to continuous media, granular materials exhibit non-uniformity and anisotropy [7]. It is unsatisfactory to solve relevant engineering problems using the theory of continuous media. Therefore, non-continuous mechanics methods have been introduced, including discontinuous deformation analysis (DDA) [8, 9], the finite-discrete element coupling method (FDEM) [10, 11], the manifold method (NMM) [12], the material point method (MPM) [13, 14], and the discrete element method (DEM) proposed by [PERSON] [15]. In the calculation of DEM, the analysis of the contact interaction between two particles is divided into two steps. The first step is usually called contact detection, which calculates the geometric features of overlap between two particles, including contact points, contact normal directions, contact depth, or contact volume. The second step is to calculatethe contact force based on the geometric features of contact. DEM is further divided into particle-based DEM and block-based DEM. The main shape elements of particle-based DEM are discs and spherical particles, and the contact between particles is relatively simple [16]. The shape elements of block-based DEM are mainly polyhedrons, and the contact modes include point-point contact, point-edge contact, point-face contact, edge-edge contact, edge-face contact, and face-face contact [17; 18]. Therefore, compared with particle-based DEM, the complexity of contact modes for polyhedral elements increases the difficulty of contact detection and contact force calculation [19]. Regarding the interaction between two particles, theoretical calculation methods for the contact force between particles have been proposed. [PERSON] proposed a block-based DEM model that regards polyhedral elements as rigid bodies that are connected by virtual springs [20]. When subjected to loads, force is transmitted through the deformation of springs, and the contact force between elements is divided into normal contact force and tangential contact force, considering the effect of damping. [PERSON] et al. incorporated the surface adhesion force of elements and the elastic and plastic deformation generated during the loading process into their consideration based on the basic theory of contact mechanics and proposed the spring-dashpot model [21]. [PERSON] added relevant calculation methods for potential functions in the finite-discrete element coupling algorithm. Similar to [PERSON]'s model, it is assumed that polyhedral elements can overlap, and the contact force is calculated by integrating the potential function over the outer surface of the overlap region [10]. [PERSON] et al. applied the energy method to the calculation of contact force between 2D polygons. Based on the energy method, the corner-corner contact between polygons is considered the basic contact mode, and other contact modes are special cases of corner-corner contact, which can be extended from the contact force calculation formula of corner-corner contact [22; 23]. [PERSON] et al. proposed a method for calculating contact forces that utilizes a discrete element model with contact forces equivalent to given macroscopic virgin stresses [24].