# An Improved Contact Force Model of Polyhedral Elements for the Discrete Element Method
## 1 Introduction
In addition, experimental studies on the contact behavior between particles have been conducted. [PERSON] et al. used a pendulum device to collide two steel balls in order to study the effect of the strain rate on ball impact, and they obtained the coefficient of restitution (COR) and contact duration through impact tests [25]. In order to obtain the contact law applicable to discrete element numerical calculations, [PERSON] et al. conducted a series of contact experiments on spherical crystal fragments with a diameter of 14.72 mm using a micro-mechanical model [26]. [PERSON] et al. conducted particle collision experiments with particles of different shapes and contact types to study the contact behavior between particles and proposed a particle contact detection algorithm based on the principle of energy conservation [21]. [PERSON] et al. investigated the relationship between micro parameters such as contact force in DEM simulations and the macro parameters of specimens by conducting uniaxial compression tests and direct tension tests [27]. [PERSON] et al. designed a series of experiments to determine the sliding friction coefficients between particles with different shapes and contact forms, which improved the accuracy of calculating tangential contact forces between particles [28]. Unlike spherical particles, the contact forces of which can accurately be calculated based on simple contact depth using Hertz theory or other methods, the calculation of contact forces for non-spherical particles strongly depends not only on the overlapping area but also on the specific geometric configuration of the overlapping area [29]. Both of these factors are difficult to determine in numerical simulations, and the contact forces for different contact modes for polyhedral elements are still widely debated issues. Therefore, the mathematical relationship between the contact force and the contact overlap depth for polyhedral particles under different contact modes has not been well established. In addressing these challenges, this study aims to establish a comprehensive understanding of the mathematical relationship between contact forces and contact overlap depth for polyhedral elements under different contact modes. By employing a combined approach of experimental investigation and numerical simulation, this work seeks to provide solutions to the difficulties posed by the unique geometric configurations of non spherical particles in determining contact forces. Ultimately, this research endeavors to provide an improved contact force model and improve the accuracy of DEM simulations in polyhedron-based granular material systems. In the following, this paper introduces an impact experimental method designed for polyhedral specimens in Section 2 in order to measure inter-element contact forces under different contact modes. Subsequently, in Section 3, based on [PERSON]'s contact model and utilizing experimental results, a stiffness correction coefficient is proposed, establishing the relationship between inter-element contact forces, stiffness, and contact depth under different contact modes. In Section 4, the improved contact force model is integrated into discrete element simulations, and the feasibility of the contact force model is validated by comparing contact forces for polyhedral elements under different contact modes in both experiments and simulations. In Section 5, three sets of packing experiments with cubic blocks of different sizes are designed. Using the DEM simulation based on the improved contact force model, these experiments are simulated to verify the accuracy of the contact force model and extend this approach to simulate granular materials composed of arbitrarily shaped polyhedral particles, demonstrating the feasibility of this method. Finally, in Section 6, a brief summary of the conclusions of this work is presented, and possible extensions are outlined.