# Analytical Modeling of Particle Scratching Process
## 1 Introduction  
The scratching process of particles is a complex material removal process involving cutting, plowing, and rubbing, and it widely exists in abrasive wear, abrasive machining, and other industry applications. Mechanisms of particle-workpiece interaction and material removal at the microscopic level are very important for the comprehensive understanding of abrasive wear, abrasive machining, and other industry applications. A number of cutting experiments with a single particle were performed to study the mechanism of particle-workpiece interaction [1]. The acting forces and the profile of the generated grooves were measured to analyze the effect of cutting speed, depth of cut [2], abrasive type [3], and particle geometry [4] on the process of material removal. The mechanical models of specific energy [5] and cutting forces [6] during particle cutting were proposed. Even the cutting forces of a single particle were integrated to predict the grinding forces [7]. The scratching test is a common test method for abrasive wear, including parallel scratches [8] and intersecting scratches [9].  
Modeling is essential for understanding the particle scratch process. A scratching model of the Berkovich diamond indenter has been presented to predict the machined depth [10]. A material removal model has been established to estimate wear rate [11]. [PERSON] et al. presented a model based on the theories of elastic-plastic micro-contact mechanics and abrasive wear mechanism to investigate the effect of particle deformation in chemical mechanical polishing [12]. Modeling of wear processes is the most important factorinfluencing wear. A model of abrasive has been given to link different effects together to better understand their influence [13]. A 3D numerical modeling has been established to determine contact pressure, normal and friction force, local friction of abrasive, and coefficient distribution [14]. The meshless smoothed particle hydrodynamics method has been used to establish numerical modeling of abrasive wear [15]. In the existing research, most analytical models are 2D models. Three-dimensional simulated models have been applied to the scratching process, but the method of simulation is difficult to achieve high accuracy and takes a long time, which is not convenient for investigating the physical phenomena of the scratching process.  
In this study, physical and analytical models of the scratching process have been proposed. The particle is simplified as a sphere, and the direction of material flow and removal are considered. The contact zone is divided into the elastic deformation zone and the plastic deformation zone. In the elastic deformation zone, the calculation of contact stress follows the Hertz stress distribution. The flow stress in the plastic deformation zone is determined by the corresponding strain, and the strain is the function of the vertical distance from a specific location to the interface of elastic deformation and plastic deformation zone. Tangential and normal forces applied to the particle have been calculated by integrating contact or flow stress on the contact zone. The stress distribution on the plastic deformation zone according to the analytical model has also been given. The force ratio has been presented, and the parameters that may affect it have been discussed.