# Analytical Modeling of Particle Scratching Process
## 2 Physical Model and Analytical Model of Scratching Process  
### Physical Model  
In this paper, a new physical and analytical model about particle scratching is proposed, in which the physical process and physical model are shown in Figure 1. The particle is simplified as a sphere. When the particle engages with the workpiece at an extremely small depth of cut, the work material undergoes elastic deformation. It is defined as the rubbing phase. With increasing depth of cut, plastic deformation occurs, and the work material piles up at the front and flows to the sides of the particle to form a groove. It is defined as the plowing phase. As the depth of cut continues to increase, the work material can no longer withstand the tearing stress, and a chip is formed. The chip formation stage is defined as the cutting phase. Therefore, there exist three types of material deformation in the single-particle cutting process, namely elastic deformation, wedge of side flow, and chips. However, the three types of material deformation may take place simultaneously. As shown in Figure 2, direction 1 represents material flowing along the flank of the particle to ward the wedge, direction 2 represents material in zone _F-HEG_ flowing under the particle and rebounding back at point \(G\) behind the particle. Figure 1: Physical schematic of the scratching process. (**a**) The particle scratching physical process (Scanning Electron Microscopy (SEM)) [16]. (**b**) Physical model of the scratching process. formation of chips. The separation point \(E\) is shown in the side view, which divides the contact zone into the plastic zone and the elastic zone. The material in zone _F-HEI_ deforms elastically. The maximum stress occurs at point \(F\), and the value is almost equal to the yield stress \(\sigma_{y}\) of the material. The minimum stress occurs on the circumference of circle \(O_{1}\), and the value is 0. As shown in Figure 2b, the material in the zone _HEI-JBK_ deforms plastically. In the zone _HEI-JBK_, the materials experience tensile stress along the particle surface and compressive stress perpendicular to the particle surface. In order to state the material flow clearly, the \(x\)-axis is built with its origin at the \(F\) point, and the positive direction is the same as the direction of the \(Y\)-axis. In any half-circular, the material was considered to flow from half-circular _HEI_ along the surface of the particle, and the half-circular in a plane always perpendicular to the \(x\)-axis in the material flow process. The plastic deformation of the material begins in the arc _HIE_ and ends in the arc _JLK_. In the \(t\) coordinate, an arbitrary plane which is perpendicular to the \(t\)-axis intersects with the particle surface, and the intersection line is a half circle named \(C(t)\). It is obvious that the length of arc \(C(t)\) is a function of \(t\). In particular, the half-circular was named the origin length of plastic deformation \(C_{HEI}\) when the plane through \(O_{1}\) point, the half-circular was named the maximum length of plastic deformation \(C_{JBK}\) when the plane through the \(B\) point. The material flow and deformation process is compared to a circular rubber band set into a sphere in order to describe the flow and deformation process of the plastic deformation zone vividly. It is worth pointing out that only half of the circular rubber band contacts the sphere because the material contacts the particle with a part of the hemispherical surface. The initial contact position of the semicircle rubber band and sphere corresponds to \(C_{HEI}\) in the physical model. The arbitrary position of the semicircular rubber band corresponding to \(C(t)\) in the physical model. The tensile strain in plastic deformation can be calculated similarly to the tensile strain of a rubber band being stretched over a sphere.