# Modelling Wear Phenomena Specific to Mixer Blades in Concrete Production Plants
## 2 Materials and Methods
The mechanical impact of the abrasive particle with the worn surface causes elastic or plastic deformation of the material, induces specific friction processes caused by the sliding of the particle on the surface, and develops characteristic fatigue processes on repeated contact. Figure 2: The steps of the research program. This is synthetically expressed by the Nepomnyashchi model describing the process of wear particle formation by erosion and underlying the calculation of the wear intensity of this process [30; 31]. According to [PERSON]'s model, the apparition of the wear particle is primarily caused by the fatigue that follows the solid particle's friction in the impact process and surface deformation. The erosion wear intensity is defined as the ratio between the cumulative mass loss of the removed material through wear of the target surface (\(m_{uz}\)) and the mass of the erosive particles that produced this wear (\(m_{ab}\)) and is calculated using the equation:  
\[I_{er}=\frac{m_{uz}}{m_{ab}}=\frac{V\cdot\rho_{m}}{\frac{4}{3}\pi r^{3}\rho_{ ab}n_{ab}} \tag{1}\]  
Considering the Nepomnyashchi model and the Kraghelski relations, the number of cycles at which particle detachment occurs is determined, depending on the deformation state of the worn surface material. For elastic and plastic deformations, the calculation relations are  
\[n_{el}=\left[\frac{3\ \pi\ \sigma_{0}\ \theta}{4\ k\ \mu}\sqrt{\frac{r}{h}} \right]^{t} \tag{2}\]  
\[n_{pl}=\left(\frac{e_{0}}{2}\sqrt{\frac{\sigma_{c}-2\ \mu\ HB}{\sigma_{c}+2\ \mu\ HB}} \right)^{t}\left(\frac{r}{h}\right)^{\frac{t}{2}} \tag{3}\]  
Under the effect of the kinetic energy, the solid particle deforms the surface and it slides relatively to it, generating a friction force. The elementary path of friction is determined on the basis of the contact surface's dimensions; thus, the elementary volume of the worn material (the volume of material worn in the elastic deformation regime, \(V_{e}\) or the plastic one, \(V_{p}\)) over the surface is determined via the equation:  
\[V_{e,p}=\frac{\pi}{2}n_{ab}\ k_{e,p}^{-t}\ r^{\frac{1-t}{2}}{\int_{0}^{x*}h^{ \frac{t+3}{2}}dx} \tag{4}\]  
The differences in the considered deformation depend on the number of cycles at which the material is detached. The integral in the relation is calculated via the law of variation in surface deformation and the limit of integration (for which the coefficients are highlighted in Table 1). Figure 3: Particle interaction with the surface under erosion: (**a**) successive positions of the particle upon impact with the surface; (**b**) detail of the trace left by the particle on the impact surface, for an intermediate position. The calculation of the erosive wear intensity \(I_{cr}\) is performed using the equations from Figure 4, considering three distinct cases (\(H_{1}\), \(H_{2}\), and \(H_{3}\)) related to the kinematics of the particle with the surface:  
1. The tangential component of the particle velocity cancels with increasing surface deformation (\(\mu\cdot tga_{0}>1\))--case \(H_{1}\);
2. The tangential component of the particle velocity cancels as the surface deformation does (\(0.5\leq\mu\cdot tga_{0}\leq 1\))--case \(H_{2}\);
3. The tangential component of the particle velocity does not cancel in the process of surface deformation (\(\mu\cdot tga_{0}<0.5\))--case \(H_{3}\). Under certain conditions, erosion manifests itself as pure abrasion; the transition can be judged by the relationship between the angle of incidence (\(\alpha_{0}\)) of the particle on the target surfaces and the critical angle of incidence (\(\alpha_{crt}\)) at which microbursting occurs:  
\[\alpha_{0}>\alpha_{crt} \tag{5}\]