# Modelling Wear Phenomena Specific to Mixer Blades in Concrete Production Plants
## 2 Materials and Methods
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & \multicolumn{1}{c}{**Meaning of Terms**} \\ \hline \(\mu\) & the particle–target-surface friction coefficient \\ \(\alpha_{0}\) & angle of incidence \\ \(\tau_{0}\) & particle velocity \\ \(r\) & particle radius (spherical particle) \\ HB & the hardness of the target surface \\ \(\rho_{m}\) & the material density of the target surface \\ \(\rho_{ab}\) & the density of the particle material \\ \(\theta=\frac{1-\mu^{2}}{\epsilon}\) & the elasticity parameter of the surface material \\ \(\sigma_{0}\) & the tensile strength of the target surface material \\ \(\sigma_{c}\) & the flow resistance of the surface material \\ \(\epsilon_{0}\) & the relative deformation at break through stretching of the surface material \\ \(k\) & a constant depending on the assumed breaking assumption \\ \(n\) & the number of cycles at which the wear particle occurs—\(n_{e}\)—elastic and \(n_{p}\)—plastic \\ \(V_{d}\) & the volume of material deformed by the abrasive particle \\ \(V_{d0}\) & the volume of material deformed by a particle \\ \(a\) & the diameter of the contact surface \\ \(m_{az}\) & the mass of the used material \\ \(m_{ab}\) & the mass of the particles that caused the wear \\ \(V\) & the volume of used material \\ \(n_{ab}\) & the number of abrasive particles hitting the surface \\ \(h\) & deformation—the distance from the original surface to the deformation trace \\ \(\varepsilon=\frac{h}{k_{max}}\) & note \\ \(t\) & friction fatigue parameter [30] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meaning of terms in relationships (1)–(7). Figure 4: Calculation of erosive wear intensity. \(\alpha_{crit}\)--is determined from the wear particle production condition at a single stress cycle (\(n=1\)). Angle \(\alpha_{crit}\) is determined using the following equations:  
* for elastic deformation: \[\alpha_{crit}=arcsin\left(\frac{3\;\sigma_{0}}{4\;k\;\mu}\right)^{\frac{5}{2}} \frac{\pi^{2}\;\theta^{2}}{V_{0}\sqrt{5\;\rho_{ab}}}\] (6)
* for plastic deformation: \[\alpha_{cr}=arcsin\left[\frac{e_{0}^{2}}{4V_{0}}\frac{1-\frac{2\;\mu\;HB}{ \sigma_{c}}}{1+\frac{2\;\mu\;HB}{\sigma_{c}}}\sqrt{\frac{3\;c\;\sigma_{c}}{ \rho_{ab}}}\right]\] (7)  
Based on the working hypotheses mentioned above, the analysis of erosive-abrasive phenomena for concrete mixer blades (Figure 1) was carried out by applying the complex Kraghelsky-Nepomnyashchi model. This model assumes that the spherical abrasive particle deforms the surface it strikes either elastically or plastically, dependent on the motion characteristic (sliding or rolling) of the particle upon contact. Thus, the erosive wear intensity can be calculated by taking into account the type of deformation in the surface, with the equations shown in Figure 4.  
Table 2 lists the mixer blade specific parameters involved in the erosive wear intensity calculation relationships, with the complex model and their values. With the specified relationships and parameters, the variation in erosive intensity in the following cases was analyzed:  
1. \(I_{er}=f(\alpha_{0})\)--Variation as a function of the angle of incidence, for the velocity v\({}_{0}\) = 2.5 m/s, for three considered values of the friction coefficient (\(\mu\): 0.25, 0.3, and 0.35);
2. \(I_{er}=f(v_{0})\)--Variation as a function of impact velocity, for \(\mu\) = 0.25, for three values of the angle of incidence (\(\alpha_{0}\): 30\({}^{\circ}\), 45\({}^{\circ}\), and 60\({}^{\circ}\));
3. \(I_{er}=f(\mu)\)--Variation as a function of the friction coefficient, for the velocity v\({}_{0}\) = 2.5 m/s, for three values of the angle of incidence (\(\alpha_{0}\): 30\({}^{\circ}\), 45\({}^{\circ}\), and 60\({}^{\circ}\)).