**3. Modelling of wear process of polyurethane liners**

### **3.1. Objectives in modelling of the wear process**

Wear caused by impact and sliding of solid particles is a complicated process due to the high number of factors that affect the wear mechanism and final wear rate. Wear resistance is a function of (a) the properties of the erodant particles: shape, density, size, and hardness, (b) the properties of the target material: Young's modulus, plastic behavior, and final strength, and (c) wear testing parameters: velocity of impacting particle, angle of impacts, testing temperature, flow rate, sliding velocity, and pushing force. Accordingly, the experimental investigation of the effect of different parameters on the wear resistance is time consuming and costly. Moreover, wear caused by impact and sliding of erodant particles occurs within microseconds, and hence, the experimental study of the physics underlying the problem is not a trivial task [27]. The simulation of the wear process by analytical and numerical approaches not only enables the understanding of the fundamental principles of the wear mechanisms; the models developed can be employed after verification as predictive tools to study the effect of different parameters on the wear rate. Through different analytical and numerical models developed to date to simulate the wear of metals, ceramics, and elastomers, the finite element (FE) formulation has received great attention due to the potent formulation of this technique that enables the modelling of complex geometries, material models, and contact algorithms [27, 49, 50]. In the following section, the research works conducted for FE modelling of wear of PU elastomers will be reviewed and discussed.

### **3.2. Finite element modelling of the wear process**

by hysteresis and friction forces increased the temperature of the PU in the layer beneath the surface. The increase in temperature negatively affected the strength of the PU material leading to lower erosion resistance. Even though, the effect of temperature on abrasive and erosive wear of PU elastomers was addressed, in these studies no external heat source for accurate and uniform control of the temperature during wear testing was employed. Accordingly, in some previous studies, testing assemblies capable of erosion testing at controlled temperatures by employing an external heat source has been developed. Zuev et al. [36] conducted erosion testing at elevated temperatures by controlling the slurry temperature. The increase of slurry temperature from 20 to 70°C improved the erosion resistance of the rubber owing to the improvement in elasticity and softness of rubber at the elevated temperature of 70°C. Marei et al. [37] also reported improvement in erosion resistance of rubber at elevated temperatures. In this study, an air blasting test scheme with the controlled temperature on the input gas was developed. It was found that at testing temperatures with greater difference from the glass transition temperature, the erosion rate of rubber was lower. In a more recent study by Ashrafizadeh et al. [21, 42], a test assembly for erosion testing at controlled temperatures was designed and developed. A cold spray system with controlled gas temperature and temperature controller and cartridge heaters were employed to heat the samples from the exposed and unexposed surfaces, respectively. The accurate temperature field within the samples during the erosion testing was further determined by a finite element numerical heat transfer model. In this study, the effect of temperature on strength, elongation at break, and elastoplastic behavior of PU elastomers was also studied and compared with their wear resistance. This comprehensive study showed that the increase in temperature may improve the erosive resistance of PU elastomers in two ways. First, the increase in softness of PU at elevated temperatures would allow for deceleration of the erodant particles at a longer time and, therefore, the stresses generated will be smaller, which means less damage to the substrate. Second, the increase in temperature can affect the elastoplastic response of PU in such a way as to revert to its initial condition with less plastic deformation after the loading caused by the impact force. Thus, a higher number of impacts will be required to deform the PU to the detachment threshold, which means improved resistance to erosive wear. It was further shown that the increase in temperature can negatively affect the wear resistance in conditions where the final strength of PU becomes smaller than the stresses produced by the impact of erodant particles [21].

**3. Modelling of wear process of polyurethane liners**

Wear caused by impact and sliding of solid particles is a complicated process due to the high number of factors that affect the wear mechanism and final wear rate. Wear resistance is a function of (a) the properties of the erodant particles: shape, density, size, and hardness, (b) the properties of the target material: Young's modulus, plastic behavior, and final strength, and (c) wear testing parameters: velocity of impacting particle, angle of impacts, testing temperature, flow rate, sliding velocity, and pushing force. Accordingly, the experimental investigation of the effect of different parameters on the wear resistance is time consuming and costly. Moreover, wear

**3.1. Objectives in modelling of the wear process**

146 Aspects of Polyurethanes

Several studies have focused so far on developing FE models for simulating the erosion caused by solid particle impact of ductile metals such as AISI 4140 steel and nickel (Ni), Al6061-T6, Ti-6Al-4V and brittle ceramics such as tungsten carbide (WC), Cr3C2, and SiC [50–53]. The models developed enabled an in-depth study of the stresses and strains produced during the erosion process and also assess the effect of testing factors such as particle size, shape, velocity, and impact angle on the erosion rate. On the other hand, fewer studies have focused on FE modelling of the solid particle erosion of soft elastomeric materials such as PU. In a recent study, Zhang et al. [43] simulated the impact of a single particle on PU liners by a FE modelling approach. In this model, an isotropic hardening elastic-plastic constitutive law was selected for the material formulation, and material removal was modelled by deleting the elements that exceeded the failure strain of the PU. The defined element removal criterion enabled to calculate the wear rate as a result of impact of a single erodant particle. The model developed was employed to study the effect of liner thickness on the erosion resistance of PU. The results obtained by the FE model were in good agreement with experiments only up to the liner thickness in which the effect of temperature was negligible. The model failed to correctly predict the same trend as experiments for the erosion rate versus liner thickness due to the fact that the model did incorporate the temperature rise caused by the repeated impact of particles.

The high elongation at break of PU elastomers can lead to significant deformation and distortion of elements in FE modelling of the wear process. To that end, FE-mesh–free techniques may be employed to eliminate the adverse effects of element distortion while modelling the erosive wear of soft substrates such as elastomers [54]. In mesh-free techniques, there is no connection between the nodes, and the model is discretized with scattered particles. For example, Gong et al. [54] developed a 3D combined FE-mesh–free model with smoothed hydrodynamics (SPH) particles. The viscoplastic material model of Johnson-Cook formulation was selected as for the material model. Even though the elements experienced extensive deformation in the FE model, the results obtained by the combined FE-SPH model predicted similar equivalent stresses at the impact point to that of the FE model with negligible difference [54]. Since the computation time by the FE model was approximately four times shorter than the combined the FE-SPH model, the FE formulation can be considered as the superior technique.

In a recent study by Ashrafizadeh [49], a more comprehensive material formulation capable of accounting for hyperelastic, elastoplastic, and stress softening of the elastomer was developed using the FE technique. The material model formulation successfully predicted the elastoplastic and stress softening response of the PU as validated by conducting cyclic deformations of a single element and comparing the stress-strain behavior of the element with that of experiments. This research allowed for an in-depth evaluation of the effect of temperature, material softness, final strength, and elastoplastic behavior on the stresses produced as a result of the impact of erodant particles. The model successfully simulated the cutting mechanism caused by the impact of a single erodant particle (see **Figure 9**). Moreover, the impact of ten solid particles was modelled to study the mechanism of material removal by accumulation of residual strains up to the detachment of material. The model provided support for this mechanism and successfully predicted the shape of the formed asperities (see **Figure 10**) similar to those of the eroded surface of PU as was observed from experiments (see **Figure 8**).

**Figure 9.** Material removal by cutting mechanism as predicted by the FE model [49].

**Figure 10.** The ridges formed on the surface of an eroded PU as predicted by FE model [49].
