## 2 Wear Prediction Procedures
Surprisingly enough, the procedure produces useful results and has been used in numerous studies as a simpler and cheaper alternative for contact analysis. In the EFM approach, the contact pressure developed by each spring in the elastic layer is given by the following [37]:  
\[p_{i}=\frac{E_{W}}{L_{i}}\delta_{i} \tag{4}\]  
where \(L_{i}\), \(E_{w}\), and \(\delta_{i}\) refer to the elastic layer thickness, the elastic modulus of the layer, and the deformation of the spring, respectively. The elastic modulus is determined from the elastic modulus and Poisson's ratio of the bodies in contact [37]. For a contact in which one of the bodies is rigid, \(E_{w}\) is given by the following [38]:  
\[E_{W}=\frac{(1-\
u)E}{(1+\
u)(1-2\
u)} \tag{5}\]where \(\
u\) is Poisson's ratio and \(E\) is the elastic modulus of the deformable body. The contact pressure on the elastic spring may thus be determined from  
\[p_{i}=\frac{\delta_{i}}{L_{i}}\frac{(1-\
u)E}{(1+\
u)(1-2\
u)}. \tag{6}\]  
The entire load supported by the elastic layer (resultant force) may then be determined through a summation of the contact forces developed by the individual springs in contact. This resultant is given by the following:  
\[F_{N}=\sum p_{i}A_{i}, \tag{7}\]  
where \(F_{N}\) is the resultant force normal to the contact surface and \(A_{i}\) is the spring element area that depends on the number or springs, and thus, geometry discretization. In contrast to the EFM, the FEM is a more superior approach to the solution of contact problems. For contact problems, the FEM approach involves the solution to a nonlinear boundary problem where contact stresses and contact boundaries are initially unknown. The details of the contact problem and solution procedures are beyond the scope of this article. For further explanations, readers are encouraged to refer to works on nonlinear finite element methods, such as [39]. The FEM approach is more complex and expensive but yields more accurate results. In both the EFM and FEM approaches, the contact geometry is discretized. Equation (3) is thus modified to indicate the discretized geometry as follows:  
\[h_{i}^{n}=h_{i-1}^{n}+kp_{i}^{n}\Delta s_{i}. \tag{8}\]  
In Equation (8), \(n\) refers to the surface nodes in the FEM approach and the number of elastic springs in the EFM model. To achieve reasonably accurate wear prediction, the geometry needs to be updated to reflect the evolution of the contact conditions as wear occurs. The procedure for geometry update has some dependence on the chosen method for contact pressure analysis; however, update procedures generally involve moving the contact boundary in a direction normal to the contact surface by an amount equal to the wear. A more detailed discussion of geometry update procedures is presented in [40]. The iterative wear prediction procedures reported in [21; 22; 23; 24; 27; 32; 27; 41; 42] defer in their details of implementation; however, the following three steps are common to these procedures:  
* Estimation of the contact pressure using techniques such as the EFM or FEM. * Computation of the incremental wear (\(kp_{i}\Delta s_{i}\)) which appears as the last term in Equation (3). * Geometry update to reflect the wear and determination of the new geometry for the next cycle of wear analysis. What follows is a description of the implementation of the three steps using the EFM and FEM approaches to predict wear on a journal bearing of a multibody system.