Issue 33

C. Montebello et alii, Frattura ed Integrità Strutturale, 33 (2015) 159-166; DOI: 10.3221/IGF-ESIS.33.20 160 mitigate the effect of the strong stress gradient. On the other hand, a different approach consists in computing the evolution of the stress intensity factors, ∆K, as a function of the crack length to analyse whether the crack will stop or not [3, 4]. Even though both the approaches show good results in terms of life prediction accuracy some important limitations make them difficult to implement in the industrial context. For instance, the analysis of the ∆K evolution at the crack tip implies to set up a computational strategy to determine the crack path resulting from the macroscopic load application. This operation is heavily time-consuming. On the other hand, the methods based on an averaged quantity over a critical distance require an accurate description of the stress gradient evolution that is obtainable only by using a really fine mesh (micron size range), condition difficult to reproduce in industrial FEM models. M ODEL Background retting-fatigue is a special damage process that occurs at the contact area between two materials under load and subject to minute relative motion by vibration or some other force. One of the main peculiarities of this phenomenon is the fact that the contact introduces a severe and extremely localized stress gradient in the vicinity of the contact edge. The feature presented above is the same that characterizes cracks or notches where a steep stress gradient is present as well. As a consequence the mechanical fields generated close to the contact edge and the ones arising at the crack tip are comparable. This analogy can be exploited to apply the mathematical tools already developed for fracture mechanics to fretting fatigue. Giannakopoulos has been the first author to quantify from an analytical point of view this analogy [5, 6]. Two different contact configuration are studied: (i) flat punch over a planar surface which creates a stress singularity at the contact tip (crack analogue) and (ii) round punch over a planar surface characterized by a smooth transition to zero pressure at the contact area edges (notch analogue). Among the different approaches available in fracture mechanics one in particular can be really useful to describe the stress gradient effect in fretting-fatigue. An original model has been proposed by Pommier [7, 8] aiming at describing mixed- mode cyclic elastic-plastic behavior of the crack tip at the global scale. The purpose was to establish a reasonably precise model but condensed into a set of partial differential equations so as to avoid huge elastic-plastic FE computations. The model proposed hereunder exploits the results obtained by Giannakopoulos [5, 6] and Pommier [7, 8]. A description of fretting-fatigue using nonlocal quantities has been developed. The tools employed to describe the elastic-plastic behavior of the crack tip at the global scale are redefine and adapted to fretting fatigue contact problem via the crack analogue approach. Field partitioning The model presented here, is based on the post-treatment of the velocity field generated by fretting-fatigue close to the contact edges. The velocity field is partitioned into a summation of multiple terms, each one expressed as the product between nonlocal intensity factors, I s , I a , I c , depending on the macroscopic loads applied to the mechanical assembly, and spatial reference fields, ds, da, dc, depending on the local geometry of the part,               ,       s s a a c c v x t I t d x I t d x I t d x (1) From the practical point of view, the velocity field is obtained through a finite element computation. In Fig. 1 an example of a FE model used in the analysis, is presented. A cylinder-plain contact configuration is employed. This choice is driven by the fact that the numerical procedure is validated by comparing it with experimental tests [9-11], carried out with a cylinder-plain apparatus. In other word the FE model needs to be as close as possible to the experimental setting. Concerning the main parameters used in the FE model definition, it consists in an elastic quasi-static computation and plane strain linear elastic quadrilateral elements are employed. To handle the contact at the interface, the technique employed is the Lagrange multipliers, which assures the best accuracy. To modelling the friction, Coulomb’s friction law is employed. With regard to the mesh used, it is a structured mesh characterized by an average length of 5-10 microns. The loads applied to the pad to reproduce fretting-fatigue consist in a constant normal load (P) and a cyclic tangential load (Q). A cyclic bulk load (σ fa ) is applied to the specimen to reproduce the fatigue effect. Once the velocity field is computed, the following step is to partition it in order to extract the nonlocal intensity factors. Since the most critical zone in fretting-fatigue is situated close to the contact edge, only the value of the velocity field inside a circular region Ω, of radius r, centered at the contact tip is retained. F