Issue 35

E. Giner et alii, Frattura ed Integrità Strutturale, 35 (2016) 285-294; DOI: 10.3221/IGF-ESIS.35.33 293 when modeling crack propagation in fatigue problems. This is accomplished through a special mathematical formulation of the FE method that includes the enrichment of the standard finite elements with additional degrees of freedom (DOFs) at the nodes. These additional DOFs are associated with the nodes of the elements that are geometrically intersected by the crack location (called enriched nodes and elements, respectively). Thus, the discontinuity is included in the numerical model without modifying the discretization. The X-FEM formulation allows for a further type of enrichment for those nodes that surround the crack-tip. These nodes are enriched with additional DOFs to represent the first term of the classical Williams series expansion in linear elastic fracture mechanics in terms of the displacement field. We have applied the X-FEM method to two problems with different indenter materials. The aim is to predict numerically the crack paths and the analysis is carried out using the X-FEM implementation developed by the authors [11,12]. The implementation is performed as a user’s subroutine in the commercial code ABAQUS and can take into account crack face contacts along the loading cycle, which have been proved to be essential for the correct crack prediction. Fig. 11 shows the result of the crack propagation in the vicinity of the contact corner after 10 increments using the min(  ) criterion after each increment to predict the orientation of the next crack segment. In Fig. 11, left, the Young’s modulus for the indenter is E indenter = 10 3 MPa, whereas in the simulation of Fig. 11, right, E indenter =10 8 MPa. In both cases E specimen = 72·10 3 MPa. As expected, it can be observed that the crack growth path is clearly deviated inwards when the material stiffness of the indenter is large with respect to the specimen. The procedure presented in this work enables the prediction of more accurate crack paths in components that can be used for the integration of crack growth laws in order to estimate fatigue lives. Figure 11 : Crack propagation after 10 increments using X-FEM and the min(  ) criterion after each increment. Left, E indenter = 10 3 MPa; right, E indenter =10 8 MPa. E specimen = 72·10 3 MPa. C ONCLUSIONS rack propagation directions and paths have been predicted for fretting fatigue tests under complete contact conditions. This type of problem is subjected to non-proportional loading, which invalidates the application of conventional orientation criteria usual in LEFM and that are only useful for proportional loading. The criterion of the minimum shear stress range has been proposed, and it is based on the minimum value of  evaluated ahead the crack tip and along the entire cycle. The prediction has been performed numerically using FEM and X-FEM including a formulation that allows for crack face contact, which is essential to take into account the effects during the compressive part of the cycle. A parametric study has been accomplished by variation of the normal load on the indenter, the bulk load on the specimen, the stress ratio and the elasticity modulus of the indenter. It has been shown that the parameters related to the loading have no effect on the crack deflection, whereas the change in the material stiffness of the indenter has a significant effect on the predicted crack path direction. The numerical results are in good agreement with the experimental observations, validating the procedure. This can lead to more accurate fatigue life estimations once the crack path is predicted using the proposed procedure. A CKNOWLEDGEMENTS he authors gratefully acknowledge the financial support given by the SGPI of the Spanish Ministry of Economy and Competitiveness (Project DPI2013-46641-R). C T