Issue 35

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 411 where L is given in mm, Δ σ -1 is the plain fatigue limit (MPa) for a load ratio of -1, and σ u is the ultimate tensile strength (MPa). The empirical equation has a correlation factor of 0.791, among a wide range of materials data for steels and cast irons. The results for the critical direction method and the critical plane model for the 35NCD16 are shown in Fig. 8. Material max(Δ σ n ) max(Δ σ n,eff ) min(Δ τ ) Critical Plane AISI 1034 20° 4° 19° -31° 35NCD16 8° 4° 4° -40° Table 4 : Results for crack initiation orientation. Note that Fig. 8b and Tab. 4 reinforce the previous result that in this kind of load conditions the critical plane model lead to severe mistakes in the estimation of the initial crack angle. Considering max(Δ σ n ) a better response was obtained again, however, as explained before this variable lead to the violation of a basic principle of the fracture mechanics (cracks should not propagate when they are closed). Therefore, considering this aspect and the results obtained for both materials the criterion chosen to predict the first orientation of the crack propagation hereinafter will be the critical direction method combined with the minimum shear stress range. R ESULTS FOR CRACK PROPAGATION Contact modelling and stress intensity factor achievement or each plane fretting test configuration described in Tab. 2, a 2D-plain strain FEM was implemented in Abaqus. The models were composed from a cylinder submitted to normal and tangential loads pressed against a fixed plane, Fig. 9a. Contact interactions were described by Lagrange multiplier with a friction coefficient f . The crack tip zone was meshed in a round domain of 5 μm radius with a mesh of approximately 0.5 μm quadrilateral linear elements, Fig. 9b. The mesh in the contact zone was refined down to 20 μm by triangular linear elements. The coefficient of friction between crack faces was assumed to be the same one that occur in the slip zones for the cylindrical contacts in the fretting tests. The stress intensity factor was obtained by a contour integrals Abaqus routine. (a) (b) Figure 9 : FEM modelling (a) contact configuration; (b) illustration of the mesh refinement at the crick tip. Crack propagation modelling As described previously the crack propagation path was assessed only for 35NCD16 due to the fact that only in this case we have the experimental crack path profile for a long crack. Therefore, from the initial angle of propagation estimated, simulations were carried out with increments of crack length, Δ b, where in each step the stress intensity factors mode I and II along a period were extracted as well as the stress field near to the crack tip. The direction of crack propagation in each step was defined applying two different methodologies, namely, the max( Δ k 1 ( θ )) in an infinitesimally kinked crack emerging from the original one or searching the material plane where the minimum shear stress range at the crack tip is reached[7], min(Δ τ ). The first step to apply this model is to define an initial crack (length and direction) in the specimen. In this setting, it was assumed that this crack had an initial length Δ b=70µm and its initiation angle was given by the critical direction method in terms of min(Δ τ ), i.e., 4°. Fig. 10 shows the crack path simulation considering both criteria. As shown in [7] the increment size does not exert significant influence in global aspects, hence, a relatively high increment was considered to simplify the analysis. F