Issue 35

R.A. Cardoso et alii, Frattura ed Integrità Strutturale, 35 (2016) 405-413; DOI: 10.3221/IGF-ESIS.35.46 407 (a) (b) Figure 2 : (a) Stress state in the vicinity of the critical zone by means of line method in cylindrical coordinates; (b) critical plane according to the MWCM. Early stage of crack propagation-The critical plane method combined with a multiaxial fatigue criterion applied in terms of the TCD (the critical plane method) In this case, one assumes that crack initiation is governed by the range of shear stress Δ τ (crack type 1). Considering a critical plane approach, such as the Modified Wöhler Curve Method (MWCM), crack initiation is expected to take place on the material plane that experiences the highest shear stress amplitude. As the shear stress always has the same magnitude in two orthogonal planes, the critical plane between these two planes will be the one with the maximum normal stress along a cycle. Fig. 2b depicts the critical plane method in association with the point method (PM) to estimate the direction of crack propagation. Once again, in order of brevity only the PM will be assessed here. Crack propagation orientation under fretting conditions In fretting, the load conditions are complex and non-proportional, which makes the process to estimate the directions of propagation hard. Now a review of some relevant criteria already used to estimate the direction of crack propagation under fretting conditions will be presented. Notice that all these criteria are valid for pre-existing cracks (stage II). Classical crack path criteria, such as the maximum tangential stress (MTS) [8], the maximum strain energy density [9] or the maximum energy release rate [10] are not adequate for non-proportional multiaxial loading like in fretting conditions. Taking into account this non-proportionality [2] and [11] considered the following criteria based on [12]: (i) Crack propagates in the direction where k 1 ( θ,t ) is maximum during a cycle (ii) Crack propagates in the direction where Δ k 1 ( θ ) is maximum during a cycle (iii) Crack propagates in the direction where da/dN ( θ ) is maximum during a cycle where k 1 and k 2 are the stress intensity factors in mode I and in mode II, respectively, of an infinitesimally small kinked crack emerging from the pre-existent crack with an angle θ , Fig. 3. The expression that relates k 1 and k 2 with the classical mode I and mode II stress intensities K I and K II is given by Eq. (4), where the angular functions K ij ( θ ) are reported in [11] and [13]. Figure 3 : Inclusion of a small virtual crack emerging from the crack tip of a pre-existing crack. )( )( )( )( )( )( ),( ),( 22 21 12 11 2 1 t K tK K K K K t k t k II I        (4) Based on the MTS criterion, Dubourg and Lamacq [2] proposed that the direction of crack propagation may be found searching for the direction that maximizes Δ σ θ along a cycle, considering that if σ θ < 0 then σ θ = 0 , once that compressive stress does not encourage crack propagation. Recently Giner et al. [7], proposed the criterion of the minimum shear stress range, which consists in finding the plane that minimizes the shear stress range at the crack tip, Fig. 4. As the shear stress