J.A.F.O. Correia et alii, Frattura ed Integrità Strutturale, 31 (2015) 80-96; DOI: 10.3221/IGF-ESIS.31.07 83 - The actual elastoplastic stresses and strains, ahead of the crack tip, are computed using the Neuber [15] or Glinka’s approaches [23]. Multiaxial approaches may be adopted using the procedures presented by Moftakhar et al. [16] and Reinhard et al. [17]. - The residual stress distribution ahead of the crack tip is computed using the actual elastoplastic stresses computed at the end of the first load reversal and subsequent cyclic elastoplastic stress range, along the y direction: σ σ σ r Δ max (6) - The residual stress distribution computed ahead of the crack tip is assumed to be applied on crack faces, behind the crack tip, in a symmetric way with respect to the crack tip. - The compressive stress distribution, acting on crack faces, is equivalent to a residual stress intensity factor which is used to correct the applied stress intensity factor range leading to a total (effective) stress intensity factor range, which excludes the effects of the compressive stresses. The residual stress intensity factor, K r , is computed using the weight function method [24]: a r r dxaxmxσ K 0 , (7) where a,xm is the weight function [24] and x r is the residual stress field computed from the elastoplastic stress analysis (see Eq. (6)). - The applied stress intensity factors (maximum and range values) are then corrected using the residual stress intensity factor, resulting the total K max,tot and K tot values [1,10]. For positive stress R -ratios, which is the range covered by the experimental data used in this research, K max,tot and K tot may be computed as follows: r applied tot r applied tot K K K K K K Δ Δ max, max, (8) where K r assumes a negative value corresponding to the compressive stress field. For high stress R -ratios, the compressive stresses ahead of the crack tip may be neglected and the applied stress intensity factor range is assumed fully effective; for low stress R -ratios the compressive stresses increases and the effectiveness of the applied stress intensity factor range decreases accordingly. - Using the total values of the stress intensity factors, the first and second steps before are repeated to determine the corrected values for the maximum actual stress and actual strain range at the material representative elements. Then, Eq. (2) is applied together with Eq. (1) to compute the fatigue crack growth rates. The described methodology does not lead to close-form (explicit) solutions for the fatigue crack propagation rates. Nevertheless, adopting some simplified assumptions about the elastoplastic conditions, such as predominantly elastic behaviour of the material at the crack tip or predominantly plastic behaviour of the material at the crack tip, it is possible to derive those close-form solutions for the stress-strain histories at the crack tip and for the number of cycles to failure of the material representative element. In these cases, the fatigue crack propagation rates may be expressed in the following two-parameters crack driving relation [1, 10]: γq tot p tot K KC dN da Δ max, (9) where C , p , q and γ are constants to be correlated with the cyclic constants of the material in a form depending on the elastoplastic conditions at the crack tip. This two-parameters ( K max and K ) fatigue crack propagation relation allows the simulation of mean stress effects on fatigue crack propagation rates. The crack propagation models based on a two parameters crack driving force has been recently followed by several authors [25, 26]. Probabilistic ε–N and SWT–N fields The fatigue crack propagation modelling based on local approaches requires a fatigue damage relation to compute the number of cycles to fail the elementary material blocks. In this paper, probabilistic fatigue damage models are proposed rather than the deterministic SWT–N , Coffin-Manson or Morrow models often used in the literature. The probabilistic ε a – N model proposed by Castillo and Fernández-Canteli [20] is used. However, and since this probabilistic ε a –N model does not account for mean stress effects, an alternative probabilistic SWT–N field is also proposed, as an extension of the p–ε a – N field suggested by Castillo and Fernández-Canteli [20], to account for mean stress effects.

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