13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- crack tip. For certain composite material systems, fiber bridging was found unavoidable and can be enhanced by multidirectional ply orientations [16]. Several laws [2, 17, 18] have been developed based on bridging zone model to evaluate the relationship between the fiber bridging stress and the crack tip opening displacement (CTOD). FEM cohesive element considering the fiber bridging effect was developed based on the bridging zone laws [19, 20]. For fatigue delamination propagation studies on composites, a Paris Law analogous linear log-log relationship between the fatigue crack growth rate and the SERR has been established by some significant fundamental works [21-26]. Fatigue degradation laws [27-30] based on cohesive interface elements and the Paris Law are developed to perform a numerical study and predict the fatigue crack growth by FEM programs. However, the Paris Law will become unsuitable for fiber bridging cases as the fatigue crack growth rate and threshold significantly affected by the additional delamination resistance. Remarkable R-curve effects on the fatigue delamination have been observed and analyzed by Hojo et al. [5] for Zanchor-reinforced laminates, Argülles et al. [31] for unidirectional laminates with fiber bridging and Shivakumar et al. [32] for woven/braided fiber composites. A bridging model was developed specially for fatigue delamination by Gregory and Spearing [33], finding that the scatter of crack propagation data was significantly reduced by applying the model. A novel fatigue delamination resistance parameter was introduced by Peng et al. [8, 34, 35] to quantitatively evaluate the effect of R-curves on the fatigue delamination growth. Normalized fatigue crack growth rate and threshold model taking the ratio of SERR to the fatigue delamination resistance as the governing fracture parameter were subsequently developed. Excellent agreement with experiments was obtained by applying the models on experimental data from carbon/bismaleimide composite laminates. Based on the normalized fatigue delamination model, the study presented here attempt to develop a numerical model to predict the fatigue delamination by interface elements. A tri-linear bridging zone model is adopted to simulate the fiber bridging effect. The normalized SERR is taken as the damage parameter to introduce fatigue damage by a degradation law. Numerical validation is performed by UMAT subroutine in commercial FEM software ABAQUS to show the ability of the model. 2. Model Description 2.1. Fatigue delamination growth model A simple form of Paris Law for composites has developed by Wikins [36] and Singh and Greenhalgh [24], shown as: d ( ) d p a A G N (1) where da/dN is the propagation rate of delamination, ΔG the total SERR range in a fatigue cycle, A and p material constants. For fiber bridging laminates, a normalized SERR was proposed instead to evaluate the effect of R-curve on the fatigue crack growth, express as: max max cf ( ) ( ) ( ) G a g a G a (2)
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