13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The σ0 and σb0 in Fig. 2 indicate the strength at the start of fracture zone and bridging zone, respectively, while δ0 and δb0 indicate the corresponding displacement values. δc is the critical displacement for the final failure of the element. δmax is the maximum displacement for the element being loaded and σmax is the corresponding stress value. The constitutive relationship could thus be expressed as: 0 max max 0 b0 0 0 max 0 0 max b0 b0 0 max b0 b0 max b0 b0 max c c b0 max for for < for < 0 for > k c (9) where k0 = σ0/δ0, presents the initial stiffness. The stiffness of damaged element could thus be expressed as: max d max k (10) The calculation of SERR is summarized as follows: max max max 0 0 max max 0 0 0 max b0 max 0 max b0 max b0 b0 max c T 1 for 2 1 ( ) for < 2 1 ( )( ) for < 2 G G G max c for (11) where the initial and total critical SERR (G0 and GT) have the expression shown as: 0 0 b0 b0 0 0 1 ( ) 2 G (12) T 0 b0 c b0 1 ( ) 2 G G (13) 2.5. Fatigue degradation law The damage parameter of tri-linear cohesive element is defined by displacement as follows: max 0 b0 0 d (14) From the expression, d has a value from 0 to 1 while the element in fracture zone has a value from 1 to (δc-δ0)/(δb0-δ0) in bridging zone. This definition implies that the crack tip is at the end of fracture zone where d equal to 1, which is in accordance with the linear elastic fracture mechanics. For a tri-linear cohesive element, the damage is the sum of three parts: s f b d d d d (15) where the subscript “s”, “f” and “b” indicate quasi-static, fatigue and bridging damage, respectively. Fig. 3 presents the illustration of the fatigue damage model of the tri-linear cohesive element to
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