where N K is the notch stress intensity factor, (r, θ) are the polar coordinates (see Fig. 1), Nij g and Nih are the load-independent but notch geometry dependent functions. Please note that the definition for N K used here is different to the widely used K in liner elastic fracture mechanics by a factor π2 : π β( 0 ) / 2 K KN o , where β is the notch opening angle. Barenblatt cohesive zone model [7] has been used to derive the fracture criterion. There are two parts that contribute to the elastic singularity at the end of the cohesive zone: the cohesive zone driving force, Fig. 1b, and the material cohesive resistance, Fig. 1c. The cancellation of stress singularity at the end of the cohesive zone can be written ( ) 0 N CZ K K K K (2) In order to derive the cohesive zone model solution, we need to calculate the stress intensity factors at the end of the cohesive zone caused by the remote N K field and by the cohesive resistance force. In addition, we also need the corresponding crack opening displacements by the driving force and resistance. In the following, we treat the driving force and resistance separately. Cohesive Zone Driving Force For an elastic isotropic solid within a notch N K annulus with a small size crack (cohesive zone) at the notch tip, Fig. 1b, Dunn et al. [8] have recently presented an approximate, but analytical relation between K and N K for anti-plane shear case. In this study we will derive a similar relation for mode I case using a more direct formulation. By using the solution by Tada et al. [9] for point-loading at the crack face in an infinite mode I notch, and integrating the point-load solution along the whole cohesive zone length for the virtual notch opening stress acting over the cohesive zone plane ( 1 λ σ x K xN yy ), the K at the end of the cohesive zone caused by the remote N K can be written: ( ) 2 1 β λ K K l m N (3) where m is a dimensionless constant that depends only on the notch opening angle, l is the cohesive zone length. Fig. 2a plots m as a function of the notch opening angle β, which has been fitted by the following polynomial, 3 2 0.02419 0.006533 ( ) 2.503 0.05302 β β β β m (4)
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