R/b 0 5 10 15 20 0 1 2 3 4 No hydrogen, Kapp=80 c0=0.1, without softening Kapp=50,60,70,80,100 c0=0.1, with softening Kapp=50,60,70,80,100 σ22 b undeformed crack deformed crack x1 x2 R R/b 0 5 10 15 20 0 1 2 3 c0=0.1, without softening Kapp=50,60,70,80,100 c0=0.1, with softening Kapp=50,60,70,80,100 (a) (b) 0c c 22 0 σ σ Figure 1. (a) plot of the normalized stress 22 0 σ σ vs normalized distance R b along the axis of symmetry ahead of the crack tip for various applied stress intensities measured in MPa m b . The parameter denotes the distance of a point from the notch root in the undeformed configuration and is the current crack opening displacement; (b) plot of the normalized hydrogen concentration R 0 c c vs normalized distance R b. Clearly, at a given initial hydrogen concentration c , the hydrogen concentration profiles are shown in Fig. 1b to scale with the applied load in accordance with the formula 0 ( ) , p kk c c σ ε = and the well known corresponding scaling [9] of the stress and effective plastic strain in the small scale yielding solution. Lattice dilatation by H is accompanied with stress relaxation in the area ahead of the tip. As shown in Fig. 1a, this relaxation is more pronounced in the region 1.6 R b> where the plastic strain in the absence of hydrogen is small in comparison to the plastic strain in the region close to the crack tip ( 1.6 R b< ). Hydrogen-induced softening decreases the local flow stress ahead of the crack tip, thus resulting in a reduced hydrostatic stress (Fig. 1a) and increased plastic strain. Since the hydrogen concentration profiles are dominated by the hydrostatic stress, the hydrogen concentration profile ahead of the tip is lower for a material undergoing softening due to hydrogen than for a material in the absence of any softening effect (Fig. 1a). It should also be mentioned here that the reduced hydrogen populations in the softened material are associated with a stronger stress relaxation than in the material without softening under the same applied load. Thus, the overall stress relaxation ahead of a crack tip is dictated by the synergism between material softening and relaxation due to hydrogen-induced dilatation. J (N/m) 0 20000 40000 60000 80000 0 4 8 12 no hydrogen c0=0.1, softening c0=0.1, no softening c0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1 1.5 2 With softening Without softening (a) (b) 0 0 a c a ( ) ( ) 0b b Figure 2: (a) Plot of normalized crack opening displacement against the applied 0 b b/ ( ) 2 2 1 app J K E ν = − 0 1 b b a c J = + / ( ) integral. The parameter denotes the crack opening displacement in the undeformed configuration; (b) Plot of against the initial hydrogen concentration , where is defined through 0b 0 0 a c a ( )/ ( ) 0c a c( )
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