0 0.08 0.16 0.24 c0=0.0 c0=0.1, with softening c0=0.1, no softening 0.1 0.2 Figure 3: Superposed plastic zones ahead of a crack tip: for the hydrogen free material, 0 0.1 c = and with no softening, and and with softening. 0 0.1 c = In the absence of hydrogen, there exists a linear relationship between the crack tip opening displacement (CTOD) and the applied J integral, namely [9], where a is a constant. In the presence of hydrogen, the numerical results of Fig. 2a show that for both hydrogen-induced softening and in the absence of any softening effect, this linear relationship between b and J continues to hold. This is a direct consequence of the fact that scaling of the stress and deformation fields ahead of the tip with the applied load continues to hold in the present case of equilibrium hydrogen concentrations. Figure 2a shows that at a certain applied load and initial hydrogen concentration, hydrogen tends to close the crack, i.e., decrease the parameter a. This is because hydrogen-induced lattice dilatation relaxes the stresses around the crack tip, thus leading to reduced plastic strains behind the tip and smaller plastic zones (see Fig. 3). Furthermore, hydrogen-induced softening causes to be greater than that without softening because the plastic zone and strains are larger in the former case than in the latter (Fig. 3). It can be deduced that softening increases the parameter a (i.e. increases the CTOD) whereas relaxation due to dilatation decreases (i.e. decreases the CTOD) it. Also, comparing the cases with no hydrogen and hydrogen-induced softening, one sees that the softening effect is overridden by the relaxation effect at the concentration of 0.1 H/M as shown in Fig. 2a. 0 b b aJ / = 1+ 04 a 2 0 / ) ( σ app K y 2 0 / ) ( σ app K x The dependence of the parameter (i.e., the CTOD) on the initial hydrogen concentration is shown in Fig. 2b for the cases with and without hydrogen-induced softening. For the case without softening, decreases with increasing c and asymptotes a nearly constant value at large initial hydrogen concentrations. This is because for , numerical results show that the hydrogen concentration enhancement a 0c a 0 0 0 c . < < 0 c c c ∆ = − 06. c ahead of the crack tip increases with , thereby resulting in increased stress relaxation and in turn, in decreasing (CTOD) with increasing . However, for initial concentrations , 0c 0 0 a c a( ) ( )/ 0c 0c .04< < ∆ is almost insensitive to , in particular for R/b>20, and hence a c is nearly constant. 0c 0 0 a ( )/ ( ) 0 0.08 0.16 0.24 No hydrogen c0=0.1, no softening c0=0.2, no softening c0=0.4, no softening c0=0.6, no softening 0.1 0.2 0 0.08 0.16 0.24 c0=0.0 0.1 0.2 0.4 0.6 0.1 0.2 (a) (b) 2 0 / ) ( σ app K y 2 0 / ) ( σ app K y 2 0 / ) ( σ app K x 2 0 ( app Figure 4: (a) Superposed plastic zones ahead of the crack tip at various initial hydrogen concentrations c for the case (a) without softening; and (b) with softening. 0 In the case of hydrogen-induced softening and 0 02 c ≤ . , decreases with increasing because the relaxation due to overrides the relatively small softening effect associated with concentrations . In contrast, for the corresponding relaxation is mild due to the small increases in 0 0 a c a ( )/ ( ) 0c 0 c c c ∆ = − 0c > 0 02 c < . 02. c∆ , and
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