this mild relaxation is overridden by the strong softening effect associated with these large initial concentrations. The numerical results for the effect of hydrogen on the shape and size of the plastic zone in the neighborhood of the crack tip are shown in Fig. 4. First, hydrogen-induced lattice dilatation relaxes the stresses ahead of the tip and as a result, the plastic zones are shown spread and confined directly ahead of the tip and smaller (Fig. 4a) than in the absence of hydrogen. In the case of no-softening, Figure 4a shows that as the initial hydrogen concentration increases, the plastic flow of the material continues to concentrate in the region ahead of the crack tip while the plastic zone shrinks in the directions along the normal to the axis of symmetry and behind the crack tip. Shrinking of the plastic zone behind the crack tip with increasing initial hydrogen concentration (Fig. 4a) yields reduced crack opening displacements in agreement with the trend shown in Fig. 2b. For 0 04 c . = and , both the plastic zone shape and size are almost the same (Fig. 4a), and this also corresponds to the nearly constant shown in Fig. 2b in the absence of softening. In the presence of hydrogen-induced softening, the plastic zones are shown expanding in every direction relative to the case with no softening (Figs. 3, 4b) as the initial concentration increases. Since 0 06 c . = 0 0 a c a ( )/ ( ) 0 01 c . = is a relatively low initial concentration at which the softening effect on spreading the plasticity is dominated by the local expansion and confinement of the plastic flow ahead of the tip due to the dilatation-induced relaxation, the plastic zone size continues to be smaller than that in the absence of hydrogen (Fig. 3). However, at much larger initial concentrations (e.g. 0 06 c = . ), the softening effect results in much larger plastically deforming regions (Fig. 4b) and this is in accordance with the behavior of a c shown in Fig. 2b. 0 ( ) CONCLUDING DISCUSSION The present finite element calculations of coupled elastoplasticity with hydrogen concentration development in equilibrium with local stress and plastic strain show that that hydrogen concentration profiles ahead of a crack tip scale with the applied load for both hydrogen-induced softening and with no softening effects (Fig. 1). Dilatation-induced relaxation causes the plastic zone to expand and be confined ahead of the crack while it shrinks in all other directions. This reduces the crack tip opening displacement. In contrast, hydrogen-induced softening causes the plastic zone to expand in all directions and the CTOD to increase. At small initial concentrations ( ), stress relaxation dominates whereas softening prevails at larger initial concentrations ( ). As a result, the plastic zones are smaller in the former case than in the latter. However, in all cases, there is substantial plastic flow that takes place directly ahead of the crack tip which is not the case in the hydrogen free material (Fig. 4b). Therefore, one may identify the role of hydrogen with promoting intensification of the ductile fracture processes (e.g. void opening and inter-void ligament fracture by shear localization) that occur directly ahead of the crack tip. This is particularly true at small c (~0.1H/M) at which the hydrostatic stress, assisting void growth, is not substantially relaxed even in the case with softening. 0 0.2 c < 0 0.2 c ≥ o ACKNOWLEDGEMENTS This work was supported by NASA through grant NAG 8-1751. REFERENCES 1. Birnbaum, H.K. and Sofronis, P. (1994) Mater. Sci. & Eng. A176, 191. 2. Sirois, E. and Birnbaum, H.K. (1992) Acta Metall. 40, 1377. 3. Sofronis, P. and H. K. Birnbaum (1995) J. Mech. Phys. Solids 43, 49. 4. Robertson I.M. and Birnbaum, H.K. (1986) Acta Metall. 34, 353. 5. Sofronis, P., Liang, Y. M. and Aravas, N. (2001) Submitted to European Journal of Mechanics A: Solids. 6. Tabata, T. and Birnbaum, H.K. (1983) Scripta Metall. 17, 947. 7. Oriani, R.A. (1970) Acta Metall. 18, 147. 8. Sofronis, P. and McMeeking, R.M. (1989) J. Mech. Phys. Solids 37, 317. 9. McMeeking, R.M. (1977) J. Mech. Phys. Solids 25, 357.
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