of these questions, the prime goal of this paper is to address the effect of hydrogen-induced softening on the stress and deformation fields around a blunting crack tip. In a continuum sense material softening can be described through a local flow stress that decreases with increasing hydrogen concentration. It is important to emphasize that the term “flow stress” denotes the intrinsic flow characteristics of a small volume of material at the microscale around a crack tip where hydrogen concentrates as the material deforms. The amount of hydrogen concentration in the specimen is calculated by considering the effect of plastic straining (trapped hydrogen) and hydrostatic stress (normal interstitial lattice site hydrogen). In view of the very high mobility of the hydrogen solute, hydrogen concentration in trapping sites is assumed always in equilibrium with hydrogen in interstitial sites, which is also assumed to be in equilibrium with local hydrostatic stress. The calculated total hydrogen concentration is then used to estimate the material softening along the lines proposed in the work of Sofronis et al. [5] on the basis of the experimental observations of Tabata and Birnbuam [6]. It is emphasized that the present equilibrium calculation of the hydrogen concentrations, stress, and deformation fields is fully coupled. It should also be pointed out that the numerical predictions for our chosen model system, i.e. niobium, turn out to be independent of the amount of trapped hydrogen. Therefore, the present numerical results can be considered as an assessment of the synergism between the local hydrostatic stress and hydrogen-induced softening ahead of a crack tip. However, in view of the generality of the present approach, the current treatment can be easily applied to other systems with different trapping characteristics. HYDROGEN CONCENTRATION AND CONSTITUTIVE LAW Hydrogen is assumed to reside either at normal interstitial lattice sites (NILS) or reversible trapping sites at microstructural defects generated by plastic deformation. Hydrogen concentration in NILS is studied under equilibrium conditions with local stress ijσ , and the occupancy of NILS sites, Lθ , is calculated through the Fermi-Dirac form in terms of the stress-free lattice concentration and stress [5]. Hydrogen atoms at trapping sites are assumed to be always in equilibrium with those at NILS according to Oriani's theory [7], and the trapping site occupancy is given by 0c ( ) / 1 T L L L K K θ −θ +θ θ = , where , W ( / B R = ) T exp K W B is the trap binding energy, R is the gas constant, and T is the temperature. Thus the total hydrogen concentration (in trapping and NILS) measured in hydrogen atoms per solvent atom (H/M) is calculated as ( ) ( ) ( )/ p L kk T L T L c N βθ σ αθ θ ε = + N , where is the trap density which is a function of the effective plastic strain TN pε , LN is the number of host metal atoms per unit volume, and α, β are material constants. Sofronis et al. [5] based on the calculations of Sofronis and Birnbaum [3] and on microscopic studies of the effect of hydrogen on dislocation behavior in iron [6] argued that a continuum description of the hydrogen effect on the local flow stress Yσ can be stated as ( ) 1/ 0 1 n H p Yσ σ ε ε = + 0 , where 0 Hσ is the initial yield stress in the presence of hydrogen that decreases with increasing hydrogen concentration, 0ε is the initial yield strain in the absence of hydrogen, and is the hardening exponent that is assumed unaffected by hydrogen. In this equation, the hydrogen effect on the local continuum flow characteristics is modeled through the initial yield stress which is assumed to be given by n 0 0 ( ) H c σ φ = σ ( )c , where φ is a monotonically decreasing function of the local hydrogen concentration and c 0σ is the initial yield stress in the absence of hydrogen. A possible suggestion for ( )cφ is a linear form ( ) ( 1) c 1 c φ ξ = − + , where the parameter ξ which is less than 1 denotes the ratio of the yield stress in the presence of hydrogen, 0 Hσ , to that in the absence of hydrogen, 0σ , at the maximum hydrogen concentration of 1. The total deformation rate tensor (symmetric part of the velocity gradient in spatial coordinates) is written as the sum of an elastic part (which is modeled as hypo-elastic, linear and isotropic), a part due to the presence of hydrogen, and a plastic part: . The mechanical effect of the hydrogen solute atom is purely dilatational and is phrased in terms of the deformation rate tensor as e h ij ij ij ij D D D D = + + p ( ) 3 h ij ij D c c δ =Λ & , where is the time rate of change of concentration , c& c ( )0 3 c− ( ) 1 c c λ λ Λ = + , v λ=∆ Ω, v∆ is the volume change per atom of
RkJQdWJsaXNoZXIy MjM0NDE=