hydrogen introduced into solution that is directly related to the partial molar volume of hydrogen VH A vN =∆ in solution, is the mean atomic volume of the host metal atom, is the corresponding initial concentration in the absence of stress, and Ω 0c ijδ is the Kronecker delta. The material is assumed to be rate independent, flow according to the von Mises flow theory, and harden isotropically under plastic straining. Thus, 2J 3 2 2 σ ) = c c σ σ ∂ ∂ = + =− ∂ ∂ L b cm 10− 2 /ρ pε 3 29.2 α 10− = σ H M 1 3 ij p kl kl ij km lm e e D h σ σ µδ δ σ σ ∇ ′ ′ = + (1) where ( 1 2 3 2 e ij ij σ σ σ ′ ′ is the von Mises equivalent stress, 3 ij ij kk ij σ σ σ δ ′ = − is the deviatoric stress, , Y Y Y p p kk c c h σ µ σ ε ε ∂ ∂ ∂ ∂ ∂ ∂ , 2 p p p ij ij D D dt ε = 3 ∫ is the effective plastic strain, and the superposed denotes the Jaumann stress rate that is spin invariant. ∇ NUMERICAL RESULTS Solutions to the boundary value problem for the equilibrium hydrogen concentration coupled with material elastoplasticity are presented in the neighborhood of a blunting crack tip under plane strain mode I opening. Small scale yielding conditions were assumed and the system’s temperature was 300K. The material used in the simulations was niobium, as this metal is a high H solubility system, suffers from embrittlement at room temperature, and experimental data are readily available. Displacement boundary conditions of the singular linear elastic field were imposed at a circular boundary at a distance 15cm L= from the tip. The ratio of 0 , where is the initial crack opening displacement, was taken equal to 30,000. The finite element mesh is described in the work of Sofronis and McMeeking [8]. Before the application of the external load, the specimen was assumed to be stress free and at a uniform initial hydrogen concentration . Upon loading, uniform redistribution of the H solute occurs within the solid so that hydrogen is always under quasi-static "local equilibrium" conditions with local stress and plastic strain as discussed in the previous section. Since hydrogen is assumed to be provided by a chemical reservoir, an arrangement corresponding to "far field concentration" kept constant at c , the calculation corresponds to a constant chemical potential for the hydrogen solute. Hydrogen was assumed to expand the lattice isotropically and its partial molar volume in solution was which corresponds to 0b 1.88 0c 0 ole 3/m HV = 0.174 λ= . The molar volume of niobium was which implies that the number of the available NILS was solvent lattice atoms per m . The hydrogen trap sites were associated with dislocations in the deforming metal. Assuming one trap site per atomic plane threaded by a dislocation, one finds that the trap site density in traps per cubic meter is given by 6 3 10.852 m / 3 mole × 28 5.55 10 N = × L TN a = , where ρ is the dislocation density and a is the lattice parameter. The dislocation density ρ, measured in dislocation line length per cubic meter was considered to vary linearly with logarithmic plastic strain so that 0 p ρ ρ γ = + ε pε < B for and for . The parameter line length/ denotes the dislocation density for the annealed material and line length/ . The trap binding energy was taken equal to W 0.5 16 10 ρ= J/mole 0.5 pε ≥ γ 10 m 0 10 ρ = 3m 16 2.0 10 = × k = [5], the parameter β was set equal to 1 and this corresponds to a maximum NILS concentration of 1 H atom per solvent lattice atom, and the parameter was also set equal to 1 which denotes 1 trapping site per trap. The lattice parameter was , Poisson's ratio 10 3.3 m a = × 0.34 ν= , Young's modulus 115GPa E , the yield stress in the absence of hydrogen 0 400MPa = , the hardening coefficient 10 =n , and the softening parameter ξ was set equal to 0.1. In Figure 1, the normalized stress 22 0 / σ σ and normalized hydrogen concentration are plotted against normalized distance 0 c c/ R b from the crack tip along the axis of symmetry at an initial hydrogen concentration 0 0.1 c = .
RkJQdWJsaXNoZXIy MjM0NDE=