13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- crack surface, and the fourth term is the interaction between other dislocations in the same dislocation array. Therefore the effective stress exerted on a dislocation source is written as follows: ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + + − = ∑ = n j j j x x x x x x A x a t 1 s s s s * s eff,s 1 1 2 1 τ τ & , (4) In this analysis, when the effective stress on dislocation source equals to the activation stress τs, then a new dislocation is introduced at the source. 3. Material Properties Obtained by Atomistic Simulations 3.1. Equilibrium Hydrogen Concentration In order to evaluate the material properties (critical stress intensity factor for dislocation emission from mode II crack tip, dislocation velocity) in the presence of hydrogen, the realistic hydrogen concentration should be adopted. According to Sievert’s law, the equilibrium concentration of hydrogen is proportional to p1/2 and exp(−ΔH/k B T), where p is the hydrogen gas pressure, ΔH is the heat of the solution, kB is the Boltzman’s constant and T is the temperature. Hirth [8] has reported the hydrogen concentration (atom fraction of hydrogen) for alpha iron under the gaseous hydrogen condition based on Sievert’s law. The hydrogen atom stably exists at the tetrahedral site (T-site) within the non-deformed bcc structured alpha iron [9]. Therefore, the hydrogen occupancy at the T-site under thermal equilibrium conditions is given by equation (5) as a function of hydrogen gas pressure p (Pa) and temperature T (K). θT−site =0.9686×10 −6 pexp − 3440 T ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , (5) A close relation was also obtained by our evaluation using first-principles calculations [10]. Using the hydrogen occupancy at T-site, θT-site, the hydrogen occupancy at a specific trap site θi with hydrogen-trap energy ETrap i under thermal equilibrium condition is given by the following equation: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = − k T E i i i B Trap T-site T-site exp 1 1 θ θ θ θ , (6) where, the hydrogen-trap energy corresponds to the energy difference between the system with a hydrogen atom trapped at the specific trap site and the system with a hydrogen atom at a T-site within the non-deformed perfect lattice; this definition is the same as in our previous study [11]. To estimate the number of hydrogen atoms exist around the crack/dislocation at a thermal equilibrium conditions, these hydrogen occupancies are employed as same as our previous study[4,12].Using the equations (5) and (6), the hydrogen concentration around a {112}<111> edge dislocation (number of hydrogen atoms per unit length of dislocation line); CH = 0.49 /nm under thermal equilibrium conditions can be realized at 300 K and 0.01 MPa hydrogen gaseous conditions. And CH = 1.24 /nm can be realized at 300 K and 0.32MPa hydrogen gas. 3.2. Dislocation velocity Dislocation motion is considered as stress-dependent thermal activation process. Our previous studies [4-5] showed that {112}<111> edge dislocation velocity as a function of applied shear stress via the estimation of energy barrier for dislocation motion. The dislocation velocity especially for the case of hydrogen free condition is written as following equation: b k T E l v ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Δ − = B w/oH d * exp ν , (7) where, ΔEw/oH is the energy barrier for dislocation motion without hydrogen as a function of shear
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