PSI - Issue 5

296 Kulbir Singh et al. / Procedia Structural Integrity 5 (2017) 294–301 Author name / Structural Integrity Procedia 00 (2017) 000 – 000 ) and l min is minimum/limiting value of average length of screw dislocation calculated based on = √ ℎ/8 value. Quantity l min is defined as f c ×L c , where f c is constant which is estimated based on the experimental results of pure iron for activation volume reported in literature (Spitzig 1973) (Quesnel et al. 1975) (Kuramoto et al. 1979). The shear stress must reach a critical value called critical resolved shear stress ( τ c ) for overcoming resistance to the dislocation mobility, hence effective shear stress τ eff which is the stress required to impart velocity v to the screw dislocation segment and can be expressed as τ eff = | τ RSS |- τ c ( τ c comprises of all the resistance to the dislocation glide). In present model τ c is defined as 2 = 2 + 2 , τ self is stress due to interaction between all dislocations present on the same slip system defined as = √ and τ LT is line tension resistance on dislocation, it contributes in case of smaller spacing between obstacles (Monnet et al. 2011) and is defined based on radius of curvature as = 2 ⁄ − ⁄2 with R c = λ/ 2 α and α depends upon the forest dislocation density. Critical stress on any slip system is dependent upon the dislocation density and strength of interaction of all systems with respect to the reference system (Taylor 1934a) (Besinski 1974) (Devincre et al. 2006) (Devincre et al. 2005). Based on relative strength between each slip system, matrix form of interaction coefficient [α AF ] is adapted (Devincre et al. 2006) (Devincre et al. 2005) (Franciosi et al. 1980) (Peirce et al. 1981) (Madec et al. 2003) (Khadyko et al. 2014) (Siddiq et al. 2006) (Alanakar et al. 2011) . [α AF ] is interaction matrix with size dependent upon the number of interacting slip systems considered and each coefficient represent the strength of the interaction. In current work 12 s lip systems (size [α AF ] = 12×12) are considered which can be defined with interaction matrix (having 6 independent coefficients). Dislocation density evolution is dependent upon two oppositely acting phenomenon viz. storage and annihilation of dislocation related by / = 1 1/2 − 2 (Estrin et al. 1984). Storage term (first) is responsible for the increase in flow stress with strain and can be expressed using dislocation mean free path Ʌ , annihilation term (second) is responsible for dynamic recovery and depends linearly on dislocation density. The final expression for dislocation density evolution as a function of strain can be written as (Kocks et al. 2003) (Devincre et al. 2008) (Monnet et al. 2013): = 1 [(1 − 0 ) √ + − ] 3. Constitutive model for irradiated case In ferritic materials due to irradiation material hardens with reduction in ductility and most importantly rise in DBTT (Lechtenberg 1985) (Dubuisson et al. 1993) (Kayano et al. 1988). As dislocations are primary carrier of the plasticity in crystalline materials, interaction of dislocations with other dislocations, irradiation induced defects, precipitates, point defects and surface defects etc may lead to change in the dislocation velocity field and eventually lead to strain localization. Defects dispersions produced due to irradiation (dislocation loops) is considered to account for radiation effects in material. Defects in the form of interstitial loops are produced during high temperature (300 - 500 °C) irradiations (Jenkins et al. 2009) which can subsequently restrict the motion of the mobile dislocation. This leads to irradiation hardening which results in an increase in the yield strength and decrease in the ductility. To account for the loop effect at continuum scale irradiation defects are modeled as obstacles (with their strength value) to the mobility of the dislocation. Interstitial loop are treated as an obstacles with density = and average strength α irr . Molecular dynamics simulation (Terentyev et al. 2010) shows that interaction of interstitial loops with screw dislocations leads to loop absorption by mobile dislocations in the form of helical jogs. This phenomenon leads to reduction in irradiation defect density on active slip system where sufficient screw-loop interactions take place. To account for this a phenomenological relation is adapted which defines the rate of change of irradiation defect density ( ) in terms of strain rate as ̇ = − ̇ . Presence of irradiation defects is accounted appropriately in terms of 3 3.1. Irradiation Defect Number Density