PSI - Issue 14

Rakesh Kumar et al. / Procedia Structural Integrity 14 (2019) 668–675 Author name / Structural Integrity Procedia 00 (2018) 000–000 In the previous equation, and are the orthonormal unit vectors defining the slip direction and slip plane normal for α slip system. Atomic hydrogen in the material causes elastic distortion which is recoverable after hydrogen removal from metal lattice. The elastic dilatation due to presence of hydrogen atom in the atomic lattice is given by (Birnbaum and Sofronis 1994)   0 1 3 c c I           h F (4) where is current and � is initial stress free hydrogen concentration (hydrogen atoms per lattice atoms) in the material (Sofronis et al. 2001). Defining / v     , where v  is the net volume changed by a hydrogen atom in solid solution, and Ω is mean atomic volume of host atom. The stress in the intermediate configuration, second Piola Kirchhoff stress S, is operating with Schmidt matrix of a particular slip system providing the resolved shear stress along the slip system in the intermediate configuration as   .     α α m n S . (5) The evolution of statistically stored dislocations � �� �� � is governed by dislocation multiplication and annihilation of random encounters between dislocations as proposed by Mecking and Kocks (1981). For a glide system the statistically stored dislocation evolutions rate is   1 2 γ SSD c c            (6) Where � and � are dislocation multiplication and annihilation governing terms. SSDs along a slip system at current time step are evolving according to the SSDs present in the same slip system at previous time step. The geometrically necessary dislocation density � �� �� � is calculated by taking the curl of p F and resolving that along the slip systems. Using the approaches defined by Fleck et al. (1994) and Schebler (2011), the hardening evolution of slip systems in presence of hydrogen is expressed by updating the Taylor type hardening rule as     3 1 c SSD GND T c c b c T            (7) where � is dimensionless parameter governing the hydrogen effect on initial yielding and �� is trapped hydrogen concentration per lattice atom along slip system α. The already available solute strengthening theories in literature relate strengthening of material with the concentration, shear modulus and size of solute atoms (Kocks 1985). In the present work we consider the effect of trapped hydrogen concentration in the dislocation cores on α slip system � �� � in raising the yield strength of the material. Hydrogen causes pinning of the dislocation and, therefore, requires high Peierls stress for dislocations motion, and find the approach by Schebler (2011) suitable for this. The � � is slip rate along α slip system and is defined by using classical Orowan-law m bv        (8) where b is Burgers vector, �� is the mobile dislocation density on slip system which is assumed to be one order less than the statistically stored dislocations �� �� (Ma 2006). The � is the dislocation velocity along glide system expressed as a function of resolved shear stress � , the critical resolved shear stress �� , strain-rate sensitivity and reference velocity � as   0 , , , c v v n v        (Engels et al. 2012). The slip rate is defined as   0 GND n m k c bv sgn                (9) where �� ��� is the back stress due to the pileup of geometrically necessary dislocations by taking the gradient of geometrically necessary dislocations. More explanation about the procedure is given in Ma and Hartmaier (2014) 671 4