PSI - Issue 14

C. Praveen et al. / Procedia Structural Integrity 14 (2019) 798–805 C. Praveen t al. / Structural Integrity Procedia 00 (20 8) 0 0–000

801

where 0  is the strength contribution arising from solutes, precipitates/dispersoids and grain boundaries, and the term f M Gb   represents flow stress contribution from dislocations. By differentiating Eq. (10), the work hardening rate   f d dp  can be obtained as

d



d M Gb 

f 

f

(11)



dp

dp



2

f

The evolution of dislocation density with respect to plastic strain can be represented as

d

f 

  

  

1

M k 



 .

(12)

f

2

dp

bL

The above relationship (i.e. Eq. 12) is the generalized formulation proposed for the rate of change of dislocation density with plastic strain by Kocks-Mecking-Estrin approach (Mecking and Kocks, 1981; Estrin and Mecking, 1984). Barlat et al. (2002) introduced the additional evolution relationship to capture the variations in mean free path ( L ) with the plastic strain (p) in the framework of two-internal-variable formulation. According to Barlat et al. (2002), mean free path evolves from the initial value ( I L ) and it approaches towards saturation ( S L ) during deformation. The rate equation for mean free path is represented as   L dL k L L s dp    . (13) The coupled differential equations i.e. Eqs (11)-(13) have been used in this study to evaluate the flow stress with equivalent plastic strain. The differentials equations have been numerically integrated using fourth-order Runge Kutta algorithm. There are five unknown constants such as 0  , L I , L S , k 2 and k L in the model. These constants were evaluated by fitting with experimental flow stress vs. true plastic strain data for 316LN SS at 300 K based on least-square algorithm (Christopher and choudhary, 2015). The constants required for the present analysis are presented in Table 1. By using Eq. (10), the yield function f in Eq. (7) can be represented as

f f M Gb    0     e

(14)

The above Eq. (14) has been used for the evaluation of yield criteria in this analysis. 2.1.5 Consistency condition

In the rate-independent plasticity theory for a given incremental total strain, the stress state must not attain values beyond the yield surface and it should stay back on the yield surface f = 0. In order to calculate plastic multiplier Δp given in Eq. 9 consistency condition (  0  f ) has been employed in the present formulation. Further radial return algorithm was used to estimate the value of incremental plastic stain at elasto-plastic condition. This algorithm consists of two steps. First step is the estimation of trail stress by assuming total strain increment as elastic strain increment i.e. e Δε = Δε Δε p and  0 . In second step, the trail stress is relaxed back onto the yield surface in the direction of plastic strain increment. 2.2. Plastic Multiplier calculation Equivalent stress (σ e ) in terms of equivalent trail stress ( tr e  ) and incremental plastic strain is given as

tr

(15)

G p      3

e

e

Substituting Eq. (15) in Eq. (14), yield function ( f ) is obtained as a function of tr

e  , p  and  f .