PSI - Issue 14

C. Praveen et al. / Procedia Structural Integrity 14 (2019) 798–805 C. Praveen et al. / Structural Integrity Procedia 00 (2018) 000–000

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have been used to predict the evolution of dislocation density (Christopher and Choudhary,2015). Though I-S-V models are extensively used for the description of tensile deformation behaviour of materials, their implementation into the finite element scheme is generally limited. In view of this, a preliminary investigation has been performed towards the implementation of rate-independent two-internal-state-variable model proposed by Barlat et al., (2002) into the finite element scheme. In the modeling framework, kinetics of evolution of forest dislocation density as well as mean free path have been considered as evolving internal-variables and these variables have been interrelated to the evolution of flow stress. In addition to above, Euler backward implicit scheme has been employed to update the stress and internal-variables during strain increment. The applicability of implementation procedure is demonstrated by simulating the tensile behaviour of type 316LN with different U-notch radii as 1.25, 2.5 and 5 mm at 300 K.

Nomenclature

Yield function

f

Strain increment

Δε

Young s modulus 

E

e Δε p Δε

ncremental e I lasti tr c s ain ncremental plastic str in I a

Shear modulus Bulk modulus Lame's constant

G K

e ε

Total elastic strain

e

λ v

Total elastic strain till previous step

t ε

Equivalent plastic strain

p

Poisson ratio

Forest dislocation density

f ρ

Plastic multiplier

p 

Mean free path

Stress tensor

σ

L

In

itia

ean l th m pa fr ee

I L s L

Stress increment

Δσ tr σ  tr σ

Saturation mean free path

Trial stress

k

Rate cons

tant

Deviatoric trial stress

L

Equivalent stress

k

Recovery coeffi

cient

e 

2

tr

Length of Burgers vector

b

Equivalent trial stress

e σ

low st ss F re

f  0 

onstant C



Taylor factor

M

Initial flow stress

Note: Bold symbols are used to denote tensor quantities. 2. Modeling framework 2.1. Constitutive equations In this study, the constitutive equations developed for the isotropic hardening with rate-independent plasticity (Dunne and Petrinic, 2005) have been employed. 2.1.1 Stress-strain relation In general, strain tensor decomposition is written as p e Δε = Δε + Δε (1) In multiaxial form, Hooke's law i.e. stress-elastic strain relationship is given as   G Tr I    2 e e σ ε ε (2)