PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 1237–1244 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

1240

4

This full-field yield function can be developed and the following expression is obtained: ( ) = ̅ = [ 2 + 2 + 2 + + + + 3 2 + 3 2 + 3 2 ] 0.5 (4) In this work only the two-dimensional analysis is performed which permits to simplify the problem formulation. If the problem is analysed assuming the plane stress deformation theory then, = = = 0 . Since the plastic flow is associated with the yield criterion, the associated Prandtl-Reuss flow rule defines the plastic strain as, = = (5) where and denote the plastic strain rate and plastic rate multiplier, respectively and is the flow vector, normal to the adopted yield function, , defined by Equation (4). The flow vector can be presented as, = = [ ] (6) Based on the linear elastic Hooke ’s law, the stress rate and the elastic strain rate is related as follows: = = ( − ) (7) In which is the total strain rate and the material constitutive tensor is identified as . Considering the associated flow rule, Equation (5), and considering that the yield surface, ( , ) , only depends on the magnitude of the applied principal stresses and of a hardening parameter , ( , ) = ( ) − ( ) = 0 , Equation (7) can be rewritten as: = ( − ) (8) The stress must remain on the yield surface in order to occur plastic flow. Therefore,

= − = 0 = − = 0

(9)

(10)

where defines as an hardening parameter depending on the hardening rule (Owen & Hinton 1980), = 1

(11)

Applying Equation (8) on Equation (10), = + Introducing into Equation (8), the stress rate can be written as, = − + = − + =

(12)

(13) where presents the tangential constitutive tensor. In order to define explicitly the hardening parameter , the work hardening supposition is considered (Owen & Hinton 1980) assuming the associated flow rule. Since the non-linear