PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 468–475 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000 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, = = (3) 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 (2). The flow vector can be presented as, = = [ ] (4) Based on the linear elastic Hooke ’s law, the stress rate and the elastic strain rate is related as follows: = = ( − ) (5) In which is the total strain rate and the material constitutive tensor is identified as . Considering the associated flow rule, Equation (3), and considering that the yield surface, ( , ) , only depends on the magnitude of the applied principal stresses and of a hardening parameter , ( , ) = ( ) − ( ) = 0 , Equation (5) can be rewritten as: = ( − ) (6) The stress must remain on the yield surface in order to occur plastic flow. Therefore, where = 1 defines an hardening parameter depending on the hardening rule (Owen & Hinton 1980), applying Equation (6) on Equation (7), = + (8) Introducing into Equation (6), the stress rate can be written as, = − + = − + = (9) where presents the tangential constitutive tensor. In order to define explicitly the hardening parameter , the work hardening supposition is assumed (Owen & Hinton 1980) considering the associated flow rule. Since the non linear material used in this work under goes a “linear elastic - linear plastic” hardening behaviour, the hardening parameter can be defined as = 0 /[1 − ( 0 0 ⁄ )] (Owen & Hinton 1980) where 0 and 0 represent the elastic and the tangential modulus in the reference direction, respectively. In the perspective of the NNRPIM meshless approach, the material behaviour is modelled in the form of an incremental relation between the incremental stress vector and the strain increment. In order to force the stress to return to the yield surface, the “backward - Euler” procedure (Crisfield 1991) is considered. This work also considers a variation of the Newton-Raphson non-linear solution method - initial stiffness method combined with an incremental solution (KT0) - in order to solve the non linear equations (Crisfield 1991). Within this approach, the stiffness matrix is calculated only once: in the first iteration of the first load increment. Considering N as the total number of discretised points, the non-linear solution algorithm = − = 0 or = − = 0 (7) 471 4