PSI - Issue 23

Atri Nath et al. / Procedia Structural Integrity 23 (2019) 263–268 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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2

However, for CSMs, as the maximum stress remains unchanged with cycles, the estimation of isotropic hardening parameters by this approach is not possible. Often the earlier investigators (Agius et al., 2017; Kourousis, 2013; Paul et al., 2010; Ramezansefat and Shahbeyk, 2015) have introduced empirical modifications to the hardening models for CSMs to achieve closer fit to the experimentally observed responses under different loading scenarios; in these simulations the sets of parameters that are used for monotonic and that for cyclic loading conditions are usually different. These inconsistencies are considered here to arise possibly due to the neglect of isotropic hardening component in the constitutive models for CSMs. This work presents a generalized approach to simulate the deformation behavior of cyclically stable materials following CIKH model. Reported experimental results (Agius et al., 2017; Hassan and Kyriakides, 1992; Kan et al., 2011; Paul et al., 2010) on two ferrous (CS1026 steel and SA333 C-Mn steel) and two non-ferrous (TA16 titanium alloy and AA 7075-T6 aluminum alloy) materials exhibiting cyclically stable behaviour are used to demonstrate the potential of the proposed methodology. The accuracy of prediction of the monotonic and cyclic deformations is assessed by using a modified root mean square error function and the results are compared with that estimated by some reported approaches.

2. Methodology

The mathematical formulation of the CIKH model used in the present study is summarized in Table 1. The non linear Voce isotropic hardening rule (which does not account strain memory effect) is considered in this study because it is focused on the behavior of CSMs only. Chaboche (1991) kinematic hardening consisting of four nonlinear backstress components is used here; the fourth backstress component incorporates a threshold term, below which dynamic recovery of plastic flow is non-existent.

Table 1: Mathematical formulation of the CIKH model used in the present study

Von Mises yield criteria = √ 3⁄2 ( − ): ( − ) − (

0 + ) = 0 (1)

Evolution of isotropic hardening ̇ = ( − ) ̇ Evolution of kinematic hardening = ∑ 4 =1

(2)

Plastic strain rate ̇ = ( 2⁄3 ̇

: ̇ )

̇ = 2⁄3 ̇ − ̇ = 1,2,3̇ ̇ = 2⁄3 ̇ − 〈1 − ( ) 〉 ̇ = 4̇

(3)

(4)

s is the deviatoric part of the stress tensor;  y0 is the initial radius of the yield surface; ̇ is the accumulated equivalent plastic strain rate, ̇ is the plastic strain rate; R, Q, b are the isotropic hardening parameters;  is the deviatoric backstress tensor related to kinematic hardening described as combination of individual backstress components ( α i ), C i and γ i are the kinematic hardening parameters, J(α i ) is the second invariant deviatoric of the i th backstress component, a i is the threshold for dynamic recovery of the i th backstress component Analysis of cyclic-plastic behavior of a material examined using CIKH model as summarized in Table 1 demands determination of 11 parameters. The initial estimate of the CIKH model parameters is obtained from stabilized hysteresis loops using the methodology proposed by Nath et al. (2019); the parameters are considered in a manner to achieve rapid saturation of isotropic hardening component for the symmetric strain-controlled loading. The initial estimates of the model parameters are next calibrated to obtain closer cyclic-plastic response under asymmetric stress controlled cycles by using a genetic algorithm optimization technique. The optimized parameters thus obtained are then used to simulate the response of CSMs under both monotonic and different cyclic loading conditions. The accuracy of the prediction obtained by the current approach is compared with that of other reported predictions by using a modified root mean square error ( F error ) function given by : = { 1⁄ ∑ [( − )/ ] 2 =1 } 0.5 − (5)