PSI - Issue 43

Atri Nath et al. / Procedia Structural Integrity 43 (2023) 246–251 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

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1. Introduction The use of conventional and advanced structural materials for various engineering applications demands an understanding of the material behaviour under different loading conditions, particularly in the elastic-plastic regime. Several phenomena like Bauschinger effect, ratcheting, shakedown, and mean stress relaxation need careful consideration with emphasis on the elastic-plastic analysis subjected to cyclic loading, unlike that for monotonic loading. Chaboche’s isotropic -kinematic hardening (CIKH) model and subsequently modified versions of this model have been used by several researchers to simulate the cyclic plastic behavior of materials, especially to simulate the hysteresis loops and ratcheting characteristics. But a generalized approach to replicate sets of experimental results with a single set of CIKH model parameters is yet to emerge or get established. This study attempts to examine the generalized framework proposed by Nath et al. (2019a) considering Chaboche's (1986) model to analyze the cyclic plastic behaviour of a wide variety of materials; the adopted methodology has been applied to different ferrous and non-ferrous metallic materials exhibiting cyclic hardening or softening or stabilized behaviour. 2. Methodology The mathematical formulation of the CIKH model used in the present study is summarized in Table 1 The adopted methodology considers a non-linear Voce isotropic hardening rule along with Chaboche's (1991) kinematic hardening (KH) as a combination of four nonlinear backstress components ( α ). The evolution of the i th backstress is controlled by the parameters C i , γ i, and a i according to the evolution rules given in Eq.4 (Table 1) and is schematically shown in Figure 1a. Typical responses of the first and the second backstresses are non-linear, the evolution of the third backstress component is linear, and the fourth backstress component has a linear portion depending on the parameter a 4 , followed by a non-linear portion (Bari and Hassan, 2000). The parameter C i is the slope at the initial portion for the i th backstress component, while the parameter γ i is the dynamic recovery parameter, resulting in a stabilized value of the backstress to be C i / γ i , as demonstrated in Figure 1a. Table 1: Mathematical formulation of the combined isotropic-kinematic (CIKH) model used in the present study (Nath et al. 2019b) Von Mises yield criteria = √ 3⁄2 ( − ): ( − ) − ( 0 + ) = 0 (1) Evolution of isotropic hardening ̇ = ( − ) ̇ (2) Plastic strain rate ̇ = ( 2⁄3 ̇ : ̇ ) (3) Evolution of kinematic hardening = ∑ 4 =1 ̇ = 2⁄3 ̇ − ̇ = 1,2,3̇ ̇ = 2⁄3 ̇ − 〈1 − ( ) 〉 ̇ = 4̇ (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, and b are the isotropic hardening parameters;  is the deviatoric backstress tensor related to kinematic hardening described as a 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 The CIKH model typically considers the first backstress component ( α 1 ) to be responsible for the prediction of the large modulus at the initial part of the hardening behaviour just after the onset of yielding (Fig 1); α 1 typically stabilizes rapidly. The second ( α 2 ) and the fourth backstress component ( α 4 ) primarily control the simulation of the transient nonlinear part of the stress-strain response in the plastic range of material. The third backstress component ( α 3 ) controls the linear hardening behaviour of the hysteresis loop throughout the entire strain range. The dominant regions of the four backstress components for strain-controlled loading are demonstrated in Figure 1b. A total of 11 parameters of the CIKH model, as summarized in Table 1, are to be determined for the analysis of the cyclic-plastic behavior of a material using the proposed approach. The initial estimate of the CIKH model parameters is obtained from stabilized hysteresis loops using the methodology proposed by Nath et al. (2019a). The