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Article History

Received: 1 July 2024

Accepted: 1 October 2024

First Online: 22 October 2024

Declarations

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: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

: The return mapping algorithm was used in this study, including elastic trial and plastic correction. The elastic trial stress isThe yield condition isThe concept of elastic trial stress is shown in Appendix Fig. (a). If , it indicates that the elastic trial stress is in or on the yield surface. If , it indicates that the elastic trial stress is outside the yield surface. In this case, the plastic correction step should be applied, and the corrected stress is as followsThe multi-step implicit format was used to implement the return mapping, each iteration updates the stress through a series of sub-steps. The principle is shown in Appendix Fig. (b). The dashed line in the figure shows the allowable range of stress error, which is generally 1E-6. The nonlinear yield criteria for the sub-steps are given aswhere n = , B= for the non-AFR. Where isThis procedure is finished when . The solution procedure for is as follows. The subsequent yield considering anisotropic hardening can be expressed asFor the non-AFR, according to flow rule and equivalent plastic work principle, there isBy differentiating equation (), there isFrom equation () and , the explicit expression of the plastic multiplier is derived asIf the AFR is applied, there is , B = 1.0 and . The anisotropic parameters of yield models are uniformly represented by P, there isAccording to the , all variables in the stress integration algorithm can be obtained aswhere j represents the iterative step. The anisotropic parameters P is a function of , which can be determined by the stresses and the evolutionary r-values. The detailed function expressions and parameters are summarized in Appendix 2.

: According to the principle of equivalent plastic work, the relationship between yield stress and r-values with equivalent plastic strain can be calculated. The functional relationship between yield stress and equivalent plastic strain is shown in equation (), and the fitting coefficients are summarized in Table . The r-values have different expressions for three solution methods, as shown in Appendix Table . The fitting coefficients under different r-value solution methods are shown in Appendix Table . Through the above function of stresses and r-values with respect to the equivalent plastic strain, the anisotropic parameters can be expressed by the equivalent plastic strain. For Hill48 and non-Hill48 model, the anisotropic parameters have display expressions, as shown in equations () and (). Therefore, the function of the required stress or r-values with respect to the equivalent plastic strain can be directly substituted to obtain the relationship between the anisotropic parameters and the equivalent plastic strain.For Yld2000 yield model, there is no analytical solution for anisotropic parameters. In this paper, the function of anisotropic parameters and equivalent plastic strain are obtained by fitting the anisotropic parameters under different equivalent plastic strain. The Poly9 was selected as the fitting function as shown in equation (), and the fitting coefficients are shown in Appendix Table .