PSI - Issue 38

Di Song et al. / Procedia Structural Integrity 38 (2022) 546–553 Di Song and Chao Yu/ Structural Integrity Procedia 00 (2021) 000–000

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Fig. 3 Non-proportional multiaxial paths used in the fatigue life prediction model: (a) square; (b) hourglass-typed; (c) butterfly-typed; (d) rhombic; (e) octagonal.

3. Models and results 3.1. The prediction model for residual strains per cycle

Based on the experimental observations in Song et al. (2015c, 2017a, 2019), it can be illustrated that the residual strain gradually accumulated during the cyclic loadings, and approached its stabilized value after tens of cycles. It is verified in Song et al.(2014) that in most of the loading cases the stabilized strain occurs within 50 cycles, and it usually appears much earlier in non-proportional multiaxial loadings (that is, less than 20 cycles). The definition of the stabilized value in the uniaxial and multiaxial loadings can be seen in Song et al. 2014 and Song et al. 2015, respectively. Meanwhile, it can be also observed in the experiments that the accumulated residual strains are affected by peak stresses and martensite transformation extent, the higher peak stress and transformation extent both lead to a faster accumulation rate of the residual strains, this effect should be considered in the prediction model. Additionally, the influence of the non-proportionality in different multiaxial paths can also result in the variation of residual strains, it should be also taken into consideration herein. Thus, the accumulated residual strains in the cyclic loadings with various non-proportional multiaxial paths can be written as: ��� ( ) = � � ���� � � � � � � � ∙ � � ���� �� �� � � � � � �� �� � � � � ∙ �1 − (�� � � � ) � ∙ � � (1) where ��� ( ) is the residual strain of the N th cycle, ���� is the peak stress, here in the multiaxial loadings it is represented by the von Mises equivalent stress. � � � is the start stress of martensite transformation, � � � is the finish stress of martensite transformation, is the non-proportionality factor, which follows the definition in Song et al.(2019) that it is the ratio of the path-length to the perimeter of its circumscribed cycle for each non-proportional multiaxial loading path. The value of butterfly-typed and square paths can be obtained as 1.0873 and 0.9750, respectively. , � , � , � are material parameters. In this equation, the fraction � ���� � �� � represents the effect of the peak stress on the residual strains, which is indicated as the ratio of the peak stress and the martensite finish stress; and the fraction � ���� �� �� � � �� � �� �� � represents the effect of the martensite transformation on the residual strains, the higher peak stress corresponds to a more complete martensite transformation progress for NiTi shape memory alloys in cyclic loadings. Moreover, it can be also observed that the non-proportionality factor is inversely proportional to the residual strain per cycle, which is according to the re-orientation induced plasticity of NiTi SMAs (Song et al. 2017b), and performs as the non proportional softening behavior in the stress-strain curves of multiaxial cyclic loadings.