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A two-dimensional space-time absolute nodal coordinates cable element and its application in shape memory alloy

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Abstract

In this paper, a SMA space-time ANCF cable element which based on an asymmetric Brinson constitutive model (Poorasadion model) is proposed. To solve the discontinuity of stress in cross section, 5-order polynomial interpolation in neutral axis is used in shape function to construct the displacement formulation of the presented element. The continuity of position and velocity is also guaranteed by 3-order polynomial interpolation in the time direction. The stress equation based on the stress distribution in the neutral axis and cross section is derived. The phase transition regions are given in detail, and the update strategy of initial state parameter for the constitutive law is determined. In the space-time discretization, a solver based on Hamilton’s law of varying action and P2 method is established. As for constrains, the application methods of replaceable constraint and supplementary constraint are proposed to solve the singular problems in Jacobian matrix. Numerical simulation verifies that this method is capable of simulating the cyclic loading of thermo-mechanic problems. A free pendulum simulation shows the difference between the variation of stress and component of detwinned martensite. Simulation results also show that this element could be used for capturing the shape memory effect and pseudo-elasticity effect by SMA cable and the asymmetric effect in tension and compression by adopting an asymmetric constitutive law.

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Acknowledgements

The work is supported by independent research project of State Key Laboratory of Green Building in West China [No. LSZZ202209] and the Technology Innovation Project of Hunan Province [No. 2018GK1040]

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Authors and Affiliations

  1. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150006, China

    Dekun Chen, Nianli Lu & Yuan Xue

  2. INTESIM(Dalian) CO., LTD., Dalian, 116023, Liaoning, China

    Yaqi Cui

  3. School of Mechanical and Electrical Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Peng Lan

  4. State Key Laboratory of Green Building in Western China, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Peng Lan

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Appendix A: The martensite fraction calculation equations

Appendix A: The martensite fraction calculation equations

Equation (27) consists of three equations for martensite calculation fraction, which are used to calculate \(\xi_{T}\), \(\xi_{S}^{ + }\) and \(\xi_{S}^{ - }\). The martensite fraction equations are piecewise equations. The martensite fraction obeys the relation given in Table 6, when it is in the loading range given by PTR under normal loading stage, which means that the total martensite fraction \(\xi\) is not more than 1, and three martensite components are no less than 0. Several kinds of abnormal loading conditions are given in Table 7, which correspond to the abnormal conditions of the total martensite fraction and each martensite fraction, respectively. In particular, when the last three cases in Table 7 occur, only the constraint martensite component is constrained to 0, and the other martensite components still need to be calculated according to the corresponding calculation equations of the interval in Table 6.

Table 6 The martensite fraction calculation equations under normal loading condition
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Table 7 The martensite fraction calculation equations under abnormal loading condition
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Chen, D., Cui, Y., Lan, P. et al. A two-dimensional space-time absolute nodal coordinates cable element and its application in shape memory alloy. Acta Mech 234, 3687–3707 (2023). https://doi.org/10.1007/s00707-023-03580-9

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