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An improved kinematic hardening rule describing the effect of loading history on plastic modulus and ratcheting strain

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Abstract

This paper focuses on the effect of loading history on the ratcheting behavior and stress–strain hysteresis loops under uniaxial loading. A nonlinear constitutive model for cyclic elastoplastic behavior of metals is developed in the rate-independent framework of small deformation plasticity theory. The effect of applied loading history on plastic modulus under cyclic loading is analyzed and considered in the model, and a linear function describing the evolution of plastic modulus with the equivalent peak stress and stress amplitude under cyclic loading is introduced into a modified Chaboche kinematic hardening rule, to address yield surface translation under different historical stress levels. Furthermore, a maximum stress memory surface is defined to describe the ratcheting effect under multi-step loading conditions. Compared to experiments and the existing models, the quantitative results demonstrate that this proposed model can well capture the cyclic hardening, ratcheting strain and stress–strain hysteresis loops.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Nos. 12072246, 11872286 and 11972272) and National Numerical Wind tunnel project (NNW2020ZT3-A23). The authors gratefully acknowledge these supports.

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Authors and Affiliations

  1. School of Aerospace Science and Technology, Xidian University, Xi’an, 710071, China

    Jiawei Bai, Ke Jin & Yong Kou

  2. Shaanxi Key Laboratory of Space Extreme Detection, Xi’an, 710071, China

    Jiawei Bai, Ke Jin & Yong Kou

  3. Key Laboratory of Equipment Efficiency in Extreme Environment, Ministry of Education, Xi’an, 710071, China

    Jiawei Bai, Ke Jin & Yong Kou

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  1. Jiawei Bai
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Appendices

Appendix A. Finite element implementation of equivalent maximum stress

Let us consider the interval of the state [n, n + 1], assuming that \({\text{s}}_{n + 1}\) has been obtained using the method given by reference [62]. We are given \(\left( {s_{{{\text{MAX}}}} } \right)_{n}\), \(\left( {s_{{{\text{MAX}}}}^{e} } \right)_{n}\) and \(s_{n + 1}^{e}\), then the maximum stress \(\left( {s_{{{\text{MAX}}}}^{e} } \right)_{n + 1}\) can be obtained by the following equation:

$$\left( {\Delta s_{{{\text{MAX}}}}^{e} } \right)_{n + 1} = \left\langle {s_{n + 1}^{e} - \left( {s_{{{\text{MAX}}}}^{e} } \right)_{n} } \right\rangle$$
(A1)
$$\left( {\Delta s_{{{\text{MAX}}}} } \right)_{n + 1} = \left( {\Delta s_{{{\text{MAX}}}}^{e} } \right)_{n + 1} \frac{{s_{n + 1} }}{{s_{n + 1}^{e} }}$$
(A2)
$$\left( {s_{{{\text{MAX}}}} } \right)_{n + 1} = \left( {s_{{{\text{MAX}}}} } \right)_{n} + \left( {\Delta s_{{{\text{MAX}}}} } \right)_{n + 1}$$
(A3)
$$\left( {s_{{{\text{MAX}}}}^{e} } \right)_{n + 1} = \sqrt{\frac{3}{2}} \left( {s_{{{\text{MAX}}}} } \right)_{n + 1}$$
(A4)

Appendix B. Discussion on limitation of material parameter \(r_{c}^{\left( i \right)}\)

The cyclic activation function \(\left[ {1 - H\left( {\varepsilon^{p} - p} \right)} \right]\) in Eq. (8) is removed in order to illustrate the limitation of \(r_{c}^{\left( i \right)}\) describing the variation of plastic modulus, so that \(r_{c}^{\left( i \right)}\) also comes into play in the monotonic loading stage. As shown in Fig. 

Fig. 17
figure 17

Strain range effected by \(r_{c}^{\left( 1 \right)}\)

Full size image

17, it has been observed that the plastic modulus can be reduced in a small strain range by changing \(r_{c}^{\left( 1 \right)}\). If the limitations of \(r_{c}^{\left( 2 \right)} = 0\) and \(r_{c}^{\left( 3 \right)} = 0\) given in Sect. 2.4 are removed, the model may be able to describe the change of plastic modulus over a larger strain range, but this will make it difficult to determine material parameters. Therefore, if there is a material whose plastic modulus varies in a large strain range, it will be difficult to adopt the model in this paper.

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Bai, J., Jin, K. & Kou, Y. An improved kinematic hardening rule describing the effect of loading history on plastic modulus and ratcheting strain. Acta Mech 234, 1757–1776 (2023). https://doi.org/10.1007/s00707-022-03469-z

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