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Micromechanical modeling of time-dependent and nonlinear responses of magnetostrictive polymer composites

Abstract

The overall time-dependent and nonlinear responses of two-phase magnetostrictive polymer composites are obtained by coupling micromechanical analysis for magnetoelastic coupled composites with a time-integration algorithm for thermorheologically complex materials. The nonlinear magnetoelastic behavior is due to large magnetic driving fields while the nonlinear viscoelastic response is associated with stress and temperature. Because of the material nonlinearity of these constituents, linearized constitutive relations are first defined for obtaining the trial overall responses of the magnetostrictive composites followed by an iterative scheme in order to correct errors from linearizing the nonlinear responses. The presented micromechanical formulation is applicable to magnetostrictive composites reinforced by continuous fiber, particle, and lamina reinforcements. The predicted responses of the composites are first validated with the experimental data available in the literature. Numerical results are then presented for the magnetostrictive composites with 1–3, 0–3, and 2–2 connectivity in terms of their strain and magnetic flux density responses. Time-dependent and nonlinear behaviors show the different degrees of the dependency on microstructural geometry, reinforcement volume fraction, environmental temperature, and loading rate of magnetic and mechanical inputs.

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Acknowledgements

This research was sponsored by the Ministry of Science and Technology (MOST), Taiwan, R.O.C. under the Grant MOST 107-2218-E-006-021-MY2, and was partly supported by Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at National Cheng Kung University. The authors also greatly appreciate the help from their colleague Ying-Zhao Lin for providing Mori–Tanaka and Hashin–Shtrikman results for the experimental validation in Sect. 4.

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Appendix

Appendix

The micromechanical relations for the strain relations among subcells are written as:

$$\begin{aligned} \varepsilon_{11}^{(1)} = \varepsilon_{11}^{(3)} = \varepsilon_{11}^{(5)} = \varepsilon_{11}^{(7)} , \hfill \\ \varepsilon_{11}^{(2)} = \varepsilon_{11}^{(4)} = \varepsilon_{11}^{(6)} = \varepsilon_{11}^{(8)} , \hfill \\ \end{aligned}$$
(39)
$$\begin{aligned} \varepsilon_{22}^{(1)} = \varepsilon_{22}^{(2)} = \varepsilon_{22}^{(5)} = \varepsilon_{22}^{(6)} , \hfill \\ \varepsilon_{22}^{(3)} = \varepsilon_{22}^{(4)} = \varepsilon_{22}^{(7)} = \varepsilon_{22}^{(8)} , \hfill \\ \end{aligned}$$
(40)
$$\begin{aligned} \varepsilon_{33}^{(1)} = \varepsilon_{33}^{(2)} = \varepsilon_{33}^{(3)} = \varepsilon_{33}^{(4)} , \hfill \\ \varepsilon_{33}^{(5)} = \varepsilon_{33}^{(6)} = \varepsilon_{33}^{(7)} = \varepsilon_{33}^{(8)} , \hfill \\ \end{aligned}$$
(41)
$$\begin{aligned} 2\varepsilon_{23}^{(1)} = 2\varepsilon_{23}^{(2)} , \hfill \\ 2\varepsilon_{23}^{(3)} = 2\varepsilon_{23}^{(4)} , \hfill \\ 2\varepsilon_{23}^{(5)} = 2\varepsilon_{23}^{(6)} , \hfill \\ 2\varepsilon_{23}^{(7)} = 2\varepsilon_{23}^{(8)} , \hfill \\ \end{aligned}$$
(42)
$$\begin{aligned} 2\varepsilon_{13}^{(1)} = 2\varepsilon_{13}^{(3)} , \hfill \\ 2\varepsilon_{13}^{(2)} = 2\varepsilon_{13}^{(4)} , \hfill \\ 2\varepsilon_{13}^{(5)} = 2\varepsilon_{13}^{(7)} , \hfill \\ 2\varepsilon_{13}^{(6)} = 2\varepsilon_{13}^{(8)} , \hfill \\ \end{aligned}$$
(43)
$$\begin{aligned} 2\varepsilon_{12}^{(1)} = 2\varepsilon_{12}^{(5)} , \hfill \\ 2\varepsilon_{12}^{(2)} = 2\varepsilon_{12}^{(6)} , \hfill \\ 2\varepsilon_{12}^{(3)} = 2\varepsilon_{12}^{(7)} , \hfill \\ 2\varepsilon_{12}^{(4)} = 2\varepsilon_{12}^{(8)} . \hfill \\ \end{aligned}$$
(44)

The micromechanical relations related to the homogenized stresses are given as:

$$\overline{{\sigma_{11} }} = \frac{{V^{(1)} \sigma_{11}^{(1)} + V^{(3)} \sigma_{11}^{(3)} + V^{(5)} \sigma_{11}^{(5)} + V^{(7)} \sigma_{11}^{(7)} }}{{V^{(1)} + V^{(3)} + V^{(5)} + V^{(7)} }} = \frac{{V^{(2)} \sigma_{11}^{(2)} + V^{(4)} \sigma_{11}^{(4)} + V^{(6)} \sigma_{11}^{(6)} + V^{(8)} \sigma_{11}^{(8)} }}{{V^{(2)} + V^{(4)} + V^{(6)} + V^{(8)} }},$$
(45)
$$\overline{{\sigma_{22} }} = \frac{{V^{(1)} \sigma_{22}^{(1)} + V^{(2)} \sigma_{22}^{(2)} + V^{(5)} \sigma_{22}^{(5)} + V^{(6)} \sigma_{22}^{(6)} }}{{V^{(1)} + V^{(2)} + V^{(5)} + V^{(6)} }} = \frac{{V^{(3)} \sigma_{22}^{(3)} + V^{(4)} \sigma_{22}^{(4)} + V^{(7)} \sigma_{22}^{(7)} + V^{(8)} \sigma_{22}^{(8)} }}{{V^{(3)} + V^{(4)} + V^{(7)} + V^{(8)} }},$$
(46)
$$\overline{{\sigma_{33} }} = \frac{{V^{(1)} \sigma_{33}^{(1)} + V^{(2)} \sigma_{33}^{(2)} + V^{(3)} \sigma_{33}^{(3)} + V^{(4)} \sigma_{33}^{(4)} }}{{V^{(1)} + V^{(2)} + V^{(3)} + V^{(4)} }} = \frac{{V^{(5)} \sigma_{33}^{(5)} + V^{(6)} \sigma_{33}^{(6)} + V^{(7)} \sigma_{33}^{(7)} + V^{(8)} \sigma_{33}^{(8)} }}{{V^{(5)} + V^{(6)} + V^{(7)} + V^{(8)} }},$$
(47)
$$\overline{{\sigma_{23} }} = \frac{{V^{(1)} \sigma_{23}^{(1)} + V^{(2)} \sigma_{23}^{(2)} }}{{V^{(1)} + V^{(2)} }} = \frac{{V^{(3)} \sigma_{23}^{(3)} + V^{(4)} \sigma_{23}^{(4)} }}{{V^{(3)} + V^{(4)} }} = \frac{{V^{(5)} \sigma_{23}^{(5)} + V^{(6)} \sigma_{23}^{(6)} }}{{V^{(5)} + V^{(6)} }} = \frac{{V^{(7)} \sigma_{23}^{(7)} + V^{(8)} \sigma_{23}^{(8)} }}{{V^{(7)} + V^{(8)} }},$$
(48)
$$\overline{{\sigma_{13} }} = \frac{{V^{(1)} \sigma_{13}^{(1)} + V^{(3)} \sigma_{13}^{(3)} }}{{V^{(1)} + V^{(3)} }} = \frac{{V^{(2)} \sigma_{13}^{(2)} + V^{(4)} \sigma_{13}^{(4)} }}{{V^{(2)} + V^{(4)} }} = \frac{{V^{(5)} \sigma_{13}^{(5)} + V^{(7)} \sigma_{13}^{(7)} }}{{V^{(5)} + V^{(7)} }} = \frac{{V^{(6)} \sigma_{13}^{(6)} + V^{(8)} \sigma_{13}^{(8)} }}{{V^{(6)} + V^{(8)} }},$$
(49)
$$\overline{{\sigma_{12} }} = \frac{{V^{(1)} \sigma_{12}^{(1)} + V^{(5)} \sigma_{12}^{(5)} }}{{V^{(1)} + V^{(5)} }} = \frac{{V^{(2)} \sigma_{12}^{(2)} + V^{(6)} \sigma_{12}^{(6)} }}{{V^{(2)} + V^{(6)} }} = \frac{{V^{(3)} \sigma_{12}^{(3)} + V^{(7)} \sigma_{12}^{(7)} }}{{V^{(3)} + V^{(7)} }} = \frac{{V^{(4)} \sigma_{12}^{(4)} + V^{(8)} \sigma_{12}^{(8)} }}{{V^{(4)} + V^{(8)} }}.$$
(50)

The micromechanical relations related to the homogenized magnetic fields are summarized in the following equations:

$$\overline{{H_{1} }} = \frac{{V^{(1)} H_{1}^{(1)} + V^{(2)} H_{1}^{(2)} }}{{V^{(1)} + V^{(2)} }} = \frac{{V^{(3)} H_{1}^{(3)} + V^{(4)} H_{1}^{(4)} }}{{V^{(3)} + V^{(4)} }} = \frac{{V^{(5)} H_{1}^{(5)} + V^{(6)} H_{1}^{(6)} }}{{V^{(5)} + V^{(6)} }} = \frac{{V^{(7)} H_{1}^{(7)} + V^{(8)} H_{1}^{(8)} }}{{V^{(7)} + V^{(8)} }},$$
(51)
$$\overline{{H_{2} }} = \frac{{V^{(1)} H_{2}^{(1)} + V^{(3)} H_{2}^{(3)} }}{{V^{(1)} + V^{(3)} }} = \frac{{V^{(2)} H_{2}^{(2)} + V^{(4)} H_{2}^{(4)} }}{{V^{(2)} + V^{(4)} }} = \frac{{V^{(5)} H_{2}^{(5)} + V^{(7)} H_{2}^{(7)} }}{{V^{(5)} + V^{(7)} }} = \frac{{V^{(6)} H_{2}^{(6)} + V^{(8)} H_{2}^{(8)} }}{{V^{(6)} + V^{(8)} }},$$
(52)
$$\overline{{H_{3} }} = \frac{{V^{(1)} H_{3}^{(1)} + V^{(5)} H_{3}^{(5)} }}{{V^{(1)} + V^{(5)} }} = \frac{{V^{(2)} H_{3}^{(2)} + V^{(6)} H_{3}^{(6)} }}{{V^{(2)} + V^{(6)} }} = \frac{{V^{(3)} H_{3}^{(3)} + V^{(7)} H_{3}^{(7)} }}{{V^{(3)} + V^{(7)} }} = \frac{{V^{(4)} H_{3}^{(4)} + V^{(8)} H_{3}^{(8)} }}{{V^{(4)} + V^{(8)} }}.$$
(53)

The micromechanical relations for the magnetic fluxes among subcells are read as:

$$\begin{aligned} B_{1}^{(1)} = B_{1}^{(2)} , \hfill \\ B_{1}^{(3)} = B_{1}^{(4)} , \hfill \\ B_{1}^{(5)} = B_{1}^{(6)} , \hfill \\ B_{1}^{(7)} = B_{1}^{(8)} , \hfill \\ \end{aligned}$$
(54)
$$\begin{aligned} B_{2}^{(1)} = B_{2}^{(3)} , \hfill \\ B_{2}^{(2)} = B_{2}^{(4)} , \hfill \\ B_{2}^{(5)} = B_{2}^{(7)} , \hfill \\ B_{2}^{(6)} = B_{2}^{(8)} , \hfill \\ \end{aligned}$$
(55)
$$\begin{aligned} B_{3}^{(1)} = B_{3}^{(5)} , \hfill \\ B_{3}^{(2)} = B_{3}^{(6)} , \hfill \\ B_{3}^{(3)} = B_{3}^{(7)} , \hfill \\ B_{3}^{(4)} = B_{3}^{(8)} . \hfill \\ \end{aligned}$$
(56)

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Shen, KJ., Lin, Ch. Micromechanical modeling of time-dependent and nonlinear responses of magnetostrictive polymer composites. Acta Mech 232, 983–1003 (2021). https://doi.org/10.1007/s00707-020-02880-8

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