在這項研究中，我們利用微觀力學模型來分析磁致伸縮高分子複合材料的整體與時間有關的非線性響應。高分子基材在機械和環境負載下表現出明顯的黏彈性行為。因此，有必要考慮磁致伸縮高分子複合材料的粘彈行為。Simplified unit‐cell微觀力學模型將微觀結構理想化成週期性排列的立方體Subcells。因為假設每個Subcell間為完美介面，微觀力學關係式必須滿足在Subcells介面間的位移的協和條件、應力連續性、磁通量的連續性和磁場的連續性。線性微觀力學關係式會違背非線性的本構方程。這將會造成殘差，但可透過迭代方法解決。我們進行實驗驗證以確認模擬的可行性並進行參數研究來討論微觀幾何結構(例如: 纖維、顆粒、層板)、體積分率、磁場加載速率、預應力和溫度的影響。這些結果強調粘彈性高分子聚合物在磁致伸縮高分子複合材料中扮演著重要腳色，並導致整體響應與時間有關。微觀力學模型可以進一步幫助設計包含磁致伸縮高分子複合材料的設備。

In this study, we utilize the micromechanical model to analyze the overall time-dependent nonlinear responses of magnetostrictive polymer composites. Polymeric matrices naturally exhibit significant viscoelastic behaviors under mechanical and environmental loading. Therefore, it is necessary to consider viscoelastic behavior of magnetostrictive polymer composites. The simplified unit‐cell micromechanical model idealize that the microstructures consist of periodically arrangement of cubical subcells. Because of the assumption of perfect interface between each subcell, the micromechanical relations must satisfy the displacement compatibility, traction continuity, magnetic flux continuity and magnetic field continuity between the interfaces of subcells. The linear micromechanical relations violate the nonlinear constitutive equations. It will cause a residual which can be solved by an iterative scheme. We carry out the experimental validation to confirm the feasibility of simulation and conduct parametric studies to discuss the influences of microstructure (e.g., fiber, particle and laminate), volume fraction, loading rate of magnetic field, prestress and temperature. These results emphasize that the viscoelastic polymer play an important role in magnetostrictive polymer composites and lead to overall time-dependent responses. The micromechanical model can further help design devices which contain magnetostrictive polymer composites.

1. Carman, G. P. and M. Mitrovic, “Nonlinear Constitutive Relations for Magnetostrictive Materials with Applications to 1-D Problems”, Intelligent Material Systems and Structures Vol. 6, Issue 5, pp. 673-683, 1995.
2. Chen, Y. and J. E. Snyder et al., “Effect of the elastic modulus of the matrix on magnetostrictive strain in composites”, Applied Physics Letters Vol. 74, Issue 8, pp. 1159-1161, 1999.
3. Dante, A. and J. D. Lopez et al., “A Compact FBG-Based Magnetostrictive Optical Current Sensor With Reduced Mass of Terfenol-D”, IEEE Photonics Technology Letters Vol. 31, Issue 17, pp. 1461-1464, 2019.
4. Fang, F., Y. T. Xu and W. Yang, “Magnetoelectric coupling of laminated composites under combined thermal and magnetic loadings”, Applied Physics Vol. 111, Issue 2, 023903, 2012.
5. Gao, X., Y. Pei and D. Fang, “Experimental Study on Magneto-thermo-mechanical Behaviors of Terfenol-D”, Solid Mechanics and Materials Vol. 4, pp. 652-657, 2010.
6. Guan, X., X. Dong and J. Ou, “Predicting performance of polymer-bonded Terfenol-D composites under different magnetic fields”, Magnetism and Magnetic Materials Vol. 321, Issue 18, pp. 2742-2748, 2009.
7. Haj‐Ali, R. M. and A. H. Muliana, “Numerical finite element formulation of the Schapery non-linear viscoelastic material model”, Numerical Methods in Engineering Vol. 59, Issue 1, pp. 25-45, 2004.
8. Herbst, J. F., T. W. Capehart, and F. E. Pinkerton, “Estimating the effective magnetostriction of a composite: A simple model”, Applied Physics Letters Vol. 70, Issue 22, pp. 3041-3043, 1997.
9. Hill, R., “A self-consistent mechanics of composite materials”, Mechanics and Physics of Solids Vol. 13, Issue 4, pp. 213-222, 1965.
10. Hogea, C. S. and W. D. Armstrong, “The time-dependent magneto-visco-elastic behavior of a magnetostrictive fiber actuated viscoelastic polymer matrix composite”, Acoustical Society of America Vol. 112, Issue 5, pp. 1928-1936, 2002.
11. Jiles, D. C. and J. B. Thoelke, “Magnetization and Magnetostriction in Terbium–Dysprosium–Iron Alloys”, Physica Status Solidi (a) Vol. 147, Issue 2, pp. 535-551, 1995.
12. Kim, Y. Y. and Y. E. Kwon, “Review of magnetostrictive patch transducers and applications in ultrasonic nondestructive testing of waveguides”, Ultrasonics Vol. 62, pp. 3-19, 2015.
13. Kim, J. S. and A. H. Muliana, “A time-integration method for the viscoelastic–viscoplastic analyses of polymers and finite element implementation”, Numerical Methods in Engineering Vol. 79, pp. 550-575, 2009.
14. Le Moal, P., D. Perreux, “Evaluation of creep compliances of unidirectional fibre-reinforced composites”, Composites Science and Technology Vol. 51, Issue 4, pp. 469-477, 1994.
15. Li, B. and T. Zhang et al., “High-performance magnetostrictive composites with large particles volume fraction”, Alloys and Compounds Vol. 805, pp. 1266-1270, 2019.
16. Lin, C. and A. Muliana, “Micromechanics models for the effective nonlinear electro-mechanical responses of piezoelectric composites” Acta Mechanica Vol. 233, Issue 4, pp. 1471-1492, 2013.
17. Lopez, J. D. and A. Dante et al., “Fiber-Optic Current Sensor Based on FBG and Terfenol-D with Magnetic Flux Concentration for Enhanced Sensitivity and Linearity”, IEEE Sens. J, vol. PP, no. c, pp. 1, 2019.
18. Mech, R. and J. Kaleta, “Influence of Terfenol-D powder volume fraction in epoxy matirx composites on their magnetomechanical properies”, Acta Mech. Autom. Vol. 11, no. 3, pp. 233-236, 2017.
19. Moffett, M. B., A. E. Clark and M. Wun-Fogle et al., “Characterization of Terfenol-D for magnetostrictive transducers”, Characterization of Terfenol-D for magnetostrictive transducers Vol. 89, Issue 3, pp. 1448-1455, 1991.
20. Muliana, A. H. and R. M. Haj-Ali, “A multi-scale constitutive formulation for the nonlinear viscoelastic analysis of laminated composite materials and structures”, Solids and Structures Vol. 41, pp. 3461-3490, 2004.
21. Nan, C. W. and G. J. Weng, “Influence of microstructural features on the effective magnetostriction of composite materials”, Phys. Rev. B Vol. 60, Issue 9, pp. 6723-6730, 1999.
22. Rosen BW, Hashin Z, “Effective thermal expansion coefficients and specific heats of composite materials”, Int J Eng Sci 8(2):157–73, 1970.
23. Schapery, R., “Viscoelastic Behavior and Analysis of Composite Materials” Compos. Mater., 5:85-168, 1974.
24. Schilling, J., W. Pachler and B. Roitner et al., “Secured miniaturized system-in-package contactless and passive authentication devices featuring NFC”, Microprocessors and Microsystems Vol. 53, no. 53, pp. 120-129, 2017.
25. Shen, K. J. and C. Lin, “Micromechanical modeling of time-dependent and nonlinear responses of magnetostrictive polymer composites”, Acta Mechanica Vol. 233, Issue 4, pp. 983-1003, 2021.
26. Taylor, R. L.,K. S. Pister and G. L. Goudreau, “Thermomechanical analysis of viscoelastic solids”, Numerical Methods in Engineering Vol. 2, Issue 1, pp. 45-59, 1970.
27. Tuttle, M. E. and H. F. Brinson, “Prediction of the long-term creep compliance of general composite laminates”, Experimental Mechanics Vol.26, Issue 1, pp. 89-102, 1986.
28. Verhoeven, J. D., J. E. Ostenson, and E. D. Gibson, “The effect of composition and magnetic heat treatment on the magnetostriction of TbxDy1−xFey twinned single crystals”, Applied Physics Vol. 66, Issue 2, pp. 772-779, 1989.
29. W. D. Armstrong, “Nonlinear behavior of magnetostrictive particle actuated composite materials”, Applied Physics Vol. 87, Issue 6, pp. 3027-3031, 2000.
30. Xu, S. and Q. Peng et al., “Magnetostrictive current sensor with high sensitivity and a large linear range for the subway”, Applied Optics Vol. 60, Issue 31, pp. 9741-9747, 2021.
31. Zhan, Y. and C. Lin, “A constitutive model of coupled magneto-thermo-mechanical hysteresis behavior for giant magnetostrictive materials”, Mechanics of Materials Vol. 148, 2020.
32. Zhan, Y. and C. Lin., “Micromechanics-based constitutive modeling of magnetostrictive 1-3 and 0-3 composites”, Composite Structures Vol. 260, 113264, 2021.
33. Zhou, Y. and F.G. Shin, “Modeling of magnetostriction in particulate composite materials”, IEEE Transactions on Magnetics Vol. 41, Issue 6, pp. 2071-2076, 2005.