本研究提出了微觀力學模型，用於分析三相複合材料之非線性全域磁-電-熱-彈耦合響應。磁電耦合響應是指材料在外加磁場作用下產生電場，或在電場作用下產生磁場的現象。雖然磁電效應可以在單相材料中取得，但其效應較低。而在兩相複合材料中，雖然可以獲得較高的磁電效應，但單相的壓電或壓磁材料通常是脆性陶瓷，因此由這兩相組成的複合材料容易發生脆性斷裂。為了改善這一問題，本研究引入了彈性材料作為基材，磁致伸縮材料及壓電材料做為加強材，使複合材料具有更好的延展性和可成形性。研究中使用了unit-cell微觀力學模型來模擬三相磁-電-熱-彈複合材料的全域響應。我們將加強材在基材內的分佈視為理想化的週期性排列，並從中取出代表整體複合材料的最小體積元素，建立unit-cell模型。由於本研究涉及三相材料，因此unit-cell模型由64 個subcells組成。為了進行比較，本研究還引入了Mori-Tanaka微觀力學模型。為了驗證模型的可行性，將模型的模擬結果與文獻中現有的實驗數據進行比對。提出的微觀力學模型還進行了參數研究，包括加強材料的體積分率、施加預應力、溫度變化、施加磁場的方向以及不同的磁致伸縮材料，並針對0-3顆粒、1-3纖維和2-2層板的磁-電-熱-彈複合材料，通過這些參數的研究揭示了三相複合材料的全域響應。

This study proposes a micromechanical model to analyze fully coupled nonlinear electro-magneto-thermo-elastic three-phase composites. The magnetoelectric coupling response refers to the phenomenon where a material generates an electric field when subjected to an external magnetic field or generates a magnetic field when subjected to an electric field. Although the magnetoelectric effect can be achieved in single-phase materials, its effectiveness is relatively low. In two-phase composites, higher magnetoelectric effects can be obtained, but single-phase piezoelectric or magnetostrictive materials are usually brittle ceramics, making the composites prone to brittle fracture. To address this issue, this study introduces elastic materials as the matrix and uses magnetostrictive and piezoelectric materials as reinforcements, thereby enhancing the ductility and formability of the composites. A unit-cell micromechanical model is employed to simulate the fully coupled response of the three-phase electro-magneto-thermo-elastic composites. The distribution of the reinforcement materials within the matrix is considered as an idealized periodic arrangement, and the representative volume element (RVE) of the overall composite is extracted to establish the unit-cell model. Since the study involves three-phase materials, the unit-cell model consists of 64 subcells. For comparison purposes, the Mori-Tanaka micromechanical model is also introduced. To verify the feasibility of the model, the results are compared with experimental data from existing literature. The proposed micromechanical model also conducts parameter studies, including the volume fraction of the reinforcement materials, the application of pre-stress, temperature variations, the direction of the applied magnetic field, and different magnetostrictive materials. The study explores the fully coupled responses of 0-3 particle, 1-3 fiber, and 2-2 laminate electro-magneto-thermo-elastic composites through these parameter studies.

1. Crawley, E. F., and Anderson, E. H.: Detailed Models of Piezoceramic Actuation of Beams. Journal of Intelligent Material Systems and Structures 1990; 1(1): 4–25.
2. Zhan, Y.-S., Lin, C.-H.: A Constitutive Model of Coupled Magneto-thermo-mechanical Hysteresis Behavior for Giant Magnetostrictive Materials. Mechanics of Material 2020; 148: 103477.
3. Mech, R., Kaleta, J.: Influence of Terfenol-D powder volume fraction in epoxy matirx composites on their magnetomechanical properties. acta mechanica et automatica 2017; 11(3).
4. Van Suchtelen, J.: Product Properties: A New Application of Composite Materials. Philips Research Reports 1972; 27: 28–37.
5. Harshe, G., Dougherty, J. P., and Newnham, R. E.: Theoretical Modelling of 3-0/0-3 Magnetoelectric Composites. International Journal of Applied Electromagnetics in Materials 1993; 4: 161–171.
6. McDannald, A., Staruch, M., Sreenivasulu, G., Cantoni, C., Srinivasan, G., and Jain, M.: Magnetoelectric coupling in solution derived 3–0 type PbZr0.52Ti0.48O3:xCoFe2O4 nanocomposite films. Appl. Phys. Lett. 2013; 102: 122905.
7. Lee, J., Boyd, J. G., and Lagoudas, D. C.: Effective properties of three-phase electro-magneto-elastic composites. Int. J. Eng. Sci. 2005; 43(10): 790–825.
8. Chau, K. H., Wong, Y. W., and Shin, F. G.: Magnetoelectric effect of polymer electrolyte composites with Terfenol-D and lead zirconate titanate inclusions. Appl. Phys. Lett. 2009; 94: 202902.
9. Pakam, N., Arockiarajan, A.: An analytical model for predicting the effective properties of magneto-electro-elastic (MEE) composites. Comput. Mater. Sci. 2012; 65: 19–28.
10. Silva, M. et al.: Optimization of the magnetoelectric response of poly(vinylidene fluoride)/epoxy/vitrovac laminates. ACS Appl. Mater. Interfaces 2013; 5: 10912–10919.
11. Lin, C.-H., Liu, F.-Y.: Effective nonlinear responses of three-phase magnetoelectric composites. Sci. Rep. 2022; 12: 15101.
12. Nan, C. W.: Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys Rev B Condens Matter 1994; 50(9): 6082–6088.
13. Wu, T.-L., Huang, J. H.: Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases. International Journal of Solids and Structures 2000; 37: 2981–3009.
14. Aboudi, J.: Micromechanical Analysis of Fully Coupled Electro-Magneto-Thermo-Elastic Multiphase Composites. Smart Materials and Structures 2001; 10: 867–877.
15. Lin, C.-H., Lin, Y.-Z.: Nonlinear magnetoelectric coupling in magnetostrictive-piezoelectric composites. Composite Structures 2021; 276: 114558.
16. Tiersten, H. F.: Electroelastic equations for electrode thin plates subject to large driving voltages. J. Appl. Phys. 1993; 74(5): 3389–3393.
17. Rosen, B. W., Hashin, Z.: Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 1970; 8: 157–173.
18. Maksimov, R. D., Kochetkov, V. A.: Thermal Deformation Prediction of Hybrid Composites With Viscoelastic Components. Science and Engineering of Composite Materials 1989; 1(4).
19. Zhan, Y.-S., Lin, C.-H.: Micromechanics-based constitutive modeling of magnetostrictive 1–3 and 0–3 composites. Composite Structures 2021; 260: 113264.
20. Maksimov, R. D., Plume, É. Z.: Elasticity of a hybrid composite material derived from organic and boron fibers. Mechanics of Composite Materials 1980;16: 279–283.
21. Kwok, K., Pellegrino, S.: Micromechanics models for viscoelastic plain-weave composite tape springs. AIAA. J. 2017; 55(1): 309–321.
22. Hatta, H., Takei, T., and Taya, M.: Effects of Dispersed Microvoids on Thermal Expansion Behavior of Composite Materials. Mater. Sci. Eng. A 2000; 285: 99–110.
23. Sagar, S. N. K., Sreekumar, M.: Miniaturized flexible flow pump using SMA actuator. Procedia Eng 2013; 64: 896–906.
24. Brito-Pereira, R., Ribeiro, C., Lanceros-Mendez, S., and Martins, P.: Magnetoelectric response on Terfenol-D/P(VDF-TrFE) two-phase composites. Composites Part B: Engineering 2017; 120: 97–102.
25. Lin, C.-H., Muliana, A.: Micromechanics models for the effective nonlinear electro-mechanical responses of piezoelectric composites. Acta Mech 2013; 224(7): 1471–1492.
26. Nan, C. W., Li, M., and Huang, J. H.: Calculations of giant magnetoelectric effects in ferroic composites of rare-earth-iron alloys and ferroelectric polymers. Physical Review B 2001; 63(14): 9.
27. Veerannan, K., Arockiarajan, A.: Analytical, numerical and experimental studies on effective properties of layered (2–2) multiferroic composites. Sensor Actuat A-Phys 2015; 236: 380–393.
28. Kim, J.: Micromechanical Analysis of Effective Properties of Magneto-electro-thermo-elastic Multilayer Composites. Int. J. Eng. Sci. 2011; 49(9): 1001–1018.
29. Choi, I., Lee, D.-G.: Surface modification of carbon fiber/epoxy composites with randomly oriented aramid fiber felt for adhesion strength enhancement. Composites Part A: Appl Sci Manuf 2013; 48: 1–8.
30. Fang, F., Xu, Y., and Yang, W.: Magnetoelectric coupling of laminated composites under combined thermal and magnetic loadings. Journal of Applied Physics 2012; 111(2): 023906.
31. Martins, P., Silva, M., Reis, S., Pereira, N., Amorín, H., and Lanceros-Mendez, S.: Wide-range magnetoelectric response on hybrid polymer composites based on filler type and content. Polymers 2017; 9: 62.
32. Karaagac, O., Yildiz, B. B., and Köçkar, H.: The influence of synthesis parameters on one-step synthesized superparamagnetic cobalt ferrite nanoparticles with high saturation magnetization. J. Magn. Magn Mater 2019; 473: 262–267.
33. Anantharamaiah, P. N., Shashanka, H. M., Kumar, R., Chelvane, J. A., and Sahoo, B.: Chemically enabling CoFe2O4 for magnetostrictive strain sensing applications at lower magnetic fields: Effect of Zn substitution. Mater. Sci. B 2021; 266: 115080.
34. Fiocchi, S., Chiaramello, E., Marrella, A., Suarato, G., Bonato, M., Parazzini, M., and Ravazzani, P.: Modeling of core-shell magneto-electric nanoparticles for biomedical applications: Effect of composition, dimension, and magnetic field features on magnetoelectric response. PLoS One 2022; 17: 36149898.
35. Barbosa, J., Gomes, I., Pereira, M., Moura, C., Mendes, J., and Almeida, B.: Structural and dielectric properties of laser ablated BaTiO3 films deposited over electrophoretically dispersed CoFe2O4 grains. J. Appl. Phys. 2014; 116: 164112.
36. Tsukrov, I., and Novak, J.: Effective elastic properties of solids with defects of irregular shapes. International Journal of Solids and Structures 2002; 39: 1539–1555.