A microstructure-dependent anisotropic magneto-electro-elastic Mindlin plate model based on an extended modified couple
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O R I G I NA L PA P E R
Y. L. Qu · P. Li · G. Y. Zhang
· F. Jin · X.-L. Gao
A microstructure-dependent anisotropic magneto-electroelastic Mindlin plate model based on an extended modified couple stress theory
Received: 5 February 2020 / Revised: 12 April 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract A new model for anisotropic magneto-electro-elastic Mindlin plates is developed by using an extended modified couple stress theory. The equations of motion and complete boundary conditions are simultaneously obtained by a variational formulation based on Hamilton’s principle. The new anisotropic magneto-electro-elastic plate model includes the models for orthotropic and transversely isotropic magnetoelectro-elastic Mindlin plates and the model for isotropic Mindlin plates, all incorporating the microstructure effect, as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported transversely isotropic magneto-electro-elastic plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection, rotation, electric potential, and magnetic potential of the simply supported plate predicted by the current non-classical model are always smaller than those predicted by the classical elasticity-based model, and the differences are significant when the plate thickness is very small but is diminishing as the thickness increases. For the free vibration problem, it is found that the natural frequency predicted by the new plate model is higher than that predicted by the classical model, and the difference is quite large for very thin plates.
Y. L. Qu · F. Jin (B) State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China E-mail: [email protected] P. Li Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto 615-8540, Japan P. Li Department of Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China G. Y. Zhang (B) Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 210096, China E-mail: [email protected] F. Jin MOE Key Laboratory for Multifunctional Materials and Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China X.-L. Gao Department of Mechanical Engineering, Lyle School of Engineering, Southern Methodist University, Dallas, TX 75275-0337, USA
Y. L. Qu et al.
Nomenclature a, b bh ci di j eikl , eiα fi h ks l m0, m2 mi j , mα nx , n y p0 qikl , qiα si ti ui u, v, w Ai jkl , Aαβ , Bi jkl , Bαβ , Ci jkl , Cαβ Bi Di Ei Hi K M Mx , M y Ni j , Mi j , Yi j , Hi j UT W σi j , σα εij , εβ χij , χβ θi ωf
i j μi j φx , φ y λ, μ γ ζ ρ ν i , i Φ ΨB
Length and width of the plate Higher-order bending parameter Components of the body couple resultant Magneto-dielectric coefficients Piezoelectric coefficients Components
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