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On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams

On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko... Article On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams Jun Hong, Shaopeng Wang, Gongye Zhang * and Changwen Mi * Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 210096, China; junhong@seu.edu.cn (J.H.); wsp@seu.edu.cn (S.W.) * Correspondence: gyzhang@seu.edu.cn (G.Z.); mi@seu.edu.cn (C.M.) Abstract: In this paper, a new magneto-electro-elastic functionally graded Timoshenko microbeam model is developed by using the variational formulation. The new model incorporates the extended modified couple stress theory in order to describe the microstructure effect. The power-law variation through the thickness direction of the two-phase microbeams is considered. By the direct application of the derived general formulation, the static bending and free vibration behavior of the newly developed functionally graded material microbeams are analytically determined. Parametric studies qualitatively demonstrate the microstructural effect as well as the magneto-electro-elastic multi-field coupling effect. The proposed model and its classic counterpart produce significant differences for thin graded magneto-electro-elastic Timoshenko microbeams. The thinner the microbeam is, the larger the difference becomes. Keywords: Timoshenko beam; functionally graded material; magneto-electro-elastic beam; microstructure effect; modified couple stress theory Citation: Hong, J.; Wang, S.; Zhang, G.; Mi, C. On the Bending and 1. Introduction Vibration Analysis of Functionally Currently, magneto-electro-elastic (MEE) materials have attracted more and more Grad-ed Magneto-Electro-Elastic attention. MEE materials can realize the mutual conversion between magnetic, electrical, Timoshenko Microbeams. Crystals and mechanical energies. Such characteristics have found important applications in 2021, 11, 1206. https://doi.org/ stability controlling, actuating, health monitoring, medical ultrasonic, and some smart 10.3390/cryst11101206 structure technologies [1–3]. In addition, functionally graded materials (FGMs) are Academic Editor: Pavel Lukáč characterized by continuous changes in material properties [4–6]. The mechanical properties of MEE materials synthesized from functionally graded materials are of great Received: 17 September 2021 significance in both research and industrial fields [7,8]. In recent years, the research on Accepted: 3 October 2021 investigating magneto-electro-elastic functionally graded materials (MEE-FGMs) on thin Published: 7 October 2021 beams and plates has become a major trend. Bhangale and Ganesan [9] studied the free vibration behavior of anisotropic and linear MEE-FGM plates. Sladek et al. [10] proposed Publisher’s Note: MDPI stays a meshless method for the bend analysis of circular MEE-FGM plates. Vinyas et al. [11] neutral with regard to jurisdictional studied the effectiveness of utilizing MEE-FGM plates in precise frequency responses claims in published maps and control. Mahesh and Harursampath [12] and Mahesh [13] evaluated nonlinear deflections institutional affiliations. of MEE-FGM porous flat panels and shells subjected to mechanical, electrical, and magnetic loads, respectively. However, numerous experiments [14,15] have proved that thin beams and plates usually exhibit size effects, (i.e. the thinner, the stiffer). Such size effects arise from non-local interactions of material particles at a very small scale, which Copyright: © 2021 by the authors. Li- cannot be described by classical theories at the micron or nanometer level due to a lack of censee MDPI, Basel, Switzerland. any material length scale parameters. Therefore, it is necessary to develop thin MEE-FGM This article is an open access article structure models based on non-classical theories. distributed under the terms and con- ditions of the Creative Commons At- In order to predict the size effects, numerous theories have been proposed with tribution (CC BY) license (http://crea- additional material parameters, such as non-local theories [16], couple stress theories [17– tivecommons.org/licenses/by/4.0/). 19], strain gradient theories [20–22], and a series of simpler versions [23–28]. These Crystals 2021, 11, 1206. https://doi.org/10.3390/cryst11101206 www.mdpi.com/journal/crystals Crystals 2021, 11, 1206 2 of 20 theories were successfully applied to develop size-dependent structure models for very small scales. For example, based on nonlocal theories, a number of MEE/MEE-FGM beam and plate models have been developed to capture non-local size effects [29–32], in which a non-local medium, including long-range material interactions, is adopted. Lim et al. [33] proposed a non-local strain gradient theory to include both non-local and strain gradient effects, and the bending, buckling, and free variation problems of FGM beams have been solved [34,35]. In addition, the modified couple stress theory (MCST) [24,25] contains only one additional parameter for isotropic materials. This MCST and its extended versions only consider the symmetrical part of the curvature tensor, which leads to fewer material parameters than their classical counterparts. In view of the great difficulties for determining additional parameters and interpreting the relevant microstructures, these modified theories have been applied to build micro/nano-beam and periodic composite pipe models [36–41], from which a microstructure-dependent stiffness is revealed. Recently, three such models have been proposed for MEE Timoshenko homogeneous beams [39] and MEE homogeneous plates [42,43] based on the extended modified couple stress theory. However, to the best of our knowledge, the extended modified couple stress theory is not applicable to MEE-FGM microbeams, which are inhomogeneous and might be helpful for smart devices miniaturization [44–48]. This motivated the present work. The present work uses the extended modified couple stress theory to develop a MEE- FGM Timoshenko microbeam model for the first time and analytically solves the static bending and free vibration problems of the new model. 2. Materials and Methods Consider a two-phase FGM microbeam with length L, width b and thickness h under the combined electric, magnetic, and mechanical loadings, as shown in Figure 1. The effective material properties P(z) (i.e. elastic stiffness, couple stress stiffness, piezoelectric constant, piezomagnetic constant, dielectric constant, magnetic permeability constant, magneto-dielectric constant and density) of the current microbeam change continuously in the thickness direction based on a power-law distribution [36], where P1 and P2 are the material properties of material I and II, respectively. The functionally graded power-law index n determines the material distribution across the thickness. z 1  Pz ()=− (P P ) + + P , (1) 12 2 h 2  Figure 1. Functionally graded microbeam configuration. Based on the extended modified couple stress theory [42,49], the constitutive equations for transversely isotropic magneto-electro-elastic materials are given by [39,42,50,51]. Crystals 2021, 11, 1206 3 of 20 CC C 00 0  11 12 13  σ ε  00 q 00 e xx xx 31 31   CC C 00 0     12 11 13  ε σ 00 q 00 e yy  yy 31 31     H E     CC C 00 0 xx 13 13 33   ε    σ 00 q 00 e       zz zz 33 33 =− H − E  00 0 C 0 0     yy (2)  2ε σ 00 q 00 e yz yz 15 15       H E 00 0 0 C 0  z  z       σ 2ε q 00 e 00  zx zx 15 15      CC −  11 12 2ε σ 00 0 0 0  00 0 00 0   xy xy       2 AA A 00 0  11 12 13 m χ    xx xx  AA A 00 0    12 11 13 m χ yy yy     AA A 00 0 13 13 33  m  χ   zz  zz   00 0 A 0 0   (3)  m 2χ yz yz     00 0 0 A 0    m 2χ  zx zx    AA −  11 12 m 2χ 00 0 0 0    xy xy      2  ε xx  yy  De 0 000 0 s 0 0E       x 15 11 x  ε   zz      De=+ 00 0 0 0 0s 0E      y 15 11 y    2ε yz         De e e 00 0 0 0 s E  z  31 31 33   33 z  2ε zx (4)  2ε  xy  dH 00  11 x   + 0 dH 0  11 y   dH  33 z  xx  yy  Bq 00 0 0 0 μ 0 0H       x 15 11 x        zz  Bq=+ 00 0 0 0 0 μ 0H      y 15 11 y    2ε yz         Bq q q 00 0 0 0 μ H  z  31 31 33   33 z  2ε zx (5)  2ε  xy  dE 00  11 x   + 0 dE 0  11 y    00 dE  33 z where 𝜎 ij, mij, Di, Bi are the Cauchy stress tensor, the deviatoric part of the couple stress tensor, the electric displacements, and the magnetic fluxes, respectively. Cαβ (α, β = 1, 2, …, 6) is the elastic stiffness tensor, Aαβ (α, β = 1, 2, …, 6) is the couple stress stiffness tensor, eiα and qiα are the piezoelectric and piezomagnetic tensors, sij and μij are the dielectric and magnetic permeability tensors, dij is the magneto-dielectric tensor, and εij and 𝜒 ij are, respectively, the infinitesimal strain and the symmetric curvature tensors, which are defined by ε =+ uu () (6) ij i,, j j i Crystals 2021, 11, 1206 4 of 20 χε=+uuε () (7) ij ipq q,, pj jpq q pi with ui being the displacement, and εijk is the Levi-Civita symbol. In addition, Ek and Hk are, respectively, the electric field intensity and magnetic field intensity read EH =−Φ , =−M kk,, k k (8) where Φ and M are the electric and magnetic potentials. For a MEE Timoshenko beam with a uniform cross-section shown in Figure 1, the displacement field and electric and magnetic potentials can be given by [52–55] uu=− ()x,, t zϕ()xt, u=0, u=w()x,t (9) 12 3 ππ22 zz    Φ =− cos zxγγ () ,t + , M = − cos zζ()x,t + ζ  00   (10) hh h h    where u and w are the beam extension and deflection, φ represents the rotation angle, γ and ζ are the spatial variations of the electric and magnetic potentials along the x- direction, respectively. γ0 and ζ0 are, respectively, the external electric and magnetic potentials. By substituting Equations (9) and (10) into Equations (6)–(8) yields ∂∂uw ϕ 1 ∂  εε =− z , = −ϕ xx xz  (11) ∂∂ xx 2 ∂x  , others = 0, 1 ∂∂ w ϕ χ =− + xy  2 (12) 4 ∂x ∂x  , others = 0, πγ ∂ π π 2   Ez== cos , E− sin z γγ− , E= 0 xz   0y (13) hx ∂ h h h   πζ ∂ π π 2    Hz== cos , H− sin z ζζ− , H= 0 xz    0y (14) hx ∂ h h h    Based on Equations (11)–(14), the constitutive equations in Equations (2)–(5) can be obtained as σε=− CeE−qH , σ = 2Cε −eE−qH (15) xxx 11 x 31z 31 z xz 44xz 15x 15 x mA=−A χ () xyx 11 12y (16) D=+2, esεε E+dH D=e +sE+dH x 15 xz 11 x 11 x z 31 xx 33 z 33 z (17) B=+2, qH ε μ +dE B=q ε + μH+dE x 15 xz 11 x 11 x z 31 xx 33 z 33 z (18) From Equations (11)–(18), the first variation of the total strain energy in the current beam satisfying the extended modified couple stress theory over the time span [0, T] takes the form [39,42] Crystals 2021, 11, 1206 5 of 20 TTL δσ Utd2 =+δε σδε+2mδχ xx xx xz xz xy xy  000 A (19) −− Dδδ ED E−BδH−BδH dAdxdt xx z z x x z z where A is the cross-sectional area. The first variation of the kinetic energy of the Timoshenko beam over the time interval [0, T] is given by [53] TTL ∂∂uu δ ∂∂uu δ  11 33 δρ Kdt=+ dd A xdt (20)  000 A ∂∂ tt ∂t ∂t  where ρ is the mass density. Furthermore, the virtual work performed by the applied forces acting on the current Timoshenko beam over the time span [0, T] can be written as [55,56] TTL δδ Wtdd =+ f u qδw xdt [] (21)  where f and q are, respectively, the x- and z-components of the body force per unit length along the x-axis. According to Hamilton’s principle [53,56], δ[] KU −−()Wdt=0 (22) Substituting Equations (19)–(21) into Equation (22), applying the fundamental lemma of the calculus of variations [57], and considering the arbitrariness of δu, δw, and δφ yield ∂N ∂∂ u ϕ xx +=fm −m 01 (23) ∂x ∂∂ tt ∂Y ∂M 1 ∂∂ ϕ u xy xx −+Nm − = −m xz 2122 (24) ∂∂ xx 2 ∂∂ tt ∂ Y ∂N 1 ∂ w xy xz ++qm= (25) ∂∂ x 2 x ∂t ∂Λ +Λ = 0 z (26) ∂x ∂Σ +Σ = 0 z (27) ∂x as the equation of motion, and N = 0 xx uu = (28) or at x = 0 and x = L, MY+= 0 (29) xx xy ϕϕ = or at x = 0 and x = L, ∂Y xy −− N = 0 xz (30) 2 ∂x ww = or at x = 0 and x = L, Crystals 2021, 11, 1206 6 of 20 ∂∂ ww Y = 0 (31) xy ∂∂ x x or at x = 0 and x = L, Λ= 0 γγ = (32) or at x = 0 and x = L, Σ= 0 ζζ = (33) or at x = 0 and x = L as boundary conditions, where the overbar denotes the prescribed value. Note that the stress, electric, magnetic resultants, and mass inertias can be expressed as ∂∂ u ϕ eq E H Nd==σγ A A −B +A +Aζ+N+N xxxx xx xx 31 31 x x (34) ∂∂ xx ∂∂ u ϕ eq E H M== zdσγ A B −D +B +Bζ+M+M xxxx xx xx 31 31 x x  (35) ∂∂ xx ∂∂ w γζ∂ 2 eq N== kσϕ dA k A − −kA −kA xz  s xz s xz s 15 s 15 (36) ∂∂ x xx∂   ∂∂ w ϕ Ym==dA F +  xy xy xy (37)  2 ∂x ∂x   π ∂∂ w  γζ∂ es d Λ= Dz cos dA=kA − ϕ +A +A xx  s 15  11 11 (38) hx∂∂x ∂x    ππ ∂∂ u ϕ  ee s d Es Hd Λ= D sin zdA= A −B − Aγζ − A −N −N zz  31 31 33 33 33 33 (39) hh ∂∂ x x  π ∂∂ w ζγ∂    qd μ Σ= Bz cos dA=kA − ϕ +A +A xx  s 15  11 11 (40) hx∂∂x ∂x    ππ ∂∂ u ϕ  qq μμ d H Ed Σ= Bz sin dA=A −B −Aζγ −A −N −N zz  31 31 33 33 33 33 (41) hh ∂∂ x x  mm,,m = ρ z 1,z,z dA () ()() (42) 01 2 where ks denotes the shape correction factor [58,59], and (, A BC , D ) = (z)(1, z, z )dA xx xx xx 11 (43) A = Cz dA () xz 44 (44) F=− Az A z dA () () () (45) xy 12 11 ππ  ee A , Be = ()z sin z () 1, zdA () 31 31 31  (46) hh  Crystals 2021, 11, 1206 7 of 20 ππ  qq A , Bq = z sin z 1, zdA () () () 31 31 31  (47) hh  2γ EE 0 NM , = e z 1, z dA () () () (48) xx  31 2ζ HH 0 NM , = q z 1, z dA () () () (49) xx 31  ππ    eq A , Ae = z cos z , q z cos z dA () () () 15 15 15  15   (50) hh      ππ π      ss 22 A , As = ()z cos z , s ()z sin z dA () 11 33 11  33    (51) hh h        ππ π      dd 22 A , Ad = z cos z , d z sin z dA () () () 11 33 11 33      (52) hh h       22 γζ  ππ ππ Es Hd 00 NN , = s ()z sin z , d ()z sin z dA () 33 33 33  33  (53) hh h h h h     ππ π       μμ 22 A , Az =μμ () cos z , ()z sin zdA () 11 33 11 33 (54)       hh h         22 ζγ ππ π π    HE μ d NN , = μ z sin z , d z sin z dA () () () 33 33 33  33   (55) hh h h h h     Based on Equations (23)–(55), it is found that the current MEE-FGM beam model can additionally capture the effects of couple stress, piezomagnetism, piezoelectricity, and MEE coupling, when compared to the classical FGM Timoshenko beam model. 3. Analytical Solution This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn. In order to illustrate the newly developed model in Section 2, the static bending and free vibration problems of the current beam are solved in this section. According to Equations (28)–(33), the relevant boundary conditions of a simply supported beam can be identified as N = 0 xx (56) ww== 0 (57) xx == 0 L MM== 0 xx xx (58) xx == 0 L Crystals 2021, 11, 1206 8 of 20 YY== 0 xy xy (59) xx == 0 L γγ== 0 (60) xx == 0 L ζζ== 0 (61) xx == 0 L It should be noted that the boundary of the electric and magnetic conditions given in Equations (60) and (61) are for an open circuit. 3.1. Static Bending Consider Fourier solutions for u(x), 𝜑 (x), w(x), γ(x), and ζ(x):  kx π ux = U cos ()  k  (62)  k =1 kx π  ϕΦ x = cos ()  (63) k   k =1 kx π  wx () = W sin  k  (64) k =1   kx π γΓ x = sin () k  (65)  k =1 kx π  ζ ()xZ = sin (66)  k  k =1  where Uk, Φk, Wk, Гk , and Zk are the Fourier coefficients to be determined. It can be shown that the Fourier solutions in Equations (62)–(66) satisfy the boundary conditions in Equations (56)–(61). In addition, the body force f is equal to zero, and the uniform load q(x) can also be expanded in Fourier series as: kx π qx () = Q sin  k (67) k =1 where Qk is a Fourier coefficient calculated by q(x) = p0 in the current case as 2 p Qk =− 1cos() π (68)  kπ According to the Equations (23)–(27), (62)–(66), and (67), the equilibrium equations of static bending problems can be written as ∂∂ u ϕγ∂ ∂ζ eq AB−+A +A = 0 (69) xx 22 xx 31 31 ∂∂ xx ∂∂ xx ∂∂uw ϕγ∂ ∂ζ ∂ eq 2 −+ BD −B −B +kA − ϕ xx 22 xx 31 31 s xz ∂∂ xx ∂∂ xx ∂x  (70) ∂∂ γζ 1  ∂ w ∂ϕ eq −− kA k A − F + = 0 ss 15 15 xy ∂∂ xx 2 ∂∂ xx  Crystals 2021, 11, 1206 9 of 20 22 2 4 3  ∂∂ww ϕγ∂ ∂ζ 1 ∂ ∂ϕ 2 eq kA −−k A −k A + F + =−q sxz s 15 s 15 xy  22 2 4 3 (71) ∂∂ xx ∂x ∂x 2 ∂x∂x    22 2  ∂∂wu ϕγ∂ ∂ζ ∂ ∂ϕ es d e e s d kA −+ A + A + A − B − Aγζ − A = 0 s 15 11 11 31 31 33 33 22 2 (72) ∂∂ xx∂∂ xx∂x ∂x  22 2  ∂∂wu ϕζ∂ ∂γ ∂ ∂ϕ qdμμq q d kA −+ A + A + A − B − Aζγ− A = 0 s 15 11 11 31 31 33 33 22 2 (73) ∂∂ xx∂∂ xx∂x ∂x  Substituting Equations (62)–(66) into Equations (69)–(73) results in SS 0 S S U 0   11 12 14 15 k   SS S S S Φ 0 12 22 23 24 25 k    0 SS S S W = −Q 23 33 34 35 k k (74)   SS S S S Γ  14 24 34 44 45 k    SS S S S Z 0  15 25 35 45 55 k  where kk ππ kπ kπ       eq SA =− , S =B , S = 0, S =A , S =A , 11 xx 12 xx 13 14 31  15 31  LL L L       22 3 kk ππ11kπ kπ      SD =− −kA + F , S =kA + F , 22 xx s xz xy 23 s xz  xy LL22L L      kk ππ  kk ππ  ee qq SA =− −kA , SA =− −kA , (75) 24 31 s 15  25 31 s 15  LL LL       24 2 2 kk ππ 1 kπ kπ      2 eq Sk =− A + F , S =kA , S =kA , 33 sxz  xy 34 s 15 35 s 15 LL 2 L L      22 2 kk ππ kπ    ss d d μμ SA=+A , S=A +A , S=A +A . 44 11 33 45 11 33 55 11 33 LL L    According to Equation (74), the Fourier coefficients Uk, Φk, Wk, Гk, and Zk will be solved. The solutions of u(x), 𝜑 (x), w(x), γ(x), and ζ(x) for the current simple supported beam can also be given by inserting these results into Equations (62)–(66). 3.2. Free Vibration In the free vibration problem of the current beam, both the external forces are vanished (i.e. f = q = 0). Consider the following Fourier series expansions for u(x, t), 𝜑 (x, t), w(x, t), γ(x, t), and ζ(x, t): kx π  V it ω ux,c t = U os e () k  (76)  k =1 kx π  V it ω ϕΦ () x,cte = os  k  (77) k =1   kx π V it ω wx,s t = Win e ()  k  (78)  k =1 kx π  V it ω γΓ x,ste = in () k  (79)  k =1 Crystals 2021, 11, 1206 10 of 20  kx π it ω ζ x,s tZ = in e () (80) k   k =1 V V V V V where ωk is the kth vibration frequency, U , W Φ , Г ,, and Z are Fourier coefficients. It k k k k k should be noted that the Fourier series expansions in Equations (76)–(80) satisfy the boundary conditions in Equations (56)–(61). Based on Equations (76)–(80) and Equations (23)–(27), the equations of motion can be expressed as 22 2 2 ∂∂uu ϕγ∂ ∂ζ ∂ ∂ϕ eq AB−+A +A =m −m (81) xx 22 xx 31 31 0 2 1 2 ∂∂ x xt ∂∂ xx ∂ ∂t ∂∂uw ϕγ∂ ∂ζ ∂  eq 2 −+ BD −B −B +kA − ϕ xx xx 31 31 s xz ∂∂ xx ∂∂ xx ∂x  (82) 32 2 2 ∂∂ γζ 1  ∂wu ∂ϕ ∂ϕ ∂ eq −− kA k A − F + = m −m ss 15 15 xy 2 1 32 2 2 ∂∂ xx 2 ∂x ∂xt∂ ∂t  22 2 4 3 2    ∂∂ww ϕγ∂ ∂ζ 1 ∂ ∂ϕ ∂w 2 eq k A −−kA −kA + F + = m sxz s 15 s 15 xy  0 22 2 4 3 2 (83) ∂∂ x ∂x xx∂ 2 ∂x∂x ∂t    22 2  ∂∂wu ϕγ∂ ∂ζ ∂ ∂ϕ es d e e s d kA −+ A + A + A − B − Aγζ − A = 0 s 15 11 11 31 31 33 33 22 2 (84) ∂∂ xx∂∂ xx∂x ∂x  22 2  ∂∂wu ϕζ∂ ∂γ ∂ ∂ϕ qdμμq q d kA −+ A + A + A − B − Aζγ− A = 0 s 15 11 11 31 31 33 33 22 2 (85) ∂∂ xx∂∂ xx∂x ∂x  Using Equations (76)–(80) in Equations (81)–(85), yields VV SS 0 S S   0 UU mm ωω − 00 0  11 12 14 15 kk 01kk   VV 22  SS S S SΦΦ −mm ωω 00 0 0 12 22 23 24 25 kk 12kk   VV 2   0 SS S S + = WW 00 m ω 0 0 0 23 33 34 35kk 0 k (86)   VV SS S S S 0 ΓΓ 00 0 0 0 14 24 34 44 45   kk   VV    SS S S SZZ 00 0 0 0 0  15 25 35 45 55  kk   Therefore, the first natural frequency ω1 of the current beam can be solved from the smallest positive root of ωk (k = 1) of the Equation (86). 4. Numerical Results The 50%-50% BaTiO3-CoFe2O4 is adopted for material I [39,42,60–62], and the material Ⅱ is taken to be epoxy [63], as listed in Table 1. Note that the couple stress constants A11 and A12 are estimates based on the formula provided in [14,39]. The magnitude of the uniform load p0 is equal to 1/2000h N/m, the shear correction factor ks is 1/2 0.8 , and the cross-sectional shape is kept at b = 2h and L = 20h. In order to verify the correctness of the current model, a comparative study of the deflection of a simply supported microbeam subjected to uniform load between the current model (with Gradient index n = 0) and the model provided by Zhang et al. [39] are plotted in Figure 2. The beam parameters are adopted from Zhang et al. [39]. Crystals 2021, 11, 1206 11 of 20 Table 1. Material properties of the BaTiO3-CoFe2O4 [39] and epoxy [63]. Physical parameter Material Ⅱ Material I C11 (GPa) 226 4.889 C44 (GPa) 44.15 1.241 e15 (C/m ) 5.8 0 e31 (C/m ) −2.2 0 e33 (C/m ) 9.3 0 -9 2 2 s11 (10 C /(N·m )) 5.64 0 -9 2 2 s33 (10 C /(N·m )) 6.35 0 q15 (N/(A·m)) 275 0 q31 (N/(A·m)) 290.15 0 q33 (N/(A·m)) 349.85 0 −12 d11 (10 Ns/(V·C)) 5.38 0 −12 d33 (10 Ns/(V·C)) 2740 0 −6 2 2 μ11 (10 Ns /C ) 297.5 0 −6 2 2 μ33 (10 Ns /C ) 83.5 0 A11 (N) 11.7484 1.4014 A12 (N) 6.4980 0.6903 ρ (kg/m ) 5550 1180 From Figure 2, it is obvious that the results of the classical and current model are the same as those in Zhang et al. [39]. In addition, this validates the current model and shows that the microstructure effect will always cause the deflection to decrease, as expected. 0.14 0.12 0.1 0.08 0.06 MCST from current Classical from current 0.04 MCST from Zhang et al. [39] Classical from Zhang et al. [39] 0.02 h = 14.42μm, b = 2h, L = 20h 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/L Figure 2. Comparison of the deflection of the simply supported microbeam subjected to a uniform load. 4.1. Static Bending Figure 3 shows the distributions of the deformation, axial normal stress, and the electric and magnetic potentials of the current beam. In order to facilitate the observation of the deformation trend of the current beam, the x-component of the displacement vector u of a point (x, y, z) on the beam cross-section has been enlarged by 10 times. In addition, the thickness h is 20 μm, and the gradient index n is 5. Crystals 2021, 11, 1206 12 of 20 From Figure 3b, it can be observed that the axial normal stress in the middle of the current beam is relatively small, and the axial normal stress at the top of the beam is relatively large. From Figure 3c,d, it is clear that the distributions of electric and magnetic potentials in the current beam are center-symmetrical, and the maximum magnitudes both appear at the center of the beam. (a) (b) (c) (d) Figure 3. Distribution of (a) deformation, (b) axial normal stress, (c) electric potential, and (d) magnetic potential (Gradient index n = 5). Figures 4 and 5 show the deflections and rotation angles with different thicknesses predicted by current and classical models. The gradient index n is 5. The numerical results for the current model (solid lines) incorporating the couple stress effect (with A11 ≠ 0 and A12 ≠ 0) are directly calculated from Equations (62), (63) and (74), while those for the classical model (dashed lines) are obtained using the same equations but with A11 = A12 = From Figures 4 and 5, it can be found that the deflections and rotation angles of the current model are always smaller than those of the classical model in all cases. The difference between the results of the current and classical models is obvious when the beam thickness h is small, as expected. Crystals 2021, 11, 1206 13 of 20 Microstructure effect Figure 4. Deflection of the MEE-FGM simply supported beam (Gradient index n = 5). Microstructure effect Figure 5. Rotation of the MEE-FGM simply supported beam (Gradient index n = 5). Figure 6 shows the axial normal stress at the beam center (x = L/2) along the thickness direction of the current and classical models. From Figure 6, it is clear that the magnitude of the axial normal stress of the current model is always smaller than that of the classical model. The differences between the axial normal stress predicted by the two models also become smaller with the increase in the thickness h. Crystals 2021, 11, 1206 14 of 20 Microstructure effect Figure 6. Axial normal stress of the MEE-FGM simply supported beam (Gradient index n = 5). Figures 7 and 8 display the electric and magnetic potentials of the FGM simply supported beam with different thickness of the current and classical models. From Figures 7 and 8, it can be observed that the values of electric and magnetic potentials of the current model are always smaller than those of the classical model. When the beam thickness h is small, the differences between the two sets of results are very large. However, the differences become small when the beam thickness increases. This phenomenon also indicates that the microstructure effect is significant for very thin beams. Microstructure effect Figure 7. Electric potential of the MEE-FGM simply supported beam (Gradient index n = 5). Crystals 2021, 11, 1206 15 of 20 Microstructure effect Figure 8. Magnetic potential of the MEE-FGM simply supported beam (Gradient index n = 5). To illustrate the material inhomogeneity, Figure 9 shows the variation of the maximum deflections wmax (x = L/2) of the MEE-FGM beam with the different gradient index n for h = 20μm and 20mm, respectively. From Figure 9a, it can be seen that the maximum deflections wmax increases with the increase of the gradient index — for both current and classical models— and the deflection of the classical model is always larger than that of the current model. From Figure 9b, it is found that when the thickness of the beam is large enough, there is almost no difference in the prediction results of the maximum deflections predicted by the two models, which further confirms that the microstructure effect is only important for very thin beams. In addition, from Figures 9a,b, it is shown that the gradient index n does have a significant effect on the static bending response for all length scales. Microstructure effect (a) (b) Figure 9. Maximum deflections of the MEE-FGM beam for different gradient index with (a) h = 20μm, (b) h = 20mm. Figure 10 shows the variation of the axial normal stress 𝜎 xx(L/2, z) of the current MEE- FGM beam through the thickness for different gradient index n. From Figure 10, it can be found that the axial normal distribution of current MEE-FGM beam is different from that of a homogeneous beam for both h = 20μm and 20mm cases. In addition, the axial normal stress of homogenous beams on the geometric central axial (z = 0) is zero, but the zero- Crystals 2021, 11, 1206 16 of 20 valued stresses positions of the current FGM beam are varying with the n. Furthermore, the axial normal stress of a homogeneous beam is linear, while those of current MEE-FGM beam are nonlinear at all length scales. (a) (b) Figure 10. Axial normal stress of the current MEE-FGM beam through the thickness with (a) h = 20μm, (b) h = 20mm. 4.2. Free Vibration Figure 11 shows the variation of natural frequency (with k = 1) of the MEE-FGM beam of the current and classical models with different beam thickness. From Figure 11, it is obvious that the natural frequencies of both the current and classical models decrease with the thickness increases. The results also show that the current model incorporating the couple stress effect always increases the value of the natural frequency (and thus increased the beam stiffness). When the beam thickness is small enough, the couple stress effect is significant. Microstructure effect Figure 11. Natural frequency with different MEE-FGM beam thickness (Gradient index n = 5). Figure 12 shows the variation of the natural frequency ω1 (k = 1) of the current and classical models with different gradient index n for h = 20μm and 20mm. From Figure 12a, it is clear that when the thickness of the beam is small (micro scale), the prediction results of the two models are very different. However, from Figure 12b, when the thickness is big Crystals 2021, 11, 1206 17 of 20 enough (macro scale), the prediction results of the two models are almost the same. In addition, the effect of the gradient index is found to be important for all length scales. Microstructure effect (a) (b) Figure 12. Natural frequency of the MEE-FGM beam for different gradient index with (a) h = 20μm, (b) h = 20mm. 5. Conclusions Based on the extended modified couple stress theory, a new graded magneto-electro- elastic Timoshenko microbeam model is developed. The new model considers the effects of both three-field coupling and couple stress. The equations of motion and complete boundary conditions of the new microbeam model are determined through a variational approach. As two direct applications of the new model, the static bending and free vibration properties of a simply supported microbeam subjected to uniformly distributed loads are analytically obtained. For the static bending problem, parametric studies demonstrate that the deflections, rotations, axial normal stresses, electric and magnetic potentials predicted by the current model are all smaller than those of the classical theory. The differences decrease with the increase in the microbeam thickness. For the problem of free vibration, the natural frequency obtained from the current model is found to be higher than that of the classical model. The difference increases as the thickness of the beam decreases. Such a behavior also indicates that the microstructure effect tends to make the graded magneto- electro-elastic microbeam stiffer, and the current model can predict the size effect for magneto-electro-elastic functionally graded microbeam. In addition, it was demonstrated that changing the gradient index significantly affects both the static and vibrational properties of the graded magneto-electro-elastic microbeam at all length scales. These findings are helpful in guiding the engineering design and optimization of graded magneto-electro-elastic materials in MEMS and NEMS devices. Author Contributions: Conceptualization, C.M. and G.Z.; methodology, J.H., S.W.; writing— original draft preparation, J.H. and S.W. All authors have read and agreed to the published version of the manuscript. Funding: The work reported here is funded by the National Key R&D Program of China (grant number 2018YFD1100401) and the National Natural Science Foundation of China [grant numbers 12002086, 11872149 and 11772091] Conflicts of Interest: The authors declare no conflict of interest. Nomenclature L, b, h Length, width and thickness of beam P(z), P1, P2 Material properties of the current beam, material I and II n Functionally graded power-law index 𝜎 ij The components of Cauchy stress tensor (MHz) (MHz) 1 Crystals 2021, 11, 1206 18 of 20 mij The components of the couple stress tensor Di Electric displacements Bi Magnetic fluxes Cαβ The components of elastic stiffness tensor Aαβ The components of couple stress stiffness tensor eiα The components of piezoelectric tensor qiα The components of piezomagnetic tensor sij The components of dielectric tensor μij The components of magnetic permeability tensor dij The components of magneto-dielectric tensor εij The components of infinitesimal strain tensor 𝜒 ij The components of the symmetric curvature tensor ui Displacement components εijk Levi-Civita symbol Ek, Hk Electric field intensity and magnetic field intensity Φ, M Electric potential and magnetic potential u, w Beam extension and deflection φ Rotation angle γ, ζ The electric potential and magnetic potentials γ0, ζ0 External electric potential, external magnetic potential A Cross-sectional area Mass density f, q The x- and z-components of the body force per unit length ks Shape correction factor Uk, Φk, Wk, Гk , Zk, Qk Fourier coefficients ωk The kth vibration frequency V V V V V U , W Φ , Г , Z Fourier coefficients k k k k k References 1. Sahmani, S.; Aghdam, M.M. Nonlocal Strain Gradient Shell Model for Axial Buckling and Postbuckling Analysis of Magneto- Electro-Elastic Composite Nanoshells. Compos. Part B Eng. 2018, 132, 258–274, doi:10.1016/J.COMPOSITESB.2017.09.004. 2. Farajpour, M.R.; Shahidi, A.R.; Hadi, A.; Farajpour, A. Influence of Initial Edge Displacement on the Nonlinear Vibration, Elec- trical and Magnetic Instabilities of Magneto-Electro-Elastic Nanofilms. Mech. Adv. Mater. Struct. 2019, 26, 1469–1481, doi:10.1080/15376494.2018.1432820. 3. Yakhno, V.G. An Explicit Formula for Modeling Wave Propagation in Magneto-Electro-Elastic Materials. J. Electromagn. Waves Appl. 2018, 32, 899–912, doi:10.1080/09205071.2017.1410076. 4. Chen, W.; Yan, Z.; Wang, L. On Mechanics of Functionally Graded Hard-Magnetic Soft Beams. Int. J. Eng. Sci. 2020, 157, 103391, doi:10.1016/J.IJENGSCI.2020.103391. 5. Taati, E. On Buckling and Post-Buckling Behavior of Functionally Graded Micro-Beams in Thermal Environment. Int. J. Eng. Sci. 2018, 128, 63–78, doi:10.1016/J.IJENGSCI.2018.03.010. 6. Yang, Z.; Xu, J.; Lu, H.; Lv, J.; Liu, A.; Fu, J. Multiple Equilibria and Buckling of Functionally Graded Graphene Nanoplatelet- Reinforced Composite Arches with Pinned-Fixed End. Crystals 2020, 10, 1003, doi:10.3390/CRYST10111003. 7. Ghayesh, M.H.; Farokhi, H.; Alici, G. Size-Dependent Performance of Microgyroscopes. Int. J. Eng. Sci. 2016, 100, 99–111, doi:10.1016/J.IJENGSCI.2015.11.003. 8. Tang, Y.; Ma, Z.S.; Ding, Q.; Wang, T. Dynamic Interaction between Bi-Directional Functionally Graded Materials and Magneto- Electro-Elastic Fields: A Nano-Structure Analysis. Compos. Struct. 2021, 264, 113746, doi:10.1016/J.COMPSTRUCT.2021.113746. 9. Bhangale, R.K.; Ganesan, N. Free Vibration of Simply Supported Functionally Graded and Layered Magneto-Electro-Elastic Plates by Finite Element Method. J. Sound Vib. 2006, 294, 1016–1038, doi:10.1016/J.JSV.2005.12.030. 10. Sladek, J.; Sladek, V.; Krahulec, S.; Chen, C.S.; Young, D.L. Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties. Mech. Adv. Mater. Struct. 2015, 22, 479–489, doi:10.1080/15376494.2013.807448. 11. Vinyas, M.; Harursampath, D.A.; Nguyen-Thoi, T. Influence of Active Constrained Layer Damping on the Coupled Vibration Response of Functionally Graded Magneto-Electro-Elastic Plates with Skewed Edges. Def. Technol. 2020, 16, 1019–1038, doi:10.1016/J.DT.2019.11.016. 12. Mahesh, V.; Harursampath, D. Large Deflection Analysis of Functionally Graded Magneto-Electro-Elastic Porous Flat Panels. Eng. Comput. 2021, 1–20, doi:10.1007/S00366-020-01270-X. Crystals 2021, 11, 1206 19 of 20 13. Mahesh, V. Porosity Effect on the Nonlinear Deflection of Functionally Graded Magneto-Electro-Elastic Smart Shells under Combined Loading. Mech. Adv. Mater. Struct. 2021, 1–27, doi:10.1080/15376494.2021.1875086. 14. Lam, D.C.C.; Yang, F.; Chong, A.C.M.; Wang, J.; Tong, P. Experiments and Theory in Strain Gradient Elasticity. J. Mech. Phys. Solids 2003, 51, 1477–1508, doi:10.1016/S0022-5096(03)00053-X. 15. McFarland, A.W.; Colton, J.S. Role of Material Microstructure in Plate Stiffness with Relevance to Microcantilever Sensors. J. Micromech. Microeng. 2005, 15, 1060, doi:10.1088/0960-1317/15/5/024. 16. Eringen, A.C. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. J. Appl. Phys. 1983, 54, 4703–4710, doi:10.1063/1.332803. 17. Toupin, R.A. Elastic Materials with Couple-Stresses. Arch. Ration. Mech. Anal. 1962, 11, 385–414. 18. Mindlin, R.D. Influence of Couple-Stresses on Stress Concentrations. Exp. Mech. 1963, 3, 1–7, doi:10.1007/BF02327219. 19. Kolter, W.T. Couple Stresses in the Theory of Elasticity: I and II. Proc. K. Ned. Akad. Wet. B 1964, 67, 17–44. 20. Mindlin, R.D. Micro-Structure in Linear Elasticity. Arch. Ration. Mech. Anal. 1964, 16, 51–78, doi:10.1007/BF00248490. 21. Mindlin, R.D.; Eshel, N.N. On First Strain-Gradient Theories in Linear Elasticity. Int. J. Solids Struct. 1968, 4, 109–124, doi:10.1016/0020-7683(68)90036-X. 22. Polizzotto, C. A Hierarchy of Simplified Constitutive Models within Isotropic Strain Gradient Elasticity. Eur. J. Mech. A Solids 2017, 61, 92–109, doi:10.1016/J.EUROMECHSOL.2016.09.006. 23. Altan, B.S.; Aifantis, E.C. On Some Aspects in the Special Theory of Gradient Elasticity. J. Mech. Behav. Mater. 1997, 8, 231–282, doi:10.1515/JMBM.1997.8.3.231. 24. Yang, F.; Chong, A.C.M.; Lam, D.C.C.; Tong, P. Couple Stress Based Strain Gradient Theory for Elasticity. Int. J. Solids Struct. 2002, 39, 2731–2743, doi:10.1016/S0020-7683(02)00152-X. 25. Park, S.K.; Gao, X.-L. Variational Formulation of a Modified Couple Stress Theory and Its Application to a Simple Shear Problem. Z. Angew. Math. Phys. 2008, 59, 904–917, doi:10.1007/s00033-006-6073-8. 26. Zhang, G.Y.; Gao, X.L. A New Bernoulli–Euler Beam Model Based on a Reformulated Strain Gradient Elasticity Theory. Math. Mech. Solids 2020, 25, 630–643, doi:10.1177/1081286519886003. 27. Qu, Y.L.; Zhang, G.Y.; Fan, Y.M.; Jin, F. A Non-Classical Theory of Elastic Dielectrics Incorporating Couple Stress and Quadru- pole Effects: Part I—Reconsideration of Curvature-Based Flexoelectricity Theory. Math. Mech. Solids 2021, doi:10.1177/10812865211001533. 28. Zhang, G.Y.; Gao, X.L.; Zheng, C.Y.; Mi, C.W. A Non-Classical Bernoulli-Euler Beam Model Based on a Simplified Micromor- phic Elasticity Theory. Mech. Mater. 2021, 161, 103967, doi:10.1016/J.MECHMAT.2021.103967. 29. Ebrahimi, F.; Barati, M.R. Vibration Analysis of Embedded Biaxially Loaded Magneto-Electrically Actuated Inhomogeneous Nanoscale Plates. J. Vib. Control. 2018, 24, 3587–3607, doi:10.1177/1077546317708105. 30. Kiani, A.; Sheikhkhoshkar, M.; Jamalpoor, A.; Khanzadi, M. Free Vibration Problem of Embedded Magneto-Electro-Thermo- Elastic Nanoplate Made of Functionally Graded Materials via Nonlocal Third-Order Shear Deformation Theory. J. Intell. Mater. Syst. 2018, 29, 741–763, doi:10.1177/1045389X17721034. 31. Liu, H.; Lv, Z. Vibration Performance Evaluation of Smart Magneto-Electro-Elastic Nanobeam with Consideration of Nano- material Uncertainties. J. Intell. Mater. Syst. Struct. 2019, 30, 2932–2952, doi:10.1177/1045389X19873418. 32. Xiao, W.S.; Gao, Y.; Zhu, H. Buckling and Post-Buckling of Magneto-Electro-Thermo-Elastic Functionally Graded Porous Nano- beams. Microsyst. Technol. 2019, 25, 2451–2470, doi:10.1007/S00542-018-4145-2. 33. Lim, C.W.; Zhang, G.; Reddy, J.N. A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and Its Applications in Wave Propagation. J. Mech. Phys. Solids 2015, 78, 298–313, doi:10.1016/J.JMPS.2015.02.001. 34. Şimşek, M. Nonlinear Free Vibration of a Functionally Graded Nanobeam Using Nonlocal Strain Gradient Theory and a Novel Hamiltonian Approach. Int. J. Eng. Sci. 2016, 105, 12–27, doi:10.1016/J.IJENGSCI.2016.04.013. 35. Li, X.; Li, L.; Hu, Y.; Ding, Z.; Deng, W. Bending, Buckling and Vibration of Axially Functionally Graded Beams Based on Nonlocal Strain Gradient Theory. Compos. Struct. 2017, 165, 250–265, doi:10.1016/J.COMPSTRUCT.2017.01.032. 36. Reddy, J.N. Microstructure-Dependent Couple Stress Theories of Functionally Graded Beams. J. Mech. Phys. Solids 2011, 59, 2382–2399, doi:10.1016/J.JMPS.2011.06.008. 37. Gao, X.-L.; Zhang, G.Y. A Microstructure-and Surface Energy-Dependent Third-Order Shear Deformation Beam Model. Z. An- gew. Math. Phys. 2015, 66, 1871–1894, doi:10.1007/S00033-014-0455-0. 38. Yu, T.; Hu, H.; Zhang, J.; Bui, T.Q. Isogeometric Analysis of Size-Dependent Effects for Functionally Graded Microbeams by a Non-Classical Quasi-3D Theory. Thin-Walled Struct. 2019, 138, 1–14, doi:10.1016/J.TWS.2018.12.006. 39. Zhang, G.Y.; Qu, Y.L.; Gao, X.L.; Jin, F. A Transversely Isotropic Magneto-Electro-Elastic Timoshenko Beam Model Incorporat- ing Microstructure and Foundation Effects. Mech. Mater. 2020, 149, 103412, doi:10.1016/J.MECHMAT.2020.103412. 40. Hong, J.; He, Z.Z.; Zhang, G.Y.; Mi, C.W. Tunable Bandgaps in Phononic Crystal Microbeams Based on Microstructure, Piezo and Temperature Effects. Crystals 2021, 11, 1029, doi:10.3390/CRYST11091029. 41. Hong, J.; He, Z.Z.; Zhang, G.Y.; Mi, C.W. Size and Temperature Effects on Band Gaps in Periodic Fluid-Filled Micropipes. Appl. Math. Mech. 2021, 42, 1219–1232, doi:10.1007/S10483-021-2769-8. 42. Qu, Y.L.; Li, P.; Zhang, G.Y.; Jin, F.; Gao, X.L. A Microstructure-Dependent Anisotropic Magneto-Electro-Elastic Mindlin Plate Model Based on an Extended Modified Couple Stress Theory. Acta Mech. 2020, 231, 4323–4350, doi:10.1007/S00707-020-02745-0. 43. Shen, W.; Zhang, G.; Gu, S.; Cong, Y. A Transversely Isotropic Magneto-electro-elastic Circular Kirchhoff Plate Model Incorpo- rating Microstructure Effect. Acta Mech. Solida Sin. 2021, 1–13, doi:10.1007/s10338-021-00271-7. Crystals 2021, 11, 1206 20 of 20 44. Qu, Y.L.; Jin, F.; Yang, J.S. Magnetically induced charge motion in the bending of a beam with flexoelectric semiconductor and piezomagnetic dielectric layers. J. Appl. Phys. 2021, 127, 064503. 45. Zhu, F.; Ji, S.; Zhu, J.; Qian, Z.; Yang, J. Study on the Influence of Semiconductive Property for the Improvement of Nanogener- ator by Wave Mode Approach. Nano Energy 2018, 52, 474–484, doi:10.1016/J.NANOEN.2018.08.026. 46. Shingare, K.B.; Kundalwal, S.I. Static and Dynamic Response of Graphene Nanocomposite Plates with Flexoelectric Effect. Mech. Mater. 2019, 134, 69–84, doi:10.106/J.MECHMAT.2019.04.006. 47. Wang, L.; Liu, S.; Feng, X.; Zhang, C.; Zhu, L.; Zhai, J.; Qin, Y.; Wang, Z.L. Flexoelectronics of Centrosymmetric Semiconductors. Nat. Nanotechnol. 2020, 15, 661–667, doi:10.1038/s41565-020-0700-y. 48. Sharma, S.; Kumar, A.; Kumar, R.; Talha, M.; Vaish, R. Geometry Independent Direct and Converse Flexoelectric Effects in Functionally Graded Dielectrics: An Isogeometric Analysis. Mech. Mater. 2020, 148, 103456, doi:10.1016/J.MECHMAT.2020.103456. 49. Zhang, G.Y.; Gao, X.L.; Guo, Z.Y. A Non-Classical Model for an Orthotropic Kirchhoff Plate Embedded in a Viscoelastic Me- dium. Acta Mech. 2017, 228, 3811–3825, doi:10.1007/S00707-017-1906-4. 50. Han, X.; Pan, E. Fields Produced by Three-Dimensional Dislocation Loops in Anisotropic Magneto-Electro-Elastic Materials. Mech. Mater. 2013, 59, 110–125, doi:10.1016/J.MECHMAT.2012.09.001. 51. Kumar, D.; Sarangi, S.; Saxena, P. Universal Relations in Coupled Electro-Magneto-Elasticity. Mech. Mater. 2020, 143, 103308, doi:10.1016/J.MECHMAT.2019.103308. 52. Wang, Q. On Buckling of Column Structures with a Pair of Piezoelectric Layers. Eng. Struct. 2002, 24, 199–205, doi:10.1016/S0141- 0296(01)00088-8. 53. Ma, H.M.; Gao, X.L.; Reddy, J.N. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory. J. Mech. Phys. Solids 2008, 56, 3379–3391, doi:10.1016/J.JMPS.2008.09.007. 54. Ansari, R.; Gholami, R.; Rouhi, H. Size-Dependent Nonlinear Forced Vibration Analysis of Magneto-Electro-Thermo-Elastic Timoshenko Nanobeams Based upon the Nonlocal Elasticity Theory. Compos. Struct. 2015, 126, 216–226, doi:10.1016/J.COMP- STRUCT.2015.02.068. 55. Hong, J.; Wang, S.P.; Zhang, G.Y.; Mi, C.W. Bending, Buckling and Vibration Analysis of Complete Microstructure-Dependent Functionally Graded Material Microbeams. Int. J. Appl. Mech. 2021, 13, 2150057, doi:10.1142/S1758825121500575. 56. Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics, 2nd ed.; Wiley: New York, NY, USA, 2002. 57. Gao, X.-L.; Mall, S. Variational Solution for a Cracked Mosaic Model of Woven Fabric Composites. Int. J. Solids Struct. 2001, 38, 855–874, doi:10.1016/S0020-7683(00)00047-0. 58. Yang, J. An Introduction to the Theory of Piezoelectricity; Springer: New York, NY, USA, 2005. doi:10.1007/B101799. 59. Yang, J. The Mechanics of Piezoelectric Structures; World Scientific: Singapore, 2006. doi:10.1142/12003. 60. Li, J.Y. Magnetoelectroelastic Multi-Inclusion and Inhomogeneity Problems and Their Applications in Composite Materials. Int. J. Eng. Sci. 2000, 38, 1993–2011, doi:10.1016/S0020-7225(00)00014-8. 61. Sih, G.C.; Song, Z.F. Magnetic and Electric Poling Effects Associated with Crack Growth in BaTiO3–CoFe2O4 Composite. Theor. Appl. Fract. Mech. 2003, 39, 209–227, doi:10.1016/S0167-8442(03)00003-X. 62. Wang, Y.; Xu, R.; Ding, H. Axisymmetric Bending of Functionally Graded Circular Magneto-Electro-Elastic Plates. Eur. J. Mech. A Solids 2011, 30, 999–1011, doi:10.1016/J.EUROMECHSOL.2011.06.009. 63. Zhang, G.Y.; Gao, X.L. Elastic Wave Propagation in 3-D Periodic Composites: Band Gaps Incorporating Microstructure Effects. Compos. 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On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams

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Abstract

Article On the Bending and Vibration Analysis of Functionally Graded Magneto-Electro-Elastic Timoshenko Microbeams Jun Hong, Shaopeng Wang, Gongye Zhang * and Changwen Mi * Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 210096, China; junhong@seu.edu.cn (J.H.); wsp@seu.edu.cn (S.W.) * Correspondence: gyzhang@seu.edu.cn (G.Z.); mi@seu.edu.cn (C.M.) Abstract: In this paper, a new magneto-electro-elastic functionally graded Timoshenko microbeam model is developed by using the variational formulation. The new model incorporates the extended modified couple stress theory in order to describe the microstructure effect. The power-law variation through the thickness direction of the two-phase microbeams is considered. By the direct application of the derived general formulation, the static bending and free vibration behavior of the newly developed functionally graded material microbeams are analytically determined. Parametric studies qualitatively demonstrate the microstructural effect as well as the magneto-electro-elastic multi-field coupling effect. The proposed model and its classic counterpart produce significant differences for thin graded magneto-electro-elastic Timoshenko microbeams. The thinner the microbeam is, the larger the difference becomes. Keywords: Timoshenko beam; functionally graded material; magneto-electro-elastic beam; microstructure effect; modified couple stress theory Citation: Hong, J.; Wang, S.; Zhang, G.; Mi, C. On the Bending and 1. Introduction Vibration Analysis of Functionally Currently, magneto-electro-elastic (MEE) materials have attracted more and more Grad-ed Magneto-Electro-Elastic attention. MEE materials can realize the mutual conversion between magnetic, electrical, Timoshenko Microbeams. Crystals and mechanical energies. Such characteristics have found important applications in 2021, 11, 1206. https://doi.org/ stability controlling, actuating, health monitoring, medical ultrasonic, and some smart 10.3390/cryst11101206 structure technologies [1–3]. In addition, functionally graded materials (FGMs) are Academic Editor: Pavel Lukáč characterized by continuous changes in material properties [4–6]. The mechanical properties of MEE materials synthesized from functionally graded materials are of great Received: 17 September 2021 significance in both research and industrial fields [7,8]. In recent years, the research on Accepted: 3 October 2021 investigating magneto-electro-elastic functionally graded materials (MEE-FGMs) on thin Published: 7 October 2021 beams and plates has become a major trend. Bhangale and Ganesan [9] studied the free vibration behavior of anisotropic and linear MEE-FGM plates. Sladek et al. [10] proposed Publisher’s Note: MDPI stays a meshless method for the bend analysis of circular MEE-FGM plates. Vinyas et al. [11] neutral with regard to jurisdictional studied the effectiveness of utilizing MEE-FGM plates in precise frequency responses claims in published maps and control. Mahesh and Harursampath [12] and Mahesh [13] evaluated nonlinear deflections institutional affiliations. of MEE-FGM porous flat panels and shells subjected to mechanical, electrical, and magnetic loads, respectively. However, numerous experiments [14,15] have proved that thin beams and plates usually exhibit size effects, (i.e. the thinner, the stiffer). Such size effects arise from non-local interactions of material particles at a very small scale, which Copyright: © 2021 by the authors. Li- cannot be described by classical theories at the micron or nanometer level due to a lack of censee MDPI, Basel, Switzerland. any material length scale parameters. Therefore, it is necessary to develop thin MEE-FGM This article is an open access article structure models based on non-classical theories. distributed under the terms and con- ditions of the Creative Commons At- In order to predict the size effects, numerous theories have been proposed with tribution (CC BY) license (http://crea- additional material parameters, such as non-local theories [16], couple stress theories [17– tivecommons.org/licenses/by/4.0/). 19], strain gradient theories [20–22], and a series of simpler versions [23–28]. These Crystals 2021, 11, 1206. https://doi.org/10.3390/cryst11101206 www.mdpi.com/journal/crystals Crystals 2021, 11, 1206 2 of 20 theories were successfully applied to develop size-dependent structure models for very small scales. For example, based on nonlocal theories, a number of MEE/MEE-FGM beam and plate models have been developed to capture non-local size effects [29–32], in which a non-local medium, including long-range material interactions, is adopted. Lim et al. [33] proposed a non-local strain gradient theory to include both non-local and strain gradient effects, and the bending, buckling, and free variation problems of FGM beams have been solved [34,35]. In addition, the modified couple stress theory (MCST) [24,25] contains only one additional parameter for isotropic materials. This MCST and its extended versions only consider the symmetrical part of the curvature tensor, which leads to fewer material parameters than their classical counterparts. In view of the great difficulties for determining additional parameters and interpreting the relevant microstructures, these modified theories have been applied to build micro/nano-beam and periodic composite pipe models [36–41], from which a microstructure-dependent stiffness is revealed. Recently, three such models have been proposed for MEE Timoshenko homogeneous beams [39] and MEE homogeneous plates [42,43] based on the extended modified couple stress theory. However, to the best of our knowledge, the extended modified couple stress theory is not applicable to MEE-FGM microbeams, which are inhomogeneous and might be helpful for smart devices miniaturization [44–48]. This motivated the present work. The present work uses the extended modified couple stress theory to develop a MEE- FGM Timoshenko microbeam model for the first time and analytically solves the static bending and free vibration problems of the new model. 2. Materials and Methods Consider a two-phase FGM microbeam with length L, width b and thickness h under the combined electric, magnetic, and mechanical loadings, as shown in Figure 1. The effective material properties P(z) (i.e. elastic stiffness, couple stress stiffness, piezoelectric constant, piezomagnetic constant, dielectric constant, magnetic permeability constant, magneto-dielectric constant and density) of the current microbeam change continuously in the thickness direction based on a power-law distribution [36], where P1 and P2 are the material properties of material I and II, respectively. The functionally graded power-law index n determines the material distribution across the thickness. z 1  Pz ()=− (P P ) + + P , (1) 12 2 h 2  Figure 1. Functionally graded microbeam configuration. Based on the extended modified couple stress theory [42,49], the constitutive equations for transversely isotropic magneto-electro-elastic materials are given by [39,42,50,51]. Crystals 2021, 11, 1206 3 of 20 CC C 00 0  11 12 13  σ ε  00 q 00 e xx xx 31 31   CC C 00 0     12 11 13  ε σ 00 q 00 e yy  yy 31 31     H E     CC C 00 0 xx 13 13 33   ε    σ 00 q 00 e       zz zz 33 33 =− H − E  00 0 C 0 0     yy (2)  2ε σ 00 q 00 e yz yz 15 15       H E 00 0 0 C 0  z  z       σ 2ε q 00 e 00  zx zx 15 15      CC −  11 12 2ε σ 00 0 0 0  00 0 00 0   xy xy       2 AA A 00 0  11 12 13 m χ    xx xx  AA A 00 0    12 11 13 m χ yy yy     AA A 00 0 13 13 33  m  χ   zz  zz   00 0 A 0 0   (3)  m 2χ yz yz     00 0 0 A 0    m 2χ  zx zx    AA −  11 12 m 2χ 00 0 0 0    xy xy      2  ε xx  yy  De 0 000 0 s 0 0E       x 15 11 x  ε   zz      De=+ 00 0 0 0 0s 0E      y 15 11 y    2ε yz         De e e 00 0 0 0 s E  z  31 31 33   33 z  2ε zx (4)  2ε  xy  dH 00  11 x   + 0 dH 0  11 y   dH  33 z  xx  yy  Bq 00 0 0 0 μ 0 0H       x 15 11 x        zz  Bq=+ 00 0 0 0 0 μ 0H      y 15 11 y    2ε yz         Bq q q 00 0 0 0 μ H  z  31 31 33   33 z  2ε zx (5)  2ε  xy  dE 00  11 x   + 0 dE 0  11 y    00 dE  33 z where 𝜎 ij, mij, Di, Bi are the Cauchy stress tensor, the deviatoric part of the couple stress tensor, the electric displacements, and the magnetic fluxes, respectively. Cαβ (α, β = 1, 2, …, 6) is the elastic stiffness tensor, Aαβ (α, β = 1, 2, …, 6) is the couple stress stiffness tensor, eiα and qiα are the piezoelectric and piezomagnetic tensors, sij and μij are the dielectric and magnetic permeability tensors, dij is the magneto-dielectric tensor, and εij and 𝜒 ij are, respectively, the infinitesimal strain and the symmetric curvature tensors, which are defined by ε =+ uu () (6) ij i,, j j i Crystals 2021, 11, 1206 4 of 20 χε=+uuε () (7) ij ipq q,, pj jpq q pi with ui being the displacement, and εijk is the Levi-Civita symbol. In addition, Ek and Hk are, respectively, the electric field intensity and magnetic field intensity read EH =−Φ , =−M kk,, k k (8) where Φ and M are the electric and magnetic potentials. For a MEE Timoshenko beam with a uniform cross-section shown in Figure 1, the displacement field and electric and magnetic potentials can be given by [52–55] uu=− ()x,, t zϕ()xt, u=0, u=w()x,t (9) 12 3 ππ22 zz    Φ =− cos zxγγ () ,t + , M = − cos zζ()x,t + ζ  00   (10) hh h h    where u and w are the beam extension and deflection, φ represents the rotation angle, γ and ζ are the spatial variations of the electric and magnetic potentials along the x- direction, respectively. γ0 and ζ0 are, respectively, the external electric and magnetic potentials. By substituting Equations (9) and (10) into Equations (6)–(8) yields ∂∂uw ϕ 1 ∂  εε =− z , = −ϕ xx xz  (11) ∂∂ xx 2 ∂x  , others = 0, 1 ∂∂ w ϕ χ =− + xy  2 (12) 4 ∂x ∂x  , others = 0, πγ ∂ π π 2   Ez== cos , E− sin z γγ− , E= 0 xz   0y (13) hx ∂ h h h   πζ ∂ π π 2    Hz== cos , H− sin z ζζ− , H= 0 xz    0y (14) hx ∂ h h h    Based on Equations (11)–(14), the constitutive equations in Equations (2)–(5) can be obtained as σε=− CeE−qH , σ = 2Cε −eE−qH (15) xxx 11 x 31z 31 z xz 44xz 15x 15 x mA=−A χ () xyx 11 12y (16) D=+2, esεε E+dH D=e +sE+dH x 15 xz 11 x 11 x z 31 xx 33 z 33 z (17) B=+2, qH ε μ +dE B=q ε + μH+dE x 15 xz 11 x 11 x z 31 xx 33 z 33 z (18) From Equations (11)–(18), the first variation of the total strain energy in the current beam satisfying the extended modified couple stress theory over the time span [0, T] takes the form [39,42] Crystals 2021, 11, 1206 5 of 20 TTL δσ Utd2 =+δε σδε+2mδχ xx xx xz xz xy xy  000 A (19) −− Dδδ ED E−BδH−BδH dAdxdt xx z z x x z z where A is the cross-sectional area. The first variation of the kinetic energy of the Timoshenko beam over the time interval [0, T] is given by [53] TTL ∂∂uu δ ∂∂uu δ  11 33 δρ Kdt=+ dd A xdt (20)  000 A ∂∂ tt ∂t ∂t  where ρ is the mass density. Furthermore, the virtual work performed by the applied forces acting on the current Timoshenko beam over the time span [0, T] can be written as [55,56] TTL δδ Wtdd =+ f u qδw xdt [] (21)  where f and q are, respectively, the x- and z-components of the body force per unit length along the x-axis. According to Hamilton’s principle [53,56], δ[] KU −−()Wdt=0 (22) Substituting Equations (19)–(21) into Equation (22), applying the fundamental lemma of the calculus of variations [57], and considering the arbitrariness of δu, δw, and δφ yield ∂N ∂∂ u ϕ xx +=fm −m 01 (23) ∂x ∂∂ tt ∂Y ∂M 1 ∂∂ ϕ u xy xx −+Nm − = −m xz 2122 (24) ∂∂ xx 2 ∂∂ tt ∂ Y ∂N 1 ∂ w xy xz ++qm= (25) ∂∂ x 2 x ∂t ∂Λ +Λ = 0 z (26) ∂x ∂Σ +Σ = 0 z (27) ∂x as the equation of motion, and N = 0 xx uu = (28) or at x = 0 and x = L, MY+= 0 (29) xx xy ϕϕ = or at x = 0 and x = L, ∂Y xy −− N = 0 xz (30) 2 ∂x ww = or at x = 0 and x = L, Crystals 2021, 11, 1206 6 of 20 ∂∂ ww Y = 0 (31) xy ∂∂ x x or at x = 0 and x = L, Λ= 0 γγ = (32) or at x = 0 and x = L, Σ= 0 ζζ = (33) or at x = 0 and x = L as boundary conditions, where the overbar denotes the prescribed value. Note that the stress, electric, magnetic resultants, and mass inertias can be expressed as ∂∂ u ϕ eq E H Nd==σγ A A −B +A +Aζ+N+N xxxx xx xx 31 31 x x (34) ∂∂ xx ∂∂ u ϕ eq E H M== zdσγ A B −D +B +Bζ+M+M xxxx xx xx 31 31 x x  (35) ∂∂ xx ∂∂ w γζ∂ 2 eq N== kσϕ dA k A − −kA −kA xz  s xz s xz s 15 s 15 (36) ∂∂ x xx∂   ∂∂ w ϕ Ym==dA F +  xy xy xy (37)  2 ∂x ∂x   π ∂∂ w  γζ∂ es d Λ= Dz cos dA=kA − ϕ +A +A xx  s 15  11 11 (38) hx∂∂x ∂x    ππ ∂∂ u ϕ  ee s d Es Hd Λ= D sin zdA= A −B − Aγζ − A −N −N zz  31 31 33 33 33 33 (39) hh ∂∂ x x  π ∂∂ w ζγ∂    qd μ Σ= Bz cos dA=kA − ϕ +A +A xx  s 15  11 11 (40) hx∂∂x ∂x    ππ ∂∂ u ϕ  qq μμ d H Ed Σ= Bz sin dA=A −B −Aζγ −A −N −N zz  31 31 33 33 33 33 (41) hh ∂∂ x x  mm,,m = ρ z 1,z,z dA () ()() (42) 01 2 where ks denotes the shape correction factor [58,59], and (, A BC , D ) = (z)(1, z, z )dA xx xx xx 11 (43) A = Cz dA () xz 44 (44) F=− Az A z dA () () () (45) xy 12 11 ππ  ee A , Be = ()z sin z () 1, zdA () 31 31 31  (46) hh  Crystals 2021, 11, 1206 7 of 20 ππ  qq A , Bq = z sin z 1, zdA () () () 31 31 31  (47) hh  2γ EE 0 NM , = e z 1, z dA () () () (48) xx  31 2ζ HH 0 NM , = q z 1, z dA () () () (49) xx 31  ππ    eq A , Ae = z cos z , q z cos z dA () () () 15 15 15  15   (50) hh      ππ π      ss 22 A , As = ()z cos z , s ()z sin z dA () 11 33 11  33    (51) hh h        ππ π      dd 22 A , Ad = z cos z , d z sin z dA () () () 11 33 11 33      (52) hh h       22 γζ  ππ ππ Es Hd 00 NN , = s ()z sin z , d ()z sin z dA () 33 33 33  33  (53) hh h h h h     ππ π       μμ 22 A , Az =μμ () cos z , ()z sin zdA () 11 33 11 33 (54)       hh h         22 ζγ ππ π π    HE μ d NN , = μ z sin z , d z sin z dA () () () 33 33 33  33   (55) hh h h h h     Based on Equations (23)–(55), it is found that the current MEE-FGM beam model can additionally capture the effects of couple stress, piezomagnetism, piezoelectricity, and MEE coupling, when compared to the classical FGM Timoshenko beam model. 3. Analytical Solution This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn. In order to illustrate the newly developed model in Section 2, the static bending and free vibration problems of the current beam are solved in this section. According to Equations (28)–(33), the relevant boundary conditions of a simply supported beam can be identified as N = 0 xx (56) ww== 0 (57) xx == 0 L MM== 0 xx xx (58) xx == 0 L Crystals 2021, 11, 1206 8 of 20 YY== 0 xy xy (59) xx == 0 L γγ== 0 (60) xx == 0 L ζζ== 0 (61) xx == 0 L It should be noted that the boundary of the electric and magnetic conditions given in Equations (60) and (61) are for an open circuit. 3.1. Static Bending Consider Fourier solutions for u(x), 𝜑 (x), w(x), γ(x), and ζ(x):  kx π ux = U cos ()  k  (62)  k =1 kx π  ϕΦ x = cos ()  (63) k   k =1 kx π  wx () = W sin  k  (64) k =1   kx π γΓ x = sin () k  (65)  k =1 kx π  ζ ()xZ = sin (66)  k  k =1  where Uk, Φk, Wk, Гk , and Zk are the Fourier coefficients to be determined. It can be shown that the Fourier solutions in Equations (62)–(66) satisfy the boundary conditions in Equations (56)–(61). In addition, the body force f is equal to zero, and the uniform load q(x) can also be expanded in Fourier series as: kx π qx () = Q sin  k (67) k =1 where Qk is a Fourier coefficient calculated by q(x) = p0 in the current case as 2 p Qk =− 1cos() π (68)  kπ According to the Equations (23)–(27), (62)–(66), and (67), the equilibrium equations of static bending problems can be written as ∂∂ u ϕγ∂ ∂ζ eq AB−+A +A = 0 (69) xx 22 xx 31 31 ∂∂ xx ∂∂ xx ∂∂uw ϕγ∂ ∂ζ ∂ eq 2 −+ BD −B −B +kA − ϕ xx 22 xx 31 31 s xz ∂∂ xx ∂∂ xx ∂x  (70) ∂∂ γζ 1  ∂ w ∂ϕ eq −− kA k A − F + = 0 ss 15 15 xy ∂∂ xx 2 ∂∂ xx  Crystals 2021, 11, 1206 9 of 20 22 2 4 3  ∂∂ww ϕγ∂ ∂ζ 1 ∂ ∂ϕ 2 eq kA −−k A −k A + F + =−q sxz s 15 s 15 xy  22 2 4 3 (71) ∂∂ xx ∂x ∂x 2 ∂x∂x    22 2  ∂∂wu ϕγ∂ ∂ζ ∂ ∂ϕ es d e e s d kA −+ A + A + A − B − Aγζ − A = 0 s 15 11 11 31 31 33 33 22 2 (72) ∂∂ xx∂∂ xx∂x ∂x  22 2  ∂∂wu ϕζ∂ ∂γ ∂ ∂ϕ qdμμq q d kA −+ A + A + A − B − Aζγ− A = 0 s 15 11 11 31 31 33 33 22 2 (73) ∂∂ xx∂∂ xx∂x ∂x  Substituting Equations (62)–(66) into Equations (69)–(73) results in SS 0 S S U 0   11 12 14 15 k   SS S S S Φ 0 12 22 23 24 25 k    0 SS S S W = −Q 23 33 34 35 k k (74)   SS S S S Γ  14 24 34 44 45 k    SS S S S Z 0  15 25 35 45 55 k  where kk ππ kπ kπ       eq SA =− , S =B , S = 0, S =A , S =A , 11 xx 12 xx 13 14 31  15 31  LL L L       22 3 kk ππ11kπ kπ      SD =− −kA + F , S =kA + F , 22 xx s xz xy 23 s xz  xy LL22L L      kk ππ  kk ππ  ee qq SA =− −kA , SA =− −kA , (75) 24 31 s 15  25 31 s 15  LL LL       24 2 2 kk ππ 1 kπ kπ      2 eq Sk =− A + F , S =kA , S =kA , 33 sxz  xy 34 s 15 35 s 15 LL 2 L L      22 2 kk ππ kπ    ss d d μμ SA=+A , S=A +A , S=A +A . 44 11 33 45 11 33 55 11 33 LL L    According to Equation (74), the Fourier coefficients Uk, Φk, Wk, Гk, and Zk will be solved. The solutions of u(x), 𝜑 (x), w(x), γ(x), and ζ(x) for the current simple supported beam can also be given by inserting these results into Equations (62)–(66). 3.2. Free Vibration In the free vibration problem of the current beam, both the external forces are vanished (i.e. f = q = 0). Consider the following Fourier series expansions for u(x, t), 𝜑 (x, t), w(x, t), γ(x, t), and ζ(x, t): kx π  V it ω ux,c t = U os e () k  (76)  k =1 kx π  V it ω ϕΦ () x,cte = os  k  (77) k =1   kx π V it ω wx,s t = Win e ()  k  (78)  k =1 kx π  V it ω γΓ x,ste = in () k  (79)  k =1 Crystals 2021, 11, 1206 10 of 20  kx π it ω ζ x,s tZ = in e () (80) k   k =1 V V V V V where ωk is the kth vibration frequency, U , W Φ , Г ,, and Z are Fourier coefficients. It k k k k k should be noted that the Fourier series expansions in Equations (76)–(80) satisfy the boundary conditions in Equations (56)–(61). Based on Equations (76)–(80) and Equations (23)–(27), the equations of motion can be expressed as 22 2 2 ∂∂uu ϕγ∂ ∂ζ ∂ ∂ϕ eq AB−+A +A =m −m (81) xx 22 xx 31 31 0 2 1 2 ∂∂ x xt ∂∂ xx ∂ ∂t ∂∂uw ϕγ∂ ∂ζ ∂  eq 2 −+ BD −B −B +kA − ϕ xx xx 31 31 s xz ∂∂ xx ∂∂ xx ∂x  (82) 32 2 2 ∂∂ γζ 1  ∂wu ∂ϕ ∂ϕ ∂ eq −− kA k A − F + = m −m ss 15 15 xy 2 1 32 2 2 ∂∂ xx 2 ∂x ∂xt∂ ∂t  22 2 4 3 2    ∂∂ww ϕγ∂ ∂ζ 1 ∂ ∂ϕ ∂w 2 eq k A −−kA −kA + F + = m sxz s 15 s 15 xy  0 22 2 4 3 2 (83) ∂∂ x ∂x xx∂ 2 ∂x∂x ∂t    22 2  ∂∂wu ϕγ∂ ∂ζ ∂ ∂ϕ es d e e s d kA −+ A + A + A − B − Aγζ − A = 0 s 15 11 11 31 31 33 33 22 2 (84) ∂∂ xx∂∂ xx∂x ∂x  22 2  ∂∂wu ϕζ∂ ∂γ ∂ ∂ϕ qdμμq q d kA −+ A + A + A − B − Aζγ− A = 0 s 15 11 11 31 31 33 33 22 2 (85) ∂∂ xx∂∂ xx∂x ∂x  Using Equations (76)–(80) in Equations (81)–(85), yields VV SS 0 S S   0 UU mm ωω − 00 0  11 12 14 15 kk 01kk   VV 22  SS S S SΦΦ −mm ωω 00 0 0 12 22 23 24 25 kk 12kk   VV 2   0 SS S S + = WW 00 m ω 0 0 0 23 33 34 35kk 0 k (86)   VV SS S S S 0 ΓΓ 00 0 0 0 14 24 34 44 45   kk   VV    SS S S SZZ 00 0 0 0 0  15 25 35 45 55  kk   Therefore, the first natural frequency ω1 of the current beam can be solved from the smallest positive root of ωk (k = 1) of the Equation (86). 4. Numerical Results The 50%-50% BaTiO3-CoFe2O4 is adopted for material I [39,42,60–62], and the material Ⅱ is taken to be epoxy [63], as listed in Table 1. Note that the couple stress constants A11 and A12 are estimates based on the formula provided in [14,39]. The magnitude of the uniform load p0 is equal to 1/2000h N/m, the shear correction factor ks is 1/2 0.8 , and the cross-sectional shape is kept at b = 2h and L = 20h. In order to verify the correctness of the current model, a comparative study of the deflection of a simply supported microbeam subjected to uniform load between the current model (with Gradient index n = 0) and the model provided by Zhang et al. [39] are plotted in Figure 2. The beam parameters are adopted from Zhang et al. [39]. Crystals 2021, 11, 1206 11 of 20 Table 1. Material properties of the BaTiO3-CoFe2O4 [39] and epoxy [63]. Physical parameter Material Ⅱ Material I C11 (GPa) 226 4.889 C44 (GPa) 44.15 1.241 e15 (C/m ) 5.8 0 e31 (C/m ) −2.2 0 e33 (C/m ) 9.3 0 -9 2 2 s11 (10 C /(N·m )) 5.64 0 -9 2 2 s33 (10 C /(N·m )) 6.35 0 q15 (N/(A·m)) 275 0 q31 (N/(A·m)) 290.15 0 q33 (N/(A·m)) 349.85 0 −12 d11 (10 Ns/(V·C)) 5.38 0 −12 d33 (10 Ns/(V·C)) 2740 0 −6 2 2 μ11 (10 Ns /C ) 297.5 0 −6 2 2 μ33 (10 Ns /C ) 83.5 0 A11 (N) 11.7484 1.4014 A12 (N) 6.4980 0.6903 ρ (kg/m ) 5550 1180 From Figure 2, it is obvious that the results of the classical and current model are the same as those in Zhang et al. [39]. In addition, this validates the current model and shows that the microstructure effect will always cause the deflection to decrease, as expected. 0.14 0.12 0.1 0.08 0.06 MCST from current Classical from current 0.04 MCST from Zhang et al. [39] Classical from Zhang et al. [39] 0.02 h = 14.42μm, b = 2h, L = 20h 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/L Figure 2. Comparison of the deflection of the simply supported microbeam subjected to a uniform load. 4.1. Static Bending Figure 3 shows the distributions of the deformation, axial normal stress, and the electric and magnetic potentials of the current beam. In order to facilitate the observation of the deformation trend of the current beam, the x-component of the displacement vector u of a point (x, y, z) on the beam cross-section has been enlarged by 10 times. In addition, the thickness h is 20 μm, and the gradient index n is 5. Crystals 2021, 11, 1206 12 of 20 From Figure 3b, it can be observed that the axial normal stress in the middle of the current beam is relatively small, and the axial normal stress at the top of the beam is relatively large. From Figure 3c,d, it is clear that the distributions of electric and magnetic potentials in the current beam are center-symmetrical, and the maximum magnitudes both appear at the center of the beam. (a) (b) (c) (d) Figure 3. Distribution of (a) deformation, (b) axial normal stress, (c) electric potential, and (d) magnetic potential (Gradient index n = 5). Figures 4 and 5 show the deflections and rotation angles with different thicknesses predicted by current and classical models. The gradient index n is 5. The numerical results for the current model (solid lines) incorporating the couple stress effect (with A11 ≠ 0 and A12 ≠ 0) are directly calculated from Equations (62), (63) and (74), while those for the classical model (dashed lines) are obtained using the same equations but with A11 = A12 = From Figures 4 and 5, it can be found that the deflections and rotation angles of the current model are always smaller than those of the classical model in all cases. The difference between the results of the current and classical models is obvious when the beam thickness h is small, as expected. Crystals 2021, 11, 1206 13 of 20 Microstructure effect Figure 4. Deflection of the MEE-FGM simply supported beam (Gradient index n = 5). Microstructure effect Figure 5. Rotation of the MEE-FGM simply supported beam (Gradient index n = 5). Figure 6 shows the axial normal stress at the beam center (x = L/2) along the thickness direction of the current and classical models. From Figure 6, it is clear that the magnitude of the axial normal stress of the current model is always smaller than that of the classical model. The differences between the axial normal stress predicted by the two models also become smaller with the increase in the thickness h. Crystals 2021, 11, 1206 14 of 20 Microstructure effect Figure 6. Axial normal stress of the MEE-FGM simply supported beam (Gradient index n = 5). Figures 7 and 8 display the electric and magnetic potentials of the FGM simply supported beam with different thickness of the current and classical models. From Figures 7 and 8, it can be observed that the values of electric and magnetic potentials of the current model are always smaller than those of the classical model. When the beam thickness h is small, the differences between the two sets of results are very large. However, the differences become small when the beam thickness increases. This phenomenon also indicates that the microstructure effect is significant for very thin beams. Microstructure effect Figure 7. Electric potential of the MEE-FGM simply supported beam (Gradient index n = 5). Crystals 2021, 11, 1206 15 of 20 Microstructure effect Figure 8. Magnetic potential of the MEE-FGM simply supported beam (Gradient index n = 5). To illustrate the material inhomogeneity, Figure 9 shows the variation of the maximum deflections wmax (x = L/2) of the MEE-FGM beam with the different gradient index n for h = 20μm and 20mm, respectively. From Figure 9a, it can be seen that the maximum deflections wmax increases with the increase of the gradient index — for both current and classical models— and the deflection of the classical model is always larger than that of the current model. From Figure 9b, it is found that when the thickness of the beam is large enough, there is almost no difference in the prediction results of the maximum deflections predicted by the two models, which further confirms that the microstructure effect is only important for very thin beams. In addition, from Figures 9a,b, it is shown that the gradient index n does have a significant effect on the static bending response for all length scales. Microstructure effect (a) (b) Figure 9. Maximum deflections of the MEE-FGM beam for different gradient index with (a) h = 20μm, (b) h = 20mm. Figure 10 shows the variation of the axial normal stress 𝜎 xx(L/2, z) of the current MEE- FGM beam through the thickness for different gradient index n. From Figure 10, it can be found that the axial normal distribution of current MEE-FGM beam is different from that of a homogeneous beam for both h = 20μm and 20mm cases. In addition, the axial normal stress of homogenous beams on the geometric central axial (z = 0) is zero, but the zero- Crystals 2021, 11, 1206 16 of 20 valued stresses positions of the current FGM beam are varying with the n. Furthermore, the axial normal stress of a homogeneous beam is linear, while those of current MEE-FGM beam are nonlinear at all length scales. (a) (b) Figure 10. Axial normal stress of the current MEE-FGM beam through the thickness with (a) h = 20μm, (b) h = 20mm. 4.2. Free Vibration Figure 11 shows the variation of natural frequency (with k = 1) of the MEE-FGM beam of the current and classical models with different beam thickness. From Figure 11, it is obvious that the natural frequencies of both the current and classical models decrease with the thickness increases. The results also show that the current model incorporating the couple stress effect always increases the value of the natural frequency (and thus increased the beam stiffness). When the beam thickness is small enough, the couple stress effect is significant. Microstructure effect Figure 11. Natural frequency with different MEE-FGM beam thickness (Gradient index n = 5). Figure 12 shows the variation of the natural frequency ω1 (k = 1) of the current and classical models with different gradient index n for h = 20μm and 20mm. From Figure 12a, it is clear that when the thickness of the beam is small (micro scale), the prediction results of the two models are very different. However, from Figure 12b, when the thickness is big Crystals 2021, 11, 1206 17 of 20 enough (macro scale), the prediction results of the two models are almost the same. In addition, the effect of the gradient index is found to be important for all length scales. Microstructure effect (a) (b) Figure 12. Natural frequency of the MEE-FGM beam for different gradient index with (a) h = 20μm, (b) h = 20mm. 5. Conclusions Based on the extended modified couple stress theory, a new graded magneto-electro- elastic Timoshenko microbeam model is developed. The new model considers the effects of both three-field coupling and couple stress. The equations of motion and complete boundary conditions of the new microbeam model are determined through a variational approach. As two direct applications of the new model, the static bending and free vibration properties of a simply supported microbeam subjected to uniformly distributed loads are analytically obtained. For the static bending problem, parametric studies demonstrate that the deflections, rotations, axial normal stresses, electric and magnetic potentials predicted by the current model are all smaller than those of the classical theory. The differences decrease with the increase in the microbeam thickness. For the problem of free vibration, the natural frequency obtained from the current model is found to be higher than that of the classical model. The difference increases as the thickness of the beam decreases. Such a behavior also indicates that the microstructure effect tends to make the graded magneto- electro-elastic microbeam stiffer, and the current model can predict the size effect for magneto-electro-elastic functionally graded microbeam. In addition, it was demonstrated that changing the gradient index significantly affects both the static and vibrational properties of the graded magneto-electro-elastic microbeam at all length scales. These findings are helpful in guiding the engineering design and optimization of graded magneto-electro-elastic materials in MEMS and NEMS devices. Author Contributions: Conceptualization, C.M. and G.Z.; methodology, J.H., S.W.; writing— original draft preparation, J.H. and S.W. All authors have read and agreed to the published version of the manuscript. Funding: The work reported here is funded by the National Key R&D Program of China (grant number 2018YFD1100401) and the National Natural Science Foundation of China [grant numbers 12002086, 11872149 and 11772091] Conflicts of Interest: The authors declare no conflict of interest. Nomenclature L, b, h Length, width and thickness of beam P(z), P1, P2 Material properties of the current beam, material I and II n Functionally graded power-law index 𝜎 ij The components of Cauchy stress tensor (MHz) (MHz) 1 Crystals 2021, 11, 1206 18 of 20 mij The components of the couple stress tensor Di Electric displacements Bi Magnetic fluxes Cαβ The components of elastic stiffness tensor Aαβ The components of couple stress stiffness tensor eiα The components of piezoelectric tensor qiα The components of piezomagnetic tensor sij The components of dielectric tensor μij The components of magnetic permeability tensor dij The components of magneto-dielectric tensor εij The components of infinitesimal strain tensor 𝜒 ij The components of the symmetric curvature tensor ui Displacement components εijk Levi-Civita symbol Ek, Hk Electric field intensity and magnetic field intensity Φ, M Electric potential and magnetic potential u, w Beam extension and deflection φ Rotation angle γ, ζ The electric potential and magnetic potentials γ0, ζ0 External electric potential, external magnetic potential A Cross-sectional area Mass density f, q The x- and z-components of the body force per unit length ks Shape correction factor Uk, Φk, Wk, Гk , Zk, Qk Fourier coefficients ωk The kth vibration frequency V V V V V U , W Φ , Г , Z Fourier coefficients k k k k k References 1. Sahmani, S.; Aghdam, M.M. Nonlocal Strain Gradient Shell Model for Axial Buckling and Postbuckling Analysis of Magneto- Electro-Elastic Composite Nanoshells. Compos. Part B Eng. 2018, 132, 258–274, doi:10.1016/J.COMPOSITESB.2017.09.004. 2. Farajpour, M.R.; Shahidi, A.R.; Hadi, A.; Farajpour, A. Influence of Initial Edge Displacement on the Nonlinear Vibration, Elec- trical and Magnetic Instabilities of Magneto-Electro-Elastic Nanofilms. Mech. Adv. Mater. Struct. 2019, 26, 1469–1481, doi:10.1080/15376494.2018.1432820. 3. Yakhno, V.G. An Explicit Formula for Modeling Wave Propagation in Magneto-Electro-Elastic Materials. J. Electromagn. Waves Appl. 2018, 32, 899–912, doi:10.1080/09205071.2017.1410076. 4. Chen, W.; Yan, Z.; Wang, L. On Mechanics of Functionally Graded Hard-Magnetic Soft Beams. Int. J. Eng. Sci. 2020, 157, 103391, doi:10.1016/J.IJENGSCI.2020.103391. 5. Taati, E. On Buckling and Post-Buckling Behavior of Functionally Graded Micro-Beams in Thermal Environment. Int. J. Eng. Sci. 2018, 128, 63–78, doi:10.1016/J.IJENGSCI.2018.03.010. 6. Yang, Z.; Xu, J.; Lu, H.; Lv, J.; Liu, A.; Fu, J. Multiple Equilibria and Buckling of Functionally Graded Graphene Nanoplatelet- Reinforced Composite Arches with Pinned-Fixed End. Crystals 2020, 10, 1003, doi:10.3390/CRYST10111003. 7. Ghayesh, M.H.; Farokhi, H.; Alici, G. Size-Dependent Performance of Microgyroscopes. Int. J. Eng. Sci. 2016, 100, 99–111, doi:10.1016/J.IJENGSCI.2015.11.003. 8. Tang, Y.; Ma, Z.S.; Ding, Q.; Wang, T. Dynamic Interaction between Bi-Directional Functionally Graded Materials and Magneto- Electro-Elastic Fields: A Nano-Structure Analysis. Compos. Struct. 2021, 264, 113746, doi:10.1016/J.COMPSTRUCT.2021.113746. 9. Bhangale, R.K.; Ganesan, N. Free Vibration of Simply Supported Functionally Graded and Layered Magneto-Electro-Elastic Plates by Finite Element Method. J. Sound Vib. 2006, 294, 1016–1038, doi:10.1016/J.JSV.2005.12.030. 10. Sladek, J.; Sladek, V.; Krahulec, S.; Chen, C.S.; Young, D.L. Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties. Mech. Adv. Mater. Struct. 2015, 22, 479–489, doi:10.1080/15376494.2013.807448. 11. Vinyas, M.; Harursampath, D.A.; Nguyen-Thoi, T. Influence of Active Constrained Layer Damping on the Coupled Vibration Response of Functionally Graded Magneto-Electro-Elastic Plates with Skewed Edges. Def. Technol. 2020, 16, 1019–1038, doi:10.1016/J.DT.2019.11.016. 12. Mahesh, V.; Harursampath, D. Large Deflection Analysis of Functionally Graded Magneto-Electro-Elastic Porous Flat Panels. Eng. Comput. 2021, 1–20, doi:10.1007/S00366-020-01270-X. Crystals 2021, 11, 1206 19 of 20 13. Mahesh, V. Porosity Effect on the Nonlinear Deflection of Functionally Graded Magneto-Electro-Elastic Smart Shells under Combined Loading. Mech. Adv. Mater. Struct. 2021, 1–27, doi:10.1080/15376494.2021.1875086. 14. Lam, D.C.C.; Yang, F.; Chong, A.C.M.; Wang, J.; Tong, P. Experiments and Theory in Strain Gradient Elasticity. J. Mech. Phys. Solids 2003, 51, 1477–1508, doi:10.1016/S0022-5096(03)00053-X. 15. McFarland, A.W.; Colton, J.S. Role of Material Microstructure in Plate Stiffness with Relevance to Microcantilever Sensors. J. Micromech. Microeng. 2005, 15, 1060, doi:10.1088/0960-1317/15/5/024. 16. Eringen, A.C. On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves. J. Appl. Phys. 1983, 54, 4703–4710, doi:10.1063/1.332803. 17. Toupin, R.A. Elastic Materials with Couple-Stresses. Arch. Ration. Mech. Anal. 1962, 11, 385–414. 18. Mindlin, R.D. Influence of Couple-Stresses on Stress Concentrations. Exp. Mech. 1963, 3, 1–7, doi:10.1007/BF02327219. 19. Kolter, W.T. Couple Stresses in the Theory of Elasticity: I and II. Proc. K. Ned. Akad. Wet. B 1964, 67, 17–44. 20. Mindlin, R.D. Micro-Structure in Linear Elasticity. Arch. Ration. Mech. Anal. 1964, 16, 51–78, doi:10.1007/BF00248490. 21. Mindlin, R.D.; Eshel, N.N. On First Strain-Gradient Theories in Linear Elasticity. Int. J. Solids Struct. 1968, 4, 109–124, doi:10.1016/0020-7683(68)90036-X. 22. Polizzotto, C. A Hierarchy of Simplified Constitutive Models within Isotropic Strain Gradient Elasticity. Eur. J. Mech. A Solids 2017, 61, 92–109, doi:10.1016/J.EUROMECHSOL.2016.09.006. 23. Altan, B.S.; Aifantis, E.C. On Some Aspects in the Special Theory of Gradient Elasticity. J. Mech. Behav. Mater. 1997, 8, 231–282, doi:10.1515/JMBM.1997.8.3.231. 24. Yang, F.; Chong, A.C.M.; Lam, D.C.C.; Tong, P. Couple Stress Based Strain Gradient Theory for Elasticity. Int. J. Solids Struct. 2002, 39, 2731–2743, doi:10.1016/S0020-7683(02)00152-X. 25. Park, S.K.; Gao, X.-L. Variational Formulation of a Modified Couple Stress Theory and Its Application to a Simple Shear Problem. Z. Angew. Math. Phys. 2008, 59, 904–917, doi:10.1007/s00033-006-6073-8. 26. Zhang, G.Y.; Gao, X.L. A New Bernoulli–Euler Beam Model Based on a Reformulated Strain Gradient Elasticity Theory. Math. Mech. Solids 2020, 25, 630–643, doi:10.1177/1081286519886003. 27. Qu, Y.L.; Zhang, G.Y.; Fan, Y.M.; Jin, F. A Non-Classical Theory of Elastic Dielectrics Incorporating Couple Stress and Quadru- pole Effects: Part I—Reconsideration of Curvature-Based Flexoelectricity Theory. Math. Mech. Solids 2021, doi:10.1177/10812865211001533. 28. Zhang, G.Y.; Gao, X.L.; Zheng, C.Y.; Mi, C.W. A Non-Classical Bernoulli-Euler Beam Model Based on a Simplified Micromor- phic Elasticity Theory. Mech. Mater. 2021, 161, 103967, doi:10.1016/J.MECHMAT.2021.103967. 29. Ebrahimi, F.; Barati, M.R. Vibration Analysis of Embedded Biaxially Loaded Magneto-Electrically Actuated Inhomogeneous Nanoscale Plates. J. Vib. Control. 2018, 24, 3587–3607, doi:10.1177/1077546317708105. 30. Kiani, A.; Sheikhkhoshkar, M.; Jamalpoor, A.; Khanzadi, M. Free Vibration Problem of Embedded Magneto-Electro-Thermo- Elastic Nanoplate Made of Functionally Graded Materials via Nonlocal Third-Order Shear Deformation Theory. J. Intell. Mater. Syst. 2018, 29, 741–763, doi:10.1177/1045389X17721034. 31. Liu, H.; Lv, Z. Vibration Performance Evaluation of Smart Magneto-Electro-Elastic Nanobeam with Consideration of Nano- material Uncertainties. J. Intell. Mater. Syst. Struct. 2019, 30, 2932–2952, doi:10.1177/1045389X19873418. 32. Xiao, W.S.; Gao, Y.; Zhu, H. Buckling and Post-Buckling of Magneto-Electro-Thermo-Elastic Functionally Graded Porous Nano- beams. Microsyst. Technol. 2019, 25, 2451–2470, doi:10.1007/S00542-018-4145-2. 33. Lim, C.W.; Zhang, G.; Reddy, J.N. A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and Its Applications in Wave Propagation. J. Mech. Phys. Solids 2015, 78, 298–313, doi:10.1016/J.JMPS.2015.02.001. 34. Şimşek, M. Nonlinear Free Vibration of a Functionally Graded Nanobeam Using Nonlocal Strain Gradient Theory and a Novel Hamiltonian Approach. Int. J. Eng. Sci. 2016, 105, 12–27, doi:10.1016/J.IJENGSCI.2016.04.013. 35. Li, X.; Li, L.; Hu, Y.; Ding, Z.; Deng, W. Bending, Buckling and Vibration of Axially Functionally Graded Beams Based on Nonlocal Strain Gradient Theory. Compos. Struct. 2017, 165, 250–265, doi:10.1016/J.COMPSTRUCT.2017.01.032. 36. Reddy, J.N. Microstructure-Dependent Couple Stress Theories of Functionally Graded Beams. J. Mech. Phys. Solids 2011, 59, 2382–2399, doi:10.1016/J.JMPS.2011.06.008. 37. Gao, X.-L.; Zhang, G.Y. A Microstructure-and Surface Energy-Dependent Third-Order Shear Deformation Beam Model. Z. An- gew. Math. Phys. 2015, 66, 1871–1894, doi:10.1007/S00033-014-0455-0. 38. Yu, T.; Hu, H.; Zhang, J.; Bui, T.Q. Isogeometric Analysis of Size-Dependent Effects for Functionally Graded Microbeams by a Non-Classical Quasi-3D Theory. Thin-Walled Struct. 2019, 138, 1–14, doi:10.1016/J.TWS.2018.12.006. 39. Zhang, G.Y.; Qu, Y.L.; Gao, X.L.; Jin, F. A Transversely Isotropic Magneto-Electro-Elastic Timoshenko Beam Model Incorporat- ing Microstructure and Foundation Effects. Mech. Mater. 2020, 149, 103412, doi:10.1016/J.MECHMAT.2020.103412. 40. Hong, J.; He, Z.Z.; Zhang, G.Y.; Mi, C.W. Tunable Bandgaps in Phononic Crystal Microbeams Based on Microstructure, Piezo and Temperature Effects. Crystals 2021, 11, 1029, doi:10.3390/CRYST11091029. 41. Hong, J.; He, Z.Z.; Zhang, G.Y.; Mi, C.W. Size and Temperature Effects on Band Gaps in Periodic Fluid-Filled Micropipes. Appl. Math. Mech. 2021, 42, 1219–1232, doi:10.1007/S10483-021-2769-8. 42. Qu, Y.L.; Li, P.; Zhang, G.Y.; Jin, F.; Gao, X.L. A Microstructure-Dependent Anisotropic Magneto-Electro-Elastic Mindlin Plate Model Based on an Extended Modified Couple Stress Theory. Acta Mech. 2020, 231, 4323–4350, doi:10.1007/S00707-020-02745-0. 43. Shen, W.; Zhang, G.; Gu, S.; Cong, Y. A Transversely Isotropic Magneto-electro-elastic Circular Kirchhoff Plate Model Incorpo- rating Microstructure Effect. Acta Mech. Solida Sin. 2021, 1–13, doi:10.1007/s10338-021-00271-7. Crystals 2021, 11, 1206 20 of 20 44. Qu, Y.L.; Jin, F.; Yang, J.S. Magnetically induced charge motion in the bending of a beam with flexoelectric semiconductor and piezomagnetic dielectric layers. J. Appl. Phys. 2021, 127, 064503. 45. Zhu, F.; Ji, S.; Zhu, J.; Qian, Z.; Yang, J. Study on the Influence of Semiconductive Property for the Improvement of Nanogener- ator by Wave Mode Approach. Nano Energy 2018, 52, 474–484, doi:10.1016/J.NANOEN.2018.08.026. 46. Shingare, K.B.; Kundalwal, S.I. Static and Dynamic Response of Graphene Nanocomposite Plates with Flexoelectric Effect. Mech. Mater. 2019, 134, 69–84, doi:10.106/J.MECHMAT.2019.04.006. 47. Wang, L.; Liu, S.; Feng, X.; Zhang, C.; Zhu, L.; Zhai, J.; Qin, Y.; Wang, Z.L. Flexoelectronics of Centrosymmetric Semiconductors. Nat. Nanotechnol. 2020, 15, 661–667, doi:10.1038/s41565-020-0700-y. 48. Sharma, S.; Kumar, A.; Kumar, R.; Talha, M.; Vaish, R. Geometry Independent Direct and Converse Flexoelectric Effects in Functionally Graded Dielectrics: An Isogeometric Analysis. Mech. Mater. 2020, 148, 103456, doi:10.1016/J.MECHMAT.2020.103456. 49. Zhang, G.Y.; Gao, X.L.; Guo, Z.Y. A Non-Classical Model for an Orthotropic Kirchhoff Plate Embedded in a Viscoelastic Me- dium. Acta Mech. 2017, 228, 3811–3825, doi:10.1007/S00707-017-1906-4. 50. Han, X.; Pan, E. Fields Produced by Three-Dimensional Dislocation Loops in Anisotropic Magneto-Electro-Elastic Materials. Mech. Mater. 2013, 59, 110–125, doi:10.1016/J.MECHMAT.2012.09.001. 51. Kumar, D.; Sarangi, S.; Saxena, P. Universal Relations in Coupled Electro-Magneto-Elasticity. Mech. Mater. 2020, 143, 103308, doi:10.1016/J.MECHMAT.2019.103308. 52. Wang, Q. On Buckling of Column Structures with a Pair of Piezoelectric Layers. Eng. Struct. 2002, 24, 199–205, doi:10.1016/S0141- 0296(01)00088-8. 53. Ma, H.M.; Gao, X.L.; Reddy, J.N. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory. J. Mech. Phys. Solids 2008, 56, 3379–3391, doi:10.1016/J.JMPS.2008.09.007. 54. Ansari, R.; Gholami, R.; Rouhi, H. Size-Dependent Nonlinear Forced Vibration Analysis of Magneto-Electro-Thermo-Elastic Timoshenko Nanobeams Based upon the Nonlocal Elasticity Theory. Compos. Struct. 2015, 126, 216–226, doi:10.1016/J.COMP- STRUCT.2015.02.068. 55. Hong, J.; Wang, S.P.; Zhang, G.Y.; Mi, C.W. Bending, Buckling and Vibration Analysis of Complete Microstructure-Dependent Functionally Graded Material Microbeams. Int. J. Appl. Mech. 2021, 13, 2150057, doi:10.1142/S1758825121500575. 56. Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics, 2nd ed.; Wiley: New York, NY, USA, 2002. 57. Gao, X.-L.; Mall, S. Variational Solution for a Cracked Mosaic Model of Woven Fabric Composites. Int. J. Solids Struct. 2001, 38, 855–874, doi:10.1016/S0020-7683(00)00047-0. 58. Yang, J. An Introduction to the Theory of Piezoelectricity; Springer: New York, NY, USA, 2005. doi:10.1007/B101799. 59. Yang, J. The Mechanics of Piezoelectric Structures; World Scientific: Singapore, 2006. doi:10.1142/12003. 60. Li, J.Y. Magnetoelectroelastic Multi-Inclusion and Inhomogeneity Problems and Their Applications in Composite Materials. Int. J. Eng. Sci. 2000, 38, 1993–2011, doi:10.1016/S0020-7225(00)00014-8. 61. Sih, G.C.; Song, Z.F. Magnetic and Electric Poling Effects Associated with Crack Growth in BaTiO3–CoFe2O4 Composite. Theor. Appl. Fract. Mech. 2003, 39, 209–227, doi:10.1016/S0167-8442(03)00003-X. 62. Wang, Y.; Xu, R.; Ding, H. Axisymmetric Bending of Functionally Graded Circular Magneto-Electro-Elastic Plates. Eur. J. Mech. A Solids 2011, 30, 999–1011, doi:10.1016/J.EUROMECHSOL.2011.06.009. 63. Zhang, G.Y.; Gao, X.L. Elastic Wave Propagation in 3-D Periodic Composites: Band Gaps Incorporating Microstructure Effects. Compos. Struct. 2018, 204, 920–932, doi:10.1016/J.COMPSTRUCT.2018.07.115.

Journal

CrystalsMultidisciplinary Digital Publishing Institute

Published: Oct 7, 2021

Keywords: Timoshenko beam; functionally graded material; magneto-electro-elastic beam; microstructure effect; modified couple stress theory

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