Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 3, pp. 853-868, Warsaw 2017 DOI: 10.15632/jtam-pl.55.3.853 BENDING, BUCKLING, AND FORCED VIBRATION ANALYSES OF NONLOCAL NANOCOMPOSITE MICROPLATE USING TSDT CONSIDERING MEE PROPERTIES DEPENDENT TO VARIOUS VOLUME FRACTIONS OF CoFe2O4-BaTiO3 Mehdi Mohammadimehr, Rasoul Rostami University of Kashan, Department of Solid Mechanics, Faculty of Mechanical Engineering, Kashan, Iran e-mail: mmohammadimehr@kashanu.ac.ir; r rostami@grad.kashanu.ac.ir In this article, the bending, buckling, free and forced vibration behavior of a nonlocal nano- composite microplate using the third order shear deformation theory (TSDT) is presented. Themagneto-electro-elastic (MEE) properties are dependent on various volume fractions of CoFe2O4-BaTiO3.According toMaxwell’s equationsandHamilton’sprinciple, the governing differential equations are derived. These equations are discretized by using Navier’smethod for anMEEnanocompositeReddyplate.The numerical results show the influences of elastic foundation parameters such as aspect ratio, length to thickness ratio, electric andmagnetic fields and various volume fractions of CoFe2O4-BaTiO3 on deflection, critical buckling lo- ad and natural frequency. The natural frequency and critical buckling load increases with the increasing volume fraction of CoFe2O4-BaTiO3, also the amplitude vibration decreases with an increase in the volume fraction. This model can be used for various nanocomposite structures. Also, a series of new experiments are recommended for future work. Keywords: bending and buckling analysis, free and forced vibration analysis, nonlocal nano- composite microplate, various volume fractions of CoFe2O4-BaTiO3 1. Introduction In the recent years, the use of nano-technology is a subject of the main discussion in the world of engineering sciences. Nano-technology is science in which the design and application of na- nostructures relates different properties at the nanoscale. The size of nanoparticles and their dispersion in a matrix composite is one of the ways to achieve desired properties of nano- composites. According to the nanometer-scale, the reinforcement particles in nanocomposites, intermolecular forces between thematrix and reinforcing is much greater than in ordinary com- posites, which improves properties of the nanocomposites. The reinforcing phase in terms of the material can be used as polymeric, metal and ceramic, which, according to different properties of each, have different applications. Because of their magnetoelectric coupling effects, magneto- electric-elastic (MEE)materials have beenwidely employed inmany technological fields, such as sensor and actuator applications, robotics, medical instruments, structural health monitoring, energyharvesting.Many researchers have carried out static, buckling, and free vibration analysis of nanocomposites, see Sih and Yu (2005) who analyzed the volume fraction effect of a MEE composite on enhancement and impediment of crack growth. Their results showed that with the increasing electric field to normal stress ratio and the volume fraction effect of theMEE compo- site, the crack growth increased and decreased, respectively. Ke andWang (2014) examined free vibration of size-dependent magneto-electro-elastic nanobeams based on the nonlocal elasticity theory. By using theHamilton principle, the governing equations and boundary conditions were derived anddiscretizedbyusing thedifferential quadraturemethod (DQM) todeterminenatural frequencies. Their results showed that with the increasing magnetic and electric potential, the 854 M.Mohammadimehr, R. Rostami natural frequencies of nanobeams increased. Shokrani et al. (2016) employed the generalized dif- ferential quadraturemethod (GDQM) to the buckling analysis of double orthotropic nanoplates (DONP) embedded in elastic media under biaxial, uniaxial and shear loadings. Their results showed that for higher values of the non-local parameter, the shear buckling was not depen- dent on the van der Waals and Winkler moduli. Lang and Xuewu (2013) studied the buckling and vibration of functionally graded magneto-electro-thermo-elastic circular cylindrical shells. Based on using the third order shear theory (TSDT), they employedHamilton’s principle to ob- tain equations of motion and numerical solutions to find the natural frequencies. Ghorbanpour Arani et al. (2012) investigated the effect of the CNT volume fraction on the magneto-thermo- electro-mechanical behavior of a smart nanocomposite cylinder. Their results indicated that the influence of internal pressure on the radial stress was larger than thermal, magnetic and elec- tric fields. Also, their results are very useful for the optimization of nano-composite structures. Xin and Hu (2015) analyzed free vibration of multilayered magneto-electro-elastic plates ba- sed on the state space approach (SSA) and the discrete singular convolution (DSC) algorithm. The results showed that the piezoelectric effect had a tendency to increase the stiffness of the plate, and vice versa for the magnetostrictive effect. Karimi et al. (2015a) investigated surface effects and non-local two variable refined plate theories thatwere combined on the shear/biaxial buckling and vibration of rectangular nanoplates. Their results showed that by increasing the non-local parameter, the effects of surface on the buckling and vibration increased. Shooshtari andRazavi (2015) studied nonlinear free vibration behavior of a symmetrically laminatedMEE doubly-curved thin shell resting on an elastic foundation. By introducing a force function and using the Galerkin method, the nonlinear partial differential equations of motion were reduced to a single nonlinear ordinary differential equation. That equation was solved analytically by the Lindstedt-Poincaré perturbationmethod. Their results showed that the shear constant coef- ficient of the foundation hadmuch greater effect on the natural frequency when compared with the spring constant coefficient, and both of those coefficients increased the fundamental natural frequency.Ebrahimi andNasirzadeh (2016) analyzed free vibration of thick nanobeamsbased on Eringen nonlocal elasticity theory andTimoshenko beam theory. Chen et al. (2014) studied free vibration ofmultilayeredMEEplates under combined clamped/free lateral boundaryconditions. Using semi-analytical solution, they obtained the natural frequency. Their results illustrated the effect of stacking sequences andmagneto-electric coupling on natural frequencies andmode sha- pes. Karimi et al. (2015c) analyzed size-dependent free vibration characteristics of rectangular nanoplates considering surface stress effects. Numerical results demonstrated that the obtained natural frequency by considering the surface effects was lower than that without considering the surface properties. Razavi and Shooshtari (2015) employed nonlinear free vibration of sym- metric MEE laminated rectangular plates with simply supported boundary conditions. Their results for the nonlinear natural frequency ratio were compared with the available results for isotropic, laminated layers and piezo-layers and laminated MEE plates. Their results depicted that the foundation parameters, negative electric potential and positive magnetic potential in- creased the equivalent stiffness of the system. Using Bert’s model, Khan et al. (2014) studied free and forced vibration characteristics of bimodular composite laminated circular cylindrical shells. The results indicated that the relative difference of positive and negative half cycle frequ- encies was considerably less for single layer orthotropic shells, and it was significant for cross-ply shells with the axisymmetric mode of vibration. Du et al. (2014) illustrated nonlinear forced vibration analysis of infinitely long functionally graded cylindrical shells using the Lagrangian theory and the multiple scale method. Their results found that the power-law exponent had not any influence on the qualitative behavior of FG cylindrical shells, but it would change the amplitude in a complex nonlinear way. Hasani Baferani et al. (2011) presented free vibration analysis of FG thick rectangular plates resting on an elastic foundation. They obtained gover- ning equations of motion using the third order shear deformation plate theory and Hamilton’s Bending, buckling, and forced vibration analyses of nonlocal... 855 principle. Their results showed that the Pasternak elastic foundation drastically changed the natural frequency. Also some boundary conditions and in-plane displacements had significant effects on the natural frequency of FG thick plates. Arefi (2015) analyzed free vibration of a FG solid and annular circular plates with two functionally graded piezoelectric layers at the top and bottom subjected to an electric field. Sobhy (2013) investigated buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. The governing equations of plates were derived by using various shear deformation plate theories. They showed influence of the inhomogeneity parameter, aspect ratio, thickness ratio and foundation parameters on natural frequencies and critical buckling loads. Zidour et al. (2014) illustrated buckling of chiral single-walled carbon nanotubes by using the nonlocal Timoshenko beam theory. Their results showed influence of a nonlocal small-scale coefficient and the vibration mode number on the nonlocal critical buckling loads. Karimi et al. (2015b) studied influence of the nonlocal parameter, van der Waals, Winkler, shear modulus on shear vibration and buckling of double-layer orthotropic nanoplates resting on an elastic foundation. In this article, bending, buckling, free and forced vibration of a magneto-electro-elastic (MEE) microplate based on the third order shear deformation theory (TSDT) is presented. According to Maxwell’s equations and Hamilton’s principle, the governing differential equation is obtained. These equations discretized by using Navier’s method for a MEE microplate with all edges simply supported boundary enabled determination of the deflection, critical buckling load, natural frequency, response of the system as well as the electric and magnetic intensity of the microplate. The numerical results show the influence of elastic foundation parameters, aspect ratio l/b, length to thickness ratio l/h, volume fraction, normal pressure on the deflection, critical buckling load, natural frequency, response of the system and the electric and magnetic intensity. 2. Nonlocal theory of the MEE The non-local modulus of elasticity was presented by Eringen (1983). This model states that the stress of a point in the micro and nano dimension is dependent on the strain in all parts of themodel. The fundamental equations of a homogeneous and isotropic non-local elastic solid are given by Eringen (2002) σnlij (x)= ∫ V α(|x−x′|,τ)σ′ij dV (x ′) ∀x ∈ V (2.1) For the MEE solid, the nonlocal fundamental equations for magnetic induction and electric displacement can be obtained as follows Dnlij (x)= ∫ V α(|x−x′|,τ)D′ij dV (x ′) ∀x ∈ V Bnlij (x)= ∫ V α(|x−x′|,τ)B′ij dV (x ′) ∀x∈ V (2.2) where σnlij , σ ′ ij, D nl ij , D ′ ij, B nl ij and B ′ ij are the nonlocal and local stress tensor, components of the nonlocal and local electric displacements, components of the nonlocal and local magnetic inductions, respectively. α(|x−x′|,τ) is the nonlocalmodulus, |x−x′| is the Euclidean distance, τ = e0a/l is defined as the small scale parameter. According to Eringen (1983, 2002), the nonlocal elasticity theory can be simplified to partial differential equations. Thus we have [1−(e0a) 2∇2]σnlij = σ ′ ij [1−(e0a) 2∇2]Dnlij = D ′ ij [1−(e0a) 2∇2]Bnlij = B ′ ij (2.3) 856 M.Mohammadimehr, R. Rostami 3. Constitutive equations of the MEE nanocomposite microplate Consider anMEE nanocomposite microplate with length l, width b and thickness h, resting on an elastic foundation as shown in Fig. 1. A Cartesian coordinate system (x,y,z) is considered such that the z direction denotes thickness of the nanocomposite microplate. Fig. 1. Schematic of anMEE nanocomposite microplate on the elastic foundation Based on the third-order shear deformation theory (TSDT) for a nanocomposite plate, the displacements of an arbitrary point in the beam along the x, y and z axes are denoted by u1(x,y,z,t), u2(x,y,z,t) and u3(x,y,z,t), respectively. They are written as follows u1(x,y,z,t) = u(x,y,t)+z [ ψx(x,y,t)− 4 3 (z h )2 [ψx(x,y,t)+w(x,y,t),x] ] u2(x,y,z,t) = v(x,y,t)+z [ ψy(x,y,t)− 4 3 (z h )2 [ψy(x,y,t)+w(x,y,t),y] ] u3(x,y,z,t) = w(x,y,t) (3.1) whereu,v,w are themid-planedisplacements of theMEErectangular nanocompositemicroplate along the (x,y,z) coordinate directions, respectively, ψx, ψy denote rotations of the plate cross- section and t is time. The linear constitutive equations for theMEEnanocompositemicroplate in the plane stress state are expressed in the following form (Mohammadimehr et al., 2016a,b, 2017; Ghorbanpour Arani et al., 2016)              σ11 σ22 τ12 τ13 τ23              = 1 1− (eoa) 2∇2               C11 C12 0 0 0 C12 C22 0 0 0 0 0 C44 0 0 0 0 0 C44 0 0 0 0 0 C55                     ε11 ε22 γ12 γ13 γ23              −        0 0 e31 0 0 e31 0 e24 0 e15 0 0 0 0 0             Ex Ey Ez      −        0 0 f31 0 0 f31 0 f24 0 f15 0 0 0 0 0             Hx Hy Hz             Bending, buckling, and forced vibration analyses of nonlocal... 857      Dx Dy Dz      = 1 1− (eoa)2∇2           0 0 0 e15 0 0 0 e24 0 0 e31 e31 0 0 0                 ε11 ε22 γ12 γ13 γ23              +    h11 0 0 0 h22 0 0 0 h33         Ex Ey Ez      +    g11 0 0 0 g22 0 0 0 g33         Hx Hy Hz         (3.2)      Bx By Bz      = 1 1− (eoa)2∇2           0 0 0 f15 0 0 0 f24 0 0 f31 f31 0 0 0                 ε11 ε22 γ12 γ13 γ23              +    g11 0 0 0 g22 0 0 0 g33         Ex Ey Ez      +    µ11 0 0 0 µ22 0 0 0 µ33         Hx Hy Hz         where σ11, σ22 and ε11, ε22 are the normal stresses and strains, respectively. τ12, τ13, τ23 and γ12, γ13, γ23 denote the shear stresses and strains, respectively. Cij, eij, fij and gij denote elastic, piezoelectric, piezomagnetic and magnetoelectric constants, respectively; hij and µij are dielectric and magnetic permeability coefficients, respectively. Eij and Hij are the electric magnetic field intensity, respectively. The electric and magnetic fields are considered in terms of electric andmagnetic potentials φ and ϕ, respectively, which are defined as follows Ei =−φ,i Hi =−ϕ,i i =1,2,3 (3.3) 4. The governing equations of motion for the MEE nanocomposite microplate The governing differential equations of motion for the MEE nanocomposite microplate are de- rived using Hamilton’s principle which is given by (Mohammadimehr andMostafavifar, 2016) t ∫ 0 (δT −δU − δW) dt =0 (4.1) where δT , δU and δW are the variations of kinetic energy and strain energy, the work done by external applied forces, respectively. Variations of the kinetic energy for a sandwich plate can be described as follows (Ghorban- pour and Haghparast, 2017) δT = ∫ V ρi ∂ui ∂t δ (∂ui ∂t ) dV = ∫ A h 2 ∫ −h 2 ρi(u̇1δu̇1+ u̇2δu̇2+ u̇3δu̇3) dz dA (4.2) where Ii = h ∫ −h ρzi dz (i =1,2,3,4,6) C1 = 4 3h2 Variations of the strain energy for theMEE nanocomposite microplate can be expressed as 858 M.Mohammadimehr, R. Rostami δU = ∫ V (σijδεij −DiδEi−BiHi) dV = ∫ V [(σ11δε11+σ22δε22+σ33δε33+ τ12δγ12+ τ13δγ13+τ23δγ23) − (DxδEx+DyδEy +DyδEy)− (BxδHx+ByδHy +ByδHy)] dV (4.3) Variations of the work can be considered as follows δW =− ∫ P(x,y)δw dx+ ∫ (kww−kG∇ 2w)δw dx (4.4) where Kw and KG are the transverse and shear coefficients of elastic medium, respectively. By substituting Eqs. (4.2)-(4.4) into Eq. (4.1), the equilibrium equations of theMEE nano- composite microplate resting on an elastic foundation can be obtained in the following form δu : N1,x+N6,y = I0ü+ I1ψ̈x−C1I3 ( ψ̈x+ ∂ẅ ∂x ) δv : N2,y+N6,x = I0v̈+ I1ψ̈y−C1I3 ( ψ̈y+ ∂ẅ ∂y ) δψx : M1,x+M6,y −Q1− 4 3h2 (P1,x+P6,y)+ 4λ h2 R1 = I1ü+ I2ψ̈x−C1 ( I3ü+2I4ψ̈x+ I4 ∂ẅ ∂x ) +C21I6 ( ψ̈x+ ∂ẅ ∂x ) δψy : M2,y +M6,x−Q2− 4 3h2 (P2,y +P6,x)+ 4 h2 R2 = I1v̈+ I2ψ̈y −C1 ( I3v̈+2I4ψ̈y + I4 ∂ẅ ∂y ) +C21I6 ( ψ̈x+ ∂ẅ ∂y ) δw : Q1,x+Q2,y+ 4 3h2 (P1,xx+P2,yy +2P6,xy)− 4 h2 (R2,y +R1,x) +(−kww+kG∇ 2w)+P(x,y)= C1I3 (∂ü ∂x + ∂v̈ ∂y ) +C1I4 (∂ψ̈x ∂x + ∂ψ̈y ∂y ) +C21I6 (∂ψ̈x ∂x + ∂ψ̈y ∂y − ∂2ẅ ∂x2 − ∂2ẅ ∂y2 ) +I0ẅ (4.5) and ∂Dz ∂z =0 ∂Bz ∂z =0 (4.6) whereNi,Mi (i =1,2,6) denote the resultant forces andmoments, respectively.Ri,Pi arehigher order resultant shear forces andmoments, respectively, and Qi are transverse shear forces which are all defined by the following expressions         N1 N2 N6      ,      M1 M2 M6      ,      P1 P2 P6         = h 2 ∫ −h 2      σ11 σ22 τ12      (1,z,z3) dz ({ Q1 Q2 } , { R1 R2 }) = h 2 ∫ −h 2 { τ13 τ23 } (1,z2) dz (4.7) By substitutingEqs. (3.3) intoEqs. (4.6), the electric andmagnetic potential are obtainedwhich electric andmagnetic boundary conditions assumed as follows Bending, buckling, and forced vibration analyses of nonlocal... 859 φ (h 2 ) = φ ( − h 2 ) =0 φ = λ1∆1 (z2 2 − z4 3h2 − 23h2 192 ) +λ1∆2 ( h2 192 − z4 3h2 ) ϕ (h 2 ) =ϕ ( − h 2 ) =0 ϕ = λ2∆1 (z2 2 − z4 3h2 − 23h2 192 ) +λ2∆2 ( h2 192 − z4 3h2 ) (4.8) where λ1 = e31− g33f31 µ33 h33− g2 33 µ33 λ2 = f31−g33λ1 µ33 ∆1 = ψx,x+ψy,y ∆2 = w,xx+w,yy By substituting Eqs. (4.7) into Eqs. (4.5) and (4.6), the governing equations of motion for the MEE nanocomposite microplate based on TSDT are obtained as follows δu : A11u,xx+(A12+A66)v,xy +A66u,yy = I0ü−e 2 0a 2I0(ü,xx+ ü,yy)+(I1−C1I3)ψ̈x −(I1−C1I3)e 2 0a 2(ψ̈x,xx+ ψ̈x,yy)−C1I3ẅ,x+C1I3e 2 0a 2(ẅ,xxx+ ẅ,xyy) (4.9) δv : A22v,yy+(A12+A66)u,xy+A66v,xx = I0v̈−e 2 0a 2I0(v̈,xx+ v̈,yy)+(I1−C1I3)ψ̈y −(I1−C1I3)e 2 0a 2(ψ̈y,xx+ ψ̈y,yy)−C1I3ẅ,y +C1I3e 2 0a 2(ẅ,yxx+ ẅ,yyy) (4.10) δψx : ( B11− 4H11 3h2 ) ψx,xx+ ( F11− 4L11 3h2 ) ψx,yy + (4T22 3h2 −T11 ) ψx + ( F11+B12− 4H12 3h2 − 4L11 3h2 ) ψy,xy+ (4K11 3h2 −D11 ) w,xxx + (4L12 3h2 + 4K12 3h2 −D12−F12 ) w,xyy + (4T22 3h2 −T11 ) w,x =(I1−C1I3)ü− (I1−C1I3)e 2 0a 2(ü,xx+ ü,yy)+(I2−2C1I4+C 2 1I6)ψ̈x +(2C1I4+C 2 1I6− I2)e 2 0a 2(ψ̈x,xx+ ψ̈x,yy)+(C 2 1I6−C1I4)ẅ,x +(C1I4−C 2 1I6)e 2 0a 2ẅ,xxx+(C1I4−C 2 1I6)e 2 0a 2ẅ,xyy (4.11) δψy : ( B22− 4H22 3h2 ) ψy,yy + ( F11− 4L11 3h2 ) ψy,xx+ (4T22 3h2 −T11 ) ψy + ( F11+B12− 4H12 3h2 − 4L11 3h2 ) ψx,xy+ (4K22 3h2 −D11 ) w,yyy + (4L12 3h2 + 4K12 3h2 −D12−F12 ) w,xxy+ (4T22 3h2 −T11 ) w,y =(I1−C1I3)v̈− (I1−C1I3)e 2 0a 2(v̈,xx+ v̈,yy)+(I2−2C1I4+C 2 1I6)ψ̈y +(2C1I4+C 2 1I6− I2)e 2 0a 2(ψ̈y,xx+ ψ̈y,yy)+(C 2 1I6−C1I4)ẅ,y +(C1I4−C 2 1I6)e 2 0a 2ẅ,yxx+(C1I4−C 2 1I6)e 2 0a 2ẅ,yyy (4.12) δw : ( T11− 4T22 h2 ) ψx,x+ ( T11− 4T22 h2 ) ψy,y + 4H11 3h2 ψx,xxx+ 4H22 3h2 ψy,yyy + (4H12 3h2 + 8L11 3h2 ) (ψx,xyy+ψy,xxy)+ ( T11− 4T22 h2 +e20a 2Kw−KG ) (w,xx+w,yy) + ( e20a 2KG− 4K11 3h2 ) w,xxxx+ ( e20a 2KG− 4K22 3h2 ) w,yyyy − (8K12 3h2 + 8L12 3h2 ) w,xxyy −Kww+P(x,y) =C1I3ü,x−C1I3e 2 0a 2(ü,xxx+ ü,xyy)+C1I3v̈,x −C1I3e 2 0a 2(v̈,xxx+ v̈,xyy)+(C1I4+C 2 1I6)ψ̈x,x− (C1I4+C 2 1I6)e 2 0a 2(ψ̈x,xxx+ ψ̈x,xyy) +(C1I4+C 2 1I6)ψ̈y,y − (C1I4+C 2 1I6)e 2 0a 2(ψ̈y,yxx+ ψ̈y,yyy)− (C 2 1I6+e 2 0a 2)ẅ,xx −(C21I6+e 2 0a 2)ẅ,yy +e 2 0a 2C1I6(ẅ,xxxx+ ẅ,yyyy)+2e 2 0a 2C1I6ẅ,xxyy+ I0ẅ (4.13) 860 M.Mohammadimehr, R. Rostami where the above coefficients are defined in Appendix A. Substituting Eqs. (4.8) into Eq. (3.3), the electric andmagnetic field is written as Ez = λ1 ( z− 4z3 3h2 ) (ψx,x+ψx,x)− 4z3 3h2 λ1(w,xx+w,yy) Hz = λ2 ( z − 4z3 3h2 ) (ψx,x+ψx,x)− 4z3 3h2 λ2(w,xx+w,yy) (4.14) 5. Navier’s type solution for the MEE nanocomposite microplate Analytical solutions for a simply supported rectangular MEE nanocomposite microplate are obtained using Navier’s solution technique. Using Navier’s solution, the displacements of the microplate can be written as follows (Mohammadimehr et al., 2016a) u(x,y,t) = ∞ ∑ m=1 ∞ ∑ n=1 Umncos(αx)sin(βy)e iωt v(x,y,t) = ∞ ∑ m=1 ∞ ∑ n=1 Vmn sin(αx)cos(βy)e iωt ψx(x,y,t)= ∞ ∑ m=1 ∞ ∑ n=1 Ψxmncos(αx)sin(βy)e iωt ψy(x,y,t) = ∞ ∑ m=1 ∞ ∑ n=1 Ψymnsin(αx)cos(βy)e iωt w(x,y,t) = ∞ ∑ m=1 ∞ ∑ n=1 Wmn sin(αx)sin(βy)e iωt (5.1) where α and β are equal to mπ/l, nπ/b, respectively. 5.1. Free vibration analysis of the nanocomposite microplate Thematrix form of free vibration equations of the microplate is written as (S−ω2M)U=0 (5.2) where the non-zero elements of the mass and stiffness matrix are given in Appendix B. 5.2. Buckling analysis of the nanocomposite microplate Thematrix form of buckling equations for the nanocomposite microplate can be written as follows        S11 S12 S13 S14 S15 S21 S22 S23 S24 S25 S31 S32 S33 S34 S35 S41 S42 S43 S44 S45 S51 S52 S53 S54 S55−N0(α 2+kβ2)                     Umn Vmn Ψxmn Ψymn Wmn              =              0 0 0 0 0              C=      S11 S12 S13 S14 S21 S22 S23 S24 S31 S32 S33 S34 S41 S42 S43 S44      −1 k = Nxx Nyy (5.3) Bending, buckling, and forced vibration analyses of nonlocal... 861 Using Eq. (5.3), we obtain an expression for the critical buckling load N0 of the MEE nanocomposite microplate N0 = 1 α2+kβ2 ( S55− { S51 S52 S53 S54 } C { S15 S25 S35 S45 }T ) (5.4) 5.3. Forced vibration of the nanocomposite microplate The load P(x,y,t) can be exoressed in the form of series p(x,y,t)= ∞ ∑ m=1 ∞ ∑ n=1 P0 sin(Ωt)sin(αx)sin(βy) (5.5) whereΩ is the frequencyof forcedvibration.Theequationofmotion for theMEEnanocomposite microplate will then include a variable, time-dependent, transverse load p(x,y,t). The matrix form of the response system equations for the MEE microplate is obtained as follows { Um Vm Ψxmn Ψymn Wmn }T = 1 ω2n−Ω 2 M −1 [ 0 0 0 0 P0 ]T (5.6) 5.4. Dimensionless parameter of the nanocomposite microplate Thedimensionlessdeflection,natural frequencyandbuckling loadof theMEEnanocomposite microplate is written as follows W = Cijmaxh 3w P0l 4 ω = √ ρl4ω Cijmaxh 2 N = l2N0 Cijmaxh 3 (5.7) 6. Numerical results and discussions The piezoelectric and piezomagentic properties of the BaTiO3 (inclusion)-CoFe2O4 (matrix) nanocomposite microplate with different volume fractions Vf of the inclusions can be found in Sih and Sog (2002), Song and Sih (2002). They are listed in Table 1. Numerical results for bending, buckling, free and forced vibration are presented for theMEE nanocomposite microplate resting on a two-parameter elastic foundations with all edges simply supported. To validate the results of this research with the literature, a single-layered MEE square thick plate, with l = b = 1m, h = 0.3m, simply-supported boundary conditions, and material properties given byTable 2 is considered.Thedimensionless fundamental frequency is calculated as ω = √ ρmax/Cijmaxlω, where Cijmax and ρmax are the maximum values of the stiffness coefficient and density of the layers, respectively. The results are shown in Table 2 along with some other published results. Table 3 indicates the dimensionless biaxial buckling load of simply-supported square nano- plates. From this Table, it is observed that the presented results are in good agreement with those reported in the literature. Table 4 presents the dimensionless center deflections of isotropic square plates under uni- form loading. They are calculated with various side-to-thickness ratios up to a/h =10000, and compared to earlier studies. The natural frequencies of the simply supportedMEE nanocomposite microplate are obta- ined using Eq. (5.2). From Fig. 2a, it is seen that the volume fraction plays an important role for the MEE nanocomposite microplate in terms of the natural frequency, and its effects can 862 M.Mohammadimehr, R. Rostami Table 1.Properties of the BaTiO3, CoFe2O4 and BaTiO3-CoFe2O4 nanocomposite microplate with different volume fractions Piezoelectric (BaTiO3) Piezomagnetic (CoFe2O4) Vf (volume fraction for CoFe2O4 in Properties BaTiO3-CoFe2O4 nanocomposite) 0.1 0.3 0.5 0.7 0.9 C11 [Gpa] 166 286 178.0 202 226 250.0 274 C12 [Gpa] 77 173 87.2 105.7 124 142.7 161 C22 [Gpa] 166 286 172.8 194.2 216 237.3 259 C44 [Gpa] 43 45.3 43.2 43.7 44 44.6 45 e31 [c/m 2] 43 45.3 −3.96 −3.08 −2.2 −1.32 −4.4 e33 [c/m 2] 44.5 56.5 16.74 13.02 9.3 5.58 1.86 e15 [c/m 2] −4.4 0 10.44 8.12 5.8 3.48 1.16 h11 [×10 −10C2/(Nm2)] 0 580.3 100.9 78.6 56.4 34.2 11.9 h33 [×10 −10C2/(Nm2)] −4.4 0 113.5 88.5 63.5 38.5 13.4 f31 [N/(Am)] 0 580.3 58.03 174.1 290.2 406.2 522.3 f33 [N/(Am)] 11.6 0 69.97 209.9 350.0 489.8 629.7 f15 [N/(Am)] 0 550 55.00 165.0 275.0 385.0 495.0 µ11 [×10 −6NS2/C2] 11.6 0 63.5 180.5 297.0 414.5 531.5 µ33 [×10 −6NS2/C2] 0 550 24.7 541.0 83.5 112.9 142.3 ρ [kg/m3] 126 0.93 5750 5650 5550 5450 5350 Table 2.Dimensionless fundamental frequencies of MEE plates Material Method Piezoelectric Piezomagnetic BaTiO3 CoFe2O4 Wu and Lu (2009) 1.2523 1.0212 Shooshtari and Razavi (2015) 1.2426 1.1023 Present study 1.2952 1.1130 Table 3. Comparison of dimensionless biaxial buckling load (Ncr = N0a/D, D = Eh 3/[12(1− υ2)] for square nanoplates with all edges simply-supported (a =10nm, a/h =2) Method e0a [nm] 0 1 Malekzadeh and Shojaee (2013) 8.5249 7.1039 Wang andWang (2011) 8.4543 7.1533 Karimi et al. (2015) 8.6052 7.2204 Present study 8.5232 7.1138 not be ignored for microplate. It is shown that by increasing the volume fraction, the dimen- sionless natural frequency increases. The reason is that a greater volume fraction makes the microplate stiffer. Figure 2b depicts the effects of the Pasternak shear constant on the natural frequency. From this figure, it can be found that by increasing this parameter, the stiffness of the nanocomposite microplate increases and this result is similar to the dimensionless natural frequency. The effect of volume fraction on the deflection is shown inFig. 3a. It is shown that an increase in the volume fraction will decrease the dimensionless deflection. The critical buckling loads of theMEEnanocompositemicroplate are obtained using Eq. (5.4). Figure 3b depicts the variation of critical buckling load versus volume fraction. From this figure, it can be seen that with an increase in the volume fraction, the critical buckling load for all the length to width Bending, buckling, and forced vibration analyses of nonlocal... 863 Table 4. Comparison of dimensionless center deflection W ( a 2 , b 2 ) D/[(P0a 4), D = Eh3/[12(1− υ2)] for simply-supported square isotropic plates under uniform loads Method a/h 10 100 1000 10000 Nguyen et al. (2016) 0.4272 0.4064 0.4062 0.4062 Nguyen-Xuan et al. (2008), MITC4 0.4273 0.4064 0.4062 0.4062 Nguyen-Xuan et al. (2008), MISC1 0.4273 0.4065 0.4063 0.4063 Taylor and Auricchio (1993) 0.4273 0.4064 0.4062 0.4062 Present study 0.4266 0.4055 0.4053 0.4053 Fig. 2. (a) The effect of volume fraction on the dimensionless natural frequency: l =4µm, b =4µm, h =0.04µm, Kw =0, KG =0. (b) The effect of the Pasternak shear constant on the dimensionless natural frequency: l =400µm, h =80µm, Vf =0.5, Kw =0, e0a =1nm Fig. 3. (a) The effect of volume fraction on the dimensionless deflection: l =400µm, b =400µm, h =80µm, Kw =0, KG =0, P =100N/m 2, e0a =2nm. (b) The effect of volume fraction on the dimensionless critical buckling load: l =400µm, h =80µm, Kw =0, KG =0, e0a =1nm ratios l/b will increase. The influence of the length to thickness ratio l/h is shown in Fig. 4a. This figure shows that by increasing the length to thickness ratio l/h, the dimensionless critical buckling load decreases. The response systemof theMEEnanocompositemicroplate is obtained using Eq. (5.6). Figure 4b indicates the response system of theMEE nanocomposite microplate and different values of the volume fraction. It is seen from the results that by increasing the excitation frequency to the natural frequency ratio Ω/ω, the amplitude of the nanocomposite 864 M.Mohammadimehr, R. Rostami microplate reinforced by CoFe2O4-BaTiO3 increases. Also, by increasing the volume fraction, the deflection to thickness ratio w/h decreases. Figure 5 depicts the effects of volume fraction on themaximumdeflection to thickness ratio wmax/h. From this figure, it can be found that by increasing the volume fraction, the maximum deflection to thickness ratio wmax/h decreases. Fig. 4. (a) The effect of the length to thickness ratio l/h on the critical buckling load: l =400µm, Vf =0.5, Kw =0, KG =0, e0a =1nm. (b) The effect of volume fraction on the response system: l =400µm, b =400µm, h=20µm, Kw =0, KG =0, P0 =1N/m 2, e0a =1nm Fig. 5. The effect of volume fraction on the maximum deflection to thickness ratio Wmax/h: l =4µm, b =4µm, h =0.1µm, Kw =0, KG =0, P0 =1N/m 2, e0a =1nm Figure 6a shows that by increasing the spring constant of the Winkler type, the intensity of electric field decreases. Figure 6b presents the influence of the Pasternak shear constant on the magnetic field, respectively. The results show that by increasing the elastic constant, the intensity of magnetic and electric field decreases. 7. Conclusions A theoretical analysis on bending, buckling, free and forced vibration characteristics of anMEE nanocomposite microplate are carried out in the present work. The Hamilton principle, higher order shear deformation theory andMaxwell’s equations are considered to derive the equations of motion and distribution of electrical potential, magnetic field along the thickness direction of theMEE nanocompositemicroplate. Some conclusions of this research can be listed as follows: • For theMEE nanocomposite microplate, the natural frequency and critical buckling load increases with the increasing volume fraction of CoFe2O4O4-BaTiO3, because the nano- composite microplate becomes stiffer in such a case. Bending, buckling, and forced vibration analyses of nonlocal... 865 Fig. 6. (a) The effect of theWinkler spring constant on the intensity of electric field: l =40µm, b =40µm, h =1µm, KG =0, P =1N/m 2, e0a =1nm. (b) The effect of the Pasternak shear constant on the intensity of magnetic field : l =40µm, b =40µm, h =1µm, Kw =0, P =1N/m 2, e0a =1nm • The natural frequency and critical buckling load decreases, and also themaximum deflec- tion, whereas the intensity of magnetic and electric fields increases with the decreasing Winkler and Pasternak shear constants of theMEE nanocomposite microplate. • For the MEE nanocomposite microplate, the amplitude of vibration decreases with the increasing volume fraction. Appendix A A11 = C11h A12 = C12h A22 =C22h A66 = C66h B11 = h3 15 (C11+e31λ1+f31λ2) B12 = h3 15 (C12+e31λ1+f31λ2) B22 = h3 15 (C22+e31λ1+f31λ2) D11 = h3 60 (C11+e31λ1+f31λ2) D12 = h3 60 (C12+e31λ1+f31λ2) D22 = h3 60 (C22+e31λ1+f31λ2) F11 = C66 h3 15 F12 = C66 h3 30 H11 = (h5 80 − h5 336 ) (C11+e31λ1+f31λ2) H12 = (h5 80 − h5 336 ) (C12+e31λ1+f31λ2) H22 = (h5 80 − h5 336 ) (C22+e31λ1+f31λ2) K11 = h5 336 (C11+e31λ1+f31λ2) K12 = h5 336 (C12+e31λ1+f31λ2) K22 = h5 336 (C22+e31λ1+f31λ2) L11 = (h5 80 − h5 336 ) C66 L12 = h5 168 C66 T11 = 2h 3 C44 T22 = (h3 12 − h3 20 ) C44 Appendix B S11 =−A11α 2−A66β 2 S12 =−(A12+A66)αβ S21 =−(A12+A66)αβ S22 =−A22β 2−A66α 2 S33 = ( B11− 4H11 3h2 ) (−α2)− ( F11− 4L11 3h2 ) β2+ 4T22 3h2 −T11 866 M.Mohammadimehr, R. Rostami S34 = ( B12+F11− 4H12 3h2 − 4L11 3h2 ) (−αβ) S35 = (4K11 3h2 −D11 ) (−α3)− (4L12 3h2 + 4K12 3h2 −D12−F12 ) αβ2+ (4T22 3h2 −T11 ) α S43 =− ( B12+F11− 4H12 3h2 − 4L11 3h2 ) αβ S44 =− ( B22− 4H22 3h2 ) β2− ( F11− 4L11 3h2 ) α2+ 4T22 3h2 −T11 S45 =− (4K22 3h2 −D22 ) β3− (4L12 3h2 + 4K12 3h2 −D12−F12 ) α2β + (4T22 3h2 −T11 ) β S53 = ( T11− 4T22 h2 ) (−α)− 4H11 3h2 α3+ (4H12 3h2 + 8L11 3h2 ) αβ2 S54 = ( T11− 4T22 h2 ) (−β)− 4H22 3h2 β3+ (4H12 3h2 + 8L11 3h2 ) α2β S55 =− ( T11− 4T22 h2 +e20a 2Kw−KG ) (α2+β2)+ ( e20a 2KG− 4K11 3h2 ) α4 + ( e20a 2KG− 4K22 3h2 ) β4− (8K12 3h2 + 8L12 3h2 ) α2β2−Kw m11 =−I0−e 2 0a 2I0(α 2+β2) m13 =(−I1−C1I3)−e 2 0a 2(I1−C1I3)(α 2+β2) m15 = C1I3α+C1I3e 2 0a 2(α3+αβ2) m22 =−I0− I0e 2 0a 2(α2+β2) m24 =−(I1−C1I3)−e 2 0a 2(I1−C1I3)(α 2+β2) m25 = C1I3β +e 2 0a 2C1I3(α 2β +β3) m31 =−(I1−C1I3)+(C1I3− I1)e 2 0a 2(α2+β2) m33 =−(I2−2C1I4+C 2 1I6)+(2C1I4−C 2 1I6− I2)e 2 0a 2(α2+β2) m35 =−(C 2 1I6−C1I4)α+(C1I4−C 2 1I6)e 2 0a 2(α3+αβ2) m42 =−(I1−C1I3)+(C1I3− I1)e 2 0a 2(α2+β2) m44 =−(I2−2C1I4+C 2 1I6)+(2C1I4−C 2 1I6− I2)e 2 0a 2(β2+α2) m45 =−(C 2 1I6−C1I4)β +(C1I4−C 2 1I6)e 2 0a 2(β3+α2β) m51 = C1I3α+C1I3e 2 0a 2(α3+αβ2) m52 = C1I3β +C1I3e 2 0a 2(β3+α2β) m53 =(C1I4+C 2 1I6)α+(C1I4+C 2 1I6)e 2 0a 2α3+(C1I4+C 2 1I6)e 2 0a 2αβ2 m54 =(C1I4+C 2 1I6)β +(C1I4+C 2 1I6)e 2 0a 2β3+(C1I4+C 2 1I6)e 2 0a 2α2β m55 =−(C 2 1I6+e 2 0a 2)(α2+β2)−C21I6e 2 0a 2(α4+β4)−2C21I6e 2 0a 2α2β2− I0 Acknowledgments The authors would like to thank the referees for their valuable comments. 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