On the Buckling the Behavior of a Multiphase Smart Plate based on a Higher-order Theory

On the Buckling the Behavior of a Multiphase Smart Plate based on a Higher-order Theory

S. Razavi

Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran

1. Introduction

Magneto-electro-elastic (MEE) materials are a type of smart materials that exhibit coupling among mechanical, electric and magnetic fields, which makes them suitable to be used in sensors and actuators, to control vibrations and to harvest energy, etc.

Various researchers have extensively studied the buckling of isotropic and composite plates. Brunelle [1] studied the buckling of transversely isotropic rectangular plates based on Mindlin’s plate theory. Mizusawa [2] and Dawe and Wang [3] used spline strip methods to study the buckling behavior of rectangular plates. Various researchers have used the differential quadrature (DQ) method to investigate the buckling of rectangular plates [4-7]. Wang et al. [8] analyzed the buckling of a thin rectangular plate with nonlinearly distributed loadings along two opposite edges by using the DQ method. Cui et al. [9] investigated the dynamic buckling of imperfect rectangular plates subjected to impact loads. Xiang and Wang [10] determined the exact buckling loads and vibration frequencies of stepped plates by using the classical plate theory (CPT) and the Levy method. Chen and Liew [11] and Wang and Peng [12] used mesh-free approaches to analyze the buckling behavior of functionally graded (FG) and thin isotropic rectangular plates, respectively. Javaheri and Eslami [13] and Matsunaga [14] studied the thermal buckling of FG plates based on the higher-order shear deformation theory (HSDT). Some authors have used finite element models (FEMs) to study the buckling of various plates. Selim and Akbarov [15] studied the buckling instability of a clamped viscoelastic plate by using a three-dimensional (3D) FEM. Ghorbanpour Arani et al. [16] used the FEM and analytical method to analyze the buckling of composite plates reinforced by single-walled carbon nanotubes. Many authors have used analytical methods with different plate theories to investigate the buckling behavior of plates. Mohammadi et al. [17] used CPT and the Levy method to analyze the buckling of FG rectangular plates. Kim et al. [18] and Thai and Kim [19] used a two-variable refined theory with the Navier and Levy methods, respectively, to study the buckling of isotropic and orthotropic plates. Bodaghi and Saidi [20] presented an analytical model for analyzing the buckling of FG rectangular plates based on the HSDT and Levy methods. Sinusoidal [21], inverse trigonometric [22], hyperbolic [23], inverse hyperbolic [24], and exponential [25] shear deformation theories have also been used to model the buckling behavior of composite and FG plates. Fares and Zenkour [26] used various theories to study the buckling and free vibration of laminated rectangular plates. They concluded that the CPT is not suitable even for thin laminates. In addition, they found that the first-order shear deformation theory (FSDT) gives acceptable results for properly chosen shear correction factors. Zenkour [27] investigated the buckling behavior of fiber-reinforced viscoelastic composite plates based on CPT, FSDT, HSDT and the sinusoidal shear deformation theory and then compared the results. He showed that despite the CPT-based results, the results of the FSDT and sinusoidal shear deformation theory agree well with those of the HSDT. Ranjbaran et al. [28] and Cetkovic [29] used layerwise theory to study the buckling of sandwich and laminated plates, respectively. Some authors have also investigated the buckling of piezoelectric structures [30, 31].

Pan [32] studied the static response of the MEE plate for the first time. Since then, many researchers have investigated the static and dynamic responses of these smart structures. Free vibration [33], large deflection [34], nonlinear vibration [35] and vibration control [36] of MEE plates have been studied. Xu et al. [37] studied the surface effects on the bending, buckling and free vibration of MEE beams. Li et al. [38] investigated the buckling and free vibration of the MEE nanobeam, based on the nonlocal theory and the Timoshenko beam theory. Li [39] studied the buckling of the MEE plate resting on a Pasternak foundation by using Mindlin’s plate theory. Kumaravel et al. [40] and Lang and Xuewu [41] studied the buckling and free vibration of MEE cylindrical shells. Ansari et al. [42] developed a shear deformable nonlocal model for nonlinear forced vibration analysis of MEE nanobeams. Ebrahimi and Barati [43] presented a size-dependent beam model to study the buckling behavior of curved MEE nanobeams. Jamalpoor et al. [44] investigated the free vibration and buckling of MEE microplates based on modified strain gradient and Kirchhoff plate theories. Farajpour et al. [45] developed a nonlocal model for nonlinear free vibration of MEE nanoplates based on the Kirchhoff plate theory. Xu et al. [46] investigated the bending, buckling and free vibration of MEE beams based on the Euler-Bernoulli beam theory. The FEM [47] and third-order shear deformation theory (TSDT) [48] have also been used to analyze the static and dynamic response of MEE plates. Wenjun et al. [49] introduced a two-dimensional linear theory to analyze the response of MEE plates.

In this paper, the buckling loads of a multiphase MEE rectangular plate with simply supported boundary conditions is investigated based on Reddy’s HSDT. Gauss’s laws for electrostatics and magnetostatics are used to model the electric and magnetic behaviors of the plate. Then, the closed- form expression for the critical buckling load of the plate considered is obtained. Some examples are presented to study the effects of some parameters on the critical buckling load of these smart plates.

1. Problem Modeling

2.1. Basic relations

Consider a multiphase MEE plate with the dimensions of a×b×h that is subjected to biaxial in-plane loads along its edges, as it is shown in Fig. 1 (−1≤ ζ ≤+1). The plate is a composite smart plate made of piezoelectric material as the inclusions and magnetostrictive material as the matrix. Based on Reddy’s HSDT, the displacement field of this plate is as shown below [50]:

where u₀, v₀ and w₀ are the displacements of the midplane along the x, y and z directions, respectively, and θ_x and θ_y are the rotations of a transverse normal about the y and x directions, respectively.

The constitutive equations of a transversely isotropic MEE plate can be expressed as shown below [32]:

where {σ} is the stress vector, {ε} is the strain vector, {D} and{B} are the electric displacement and magnetic flux vectors, respectively. [C], [η] and [μ] are the elastic, dielectric and magnetic permeability coefficient matrices, respectively, and [e], [q] and [d] are the piezoelectric, piezomagnetic and magnetoelectric coefficient matrices, respectively.

Figure 1. Schematic of the MEE plate subjected to in-plane loads.

Based on the HSDT and when transversely isotropic MEE plates with electric and magnetic fields are applied along the z-direction, these vectors and matrices are expressed in the following form [32, 50]:

where ϕ and ψ are the electric and magnetic potentials, respectively.

2.2. Equations of motion

Based on Reddy’s HSDT, the equations of the static motion of the MEE plate are expressed by [50]:

where

And the resultants are determined by:

To obtain the resultants of Eq. (15) and then the parameters of Eq. (14), ϕ_,z and ψ_,z, which are introduced in Eq. (7), must be determined first. To do this, Gauss’s laws for electrostatics and magnetostatics, with the magneto-electric (ME) boundary condition on bottom and upper surfaces of the MEE plate, are used here. Thus, using

one obtains the following expressions for ϕ_,z and ψ_,z:

where

In Eq. (17), ϕ₀ and ψ₀ are the constants of integration and are obtained by using ME boundary condition, which is assumed as below:

where V₀ is the electric potential and Ω₀ is the magnetic potential that are applied to the upper surface of the MEE plate. Eqs. (17) and (20) give ϕ₀ = V₀/h and ψ₀ = Ω₀/h.

Thus, Eqs. (4), (7), (8), (15) and (17) give the resultants:

where the in-plane boundary loads have been included in the in-plane force resultants of Eq. (21).

Substituting Eqs. (14) and (21) – (28) into Eqs. (9) – (13) gives the equations of motion in terms of the displacements and rotations of the midplane of the plate:

where L_i (i=1,2,…,34) are the constant coefficients and are given in Appendix A.

2.3. Determining the critical buckling load

It can be seen that Eqs. (29) and (30) are decoupled from Eqs. (31) – (33). Therefore, to study the buckling behavior of the plate, it is sufficient to consider only Eqs. (31) – (33). For the simply supported boundary condition, the following relations hold:

Therefore, the transverse displacement and the rotations can be determined by:

where W, X, and Y are the amplitudes of the transverse displacement and rotations, and (m,n) denotes the mode of the plate.

Substituting Eq. (35) into Eqs. (31) to (33) and then using the orthogonality of the trigonometric functions, one obtains the following set of ordinary differential equations:

where the components of the coefficient matrix are given in Appendix B. To have a nontrivial solution for Eq. (36), the determinant of the coefficient matrix must equal zero. That is,

Solving Eq. (37) gives the critical buckling load of the multiphase MEE plate.

3. Examples and Discussion

In this section, numerical examples are presented to validate the proposed model and to investigate the effects of some parameters on the critical buckling load of the multiphase MEE plate. As the first comparison, an isotropic square plate is considered and its dimensionless critical buckling loads are determined for different length-to-thickness (a/h) ratios in the uniaxial (ζ = 0) and biaxial (ζ = +1) compressions. The results are shown in Table 1, with the results based on various plate theories, where RPT denotes the refined plate theory. The dimensionless critical buckling loads of orthotropic plates with different a/h and b/a ratios and different degrees of orthotropy (E₁/E₂) are also determined, and the results are shown in Tables 2 and 3. The results are compared with those based on the FSDT, HSDT and exponential shear deformation theory (ESDT), where the error is obtained by using,

The dimensionless critical buckling loads in Tables 1 to 3 are determined by using P^* = P_cra²/E₂h³. The material properties of the orthotropic plate are: E₁/E₂ = open, G₁₂ = G₁₃ = 0.5, E₂, G₂₃ = 0.2 E₂ and ν₁₂ = 0.25. It is seen that the present model accurately predicts the critical buckling loads of isotropic and orthotropic plates. As the last comparison, the dimensionless critical buckling loads of a square MEE plate are determined and compared with those by Li [39], which are based on Mindlin’s plate theory.

The material properties of the multiphase MEE plate are [39, 51]: C₁₁ = 226 GPa, C₁₂ = 124 GPa, C₂₂ = 216 GPa, C₄₄ = C₅₅ = 44 GPa, C₆₆ = 51 GPa, e₃₁ = e₃₂ = −2.2 C/m², e₁₅ = e₂₄ = 0, q₃₁ = q₃₂ = 290.2 N/Am, q₁₅ = q₂₄ = 0, d₃₃ = 2737.5×10⁻¹² Ns/VC, η₃₃ = 6.35×10⁻⁹ C²/Nm² and μ₃₃ = 83.5×10⁻⁶ Ns²/C². The critical buckling loads of the MEE plate are normalized by using P^* = P_cra²/h³C₁₁ and are shown in Table 4. The discrepancy between the results of the present approach and Li’s [39] results are computed by,

.

Table 1. Comparison of the dimensionless critical buckling load of a square isotropic plate (ν = 0.3).

^* reported by Kim et al. [18]

Table 2. Comparison of the dimensionless critical buckling load of square orthotropic plates (ζ = 0).

^* reported by Sayyad and Ghugal [25]

Table 3. Comparison of the dimensionless critical buckling load of rectangular orthotropic plates (ζ = 0, E₁/E₂ = 40, a/h = 5).

^* reported by Sayyad and Ghugal [25]

Table 4. Comparison of the dimensionless critical buckling load of a square MEE plate.

It is seen in Tables 1 to 3 that the FSDT cannot accurately predict the critical buckling loads for thick plates (a/h = 5) compared with the present HSDT-based formulation. Moreover, to obtain accurate results from the FSDT, a proper shear correction factor must be chosen [26], which is still an unresolved issue for composite structures [50].

In Table 5, the dimensionless critical buckling loads of rectangular MEE plates are presented for uniaxial and biaxial compression cases. Table 5 shows that for higher aspect ratios, the dimensionless critical buckling load increases. Moreover, in biaxial compression, the dimensionless critical buckling load is smaller.

Fig. 2 shows the dimensionless critical buckling curves for piezoelectric BaTiO₃ (barium titanate), magnetostrictive CoFe₂O₄ (cobalt ferrite) and MEE square plates. The material properties of BaTiO₃ are [52]: C₁₁ = 166 GPa, C₁₂ = 77 GPa, C₂₂ = 166 GPa, C₄₄ = C₅₅ = 43 GPa, C₆₆ = 44.5 GPa, e₃₁ = e₃₂ = −4.4 C/m², e₁₅ = e₂₄ = 11.6 C/m², η₃₃ = 12.6×10⁻⁹ C²/Nm² and μ₃₃ = 10×10⁻⁶ Ns²/C²; and for CoFe₂O₄, the material properties are: C₁₁ = 286 GPa, C₁₂ = 173 GPa, C₂₂ = 286 GPa, C₄₄ = C₅₅ = 45.3 GPa, C₆₆ = 56.5 GPa, q₃₁ = q₃₂ = 580.3 N/Am, q₁₅ = q₂₄ = 550 N/Am, η₃₃ = 0.093×10⁻⁹ C²/Nm² and μ₃₃ = 157×10⁻⁶ Ns²/C². It is seen that BaTiO₃ has the smallest stiffness coefficient among these smart plates, which is why its dimensionless critical buckling load is smaller for a fixed value of a/h ratio. The dimensionless critical buckling curves of the MEE plate and its equivalent non-magneto-electric (non-MEE) plate are shown in Fig 3. In this figure, non-MEE denotes the MEE plate with e₃₁ = e₃₂ = q₃₁ = q₃₂ = d₃₃ = η₃₃ = μ₃₃ = 0. It is seen that the ME coefficients slightly increase the dimensionless critical buckling load. In addition, the buckling behavior of the MEE plate is dominated by the elastic properties of the plate. Figs 4 and 5 show the effects of the electric and magnetic potentials on the dimensionless critical buckling loads of MEE plates. It is seen that using negative electric potentials or positive magnetic potentials leads to higher dimensionless critical buckling loads for MEE rectangular plates.

Table 5.  The dimensionless critical buckling load of a rectangular MEE plate (a/h = 5).

Figure 2. The dimensionless critical buckling load curves for different smart square plates (ζ = 0).

Figure 3. The dimensionless critical buckling load curves of an MEE and its equivalent non-ME square plates (ζ = 0).

In the last example, an MEE square plate with a fixed a/h ratio is considered with a nonzero magneto-electric boundary condition.

Fig 6 shows the result and shows that for MEE plates with smaller thicknesses, the electric and magnetic potentials have considerable effect on the dimensionless critical buckling load. However, as the plate becomes thicker, the effect of the potentials on dimensionless critical buckling decreases dramatically.

Figure 4. The effect of electric potential on the dimensionless critical buckling load of MEE and BaTiO₃ square plates (a/h = 10, ζ = 0).

Figure 5. The effect of the magnetic potential on the dimensionless critical buckling load of MEE and CoFe₂O₄ square plates (a/h = 10, ζ = 0).

1. Conclusions

The buckling behavior of a multiphase MEE rectangular plate with simply supported boundary conditions was investigated analytically, based on Reddy’s higher-order shear deformation theory, Gauss’s laws for electrostatics and magnetostatics, and the Galerkin method. Numerical examples were presented and it was found that: (a) in biaxial compression, the dimensionless critical buckling load of the MEE plate was smaller, (b) for a fixed value of the a/h ratio, the piezoelectric BaTiO₃ had a smaller dimensionless critical buckling load compared with that of an MEE plate due to its smaller stiffness coefficients, (c) the ME properties of the MEE plate increased the dimensionless critical buckling load of the plate, because the ME effects increased the effective stiffnesses of the MEE plate, and (d) for a fixed value of the a/h ratio, the dimensionless critical buckling loads of MEE plates with smaller thickness values changed considerably with the change in electric or magnetic potentials.

Nomenclature

Appendix A

Appendix B

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