Bending and buckling analysis of functionally graded annular microplate integrated with piezoelectric layers based on layerwise theory using DQM

Accepted Manuscript Bending and buckling analysis of functionally graded annular microplate integrated with piezoelectric layers based on layerwise theory using DQM

Mohammad Hadi Hajmohammad

PII: DOI: Reference:

S1270-9638(17)31344-5 https://doi.org/10.1016/j.ast.2018.05.055 AESCTE 4608

To appear in:

Aerospace Science and Technology

Received date: Revised date: Accepted date:

24 July 2017 5 February 2018 30 May 2018

Please cite this article in press as: M.H. Hajmohammad, Bending and buckling analysis of functionally graded annular microplate integrated with piezoelectric layers based on layerwise theory using DQM, Aerosp. Sci. Technol. (2018), https://doi.org/10.1016/j.ast.2018.05.055

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Bending and buckling analysis of functionally graded annular microplate integrated with piezoelectric layers based on layerwise theory using DQM

Abstract In present paper, bending and buckling analysis of functionally graded material (FGM) annular microplate integrated with piezoelectric layers is investigated. The annular sandwich microplate is subjected to radial compressive and uniform transverse load. Considering the nonlocal elasticity theory for small scale effect is developed. In order to mathematical modeling of annular sandwich microplate, layerwise theory, is employed. Furthermore, the surrounding elastic medium is simulated by Pasternak foundation model, in which both compression and tension are assumed. The material properties of FGM annular microplate are supposed to vary through the thickness according to power law. Using energy method and principle of minimum potential energy, the size dependent governing motion equations are derived. In this study, the governing motion equations are solved numerically using differential quadrature (DQ) method. The present numerical solutions are validated through comparisons against those available in open literature for the reduced cases. Also, different boundary conditions at the edges of the annular sandwich microplate are considered. A parametric study is conducted to examine the effects such as elastic foundation, small scale effect, various boundary conditions, number of grid point, external voltages of piezoelectric layers, outer-inner radius ratio on the critical buckling load, are developed. The results clarify that external voltages are effective parameter on the critical buckling load and the bending behaviour of the system. In addition, the efects of elastic foundation is very remarkable on the buckling and deflection of the annular sandwich plate.

Keywords: Buckling; Bending; Layerwise theory; Anuular sandwich microplate; FGM; Small scale effect; DQ method

*Corresponding author.

1. Introduction The FGM are the materials, those whose thermo-mechanical properties vary smoothly in one or two preferred directions. Because of their incredible features such as good resistance in high temretures and pressures, usage of these materials in recent years, in particular in spacecrafts become widespread. Mirzavand and Eslami [1] presented a closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature dependent properties. The equation of motions are derived using third-order shear deformation theory. Vibration characteristics of FGM circular cylindrical shells using wave propagation approach was analyzed by Sofiyev et al. [2]. An exact solution for thermal buckling of annular FGM plates on an elastic medium was developed by Kiani et al. [3]. Pasternak-type elastic foundation is assumed and equilibrium equations are obtained based on classical plate theory. Thermo-mechanical property of the structure is supposed to be graded across the thickness direction and exact analytical solution is presented to calculate the themal buckling load. The results shown that the boundary conditions on the inner and outer edges and elastic foundation have a great effect on the behaviour of the system. Zhang and Zhou [4] carried out mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations. High-order shear deformation theory is used with different boundary conditions. Governing equations are solved by multi-term Ritz method. This paper indicated that effect of foundation is small at the pre-buckling state. Jodaei et al. [5] investigated three-dimensional free vibration analysis of functionally graded piezoelectric (FGP) annular plates via SSDQM and comparative modeling by ANN. It is concluded that the influence of FGM valume fraction index and boundary conditions decrease for thinner plates. Hosseini-Hashemi [6] presented closed-form vibration analysis of thick annular FG plates with integrated piezoelectric layers. In this paper, various

boundary conditions such as simply supported, hard simply supported and clamped at the inner and outer edges of the annular plate are analyzed. Reddy’s third-order shear deformation theory (TSDT) is utilized and the governing equations are solved analytically. Stress, vibration and buckling analysis of FGM plates was develped by Swaminathan et al. [7]. In another work, thermo elasticity solution of FG, solid, circular and annular plates integrated with piezoelectric layers using the DQM was investigated by Alibeigloo [8]. Different edge conditions are analyzed and material properties are supposed to vary in an exponential law. Hosseini-Hashemi et al. [9] carried out DQ analysis of FG circular and annular sector plates on elastic foundation. Buckling and free vibration of radially FG circular and annular sector thin plates subjected to uniform in-plane compressive loads are investigated. Behravan and Alibeigloo [10] presented semi analytical solution for the static analysis of 2D FG solid And annular circular plates resting on elastic foundation. Governing equations are derived from 3D theory of elasticity and the modulus of elasticity varies continuously throughout the thickness. Sandwich constructions are composed of three layers, the thick core layer which is covered by two stiff and slimer face sheets. In recent decade, sandwich strucures due to the exquisite material properties such as low specific weight, good energy and sound absorption and outstanding bending rigidity are more utilized in many engineering applications such as aerospace, ship and underwater, aircraft and automotive and these constructions have become more attractive in civil and aerospace engineering. Furthermore, many researches has been investigated to recognize the mechanical behaviour of these structures. Moita et al. [11] carried out buckling and geometrically nonlinear analysis of sandwich structures. In this paper finite element model is presented for nonlinear analysis of multilayer and sandwich plates and shells. The core layer and face sheets are modelled using Reddy’s third order shear deformation theory and classic plate theory, respectively. A new

higher order shear and normal deformation theory for static and free vibration analysis of sandwich plates with functionally graded (FG) isotropic face sheets was presented by Bessaim et al. [12]. The theory used in this study, requires no shear correction factors. The boundary condition of the sandwich structure is assumed to be simply supported in all edges and the structure is subjected to a sinusoidally load and the governing equations were derived using Hamilton’s principle. The results deduced that this theory is accurate and efficient in predicting the natural frequency and bending of the structure. Ghorbanpour Arani et al. [13] developed refined Zigzag theory for vibration analysis of viscoelasic functionally graded carbon nanotube reinforced composite (FGCNTRC) microplates integrated with piezoelectric layers. The Kelvin-Voigt model is utilized to present a realistic model and the distributions of single-walled carbon nanotubes (SWCNT) are along the thick direction. Sandwich micro plate is subjected to 2D magnetic and electric fields and that is embedded in an orthotropic visco-Pasternak foundation. The governing equations are developed using Hamilton’s principle and they are solved analitically based on Navier-type solution. The results shown that the effect of valume fraction are noticable on the amount of natural frequency of the structure and electric and magnetic fields are the effective controlling parameters. Zenkour [14] carried out the magneto-thermo-elatic responses of FG annular sandwich disks. This structure is subjected to nonuniform steady-state thermal load and placed in magnetic field. The results clarified that the variations of stresses and strains are very sensitive to the variations of the thermal –load factor. Pandit et al. [15] developed buckling of laminated sandwich plates with soft core based on an improved higher oreder zigzag theory. In analysis of sandwich structures, there is an accurate plate theory namely layerwise theory. Analytical layerwise free vibration analysis of circular/annular composite sandwich plates with auxetic cores was presented by Alipour et al. [16]. In this study, a layerwise theory is employed to

ensure that the values and results are accurate. The equilibrium equations are solved using a Taylor transform and the results are compared with ABAQUS. It is indicated that influence of the core auxeticity is more noticable for plates with more rigid edges. Malekzadeh et al. [17] presented three-dimensional layerwise finite element free vibration analysis of thick laminated annular plates on the elastic foundation. Alipour et al. [18] developed analytical stress analysis of annular FGM sandwich plates with non-uniform shear and normal tractions, employing a zigzag-elasticity plate theory. The governing equations are derived based on principle of minimum potential energy and zigzag theory. The results shown that using FGM facesheets or FGM cores may prevent occurrence of failure. In another paper he [19] analyzed effects of elastically restrained edges on FG sandwich annular plates by using a novel solution procedure based on layerwise formulation. In this paper, various elastically restrained edges under distributed loads has been analyzed and stress components has been obtained too. In another work, Bayat and Ekhteraei Toussi [20] discussed exact solution of thermal buckling and post buckling of compsoite and SMA beams by layerwise theory. Using a variational approach, the layerwise theory of beams is employed to develop the governing equations and the accuracy of this theory has been shown. Thermo-mechanical bending of laminated composite and sandwich plates using layerwise displacement model has been analyzed by Cetkovic [21]. Therefore, mathematical model, based on layer-wise displacement field of Reddy, is formulated using small deflection linear-elasticity theory. On micro/nano scale, the size effect becomes remarkable and essential. Many papers [22-25] have been presented on the basis of the nonlocal elasticity theory which was invented by Eringen [26]. Actually, unlike classical continuum theory which is scale independent, the nonlocal elasticity theory regards the size effect. There are some papers that analyzed the behaviour of the annular and circular plates on micro scale. Fadaee [27] discussed buckling analysis of a defective annular

graphene sheet in elastic medium. In this work, the governing equations are solved using translational addition theorem and the effect of some parameter such as elastic foundations, different boundary conditions and eccentricity of defects are considered. It is concluded that increasing eccentricity ratio decreases critical buckling load. Thermal buckling of annular microstructure-dependent FGM plates resting on an elastic medium was investigated by Ashoori et al. [28]. The formulations is based on modified couple stress theory (MCST). Li et al. [29] developed buckling and free vibraion of magnetoelectroelastic nanoplate based on nonlocal theory. In this study, the effect of electric and magnetic fields on the critical buckling load and natural frequency are compared. Narendar et al. [30] carrird out wave propagation in SWCNT under longitudinal magnetic field using nonlocal Euler-Bernouli beam theory. The main advantage of numerical solutions such as DQ method over analytical solutions is that, the various boundary conditions canbe considered. Eshraghi et al. [31] discussed bending and free vibration of FG annular and circular microplates under thermal loading. In this paper, geverning equations are obtained using Kirchhoff, Mindlin and third order shear deformation theories and the dynamic and static problems are solved by means of DQ method. Ke et al. [32] analyzed free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. The influence of tempreture, electric field and mechanical load on the values of natural frequency are indicated and equilibrium equations are solved by DQ method and various boudary conditions are considered, consequently. Ghorbanpour Arani et al. [33] presented nonlinear vibration of nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory. The VonKarman geometric nonlinearity and charge equation for coupling of electrical and mechanical fields are considered.

However, to date, no paper has been concentrated on the critical buckling load and deflection of FG annular microplate integrated with two piezoelectric layers as face sheets so far. Actually, micro structure is subjected to electric fields and the material properties of FGM annular microplate (the core layer) are supposed to vary through the thickness according to power law. The sandwich structure is rested in a Pasternak foundation. Based on layerwise theory, equilibrium equations are obtained using principle of virtual work. Meanwhile, the nonlocal elasticity theory is utilized to consider small scale effect. DQ method is applied to calculate the critical buckling load according to various boundary conditions. At the end, the effect of elastic stiffness, external voltage, small scale effect, various boundary conditions, number of grid point, outer-inner radius ratio, power index and thickness ratio on the critical buckling load and deflection are taken into account. 2 Nonlocal elasticity theory and material properties of FGM 2.1 Nonlocal elasticity theory Based on the nonlocal piezoelasticity theory, the stress field and electric displacement at one point depend on the strains and electric component at other points of the body. The nonlocal constitutive behaviour for the piezoelectric material can be written [13]:

(

)

(

)

σ ijnl ( x ) = ³ λ x − x ' ,τ σ ijl dV (x ′), V

Dknl ( x ) = ³ λ x − x ' ,τ Dkl dV (x ′), V

∀x ∈V ,

(1a)

∀x ∈V ,

(1b)

in which σ ijl and σ ijnl are the local and nonlocal stress tensors, respectively, and D kl and Dknl

(

' represent the components of local and nonlocal electric displacement, respectively. λ x − x ,τ

)

is the nonlocal modulus, x − x

'

is the eucliden distance and τ = e0a l describes the material

constant where l is the external characteristic length, a is internal characteristic length and e0 is the Eringen’s nonlocal parameter. The constitutive equation of the nonlocal elasticity can be given as follows [13]: (1 − (e 0a ) 2 ∇ 2 ) σ ijnl = σ ijl ,

(2a)

(1 − (e 0a )2 ∇2 ) D knl = D kl ,

(2b)

where the parameter μ = (e 0a ) is the small scale effect and ∇ 2 represents the Laplacian operator. 2.2 material properties of FGM The core layer of the sandwich microplate is composed of FGM. This layer is considered to be made of combined metal-ceramic materials. The material properties of FG microplate are assumed to vary through the thickness smoothly from pure metal of the top surface to pure ceramic of the bottom surface.The volume fractions of these materials can be given as follows (3)

V m +V c = 1,

in which V m and Vc are the volume fraction of metallic and ceramic part of the annular plate, respectively. According to the power low distribution, the variation of Vc versus thickness coordinate can be described as [6]: g

§ z 1· Vc = ¨ + ¸ , © hc 2 ¹

g ≥0

(4)

where g describes the power index which takes amounts greater than or equal zero and hc is the thickness of core layer. It is supposed that the inhomogeneous material properties such as modulus

of elasticity E varies within the thickness direction, while Poisson’s ratio

υ is assumed to be

constant in thickness direction as E ( z ) = ( E m − E c )V c + E c ,

(5a)

υ (z ) = υ ,

(5b)

in which subscripts c and m are the ceramic and metal constitutive, respectively. 3 Mathematical formulations 3.1 Layerwise theory and electric potential To present a sandwich plate theory with a best compromise between the accuracy and computational costs, a layerwise theory with piecewise- defined linear local and linear global components may be proposed in the present research. This piecewise linear layerwise theory is sufficient for accurately prediction of not only the global behaviors but also the local responses. Fig. 1 illustrates a FGM anuular micro plate integrated with two piezoelectric layers in which geometrical parameters of thickness of face sheets hb and ht , thickness of core hc , inner radius

ri and outer raduis ro are indicated. The micro structure are bounded in a Pasternak foundation in which k w and k g are the Winkler and Pasternak stiffnesses, respectively. Figure 1 Annular sandwich microplate resting on Pasternak foundation According to the layerwise theory with the linear variation of in-plane displacement fields, the displacement field of the each layer of the sandwich micro plate can be written as [18]:

U t = u 0 + ζ tψ rt +

ht t hc c ψr + ψr , 2 2

−

ht h ≤ζt ≤ t 2 2

(6a)

U c = u 0 + ζ cψ rc U b = u 0 + ζ bψ rb −

hb b hc c ψr − ψr , 2 2

−

hc h ≤ζ c ≤ c 2 2

(6b)

−

hb h ≤ζb ≤ b 2 2

(6c) (6d)

W =w 0

in which u 0 and w0 are the radial and transverse displacement of mid-plane of each plate, respectively. ψ ri (i = t ,b ,c ) represents the local rotation of the ith layer. Three transverse local coordinates which are measured from the mid-plane of each layer, are defined for the bottom face b c t sheet ζ , core ζ and top face sheet ζ . The strain-displacement relationsin the cylindrical

coordinate can be expressed as:

ε ri =

∂U i , ∂r

ε θi =

U i 1 ∂V i + , r r ∂θ

∂U i ∂W ε = . + ∂z ∂r

(7)

i rz

The stress components in the core layer of the annular sandwich structure are expressed as:

σ rc =

E (z ) (ε r + υεθ ), 1 −υ 2

(8a)

σ θc =

E (z ) (εθ + υε r ), 1 −υ 2

(8b)

τ rzc =

E (z ) ε rz , 2(1 + υ )

(8c)

where the superscript c reffers to the core layer. The constitutive relations in the piezoelectric layers are written as:

σ rP = Q11P ε r + Q12P εθ − e31P E z ,

(9a)

σθP = Q12P ε r + Q22P εθ − e31P E z ,

(9b)

τ rzP = Q 44P ε rz − e15P E r ,

(9c)

in which superscript P , represents the top (t ) and bottom (b ) of the annular sandwich plate as piezoelectric layers. Q 11 and Q 12 are the elastic stiffness of piezoelectric layers, e15 and e31 are the piezoelectric coefficient, Er and E z are electric fields in radial and transverse directions, respectively and they can be written as follows [6]:

Er = −

∂φ l , ∂r

Ez = −

∂φ l ∂z

(10)

l where φ (l = b ,t ) describes electric potential at any point of piezoelectric layer. The

corresponding electric displacement, D r and D z can be written as:

Dr = e15ε rz +∈11 E r ,

(11a)

Dz = e31 (ε r + εθ )+∈33 E z ,

(11b)

in which ∈11 and ∈33 describe the dielectric constants. for buckling analysis, a half-cosine and linear variation of electrical potential which satisfies Maxwell equation is assumed that can be described as [5]:

§ πζ t © ht

· 2ζ tV 2 ϕ ( r , θ , t ) , + ¸ ht ¹

(12a)

§ πζ b © hb

· ¸ ϕ (r ,θ , t ), ¹

(12b)

φ t (r ,θ , z , t ) = − cos ¨

φ b (r ,θ , z , t ) = − cos ¨

in which V 2 represents applied external voltage. 3.2 Motion equations

Based on the minimum total potential energy principle, the governing equations of motion of sandwich structure may be obtained as [13]: δ Π = δ U − δW = 0,

(13)

where U is the strain energy and W is external work done by Pasternak foundation, external voltage and mechanical load. The strain energy for a annular sandwich plate can be expressed as: h 1 § ro 2π t U = ¨ ³ ³ ³ 2ht (σ rt ε rt + σ θt εθt + 2τ rzt ε rzt − D rt E rt − D zt E zt ) rd θ drd ζ t 2 © ri 0 − 2

+³

ro

ri

hc 2 h − c 2

³ ³ (σ 2π

0

hb 2 h − b 2

³ ³ ³ (σ ro

2π

ri

0

b r

c r

(14a)

ε rc + σ θc εθc + 2τ rzc ε rzc ) rd θ drd ζ c ·

ε rb + σ θb εθb + 2τ rzb ε rzb − D rb E rb − D zb E zb ) rd θ drd ζ b ¸ ¹

According to Eq. (13), Eq. (14a) may be rewritten by performing the integrations in the thickness direction to achieve:

§ § N rb − N θb ∂N rb N rc − N θc ∂N rc N rt − N θt ∂N rt · ht § § N rt − N θt ∂N rt · + + + + + + + δ u ¨ ¨¨ ¨ ¸ ¸+ 0 ³ r r r r 2 ©© ∂r ∂r ∂r ¹ ∂r ¹ A ©© § h § N t − N θt ∂N rt · hc § N rb − N θb ∂N rb · · M rt − M θt ∂M rt + −Qrzt ¸δψ rt + ¨ c ¨ r + + ¸− ¨ ¸+ r r r ∂r ∂r ¹ 2 © ∂r ¹ ¹ ©2© § h § N b − N θb ∂N rb · M rb − M θb ∂M rc · · M rc − M θc ∂M rc + −Qrzc ¸δψ rc + ¨ − b ¨ r + + −Qrzb ¸δψ rb ¸+ r r r ∂r ∂r ¹ ∂r ¹ ¹ © 2© § § πζ t + ¨ −Dr cos ¨ ¨ © ht © § § π § πζ b · · · § πζ b · D sin + ¨ −Dr cos ¨ − − ¸ z ¨¨ ¨ ¸ ¸¸δϕ2 ¸¸ ¨ h h h b b b © ¹ © ¹ © ¹ ¹ ©

§ ∂Qrb Qrb ∂Qrc Qrc ∂Qrt Qrt · + + + + + ¸δw ¨ r ¹ r r ∂r ∂r © ∂r

The stress resultants M , N and Q can be described as:

§ π · § πζ t · · · D s in − − ¸ z ¨¨ ¨ ¸ ¸¸δϕ1 ¸ ¹ © ht ¹ ¹ ¸¹ © ht

(14b)

(15a)

­°N qi ½° h i i ­1 ½ i ® i ¾ = ³− 2h i σ q ® i ¾d ζ , ¯ζ ¿ 2 ¯°M q ¿° hi 2 h − i 2

Q ri = ³

τ rzi d ζ i ,

i = t ,c ,b

q = r ,θ

(15b)

which these terms are described in Appendix 1. Also, the applied external work can be written as follow:

W =

1 (qw )rd θdr , 2 ³A

(16)

where § · § ∂ 2w 1 ∂w · § ∂ 2w · q1 = ¨ k w w − k g ¨ 2 + ¸ − N rr ¨ 2 ¸ + P ¸ , r ∂r ¹ © ∂r © ∂r ¹ © ¹

(17)

in which k w and k g represent Winkler and Pasternak coefficients, respectively. Also, N rr and P describe radial compressive and transverse loads, respectively. Similarly, it should be noted that the electrical force is ( N e = 2e 31PV 2 ) . Substituting Eqs. (14)-(16) into Eq. (13) and using Eqs. (6)(12) the equations of motion in the cylindrical coordinate system, may be expressed as: N rb − N θb ∂N rb N rc − N θc ∂N rc N rt − N θt ∂N rt + + + + + = 0, r r r ∂r ∂r ∂r

(18a)

ht § N rt − N θt ∂N rt · M rt − M θt ∂M rt + + − Q rzt = 0, ¨ ¸+ r r ∂r ¹ ∂r 2©

(18b)

hc § N rt − N θt ∂N rt · hc § N rb − N θb ∂N rb · M rc − M θc ∂M rc + + + − Q rzc = 0, ¨ ¸− ¨ ¸+ r r r ∂r ¹ 2 © ∂r ¹ ∂r 2©

(18c)

−

hb § N rb − N θb ∂N rb · M rb − M θb ∂M rc + + − Q rzb = 0, ¨ ¸+ r r 2© ∂r ¹ ∂r

§ ∂Q rb Q rb ∂Q rc Q rc ∂Q rt Q rt ∂ 2w ∂ 2w + + + + + − (1 − μ 2∇ 2 ) ¨ −P − N rr + 2 2 e 31V 2 2 r r r ∂r ∂r ∂r ∂r ∂r ©

(18d)

(18e)

§ ∂ 2w ∂w · · +k w w − k g ¨ 2 + ¸ ¸ = 0, r ∂r ¹ ¹ © ∂r hb 2 h − b 2

ª § πζ b · ∂D r π § πζ b · º b + sin ¨ «cos ¨ ¸ ¸ D z » d ζ = 0, h r h h ∂ b © b ¹ ¼ ¬ © b ¹

(18f)

ht 2 h − t 2

ª § πζ t · ∂D r π § πζ t · º t − sin ¨ « cos ¨ ¸ ¸ D z » d ζ = 0, h r h h ∂ t © t ¹ ¼ ¬ © t ¹

(18g)

³

³

3. 3. The mathematical forms of the edge conditions The governing Eqs. (18a)–(18g) have to be solved along with the boundary conditions. As mentioned in Sect. 3, the essential (kinematic) and natural boundary conditions may be defined based on the second integral of Eq. (14b). In this regard, the DQ method form of clamped, simply boundary conditions at the both edges of sandwich structure can be written as [18]: ™ Clamped edge:

­u 0 = 0 ° t °ψ r = 0 ° c ®ψ r = 0 ° b °ψ r = 0 °w = 0 ¯ ™ Simply supported edge:

(19a)

­u 0 = 0 °h ° t N rt + M rt = 0 °2 °° h hc b t c c ® Nr +Mr − Nr =0 2 °2 h ° b b b °− 2 N r + M r = 0 ° ¯°w = 0

(19b)

4 Solution procedure Numerical solution is developed for a annular sandwich micro plate. In this paper, DQ method is employed in the solution of the partial differential equations with different boundary conditions. In essence, this method approximates the partial derivative of a function, according to a spatial variable at a given point, as a weighted linear sum of the function values at all discrete points chosen in the solution domain of the spatial variable. In another words, the partial derivatives of a function F ( r ) are approximated with respect to specific variables at a discontinuous point in a defined domain ( ri ¢ r ¢ ro ) as a set of linear weighting coefficients and the amount represented by the function itself at that point and other points throughout the domain. In the DQ method, an nth order partial derivative can be approximated as follow [31]:

∂n F (rj ) ∂r

n

N

= ¦A jk(n )F (rk ),

n = 1,..., N −1

(20)

k =1

in which N is the total number of grid points and A jk( n ) is the weighting coefficients [31]. The Chebyshev-Gauss-Lobatto polynomials was utilized to determine the spaced position of the grid points as [28]:

rj = ri +

ro − ri § § 2 j −1 · · 1 − cos ¨ ¸π ¸. ¨ 2 © © N −1 ¹ ¹

(21)

where ri and ro are the inner radius and outer radius, respectively. 4.1. Motion equations as matrix form Based on Eq (19), the motion equations may be written in matrix form as follows (22)

­Y b ½ ­0 ½ [K s + N rr ] ® ¾ = ® ¾ , ¯Y d ¿ ¯P ¿

where [K s ] is stiffness matrix, N rr and P describe radial compressive and uniform transverse load, respectively. Y is the nodal displacement vector that can be described as follows: T T T T T T T Y = ª{u 0 } , {ψ rt } , {ψ rc } , {ψ rb } , {w } , {φ t } , {φ b } «¬

º. »¼

(23)

where superscripts b and d represent boundary and domain points. Based on an iterative method, for analysis of bending and buckling we have •

Bending

For analysis of bending, N rr = 0 . Hence

­Y b ½ ­0 ½ [K s ] ® ¾ = ® ¾ , ¯Y d ¿ ¯P ¿

(24)

The main purpose is calculating the deflection of annular sandwich microplate. •

Buckling

For analysis of bending, P = 0 . Hence

­Y b ½ ­0 ½ [K s + N rr ] ® ¾ = ® ¾ , ¯Y d ¿ ¯0 ¿ The main purpose is calculating the buckling load of annular sandwich microplate.

(25)

5 Numerical results and discussion Buckling and bending of annular sandwich microplate bounded with piezoelectric layers embedded in Pasternak foundation is studied in this paper. In this section, the influence of some noticable parameter such as small scale effect, external voltage, boundary conditions, elastic foundations, thickness ratio, power index and number of grid points on buckling and bending of micro structure are investigated. As stated, the material for the core layer is combined ceramicmetal and that of the piezoelectric layers are PZT4, which their properties are listed in Table 1. Table 1 Material properties [6].

5.1 Convergence of DQ method The accuracy and convergence of the DQ method in obtaining the critical buckling load of annular sandwich microplate are shown in Table 2. for this case the clamped support at both edges are assumed and the various radial ratios, ri ro = 0.2 , ri ro = 0.25 and ri ro = 0.3 are considered. The results are derived for different values of DQ method grid points. Fast rate of convergense of the DQ method is quite clear. As can be noticed, eleven grid points can yield accurate results. Table 2 Converegence behavior and accuracy of the DQ method for dimensionless critical buckling load

5.2 Comparison study To the best of the authors’ knowledge no published literature is available for bending and analysis of annular sandwich microplate embedded in a Pasternak foundation based on layerwise theory using DQ method. However, in an attempt to validate this work for buckling load, as far as

possible a simplified analysis of this paper is carried out without considering Pasternak foundation, layerwise theory, nonlocal theory, using piezoelectric layers as facesheets and FGM. Present results are compared with the work of Fadaee [27]. The comparison of dimensionless critical 2 3 2 buckling load parameter, N = Nro D , in which D = Ehc 12(1 −υ ) , for clamped and simply

supported circular plates for different amount of dimensionless Winkler coefficient,

Kw = Kw ro4 D , is shown in Table 3. It should be noted the material considered in this case, is graphene sheet

( E = 1765 (Gpa), υ =0.3) .

This table ascertains excellent agreement between

obtained results of present method and the obtained results by finite element method. In another attempt to validate this work for deflection, a simplified analysis of this paper is carried out without considering Pasternak foundation, nonlocal theory, layerwise theory, using piezoelectric layers as facesheets and using DQ method. Present results are compared with the work of saidi et al. [34]. The comparison of dimensionless deflection, w = w ( 64 D ro4 P ) , in which υ =0.288 and the ratio of Young’s modulus of metal component to ceramic component ( E r = 0.396 ) , for circular clamped edge plate, different values of power index and thickness-radius ratio, is shown in Table 4.

Table 3 Validation of present work with Ref. [27] for dimensionless critical buckling load of

circular plate Table 4 Validation of present work with Ref. [34] for dimensionless deflection of circular plate

5.3 Effects of various parameters on dimensionless buckling load and deflection 5.3.1 Buckling analysis

In this section, results concerning the dimensionless buckling load of an embedded annular sandwich micro-plate subjected to uniform radial compressive load. The effect of power index,

g , on the dimensionless critical buckling load, ( N cr = N ri 2 Q 11t h c3 ) , for different small scale effects is shown in Fig. 2 for clamped edges. In this figure, the coefficients of foundation k w , k g are assumed zero. It can bee seen from Figure 2 the dimensionless critical buckling load decreases as the power index increases. It should be noticed, the amount of thickness and radial ratios are

ht hc = 0.5 and ri ro = 0.2 , respectively. Figure 2 Dimensionless buckling load versus power index for different small scale effect

Figs. 3 and 4 illustrate the influences of Winkler and Pasternak coefficients on the critical buckling load versus radial ratio, that ranges from 0- 2 ( μ m ) . It is shown, increasing small scale effect abates the value of critical buckling load. Also, it is seen that in both Figs. 3 and 4, as the Winkler and Pasternak coefficients increase, the critical buckling load increases, consequently. It is due to the effect of foundation, that increasing coefficients of foundations make the system stiffer and more stable. As can be concluded, the influence of Pasternak coefficient are more effective than Winkler. It is because the both normal and transverse shear loads which are considered in Pasternak foundation while Winkler foundation is able to describe just normal loads.

Figure 3 Dimensionless buckling load versus radial ratio for different Winkler coefficients

Figure 4 Dimensionless buckling load versus radial ratio for different Pasternak coefficients

Fig. 5 demontrates the effect of dimensionless critical buckling load with respect to the different amounts of external voltage applied in upper layer versus small scale effect. This figure illustrates that imposing external voltage changes the critical buckling load of system and it is clear that negative voltage decreases the critical buckling load and positive voltage is vice versa. This is because, negative and positive voltages generate the axial compressive and tensile forces in the top layer, respectively. Therefore, the imposed external voltage is an effective controlling parameter for the critical buckling load. In this section, foundation parameters are assumed zero, g = 0

and α = ht hc = 0.5 .

Figure 5 Dimensionless buckling load versus small scale effect for different external voltage

Figure 6 presents the effect of varoius boundary conditions such as clamped-clamped, clampedsimply and simply-simply supported on the dimensionless critical buckling load versus small scale effect. As observed, using clamped support make the system stiffer and more stable, so the annular sandwich microplate can endure more loads than simply supported. Actually, choosing clamp support strengthen the system. In this figure, the amount of small scale effect is assumed zero and inner radius is 1 μ m , meanwhile k w = 0 and k g = 0 and α = ht hc = 0.5 . Figure 6 Dimensionless buckling load versus small scale effect for different boundary conditions

Fig 7 illustrates the influence of thickness ratio α = ht hc on the dimensionless critical buckling load of sandwich microplate. This figure shows that increasing the thickness ratio, increases the dimensionless critical buckling load. Because, as the thickness of structure increases, the structure

becomes thicker. The thicker structure becomes stiffer. In this section, the assumed external voltage, small scale effect, outer radius are respectively, 0, 0 and 5 μm . Figure 7 Dimensionless buckling load versus small scale effect for different thickness ratio

Finally, the effect of radial ratio, β , on the dimensionless critical buckling load versus small scale effect is shown in Figure 8. As can be seen, increasing the radial ratio, increases the dimensionless critical buckling load. This is due to the fact, as the radius of structure increases, structure becomes stiffer. In Figure 8 , foundation parameters are assumed zero, g = 0 and α = ht hc = 0.5 . Figure 8 Dimensionless buckling load versus small scale effect for different radial ratio

5.3.2 Bending analysis In this section, bending analysis of a annular sandwich micro-plates resting on elastic foundation is studied. Numerical solutions based on DQ method and layerwise theory are presented. In Figs. 9 and 10, dimensionless deflection, w = w ( E m hc3 ri 4 P ) × 10 2 , of annular sandwich micro plate with α = 0.5 , β = 0 .2 , μ = 0 and clamped edges are shown for different type of elastic medium. It

can be seen that the deflection of Pasternak medium is lower than Winkler one. It is due to the fact that in Pasternak medium, spring and shear constant are considered and the stiffness of structure is higher. However, considering elastic medium increases the stiffness of laminated nanoplates and consequently the bending deflection reduces. In this section, the assumed external voltage, small scale effect, outer radius are respectively, 0, 0 and 5 μm .

Figure 9 Dimensionless deflection versus radial ratio for different Winkler coefficients

Figure 10 Dimensionless deflection versus radial ratio for different Pasternak coefficients

The effect of power index, g , on the dimensionless deflection, for different values of thickness ratios is illustrated in Fig. 11 for clamped edges. In this figure, the coefficients of foundation

k w , k g are assumed zero. It can bee seen from Fig. 11 the dimensionless deflection increases as the power index increases. In this figure, the amount of small scale effect is assumed zero and inner radius is 1 μ m , meanwhile k w = 0 and k g = 0 .

Figure 11 Dimensionless deflection versus power index for different thickness ratios

Fig. 12 shows the effect of dimensionless deflection with respect to the different amounts of external voltage applied in upper layer versus small scale effect. This figure illustrates that imposing external voltage changes the deflection of system and it is clear that negative voltage increases the deflection and positive voltage is vice versa. It is due to the fact that negative and positive voltages generate the axial compressive and tensile forces in the top layer, respectively.

Figure 12 Dimensionless deflection versus small scale effect for different external voltage

6 Conclusion

In the present work, the critical buckling load and deflection of annular microplates integrated with piezoelectric layers, subjected to radial compressive load and uniform transverse load based on nonlocal elasticity theory was presented. Utilizing layerwise theory, the governing equations were derived using principle of minimum potential energy as well as energy method. To calculate the deflection and critical buckling load of the system, DQ method was utilized. The structure was rested in Pasternak foundation. The effects of some noticeable method such as small scale effect, Pasternak and Winkler coefficients, boundary conditions, radial ratio, thickness ratio, external voltage and FGM valume fraction index were discussed. The results demonstrated that considering elastic medium increases the dimensionless critical buckling load and decreases the dimensionless deflection. Therefore, elastic medium was a remarkable parameter to control the mechanical behaviour of the annular sandwich micoplates. Obtained results by DQ method indicated that this procedure was accurate especially at high grid points. As it is shown, the influence of external voltage in the behaviour of system is very significant. Using FGM, make the system stiffer and strengthen the system. At the end, it would be beneficial in design and manufacturing of nano and micro structures.

Appendix 1 § §∂ · ht ¨ − ¨ ψ rt ¸ ri Qa11ht © ∂r ¹ N rt = ©

· §∂ · §∂ · − ¨ ψ rc ¸ ri Qa11hc + 2¨ u ¸ ri Qa11 − (Qa12ht )ψ rt − (Qa12hc )ψ rc + ( 2Qa12 ) u ¸ © ∂r ¹ © ∂r ¹ ¹, ri

(Q h )u §∂ · N rc = 2Qb11 ¨ u ¸ h2 + 2 b12 2 , ri © ∂r ¹ §§ ∂ · · §∂ · §∂ · h3 ¨ ¨ ψ rb ¸ ri Qa11h3 + ¨ ψ rc ¸ ri Qa11h2 + (Qa12h3 )ψ rb + 2¨ u ¸ ri Qa11 + (Qa12h2 )ψ rc + ( 2Qa12 ) u ¸ © ∂r ¹ © ∂r ¹ © ∂r ¹ ¹, N rb = © ri §§ ∂ · · h1 ¨ ¨ ψ rt ¸ ri Q a11h12 +ψ rt (Q a12 h12 ) + ( 3e 31a ri ) ϕ rt ¸ 2 © © ∂r ¹ ¹, M rt = ri 3 §Q 2§ ∂ M rc = ¨¨ Q a11 ψ rc + ¨ a12 3© ∂r © ri

· c· 3 ¸ψ r ¸¸ h 2 , ¹ ¹

§§ ∂ · · h3 ¨ ¨ ψ rb ¸ ri Q a11h32 + (Q a12 h32 )ψ rb + ( 3e 31a ri ) ϕ rt ¸ 2 © © ∂r ¹ ¹ M rb = ri 3 § §∂ · · §∂ · §∂ · h1 ¨ − ¨ ψ rc ¸ ri Qa 21h2 − ¨ ψ rt ¸ ri Qa 21h1 + 2¨ u ¸ ri Qa 21 − (Qa 22h2 )ψ rc − (Qa 22h1 )ψ rt + ( 2Qa 22 ) u ¸ r r r ∂ ∂ ∂ © ¹ © ¹ © ¹ ¹, N θt = © ri

§Q h · §∂ · N θc = 2Qa 21 ¨ u ¸ h2 + 2¨ a 22 2 ¸u , © ∂r ¹ © ri ¹ §§ ∂ · · §∂ · §∂ · h3 ¨ ¨ ψ rc ¸ ri Qa 21h2 + ¨ ψ rb ¸ ri Qa 21h3 + 2¨ u ¸ ri Qa 21 + (Qa 22h2 )ψ rc + (Qa 22h3 )ψ rb + ( 2Qa 22 )u ¸ © ∂r ¹ © ∂r ¹ © ∂r ¹ ¹, N θb = © ri

M θt =

2h1 § § ∂ t · 2 2 t a t · ¨ ψ r ¸ ri Qa 21h1 + (Qa 22h1 )ψ r + ( 3e32 ri ) ϕr ¸ , ¨ 3r © © ∂r ¹ ¹

2§ § ∂ · §Q · · M θc = ¨¨ Q a 21 ¨ ψ rc ¸ + ¨ a 22 ¸ψ rc ¸¸ h 2 3 , 3© © ∂r ¹ © ri ¹ ¹

2 M θb = 3

§§ ∂ · · h3 ¨ ¨ ψ rb ¸ ri Q a 21h32 + (Q a 22 h32 )ψ rb + ( 3e 32c ri ) ϕ rt ¸ ∂ r ¹ ©© ¹, ri

§ ∂ Q rt = 2¨ w © ∂r

· t ¸ Q a 44 h1 + ( 2Q a 44 h1 )ψ r , ¹

§ ∂ · Q rc = 2¨ w ¸ Q b 44 h 2 + ( 2Q b 44 h 2 )ψ rc , © ∂r ¹ § ∂ Q rb = 2¨ w © ∂r

· b ¸ Q c 44 h 3 + ( 2Q c 44 h3 )ψ r , ¹

References

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Table 1 Material properties [6]. Table 2 Converegence behavior and accuracy of the DQ method for dimensionless critical

buckling load. Table 3 Validation of present work with Ref. [27] for dimensionless critical buckling load of

circular plate. Table 4 Validation of present work with Refs. [34] and [35] for dimensionless deflection of

circular plate.

Table 1

Property

FGM plate

PZT4

Young’s modulus (Gpa)

E c = 380

Q 11 = 132, Q 12 = 71

E m = 70

Q 33 = 115, Q 13 = 73

Poisson ratio

0.3

-

e31(C m2 )

-

-4.1

e33 (C m2 )

-

14.1

e15 (C m2 )

-

10.5

∈11 ( n F m )

-

7.124 5.841

Q 44 = 26

∈ 33 ( nF m )

Table 2

ri ro

Grid points (N)

External votage (V ) V 2 = − 50

V2 = 0

V 2 = + 50

0.20

7 9 11 13

14.6796 14.5921 14.5897 14.5897

15.0678 14.9754 14.9779 14.9779

15.4561 15.3661 15.3636 15.3636

0.25

7 9 11 13

22.4410 22.2697 22.2628 22.2628

23.0476 22.8694 22.8693 22.8693

23.6542 23.4771 23.4760 23.4760

0.30

7 9 11 13

31.8866 31.5506 31.5478 31.5478

32.7602 32.4242 32.4214 32.4214

33.6339 33.2977 33.2950 33.2950

Table 3

Boundary conditions

Simply supported

Clamped

Kw

Referrence 0

0.1

0.2

0.5

1

2

5

10

Fadaei[27]

4.1978

4.2150

4.2322

4.2839

4.3700

4.5424

5.0591

5.9203

Present

4.1888

4.2060

4.2232

4.2749

4.3590

4.5314

5.0480

5.9083

Fadaei[27]

14.6820

14.6820

14.7092

14.7501

14.8181

14.9542

15.3618

16.0359

Present

14.6670

14.6670

14.6922

14.7331

14.8011

14.9362

15.3438

16.0179

Table 4 g

Ref

Thickness radius ratio, h c ro

0.1

0.2

0.25

0.3

0.33

0

Saidi et al. [34] Ma and Wang [35] Present

2.2558 2.2546 2.2552

2.5516 2.5423 2.5465

2.7735 2.7554 2.7591

3.0447 3.0134 3.0182

3.2310 3.1893 3.1981

0.5

Saidi et al.[34] Ma and Wang [35] Present

1.6363 1.6364 1.6363

1.8498 1.8470 1.8476

2.0099 2.0027 2.0055

2.2056 2.1913 2.2014

2.3401 2.3199 2.3260

2

Saidi et al.[34] Ma and Wang [35] Present

1.3780 1.3763 1.3770

1.5450 1.5355 1.5368

1.6703 1.6535 1.6597

1.8234 1.7964 1.8174

1.9286 1.8939 1.9162

Figure Captions Figure 1 Annular sandwich micro plate resting on Pasternak foundation. Figure 2 Dimensionless buckling load versus power index for different small scale effect Figure 3 Dimensionless buckling load versus radial ratio for different Winkler coefficients Figure 4 Dimensionless buckling load versus radial ratio for different Pasternak coefficients Figure 5 Dimensionless buckling load versus small scale effect for different external voltage Figure 6 Dimensionless buckling load versus small scale effect for different boundary conditions Figure 7 Dimensionless buckling load versus small scale effect for different thickness ratio Figure 8 Dimensionless buckling load versus small scale effect for different radial ratio Figure 9 Dimensionless deflection versus radial ratio for different Winkler coefficients Figure 10 Dimensionless deflection versus radial ratio for different Pasternak coefficients Figure 11 Dimensionless deflection versus power index for different thickness ratios Figure 12 Dimensionless deflection versus small scale effect for different external voltage

‫ݎ‬௜

Fig 1

‫ݎ‬௢

20

18

Dimensionless buckling load

e0 a=0 (μm) e0 a=1.0 (μm) e0 a=1.5 (μm)

16

e0 a=2.0 (μm)

14

12

10

8

0

2

4

6

8

10

Power index

Fig 2

12

14

16

18

20

18

Dimensionless buckling load

16 kw=0*1015 (N/m3 )

14

kw=5*1015 (N/m3 ) kw=10*1015 (N/m3 )

12

kw=15*1015 (N/m3 )

10 8 6 4 2

5

6

7

8

9

10

(ro/ri) Fig 3

11

12

13

14

15

16

Dimensionless buckling load

14

kg=0 (N/m) kg=250 (N/m)

12

kg=500 (N/m) kg=750 (N/m)

10

8

6

4

2

5

6

7

8

9

10

(ro/ri) Fig 4

11

12

13

14

15

16

Dimensionless buckling load

14 V2 =-60(V) V2 =-30(V)

12

V2 =0(V) 10

V2 =+30(V) V2 =+60(V)

8 6 4 2 0

0

0.2

0.4

0.6

0.8

1

1.2

Small scale effect (μm) Fig 5

1.4

1.6

1.8

2

Dimensionless buckling load

15

C-C C-S S-S

10

5

0

0

0.2

0.4

0.6

0.8

1

1.2

Small scale effect (μm) Fig 6

1.4

1.6

1.8

2

25

Dimensionless buckling load

20

α =0.50 α =0.55 α =0.60 α =0.65

15

10

5

0

0

0.2

0.4

0.6

0.8

1

1.2

Small scale effect (μm) Fig 7

1.4

1.6

1.8

2

40 β=0.2 β=0.25 β=0.3 β=0.35

Dimensionless buckling load

35 30 25 20 15 10 5 0

0

0.2

0.4

0.6

0.8

1

1.2

Small scale effect (μm) Fig 8

1.4

1.6

1.8

2

0.28

Dimensionless deflection

0.26

0.24

0.22 kw=0*1015 (N/m3 )

0.2

kw=5*1015 (N/m3 ) kw=10*1015 (N/m3 )

0.18

kw=15*1015 (N/m3 ) 0.16

5

6

7

8

9

10

(ro/ri) Fig 9

11

12

13

14

15

0.28

Dimensionless deflection

0.26

0.24

0.22 kg=0 (N/m) kg=250 (N/m)

0.2

kg=500 (N/m) kg=750 (N/m)

0.18

0.16

5

6

7

8

9

10

(ro/ri) Fig 10

11

12

13

14

15

0.19

Dimensionless Deflection

0.18 0.17 0.16 0.15 0.14 α=0.50 α=0.55 α=0.60 α=0.65

0.13 0.12 0.11

0

2

4

6

8

10

Power index Fig 11

12

14

16

18

20

0.164 0.163

Dimensionless deflection

0.162

V2 =+60 (V) V2 =+30 (V)

0.161

V2 =-30 (V) V2 =-60 (V)

0.16 0.159 0.158 0.157 0.156 0.155

0

0.2

0.4

0.6

0.8

1

1.2

Small scale effect (μm) Fig 12

1.4

1.6

1.8

2