Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory

Author's Accepted Manuscript

Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory Woo-Young Jung, Weon-Tae Park, Sung-Cheon Han

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S0020-7403(14)00208-2 http://dx.doi.org/10.1016/j.ijmecsci.2014.05.025 MS2747

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International Journal of Mechanical Sciences

Received date: 17 August 2013 Revised date: 30 March 2014 Accepted date: 24 May 2014 Cite this article as: Woo-Young Jung, Weon-Tae Park, Sung-Cheon Han, Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2014.05.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Bending and Vibration Analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory a

Woo-Young Jung , a

b

Weon-Tae Park , Sung-Cheon Han

c, *

Department of Civil Engineering, Gangneung-Wonju National University, 7 Jukheon, Gangneung, 210-702, Republic of Korea

b

Division of Construction and Environmental Engineering, Kongju National University, 275 Budai, Cheonan, 330-717, Republic of Korea c

Department of Civil & Railroad Engineering, Daewon University College, 599 Shinwol, Jecheon, 390-702, Republic of Korea

---------------------------------------------------------------------------------------------------------------Abstract A model for sigmoid functionally graded material (S-FGM) microplates based on the modified couple stress theory with first order shear deformation is developed. The advantages of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of microplate can be viewed as a simplified couple stress theory in engineering mechanics. The present models contain one material length scale parameter and can capture the size effect, and two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton’s principle based on the modified couple stress theory, and the power law variation of the material through the thickness of the plate. Material properties of functionally graded plate are assumed to vary according to two power law distribution of the volume fraction of the constituents. The elastic medium is modeled as Pasternak elastic medium. Analytical solution of rectangular plates is derived, and the obtained results are compared well with reference solutions. Finally, the influences of power law index, material length scale parameter, thickness ratio, and elastic medium parameter on the deflection and the natural frequency of plates have been investigated.

Keywords: Modified couple stress theory; Sigmoid functionally graded material plate; Material length scale parameter; Elastic medium; Bending analysis; Vibration analysis.

---------------------------------------------------------------------------------------------------------------* Corresponding author. Tel.: +82-(0)43-649-3267; fax: +82-(0)43-649-3681. E-mail: [email protected] (S.-C. Han)

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1. Introduction Since the 1960s, experiments have shown that micro-structure has scale effects due to impurities, crystal lattice mismatch and micro cracks at micro scales. As conventional continuum theory cannot explain or solve the problems of the scale effects, theories for microstructures need to be developed. Theories for microstructures include couple stress theory and strain gradient theory. A series of research in the couple stress/ strain gradient theories have been made. For example, in the 1960s, Toupin [1], Koiter [2] and Mindlin proposed couple stress theory [3]. Between the 1980s and 1990s, Aifantis [4], Fleck and Hutchinson [5,6] developed the strain gradient theory in plasticity. Gao et al. [7] further improved the strain gradient theory in plasticity. The modified couple stress theory has recently been proposed by Yang et al. in which the couple stress tensor is symmetric and only one internal material length scale parameter is considered [8]. Several size-dependent beam and plate models have been developed to capture the size effects in small scale structures based on the modified couple stress theory. For example, Kong et al. [9] and Kahrobaiyan et al. [10] developed a Euler-Bernoulli beam model for vibration analysis of nanobeams. Ma et al. [11] studied a Timoshenko beam model and this model was adopted to study the buckling [12] and vibration [13,14] of microtubes. Tsiatas [15] first developed a Kirchhoff plate model for static analysis of microplates. This model was used by Yin et al. [16] and Jomehzadeh et al. [17] investigated the vibration of microplates using the Kirchhoff plate theory. Recently, Asghari and Taati [18] deal with Kirchhoff plate theory of FGM plates. To account for the effects of transverse shear deformation and rotary inertia in moderately thick microplates, Ma et al. [19] and Ke et al. [20] developed a plate model using shear deformation theory. It should be noted that the above mentioned studies dealt with the microbeams and microplates made of homogeneous materials only. In recent years, the modified coupled stress theory is further used to develop functionally graded Euler– Bernoulli beam [21], functionally graded Timoshenko beam [22–25], laminated Timoshenko beam [26], and functionally graded Kirchhoff and Mindlin plate [27].

Asghari [28] studied geometrically

nonlinear formulation of micro-plate using the modified couple stress theory. It is the first paper presented the boundary conditions for a plate with arbitrary shape based on the modified couple stress theory. Functionally graded materials (FGMs) are a class of composites that have continuous variation of material properties from one surface to another and thus eliminate the stress concentration found in laminated composites. The FGMs which are often isotropic and nonhomogeneous, are made from a mixture of two materials to achieve a composition that provides a certain functionality. In recent years, the application of FGMs has broadly been spread in micro- and nano-scale devices and systems such as thin films [29,30], atomic force microscopes [31], micro- and nano-electromechanical systems (MEMS and NEMS) [32,33]. In FGM, these problems are avoided or reduced by gradual variation of the constituents’ volume fraction rather than abruptly changing it across the

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interface [34]. Power-law function [35, 36], and exponential function [37, 38] are commonly used to describe the variations of material properties of FGM. However, in both power-law and exponential functions, the stress concentrations appear in one of the interfaces in which the material is continuously but rapidly changing. Therefore, Chung and Chi [39] proposed a sigmoid FGM (SFGM), which was composed of two power-law functions to define a new volume fraction. Chi and Chung [40] indicated that the use of a S-FGM can significantly reduce the stress intensity factors of a cracked body. Han et al. [41] studied mechanical vibration and buckling analysis of S-FGM plates and shells using finite element method. The non-linear analysis of anisotropic S-FGM structures was presented by Han et al. [42]. The surrounding elastic medium is generally modeled as Winkler-type elastic medium. The Winkler-type elastic foundation is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs. The elastic medium modulus is represented by stiffness of the springs. However, this model is considered as a crude approximation of the true mechanical behavior of the elastic material. This is due to inability of the model to take into account the continuity or cohesion of the medium. The interaction between the springs is not taken into account in Winkler-type elastic medium. A more realistic and generalized representation of the elastic medium can be accomplished by the way of a two-parameter elastic medium model. Thus two-parameter elastic medium model is preferred. One such physical elastic medium model is the Pasternak-type elastic medium model. The first parameter of Pasternak elastic medium model represents the normal pressure while second parameter accounts for the transverse shear stress due to interaction of shear deformation of the surrounding elastic medium. Pasternak-type model is physically realistic representation of the elastic medium. Further it is a mathematically simple model for analyzing the surrounding elastic medium. Successful use of Pasternak-type elastic medium model for simulating the interaction of the surrounding elastic medium with graphene sheet is shown by Liew et al. [43] and Pradhan et al. [44]. The FGM plates are often found embedded in the elastic medium [45-48]. Although the size-dependent beam and plate models have been developed in the aforementioned studies based on the modified couple stress theory, no literature has been reported for the sized dependent plate models on elastic medium accounting for material variation through the thickness of the microplate. Prompted by the lack of study on the bending and vibration of S-FGM microplates on elastic medium with allowance for small scale effect, this article presents a formulation of modified couple stress theory for the bending and vibration analysis of S-FGM microplates on elastic medium. The S-FGM microplates are assumed to be embedded in a Pasternak elastic medium. Both Winkler-type [49] and Pasternak-type elastic medium [50] are used to simulate the interaction between the S-FGM microplates and the surrounding elastic medium. Material properties of FGM plate are assumed to vary according to two power law distribution of the volume fraction of the constituents. The equations of motion are derived from Hamilton’s principle based on

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the modified couple stress theory, and power law variation of material through the thickness. Analytical solutions for the static bending, and free vibration problems are presented for a simply supported plate to bring out the effects of material length scale parameter on the deflection, and frequency. Equations of motion are derived using Hamilton’s principle. The validity of the present theory is verified. Finally, the effects of (i) power law index (ii) material length scale parameter (iii) stiffness of the elastic medium modeled as Winkler elastic medium, (iv) stiffness of the elastic medium modeled as Pasternak elastic medium and (v) side to thickness ratios of S-FGM microplates on the deflection and natural frequency of S-FGM microplates on elastic medium are examined and discussed.

2. Theoretical Formulation 2.1 Elastic medium models A sigmoid functionally graded material (S-FGM) microplates embedded in an elastic medium is considered (Fig. 1). The chemical bonds are assumed to be formed between the S-FGM microplate and the elastic medium. The polymer matrix is described by a Pasternak-type foundation model [50], which accounts for both normal pressure and the transverse shear deformation of the surrounding elastic medium. When the shear effects are neglected, the model reduces to Winkler- type foundation [49]. The normal pressure or Winkler elastic foundation parameter is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs. Here the foundation modulus is assumed equivalent to stiffness of the springs. The normal pressure and the incompressible layer that resists transverse shear deformation are represented by Winkler and Pasternak elastic medium models, respectively. These loadings are expressed as

qWinkler = kW u3

(1)

qPasternak = kW u3 − k P∇ 2u3

(2)

where ∇ 2 = ∂ 2 / ∂x 2 + ∂ 2 / ∂y 2 , kW and k P denote the Winkler modulus and the shear modulus of the surrounding elastic medium, respectively. Consider the origin at one corner of the plate. The x coordinate of the axis is taken along the length of the plate, y coordinate is taken along the width of the plate and z coordinate is taken along the thickness of the plate (Fig. 1).

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y

FGM microplate

x Shear Layer Winkler Layer

Fig. 1 S-FGM microplate on Pasternak’s elastic medium

2.2 Classical couple stress theory The classical couple stress theory was developed by Mindlin [3]. The strain tensor and curvature tensor can be defined respectively as

ε ij =

1 ( ui , j + u j ,i ) , 2

χ ij = ωi , j , i, j = 1, 2,3

(1)

where ui are the components of the displacement vector, ωi are the components of rotation vector and ωi =

1 eijk u k , j , ε ij is symmetric tensor and χ ij is asymmetric tensor. 2

Constitutive relations are given by

σ ij = 2με ij + λδ ijε kk , mij = 4μ A2 χij ,

(2)

where μ and

λ are the Lamé parameters (see [51]), and A is the material length scale parameter.

2.3 Modified couple stress theory The couple stress theory proposed by Yang et al. [8] is a modification of the classical couple stress theory (see [2-4]). They established that the couple stress tensor is symmetric and the

6

symmetric curvature tensor is the only proper conjugate strain measure to have a contribution to the total strain energy of the body. The two main advantages of the modified couple stress theory over the classical couple stress theory are the inclusion of a symmetric couple stress tensor and the involvement of only one length scale parameter, which is a direct consequence of the fact that the strain energy density function depends only on the strain and the symmetric part of the curvature tensor (see [11]). According to the modified couple stress theory, the virtual strain energy

δ U can be written

as

δ U = ∫ (δ ε : σ + δ χ : m ) dV = ∫ (σ ijδε ij + mijδχ ij ) dV V

V

(3)

where summation on repeated indices is implied; here σ ij denotes the cartesian components of (the symmetric part of) the stress tensor, ε ij are the strain components, mij are the components of the deviatoric part of the symmetric couple stress tensor, and χ ij are the components of the symmetric curvature tensor

χ=

1 1 ⎡⎣∇ω + (∇ω) T ⎤⎦ , ω = ∇ × u 2 2

(4)

or

1 ⎛ ∂ωi ∂ω j + 2 ⎝ ∂x j ∂xi

χij = ⎜ ⎜

⎞ ⎟⎟ , i, j = 1, 2,3 ⎠

(5)

and ωi (i = 1, 2,3) are the components of the rotation vector

1 ⎛ ∂u3 ∂u2 ⎞ 1 ⎛ ∂u1 ∂u3 ⎞ 1 ⎛ ∂u2 ∂u1 ⎞ − − − ⎟ , ω y = ω2 = ⎜ ⎟ , ω z = ω3 = ⎜ ⎟. 2 ⎝ ∂x2 ∂x3 ⎠ 2 ⎝ ∂x3 ∂x1 ⎠ 2 ⎝ ∂x1 ∂x2 ⎠

ωx = ω1 = ⎜

Thus, we have

1 ⎛ ∂ 2u3 ∂ 2u2 ⎞ − ⎟, 2 ⎝ ∂x1 x2 ∂x1 x3 ⎠

χ xx = χ11 = ⎜

(6)

7

1 ⎛ ∂ 2u1 ∂ 2u3 ⎞ − ⎟, 2 ⎝ ∂x2 x3 ∂x1 x2 ⎠

χ yy = χ 22 = ⎜

1 ⎛ ∂ 2u2 ∂ 2u1 ⎞ − ⎟, 2 ⎝ ∂x1 x3 ∂x2 x3 ⎠

χ zz = χ 33 = ⎜

1 ⎛ ∂ 2u3 ∂ 2u2 ∂ 2u1 ∂ 2u3 ⎞ − + − ⎟, 2 ⎝ ∂x22 ∂x2 x3 ∂x1 x3 ∂x12 ⎠

η xy = 2 χ xy = 2 χ12 = ⎜

(7)

1 ⎛ ∂ 2u3 ∂ 2u2 ∂ 2u2 ∂ 2u1 ⎞ − + 2 − ⎟, ∂x1 ∂x1 x2 ⎠ 2 ⎝ ∂x2 x3 ∂x32

η xz = 2 χ xz = 2 χ13 = ⎜

1 ⎛ ∂ 2u1 ∂ 2u3 ∂ 2u2 ∂ 2u1 ⎞ − + − ⎟. 2 ⎝ ∂x32 ∂x1 x3 ∂x1 x2 ∂x22 ⎠

η yz = 2 χ yz = 2 χ 23 = ⎜

For an isotropic, linear elastic material the 3-D stress–strain relations are

σ ij = 2με ij + λδ ijε kk , mij = 2μ A2 χij ,

(8)

where

λ=

Eν E , 2μ = , (1 +ν )(1 − 2ν ) (1 +ν )

with E being Young’s modulus and

(9)

ν being Poisson’s ratio. The material length scale parameter

A is the square root of the ratio of the modulus of curvature to the modulus of shear, and it is a property measuring the effect of the couple stress. For a functionally graded material, μ and

λ are

functions of z . The coefficient 4 in Eq. (2) is chosen for a couple stress theory developed by Mindlin [3]. However, in Eq. (8) the coefficient 2 is chosen for the modified couple stress theory developed by Yang et al. [8], which can ensure the coefficients of the constitutive relations the couple stress tensor is symmetric. It is means that the value of A in the two theories is difference only with a multiple.

2.3. Material variation through the thickness The functionally graded material (FGM) can be produced by continuously varying the constituents of multi-phase materials in a predetermined profile. The most distinct features of an FGM

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are the non-uniform microstructures with continuously graded properties. An FGM can be defined by the variation in the volume fractions. Most researchers use the power-law function, exponential function, or sigmoid function to describe the volume fractions. This paper uses FGM plates and shells with sigmoid function. The volume fraction using two power-law functions which ensure smooth distribution of stresses is defined.

1⎛ h/2− z ⎞ V f1 ( z ) = 1 − ⎜ ⎟ 2⎝ h/2 ⎠ 1⎛ h/2+ z ⎞ V f2 ( z ) = ⎜ ⎟ 2⎝ h/2 ⎠

p

for 0 ≤ z ≤ h / 2 ,

(10a)

p

for − h / 2 ≤ z ≤ 0 .

(10b)

where p is the power law index indication the volume fraction of material. By using the rule of mixture, the material properties of the S-FGM can be calculated by:

H ( z ) = V f1 ( z ) H1 + (1 − V f1 ( z ) ) H 2

for 0 ≤ z ≤ h / 2 ,

H ( z ) = V f2 ( z ) H1 + (1 − V f2 ( z ) ) H 2

for − h / 2 ≤ z ≤ 0 .

(11a) (11b)

Fig. 2 shows that the variation of Young’s modulus in Eqs. (11a) and (11b) represents sigmoid distributions, and this FGM structure is thus called a sigmoid FGM structure (S-FGM structures). In this paper, the volume fraction using two power-law functions by Chung and Chi [39] is used to ensure smooth distribution of stresses among all the interfaces.

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Normalized Thickness, z/h

0.5

0.0 p = 1.0 p = 2.0 p = 3.0 p = 5.0 p = 7.0 p =10.0 -0.5 E2

E1

Young's Modulus

Fig. 2. The variation of Young’s modulus of S-FGM plate

2.4. Constitutive equations Consider a plate made of two constituent functionally graded materials. The material properties of the plate such a Young’s modulus E and the mass density ρ

{Et ( z ), ρt ( z )} = V f1 ( z ) {E1 , ρ1} + (1 − V f1 ( z ) ) {E2 , ρ2 } {Eb ( z ), ρb ( z )} = V f2 ( z ) {E1 , ρ1} + (1 − V f2 ( z ) ) {E2 , ρ2 }

for 0 ≤ z ≤ h / 2 , for − h / 2 ≤ z ≤ 0 .

(12a) (12b)

Where the subscripts 1 and 2 represent the two materials used. Poisson’ ratio ν is assumed to be constant [52].

Here we represent the profile for volume fraction variation by the expression in Eq.

(12) we assume that modulus E , shear modulus G and density ρ vary according to Eq. (12). The linear constitutive relations that use the plane stress-reduced constitutive relations are

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⎡1 ⎢ν ⎧σ xx ⎫ ⎢ ⎪σ ⎪ ⎢ ⎪⎪ yy ⎪⎪ E ( z ) ⎢ 0 ⎨σ xy ⎬ = 2 ⎢ ⎪σ ⎪ 1 −ν ⎢ 0 ⎢ ⎪ xz ⎪ ⎢ ⎩⎪σ yz ⎭⎪ ⎢0 ⎣⎢

⎡1 ⎢0 ⎧ mxx ⎫ ⎢ ⎪m ⎪ ⎢0 ⎪ yy ⎪ ⎢ ⎪⎪ mzz ⎪⎪ E ( z )A 2 ⎢0 ⎨ ⎬= ⎢ ⎪ mxy ⎪ 1 +ν ⎢ ⎪ mxz ⎪ ⎢0 ⎪ ⎪ ⎢ ⎪⎩ myz ⎪⎭ ⎢ ⎢0 ⎣

ν

0 0

0 0

1 −ν 2

0

1 0 0

0

1 −ν 2

0

0

0

0 0

0

0

1 0

0

0

0 1

0

0

0 0

1 2

0

0 0

0

1 2

0 0

0

0

0 ⎤ 0 ⎥⎥ ⎧ε xx ⎫ ⎥⎪ ⎪ 0 ⎥ ⎪ε yy ⎪ ⎪ ⎪ ⎥ ⎨γ xy ⎬ 0 ⎥⎥ ⎪γ xz ⎪ ⎪ ⎪ ⎥ ⎩⎪γ yz ⎭⎪ 1 −ν ⎥ 2 ⎦⎥

0⎤ 0 ⎥⎥ ⎧ χ xx ⎫ 0 ⎥ ⎪ χ yy ⎪ ⎥⎪ ⎪ 0 ⎥ ⎪⎪ χ zz ⎪⎪ ⎥ ⎨η ⎬ ⎥ ⎪ xy ⎪ 0 ⎥ ⎪η xz ⎪ ⎥ ⎪η ⎪ 1 ⎥ ⎪⎩ yz ⎪⎭ ⎥ 2⎦

(13)

(14)

2.5 First-order plate theory The first-order plate theory accounts for the transverse shear deformation effects by the way of linear variation of in-plane displacements through the thickness. The displacement field of the firstorder plate theory can be expressed as

u1 ( x, y, z , t ) = u ( x, y, t ) + zθ x ( x, y, t ) ,

u2 ( x, y, z, t ) = v( x, y, t ) + zθ y ( x, y, t ) ,

(15)

u3 ( x, y, z , t ) = w( x, y, t )

where (u , v, w) are the displacements along the coordinate lines of a material point on the xy plane, i.e., u ( x, y, t ) = u1 ( x, y, 0, t ), v( x, y, t ) = u2 ( x, y, 0, t ), w( x, y, t ) = u3 ( x, y, 0, t ) and

⎛ ∂u ⎞

⎛ ∂u ⎞

θx = ⎜ 1 ⎟ , θ y = ⎜ 2 ⎟ ⎝ ∂z ⎠ z =0 ⎝ ∂z ⎠ z =0

(16)

The components of strain tensor, rotation vector and curvature tensor associated with the

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displacement field in Eq. (15) are obtained as

⎧ ⎫ ⎪ 0 ⎪ ⎧ε zz ⎫ ⎧ε zz(0) ⎫ ⎪ ⎪ ∂w ⎪ ⎪ ⎪ ⎪ (0) ⎪ ⎪ ⎨γ xz ⎬ = ⎨γ xz ⎬ = ⎨θ x + ⎬, ∂ x (0) ⎪γ ⎪ ⎪γ ⎪ ⎪ ⎪ ⎩ yz ⎭ ⎩ yz ⎭ ⎪ ∂w ⎪ ⎪θ y + ∂y ⎪ ⎩ ⎭

⎧ε xx(1) ⎫ ⎧ε xx ⎫ ⎧ε xx(0) ⎫ ⎪ (1) ⎪ ⎪ ⎪ ⎪ (0) ⎪ ⎨ε yy ⎬ = ⎨ε yy ⎬ + z ⎨ε yy ⎬ , ⎪γ ⎪ ⎪γ (0) ⎪ ⎪γ (1) ⎪ ⎩ xy ⎭ ⎩ xy ⎭ ⎩ xy ⎭

with

⎧ ∂u 1 ⎛ ∂w ⎞2 ⎫ + ⎜ ⎪ ⎟ ⎪ ∂x 2 ⎝ ∂x ⎠ ⎪ ⎪ ⎧ε xx(0) ⎫ ⎪ 2 ⎪ ⎪ (0) ⎪ ⎪ ∂v 1 ⎛ ∂w ⎞ ⎪ + ⎜ ⎨ε yy ⎬ = ⎨ ⎟ ⎬, ⎪γ (0) ⎪ ⎪ ∂y 2 ⎝ ∂y ⎠ ⎪ ⎩ xy ⎭ ⎪ ∂u ∂v ∂w ∂w ⎪ ⎪ + + ⎪ ⎪⎩ ∂y ∂x ∂x ∂y ⎪⎭ where

(ε

(0) xx

⎧ ∂θ x ⎫ ⎪ ⎪ ∂x ⎪ ⎧ε xx(1) ⎫ ⎪ ⎪ ∂ θ ⎪ (1) ⎪ ⎪ ⎪⎪ y ⎨ε yy ⎬ = ⎨ ⎬. ∂y ⎪γ (1) ⎪ ⎪ ⎪ xy ⎩ ⎭ ⎪ ∂θ ∂θ ⎪ y ⎪ x+ ⎪ ∂x ⎭⎪ ⎩⎪ ∂y

, ε yy(0) , γ xy(0) ) are the membrane strains,

(ε

(17)

(1) xx

, ε yy(1) , γ xy(1) ) are the flexural (bending)

strains.

⎧ χ xx ⎫ ⎧ χ xx(0) ⎫ ⎪ ⎪ ⎪ (0) ⎪ ⎪ χ yy ⎪ ⎪ χ yy ⎪ ⎨ ⎬ = ⎨ (0) ⎬ , ⎪ χ zz ⎪ ⎪ χ zz ⎪ ⎪η xy ⎪ ⎪η xy(0) ⎪ ⎩ ⎭ ⎩ ⎭

⎧⎪η xz ⎫⎪ ⎧⎪η xz(0) ⎫⎪ ⎧⎪η xz(1) ⎫⎪ ⎨ ⎬ = ⎨ (0) ⎬ + z ⎨ (1) ⎬ , ⎪⎩η yz ⎪⎭ ⎪⎩η yz ⎪⎭ ⎪⎩η yz ⎪⎭

where

χ xx(0) =

⎞ 1 ∂ ⎛ ∂w −θy ⎟, ⎜ 2 ∂x ⎝ ∂y ⎠

χ yy(0) =

1 ∂ ⎛ ∂w ⎞ ⎜θ x − ⎟, 2 ∂y ⎝ ∂x ⎠

1 ⎛ ∂θ y ∂θ x ⎞ − ⎟, ∂y ⎠ 2 ⎝ ∂x

χ zz(0) = ⎜

⎞ ∂ ⎛ 1 ⎡ ∂ ⎛ ∂w ∂w ⎞ ⎤ − θ y ⎟ + ⎜θ x − ⎜ ⎟⎥ , 2 ⎣ ∂y ⎝ ∂y ∂x ⎠ ⎦ ⎠ ∂x ⎝

η xy(0) = ⎢

η xz(0) =

1 ∂ ⎛ ∂v ∂u ⎞ 1 ∂ ⎛ ∂θ y ∂θ x ⎞ (1) − ⎜ ⎟, ⎜ − ⎟ , η xz = 2 ∂x ⎝ ∂x ∂y ⎠ 2 ∂x ⎝ ∂x ∂y ⎠

η yz(0) =

1 ∂ ⎛ ∂v ∂u ⎞ 1 ∂ ⎛ ∂θ y ∂θ x ⎞ (1) − ⎜ ⎟. ⎜ − ⎟ , η yz = 2 ∂y ⎝ ∂x 2 ∂y ⎝ ∂x ∂y ⎠ ∂y ⎠

3. Equations of motion 3.1 Equations of motion in terms of displacements

(18)

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The equations of motion can be derived using the principle of virtual displacements. The principle of virtual displacements for the dynamic case requires that (see [53])

∫ (δ K − δ U − δ U T

0

where

EM

− δ V ) dt = 0

(19)

δ K is the virtual kinetic energy, δ U is the virtual strain energy, δ U EM is the virtual

strain energy of the elastic medium, and

δV is the virtual work done by external forces. Each of

these quantities is derived next.

δ K is

The virtual kinetic energy

δK = ∫

∫

h/2

Ω −h / 2

⎛ ∂u1 ∂δ u1 ∂u2 ∂δ u2 ∂u3 ∂δ u3 ⎞ + + ⎟ dzdxdy ∂t ∂t ∂t ∂t ⎠ ⎝ ∂t ∂t

ρ⎜

(20)

The virtual strain energy is given by [see Eq. (3)]

∫ (σ

δε xx + σ yyδε yy + σ zzδε zz + σ xyδγ xy + σ xzδγ xz + σ yzδγ yz ) dzdxdy

h/2

δU = ∫

Ω −h / 2

xx

∫ (m h/2

+∫

Ω −h / 2

δχ xx + myyδχ yy + mzzδχ zz + mxyδη xy + mxzδη xz + myzδη yz ) dzdxdy

xx

(21) The virtual strain energy of the elastic medium can be expressed as

⎡

⎛ ∂u3 ∂δ u3 ∂u3 ∂δ u3 ⎞ ⎤ + ⎟ ⎥ dxdy ∂y ∂y ⎠ ⎦ ⎝ ∂x ∂x

δ U EM = ∫ ⎢ kW u3δ u3 + k P ⎜ Ω

⎣

Next, we introduce thickness-integrated stress resultants

M ij( k ) = ∫

h/2

−h / 2

σ ij ( z ) k dz , M ij( k ) = ∫

h/2

−h / 2

mij ( z ) k dz, (k = 0,1, 2,3)

Then the virtual strain energy can be expressed in terms of the stress resultants as

(22)

13

⎡

δ U = ∫ ⎢∑ ( M xx(i )δε xx + M yy(i )δε yy + M xy(i )δγ xy ) + ∑ ( M zz(i )δε zz + M xz(i )δγ xz + M yz(i )δγ yz ) Ω 3

2

⎣ i =0

i =0

2 3 ⎤ + ∑ ( M xx(i )δχ xx + M yy(i )δχ yy + M zz(i )δχ zz + M xy(i )δη xy ) + ∑ ( M xz(i )δη xz + M yz(i )δη yz ) ⎥ dxdy (23) i =0 i =0 ⎦

(0) (0) (0) Note that M xx , M yy , and M xy are the membrane forces (often denoted by N xx , N yy , and N xy ),

M xx(1) , M yy(1) , and M xy(1) are the bending moments (denoted by M xx , M yy , and M xy ), and M xz(0) (0) and M yz are the shear forces (denoted by Qx and Q y ).

The virtual work done by external forces consists of three parts: (1)The virtual work done by the body forces in V = Ω× ( −h / 2, h / 2 ) , (2) virtual work done by surface tractions acting on the top and bottom surfaces of the plate Ω + and Ω − , and (3) virtual work done by the surface tractions on the lateral surface S = Γ × ( −h / 2, h / 2 ) , where Ω + denotes the top surface of the plate, Ω − the middle surface of the plate, Ω bottom surface of the plate, and Γ is the boundary of the

middle surface (see Fig. 2).

(f ,f

Let

x

y

, f z ) be the body forces (measured per unit volume),

surface forces (measured per unit area) on S, and + per unit area) on Ω ,

(c , c , c ) x

y

(q , q , q ) b x

b y

b z

(q , q , q ) t x

t y

t z

x

y

z

be the

be the distributed forces (measured

− be the distributed forces (measured per unit area) on Ω , and

be the body couples (measured per unit volume) in the

z

(t , t , t )

( x, y , z )

coordinate directions.

Then the virtual work done by external forces is

δ V = − ⎡ ∫ ( f xδ u1 + f yδ u2 + f zδ u3 + cxδω1 + c yδω2 + czδω3 ) dV ⎣

V

+∫

Ω+

(q δ u + q δ u t x

1

t y

2

+ qzt δ u3 ) dxdy + ∫

Ω−

(q δ u + q δ u b x

1

b y

2

+ qzbδ u3 ) dxdy

(24)

+ ∫ ( tx δ u1 + ty δ u2 + tz δ u3 + ) dS ⎤ S ⎦

The equations of motion of the general third-order plate theory governing functionally graded plates accounting for modified couple stresses are obtained by substituting

δ K , δ U and

δV , from Eqs. (20), (21) and (24), respectively, into Eq. (19), applying the integration-by-parts to relieve all virtual generalized displacements of differentiations with respect to x, y and t , noting

14

that all variations at the upper and lower time limits are zero, and invoking the fundamental lemma of the variational calculus (see [54, 55]). We obtain (after a lengthy algebra and manipulations) the following equations: (0) (0) ∂M xx(0) ∂M xy 1 ∂ ⎛ ∂M xz(0) ∂M yz ⎞ 1 ∂cz(0) (0) + + + δu : ⎜⎜ ⎟⎟ + f x + ∂x ∂y ∂y ⎠ 2 ∂y ⎝ ∂x 2 ∂y

= m0 u + m1θx

(25)

(0) 1 ∂ ⎛ ∂M xz(0) ∂M yz ⎞ 1 ∂cz(0) (0) δv : + − + ⎜⎜ ⎟⎟ + f y − 2 ∂x ⎝ ∂x 2 ∂x ∂x ∂y ∂y ⎠

∂M xy(0)

∂M yy(0)

= m0v + m1θy

(26)

∂ ⎛ ∂w (0) ∂w (0) ⎞ ∂ ⎛ ∂w (0) ∂w (0) ⎞ M xx + M xy ⎟ + ⎜ M xy + M yy ⎟ ⎜ ∂x ⎝ ∂x ∂y ∂y ⎠ ∂y ⎝ ∂x ⎠

δw:

+

(0) (0) (0) (0) 1 ∂ ⎛ ∂M xx(0) ∂M xy ⎞ 1 ∂ ⎛ ∂M yy ∂M xy ⎞ ∂M xz(0) ∂M yz (0) − + + + ⎜⎜ ⎟⎟ + ⎜⎜ ⎟ + Fz 2 ∂y ⎝ ∂x ∂y ⎠ 2 ∂x ⎝ ∂y ∂x ⎟⎠ ∂x ∂y

(0) 1 ⎛ ∂c(0) ∂cy ⎞  −kW w + kP∇ 2 w + ⎜ x + ⎟ = m0 w 2 ⎜⎝ ∂y ∂x ⎟⎠

(27)

(1) (0) (0) ∂M xx(1) ∂M xy 1 ⎛ ∂M xy ∂M yy ∂M zz(0) ⎞ + − M xz(0) + ⎜ + − ⎟ 2 ⎜⎝ ∂x ∂x ∂y ∂y ∂y ⎟⎠

δθ x :

+

δθ y :

(1) 1 ∂ ⎛ ∂M xz(1) ∂M yz ⎞ 1 (0) 1 ∂cz(1) (1) + + f + cy + = m1u + m2θx ⎜ ⎟ x 2 ∂y ⎜⎝ ∂x ∂y ⎟⎠ 2 2 ∂y

∂M yy(1) ∂y

+

+

(28)

(0) 1 ⎛ ∂M xx(0) ∂M xy ∂M zz(0) ⎞ − M yz(0) + ⎜ + − ⎟ 2 ⎜⎝ ∂x ∂x ∂y ∂x ⎟⎠

∂M xy(1)

(1) 1 ∂ ⎛ ∂M xz(1) ∂M yz ⎞ 1 (0) 1 ∂cz(1) (1) + + f + cx − = m1v + m2θy ⎜ ⎟ y 2 ∂x ⎜⎝ ∂x ∂y ⎟⎠ 2 2 ∂x

(29)

where the superposed dots denote differentiation with respect to time, Fz(0) = ⎡⎣ q zt + q zb ⎤⎦ , and

mi (i = 0,1, 2) are the mass moment of inertia

mi = ∫

h/2

−h / 2

ρ ( z ) dz i

(30)

For linear S-FGM plates without body force and body couple, the equations of motion (25-29) reduce to

15

(0) (0) ∂M xx(0) ∂M xy 1 ∂ ⎛ ∂M xz(0) ∂M yz ⎞ + + + ⎜ ⎟ = m0 u + m1θx 2 ∂y ⎜⎝ ∂x ∂x ∂y ∂y ⎟⎠

(31)

(0) 1 ∂ ⎛ ∂M xz(0) ∂M yz ⎞ δv : + − + ⎜ ⎟ = m0v + m1θy 2 ∂x ⎜⎝ ∂x ∂x ∂y ∂y ⎟⎠

(32)

δu :

∂M xy(0)

∂M yy(0)

(0) (0) (0) (0) 1 ∂ ⎛ ∂M xx(0) ∂M xy ⎞ 1 ∂ ⎛ ∂M yy ∂M xy ⎞ ∂M xz(0) ∂M yz (0) − + + + δw: ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ + Fz 2 ∂y ⎝ ∂x ∂y ⎠ 2 ∂x ⎝ ∂y ∂x ⎠ ∂x ∂y

 − kW w + k P ∇ 2 w = m0 w

(33)

(1) (0) (0) ∂M xx(1) ∂M xy 1 ⎛ ∂M xy ∂M yy ∂M zz(0) ⎞ + − M xz(0) + ⎜ + − ⎟ 2 ⎜⎝ ∂x ∂x ∂y ∂y ∂y ⎟⎠

δθ x :

(1) 1 ∂ ⎛ ∂M xz(1) ∂M yz ⎞ + + ⎜ ⎟ = m1u + m2θx 2 ∂y ⎜⎝ ∂x ∂y ⎟⎠

δθ y :

∂M yy(1) ∂y

+

∂M xy(1) ∂x

(34)

1 ⎛ ∂M xx(0) ∂M xy ∂M zz(0) ⎞ + ⎜ + − ⎟ 2 ⎜⎝ ∂x ∂y ∂x ⎟⎠ (0)

−M

(0) yz

(1) 1 ∂ ⎛ ∂M xz(1) ∂M yz ⎞ + + ⎜ ⎟ = m1v + m2θy 2 ∂x ⎜⎝ ∂x ∂y ⎟⎠

(35)

The natural boundary conditions are of the form:

1 ⎛ ∂M xz(0) ∂M yz + ∂y 2 ⎝ ∂x

⎞ ⎟⎟ ny = 0 ⎠

(36)

(0) 1 ⎛ ∂M xz(0) ∂M yz ⎞ + ⎟ nx = 0 ∂y ⎟⎠ 2 ⎝ ∂x

(37)

δ u : M xx(0) nx + M xy(0) ny + ⎜⎜

(0)

δ v : M xy(0)nx + M yy(0) ny − ⎜⎜

1 ⎛ ∂M xx(0) ∂M xy + 2 ⎝ ∂x ∂y

(0)

δ w : M xz(0) nx + M yz(0) n y − ⎜⎜

δθ x : M xx(1) nx + M xy(1) ny +

(0) (0) ⎞ 1 ⎛ ∂M yy ∂M xy + ⎟⎟ n y + ⎜⎜ 2 ⎝ ∂y ∂x ⎠

⎞ ⎟⎟ nx = 0 ⎠

(38)

(1) 1 1 ⎛ ∂M xz(1) ∂M yz ⎞ + M xy(0)nx + M yy(0)ny − M zz(0) ny ) + ⎜⎜ ( ⎟ ny = 0 ∂y ⎟⎠ 2 2 ⎝ ∂x

(39)

δθ y : M xy(1)nx + M yy(1) ny −

(1) 1 1 ⎛ ∂M xz(1) ∂M yz ⎞ (0) (0) (0) + − − + n n n M M M ( xy y xx x zz x ) 2 ⎜⎜ ∂x ⎟⎟ nx = 0 ∂ y 2 ⎝ ⎠

16

(40)

3.2 Plate constitutive equations Here we relate the generalized forces generalized couples

(M

(i ) xx

(M

(i ) xx

, M yy(i ) , M zz(i ) , M xy(i ) , M xz(i ) , M yz( i ) ) and the

, M yy( i ) , M zz(i ) , M xy(i ) , M xz( i ) , M yz(i ) ) to the generalized displacements

(u , v , w, θ x , θ y ) . We have

⎧ M xx( i ) ⎫ ⎧σ xx ⎫ ⎡ A11( k ) 3 + i h/2 ⎪ ⎪ (i ) ⎪ ⎪ i ⎢ (k ) ⎨ M yy ⎬ = ∫− h / 2 ⎨σ yy ⎬( z ) dz = ∑ ⎢ A12 k =i ⎪ M (i ) ⎪ ⎪σ ⎪ ⎢ (k ) ⎩ zz ⎭ ⎩ zz ⎭ ⎣ A12 ⎧ M xy( i ) ⎫ ⎧σ xy ⎫ ⎡ B11( k ) 3+ i h / 2 ⎪ (i ) ⎪ ⎪ ⎪ i ⎢ ⎨ M xz ⎬ = ∫− h / 2 ⎨σ xz ⎬( z ) dz = ∑ ⎢ 0 k =i ⎪ M (i ) ⎪ ⎪σ ⎪ ⎢ ⎩ yz ⎭ ⎩ yz ⎭ ⎣ 0 ⎧ M xx(i ) ⎫ ⎧ mxx ⎫ ⎡ B11( k ) 2 + i h/2 ⎪ ⎪ (i ) ⎪ ⎪ i ⎢ ⎨ M yy ⎬ = ∫− h / 2 ⎨ m yy ⎬( z ) dz = ∑ ⎢ 0 k =i ⎪M (i ) ⎪ ⎪m ⎪ ⎢ ⎩ zz ⎭ ⎩ zz ⎭ ⎣ 0 ⎧ M xy(i ) ⎫ ⎧ mxy ⎫ ⎡ B11( k ) 3+ i h / 2 1 ⎢ ⎪ (i ) ⎪ ⎪ ⎪ i ⎨ M xz ⎬ = ∫− h / 2 ⎨ mxz ⎬( z ) dz = ∑ ⎢ 0 2 k =i ⎪M (i ) ⎪ ⎪m ⎪ ⎢ yz yz ⎩ ⎭ ⎩ ⎭ ⎣ 0

A12( k ) A11( k ) A12( k ) 0

B11( k ) 0 0

B11( k ) 0 0

B11( k ) 0

A12( k ) ⎤ ⎧ε xx( k −i ) ⎫ ⎥⎪ ⎪ A12( k ) ⎥ ⎨ε yy( k −i ) ⎬ , A11( k ) ⎦⎥ ⎪⎩ε zz( k −i ) ⎪⎭

(41)

0 ⎤ ⎧γ xy( k −i ) ⎫ ⎥⎪ ⎪ 0 ⎥ ⎨γ xz( k −i ) ⎬ , B11( k ) ⎦⎥ ⎪⎩γ yz( k −i ) ⎪⎭

(42)

0 ⎤ ⎧ χ xx( k −i ) ⎫ ⎥⎪ ⎪ 0 ⎥ ⎨ χ yy( k −i ) ⎬ , B11( k ) ⎦⎥ ⎪⎩ χ zz( k −i ) ⎪⎭ 0 ⎤ ⎧η xy( k −i ) ⎫ ⎥⎪ ⎪ 0 ⎥ ⎨η xz( k −i ) ⎬ , B11( k ) ⎦⎥ ⎪⎩η yz( k −i ) ⎪⎭

(43)

(44)

where

A11( k ) =

h/2 1 ⎡ 0 (k ) (k ) z Eb ( z ) dz + ∫ ( z ) Et ( z ) dz ⎤ , 2 ⎢ ∫− h / 2 ( ) 0 ⎣ ⎦⎥ (1 −ν )

A12( k ) =

h/2 ν ⎡ 0 (k ) (k ) z Eb ( z ) dz + ∫ ( z ) Et ( z ) dz ⎤ , 2 ⎢ ∫− h / 2 ( ) 0 ⎣ ⎦⎥ (1 −ν )

B11( k ) =

h/2 1 ⎡ 0 (k ) (k ) + ( ) z E z dz z ) Et ( z ) dz ⎤ , ( ) ( b ∫ ∫ ⎢ ⎥⎦ h / 2 0 − 2(1 +ν ) ⎣

B11( k ) =

h/2 A2 ⎡ 0 (k ) (k ) z ) Eb ( z ) dz + ∫ ( z ) Et ( z ) dz ⎤ . ( ∫ 0 ⎦⎥ (1 +ν ) ⎣⎢ − h / 2

(45)

Note that a shear correction factor must multiply the coefficient B11( k ) when computing the transverse

17

shear forces.

4. The analytical solutions of S-FGM micro-scale plates Here, analytical solutions for bending and free vibration of simply supported S-FGM microscale plates are presented using the modified couple stress first-order plate theory to illustrate the small scale effects on deflections of the micro-scale plates. For the static case, all time derivative terms are set to zero. For the set of simply supported boundary conditions, the analytical solution can be obtained [54, 55]. The presented analytical solutions using the Navier method are limited to simply supported rectangular plate but it is useful for the purpose of comparison with the numerical solutions. According to the Navier solution theory, the generalized displacements at middle of the plane ( z = 0 ) are expanded in double Fourier series as ∞

∞

{u ( x, y, t ) ,

θ x ( x, y, t )} = ∑∑ {U mn , X mn } Λ1 ,

{v ( x, y, t ) ,

θ y ( x, y, t )} = ∑∑ {Vmn , Ymn } Λ 2 ,

m =1 n =1 ∞

∞

m =1 n =1

∞

∞

w ( x, y, t ) = ∑∑ Wmn Λ 3 .

(46)

m =1 n =1

where Λ1 = cos ξ x sin η y ⋅ e in which ξ =

iϖ mn t

, Λ 2 = sin ξ x cosη y ⋅ e

iϖ mn t

, and Λ 3 = sin ξ x sin η y ⋅ e

iϖ mn t

mπ nπ , η= and ϖ mn is the frequency. a b

By substituting Eq. (46) into Eqs. (31)-(35), matrix form is as follow

[K ]{Δ} + [M ]{Δ} = {Q} , where

{Q}

{Δ} = {U mn ,Vmn ,Wmn , X mn , Ymn } , [K ]

(47) is the stiffness matrix, [ M ] is the mass matrix and

is the force vector. The transverse load Fz(0) is also expanded in the double-Fourier sine series

as ∞

∞

Fz(0) ( x, y ) = ∑∑ Qmn sin ξ x sin η y m=1 n =1

(48)

18

For static bending analysis, Eq. (47) takes

[M ] = 0

and for free vibration analysis, set

{Q}

to zero

in Eq. (47).

5. Numerical results and discussion In order to validate, several numerical examples are solved to test the performance in bending and free vibration analysis of S-FGM microplates on elastic medium. Examples include PFGM plates on elastic medium and P-FGM microplates to check some crucial features and to comparison with previous published analysis results. 5.1. Validation Firstly, since the results of micro-scale plate made of S-FGM (see Fig. 3) on elastic medium are not available in the open literature, homogeneous and P-FGM ( p = 1 ) plates are used herein for the verification. Table 1 shows the non-dimensional displacements of simply supported plates with various values of side-to-thickness ratio a / h in homogeneous and functionally graded ( p = 1,5,10 ) plates. The FGM plate is made of the following material properties [44]:

E1 = 380 GPa, E2 = 70 GPa, ν =0.3, ρ1 = 3,800 kg/m3 , ρ 2 = 2, 702 kg/m3 .

(49)

The following non-dimensionalizations are used:

kW =

kW a 4 kP a2 a2 k , = , ϖ = ϖ P A11(2) A11(2) h

ρ2

(50)

E2

Table 1. Non-dimensional frequency of simply supported S-FGM plate on elastic medium (A/h = 0)

p =1

A homogeneous plate

kW

100

kP

a/h

100

5 10 20 100

P-FGM (Thai et al. [45]) 15.3904 16.1728 16.4249 -

Present (S-FGM) 15.3891 16.1727 16.4249 16.5121

P-FGM (Thai et al. [45]) 14.6305 15.1887 15.3663 -

p=5

p = 10

Present (S-FGM)

Present (S-FGM)

Present (S-FGM)

14.6299 15.1886 15.3663 15.4276

14.1959 14.7019 14.8575 14.9107

14.1341 14.6340 14.7869 14.8390

For the case of a / h = 5 , there is small difference between the present results and those given by Thai et al. [45]. This is due to the fact that Thai et al. [45] employed a refined third order shear deformation

M G F

19

theory to calculate the frequency, whereas the present results are based on first order theory. Other cases, it can be observed that the present results are identical with those given by Thai et al. [45].

Secondly, In Tables 2 and 3, the calculated displacement and frequency based on FSDT plate theory with various material length scale parameters and S-FGM power law index ( p = 1,5,10 ) are compared with those reported by Thai et al. [27] with P-FGM index. The micro-scale plate is made of epoxy with the following material properties [23]:

E1 = 14.4 GPa, E2 = 1.44 GPa, ν =0.38, h = 17.6 ×10−6 m, q0 = 1.0 N/m,

ρ1 = 12.2 ×103 kg/m3 , ρ 2 = 1.22 ×103 kg/m3 .

.

(51)

The non-dimensional displacement are defined as

w=w

E2 h 3 × 102 . 4 q0 a

(52)

A shear correction factor of 5/6 is used for FSDT plate theory. It can be observed that the present results are identical with those given by Thai et al. [27] based on MPT. There is no difference between the present S-FGM results and those P-FGM results given by Thai et al. [27]. This is due to the fact that the S-FGM material properties are identical with P-FGM, when the power law index is 1.

Fig. 3 Geometry of S-FGM plate

20

Table 2. Non-dimensional displacement of simply supported micro-scale S-FGM plate ( a / h = 5, 10, kW = k P = 0)

p =1

A homogeneous plate

a/h

A/h

P-FGM (Thai et al. [27]) 0.5147 0.4479 0.3250 0.2268 0.1631 0.1230 0.4415 0.3844 0.2775 0.1907 0.1335 0.0972

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

5

10

Present (S-FGM) 0.5147 0.4479 0.3250 0.2268 0.1631 0.1230 0.4415 0.3844 0.2775 0.1907 0.1335 0.0972

P-FGM (Thai et al. [27]) 1.1536 0.9685 0.6599 0.4395 0.3073 0.2279 1.0205 0.8567 0.5798 0.3790 0.2573 0.1838

p=5

p = 10

Present (S-FGM)

Present (S-FGM)

Present (S-FGM)

1.1536 0.9685 0.6599 0.4395 0.3073 0.2279 1.0205 0.8567 0.5798 0.3790 0.2573 0.1838

1.5699 1.2343 0.7610 0.4750 0.3204 0.2331 1.4368 1.1290 0.6895 0.4209 0.2748 0.1919

1.6556 1.2849 0.7781 0.4806 0.3224 0.2338 1.5225 1.1807 0.7080 0.4274 0.2774 0.1930

Table 3. Non-dimensional frequency of simply supported micro-scale S-FGM plate ( a / h = 5, 10, kW = k P = 0)

p =1

A homogeneous plate

a/h

5

10

A/h 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

P-FGM (Thai et al. [27]) 5.3871 5.7797 6.7996 8.1595 9.6451 11.1311 5.9301 6.3559 7.4807 9.0261 10.7848 12.6360

Present (S-FGM) 5.3871 5.7797 6.7996 8.1595 9.6451 11.1311 5.9301 6.3559 7.4807 9.0261 10.7848 12.6360

P-FGM (Thai et al. [27]) 4.8744 5.3239 6.4600 7.9298 9.4998 11.0451 5.2697 5.7518 6.9920 8.6477 10.4942 12.4128

p=5

p = 10

Present (S-FGM)

Present (S-FGM)

Present (S-FGM)

4.8744 5.3239 6.4600 7.9298 9.4998 11.0451 5.2697 5.7518 6.9920 8.6477 10.4942 12.4128

4.2024 4.7420 6.0453 7.6590 9.3326 10.9481 4.4508 5.0212 6.4250 8.2217 10.1730 12.1690

4.0957 4.6516 5.9830 7.6192 9.3084 10.9343 4.3250 4.9114 6.3424 8.1609 10.1277 12.1349

It can be seen that the present results are in excellent agreement with those generated by FSDT [27] for all material length parameters. 5.2. Parameter studies Consider a simply supported square plate with the material properties of Eq. (45). Parameter

21

studies are presented to investigate the influences of elastic medium parameters ( kW , kP ), side-tothickness ratio ( a / h ), material length scale parameter ( A / h ), and power law index p on bending and vibration responses of S-FGM micro-scale plate. Firstly, to present the effect of material length scale parameter on responses of S-FGM micro-scale plate, Figs. 4 and 5 plots the dimensionless deflection and frequency versus the dimensionless material length scale parameter ( A / h ) for various values of the power law index ( p ). In these figures, for various values A / h , the material length scale parameter is varied when the microplate thickness is keep constant as h = 17.6 μm . It can be found from Figs. 4 and 5 that the dimensionless material length scale parameter has no effect on the dimensionless deflection and frequency for the classical plate model, which is unable to capture the size effect. On the other hand, the dimensionless deflection and frequency of the non-classical plate model decreases and increases, respectively, as the dimensionless material length scale parameter increases. The bending and vibration response predicted by the modified couple stress theory is always smaller and larger than those of classical plate theory. When the plate thickness is equal to the material length scale parameter ( A / h = 1 ), the difference between the two models is not significant, as seen from Figs. 4 and 5. 1.6

Nondimensinal Deflection

1.4

Classical, p= 1 Classical, p= 5 Classical, p=10

1.2

1

0.8

0.6

0.4

MCST, p= 1 MCST, p= 5 MCST, p=10

0.2

0 0.2

0.4

0.6

0.8

/

0

1

Material Length Scale Parameter (l h)

Fig. 4 Effect of material length scale parameter ( A / h ) on nondimensional deflection of S-FGM microplate ( a / h = 10, kW = k P = 0 ).

22

13

MCST, p= 1 MCST, p= 5 MCST, p=10 Classical, p= 1 Classical, p= 5 Classical, p=10

Nondimensional Frequency

12

11

10

9

8

7

6

5

4 0.2

0.4

0.6

0.8

/

0

1

Material Length Scale Parameter (l h)

Fig. 5 Effect of material length scale parameter ( A / h ) on nondimensional deflection of S-FGM microplate ( a / h = 10, kW = k P = 0 ).

To illustrate the effect of Winkler’s elastic medium parameter on responses of S-FGM micro-scale plate, Fig. 6 and 7 plot the deflection and the frequency with respect to the Winkler’s elastic medium parameter for a simply supported S-FGM plate with p = 10.0 and a / h = 10 . In this parameter study, for various values a / h , the microplate thickness is varied when the in-plane dimensions are keep constant as a = b = 17.6 ×10−5 m . The material length scale parameters are taken as

A / h = 0.0, 0.5, 1.0 . The inclusion of the micro-scale effect will increase the stiffness of the S-FGM micro-scale plate, and consequently, leads to a reduction of deflection and an enlargement of frequency. The effect of elastic medium parameter on deflection and frequency are more significant when the material length scale parameter is not considered ( A / h = 0.0 ). The nondimensional deflection is almost constant with respect to the variation of elastic medium parameter when the material length scale parameter A / h = 1.0

23

1.6 / / /

l h=0.0 1.4

l h=0.5

Nondimensinal Deflection

l h=1.0 1.2

1

0.8

0.6

0.4

0.2

0 0

200

400

600

800

1000

Elastic Medium Parameter

Fig. 6 Effect of elastic medium parameter ( kW ) on nondimensional deflection of S-FGM microplate ( a / h = 10, p = 10 ).

14

/ / /

Nondimensional Frequency

12

l h=0.0 l h=0.5

10

l h=1.0

8

6

4 0

200

400

600

800

1000

Elastic Medium Parameter

Fig. 7 Effect of elastic medium parameter ( kW ) on nondimensional frequency of S-FGM microplate ( a / h = 10, p = 10 ).

To account for the effect of Pasternak’s elastic medium parameter on responses of S-FGM micro-scale plate, Fig. 8 and 9 plot the deflection with respect to the Pasternak’s elastic medium

24

parameter for a simply supported S-FGM plate with kW = 1000 , p = 10.0 and a / h = 10 . The material length scale parameters are taken as A / h = 0.0, 0.5, 1.0 . The inclusion of the micro-scale effect will increase the stiffness of the S-FGM micro-scale plate, and consequently, leads to a reduction of deflection and an enlargement of frequency. The results show the Pasternak’s elastic medium parameter has more effect on decreasing the nondimensional deflection and increasing the nondimensional frequency than the Winkler’s elastic medium parameter.

0.8 / / /

l h=0.0 l h=0.5

0.7

Nondimensional Deflection

l h=1.0

0.6

0.5

0.4

0.3

0.2

0.1 0

20

40

60

80

100

Elastic Medium Parameter

Fig. 8 Effect of elastic medium parameter ( k P ) on nondimensional deflection of S-FGM microplate ( a / h = 10, p = 10 , kW = 1000 ).

25

15

/ / /

Nondimensional Frequency

13

l h=0.0 l h=0.5

11

l h=1.0

9

7

5 0

20

40

60

80

100

Elastic Medium Parameter

Fig. 9 Effect of elastic medium parameter ( k P ) on nondimensional frequency of S-FGM microplate ( a / h = 10, p = 10 , kW = 1000 ).

It is shown that the effects of side-to-thickness ratio on the dimensionless deflection and frequency with variable Winkler’s elastic medium parameter is presented in Figs. 10 and 11, respectively, for a simply supported square plate with A / h = 1.0 , k P = 0 and p = 10 . Figures 10 and 11 clearly show the diminishing effect of transverse shear deformation on deflections and frequencies, the effect being negligible for side-to-thickness ratios larger than 40. The increasing value of the Winkler’s elastic medium parameter leads to a decrease in the magnitude of deflection and an increase in the amplitude of frequency. The increasing value of side-to-thickness ratio leads to a decrease the deflection and an increase the frequency. As expected, the effect of Winkler’s elastic medium parameter is to decrease the deflection and increase the frequency.

26

0.25 Elastic Medium Parameter= 0 Elastic Medium Parameter= 200 Elastic Medium Parameter= 400 Elastic Medium Parameter= 600 Elastic Medium Parameter= 800 Elastic Medium Parameter=1000

Nondimensional Deflection

0.23

0.21

0.19

0.17

0.15 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 10 Effect of side-to-thickness ratio on the nondimensional deflection of S-FGM microplate ( A / h = 1.0, p = 10, k P = 0 ). 13.5

Nondimensional Frequency

13.0

12.5

12.0

11.5 Elastic Medium Parameter= 0 Elastic Medium Parameter= 200 Elastic Medium Parameter= 400 Elastic Medium Parameter= 600 Elastic Medium Parameter= 800 Elastic Medium Parameter= 1000

11.0

10.5 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 11 Effect of side-to-thickness ratio on the nondimensional frequency of S-FGM microplate ( A / h = 1.0, p = 10, k P = 0 ).

In Figs. 12 and 13, the effects of side-to-thickness ratio on the dimensionless deflection with variable Pasternak’s elastic medium parameter is presented for a simply supported square plate with

27

A / h = 1.0 , kW = 1000 and p = 10 . It is observed from Figs. 12 and 13 that the effect of shear deformation is significant when the side-to-thickness ratio is small, but it is negligible when the sideto-thickness ratio becomes larger. The increasing value of the Pasternak’s elastic medium parameter leads to a decrease in the magnitude of deflection and an increase in the amplitude of frequency. The increasing value of side-to-thickness ratio leads to a decrease the deflection and an increase the frequency. It is shown that the effect of Pasternak’s elastic medium parameter is to decrease deflection.

0.22 Elastic Medium Parameter= 0 Elastic Medium Parameter= 20 Elastic Medium Parameter= 40 Elastic Medium Parameter= 60 Elastic Medium Parameter= 80 Elastic Medium Parameter=100

Nondimensional Deflection

0.2

0.18

0.16

0.14

0.12 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 12 Effect of side-to-thickness ratio on the nondimensional deflection of S-FGM microplate ( A / h = 1.0, p = 10, kW = 1000 ).

28

Nondimensional Frequency

15

14

13

Elastic Medium Parameter= 0 Elastic Medium Parameter= 20 Elastic Medium Parameter= 40 Elastic Medium Parameter= 60 Elastic Medium Parameter= 80 Elastic Medium Parameter=100

12

11 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 13 Effect of side-to-thickness ratio on the nondimensional frequency of S-FGM microplate ( A / h = 1.0, p = 10, kW = 1000 ).

The effects of the power law index p on the dimensionless deflection is presented in Figs. 14 and 15 for a simply supported square plate with A / h = 0.0, 0.5, 1.0 and a / h = 10 . The increasing value of the power law index decreases the stiffness of the S-FGM micro-scale plates. It can be seen that increasing value of the power law index leads to an increase in the magnitude of deflection and a decrease in the amplitude of frequency. When the material length scale parameter ( A / h ) is 0, a S-FGM micro-scale plate is treated as a micro plate without size-effect. The effect of power law index on deflection and frequency is more significant when the size-dependent constitutive model is not considered ( A = 0 ). It is noticed that if a microplate is rested on Winkler-Pasternak elastic medium with material length scale parameter A / h = 1.0 , the nondimensional deflection is almost constant with respect to the variation of power law index.

29

0.40

0.30

0.25

/ / /

Nondimensional Deflection

0.35

l h=0.0

0.20

l h=0.5 l h=1.0

0.15

0.10 0

2

4

6

8

10

Power Law Index

Fig. 14 Effect of power law index on the nondimensional deflection of S-FGM microplate ( kW = 1000, k P = 100 , a / h = 10 ). 15

13 / / /

Nondimensional Frequency

14

l h=0.0 l h=0.5 12

l h=1.0

11

10

9

8 0

2

4

6

8

10

Power Law Index

Fig. 15 Effect of power law index on the nondimensional frequency of S-FGM microplate ( kW = 1000, k P = 100 , a / h = 10 ).

It can be seen that the effects of side-to-thickness ratio on the dimensionless deflection and

30

frequency with variable material length scale parameter is presented in Figs. 16 and 17, respectively, for a simply supported square plate with kW = 1000 , k P = 100 and p = 10 . It is shown that the effect of shear deformation is considered for thick microplates ( a / h < 10 ), and negligible for thin microplates. The increasing value of material length scale parameter leads to a decrease in the magnitude of deflection and an increase in the amplitude of frequency. The increasing value of sideto-thickness ratio leads to a decrease the deflection and an increase the frequency. It is observed that if a microplate is rested on Winkler-Pasternak elastic medium with material length scale parameter

A / h = 0.0 , the nondimensional deflection and frequency are almost constant with respect to the variation of side-to-thickness ratio. 0.40

0.30

0.25

/ / /

Nondimensional Deflection

0.35

l h=0.0 l h=0.5

0.20

l h=1.0

0.15

0.10 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 16 Effect of side-to-thickness ratio on the nondimensional deflection of S-FGM microplate ( p = 10, kW = 1000, k P = 100 ).

31

15

13 / / /

Nondimensional Frequency

14

l h=0.0 l h=0.5 12

l h=1.0

11

10

9

8 0

10

20

30

40

50

Side to Thickness Ratio (a/h)

Fig. 17 Effect of side-to-thickness ratio on the nondimensional frequency of S-FGM microplate ( p = 10, kW = 1000, k P = 100 ).

6. Conclusions Modified couple stress model for bending and free vibration analysis of sigmoid functionally graded materials (S-FGM) micro-scale plates are presented using a first-order shear deformation theory and Hamilton’s principle. The present models contain one material length scale parameter and can capture the size effect, and two-constituent material variation through the plate thickness with two power-law functions. Also, the present model can be reduced to the homogeneous micro-scale plates by setting E1 = E2 and S-FGM plates without size-effect by setting the material length scale parameter equal to zero. Analytical solutions for deflection and frequency of a simply supported rectangular S-FGM micro-scale plate are presented. From the present work following conclusions are drawn:

(1) The results show that the Winkler’s and Pasternak’s elastic medium parameters have effects of decreasing the nondimensional deflection and increasing the frequency, and the Pasternak’s parameter has more effect on decreasing the deflection and increasing the frequency than the Winkler’s parameter. (2) The inclusion of material length scale parameter will increase the stiffness of the microplates,

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leads to a reduction of deflection and increase the frequency. (3) The S-FGM microplate is rested on Winkler-Pasternak elastic medium with material length scale parameter A / h = 1.0 , the nondimensional deflections and frequencies are almost constant with respect to the variation of power law index. (4) The nondimensional frequency is almost constant with respect to the variation of side-to-thickness ratio, when the S-FGM microplate is rested on Winkler-Pasternak elastic medium with material length scale parameter A / h = 0.0 .

Due to the interesting features of the present theory, the present results will be used for evaluating the reliability of size-dependent plate models on elastic medium developed in the future. Further, in the analysis of S-FGM micro-structures on elastic medium it is necessary to include the modified couple stress theory for micro-scale shells.

Acknowledgements This work was supported by the National Research Foundation of Korea grant funded [NRF] by the Korea Government [MEST] (No. 2011-0028531)

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Editors-in-Chief, International Journal of Mechanical Sciences Marian Wiercigroch University of Aberdeen, Aberdeen, UK

Bending and Vibration Analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory

Highlights

1. An S-FGM composed of two power-law functions to define volume fraction is proposed. 2. A modified couple stress theory with first order shear deformation is developed. 3. The model contains one material length scale parameter and can capture the size effect.

4. The Pasternak’s parameter has more effect on variation of structural responses.